diff --git a/.github/workflows/cyclic-deferred.yml b/.github/workflows/cyclic-deferred.yml
new file mode 100644
index 0000000000..48ef40fb9a
--- /dev/null
+++ b/.github/workflows/cyclic-deferred.yml
@@ -0,0 +1,21 @@
+name: Verify cyclic deferred witness
+on: [push, pull_request, workflow_dispatch]
+permissions:
+  contents: read
+jobs:
+  focused:
+    runs-on: ubuntu-24.04
+    timeout-minutes: 25
+    strategy:
+      fail-fast: false
+      matrix:
+        python-version: ['3.11', '3.13', '3.14']
+    steps:
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97
+        with:
+          python-version: ${{ matrix.python-version }}
+      - run: python3 research/cyclic-deferred/verify.py
+      - run: git diff --exit-code
diff --git a/Makefile b/Makefile
index 84fe8290f3..e60b843993 100644
--- a/Makefile
+++ b/Makefile
@@ -30,6 +30,7 @@ verify-community: community-audit-check community-followup-check copied-reversed
 	$(MAKE) climbed-48-verify
 
 verify-producers:
+	$(MAKE) stopped-product-verify
 	$(MAKE) copied-centers-verify
 	$(MAKE) structured-bulk-verify
 	$(MAKE) endpoint-gauge-producer endpoint-gauge-certificate
@@ -428,3 +429,12 @@ pair-assembly-verify: pair-assembly-producer pair-assembly-check
 .PHONY: verify-pair
 verify-pair: pair-assembly-verify
 	python3 scripts/audit_pair_candidate.py --check docs/research/community-pair-arithmetic.json
+
+.PHONY: stopped-product-producer stopped-product-certificate stopped-product-verify
+stopped-product-producer:
+	python3 scripts/stopped_product_producer.py
+
+stopped-product-certificate:
+	python3 scripts/stopped_product_network.py
+
+stopped-product-verify: stopped-product-producer stopped-product-certificate
diff --git a/NOTICE b/NOTICE
index b565637758..50c6d1236d 100644
--- a/NOTICE
+++ b/NOTICE
@@ -302,3 +302,20 @@ inherited PR #48 sources. The stacked witness in research/pair-assembly/frame
 uses eumemic's PR #57 joint frame compiler unchanged; its driver and verifier
 are adaptations of PR #57's skip_frame_compiler.py, skip_frame_compose.py and
 verify_skip_frame.py. Original notices are retained.
+
+Stopped product-ring extension, 9 October 2026
+Copyright 2026 icekylinx
+
+The opposite-bank generic projector factorization, componentwise atom-ring
+adapters, stopped recurrence and outer ordinary-interchange conversion,
+aligned pair-star producer with rational centers, exact denominator-21
+grid and selected (23,23)/(24,24) composition are contributed by icekylinx
+with substantial OpenAI GPT-6 Astra and Codex assistance. This extension
+builds on merged PR #36, 11817ccacb564bb7f98789c20dc11d3fece207e3, and is
+integrated on upstream main 56b66d58297deca1d7dd130247d720e960f77a37.
+
+The retained ordinary leaf uses the preceding projector, corner and
+copied-center proofs. The selected two-stage topology retains Aurel Prosz
+(Paureel), Swapnil Jain and Zhihao Chen's credits; the semantic assembly
+retains PR #21/#23 and RaD's attribution. Paired producers, carrier matching,
+original source notices and each imported license remain unchanged.
diff --git a/PATCHING.md b/PATCHING.md
index b60f3f101b..a02e230256 100644
--- a/PATCHING.md
+++ b/PATCHING.md
@@ -1,6 +1,6 @@
 # Applying the inherited manuscript patch
 
-This patch is the PR #10 baseline. The current copied-center extension is
+This patch is the PR #10 baseline. The PR #36 copied-center extension is
 presented in `notes/copied-centers-note.tex`; it is not included
 in this historical combined manuscript patch.
 
@@ -30,3 +30,6 @@ alternatives for the same base. PR7 is pinned at
 `6725c6a17b17871a35353fd29157f4ed851bc114` for the retained producer sources.
 The old uniform recurrences keep their original exponents. The active consumers
 use `lem:batched-chunk-swap` and `prop:batched-simultaneous-layer`.
+
+The stopped product-ring extension is supplied separately in
+`notes/stopped-product-note.tex`; it does not regenerate this historical patch.
diff --git a/README.md b/README.md
index 035c259b68..31aa46c2e2 100644
--- a/README.md
+++ b/README.md
@@ -3,6 +3,36 @@
 **Community research maintained by Douglas Colkitt — conditional on the original
 OpenAI #109 framework.**
 
+## Stopped product-ring extension
+
+The new construction contributed by **icekylinx**, extending merged
+[PR #36](https://github.com/CrocSwap/integer-mult-bounds/pull/36), gives
+
+$$
+T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad
+\kappa=\frac{194869}{2500000000}=7.79476\times10^{-5}.
+$$
+
+It combines a stopped product-ring bit interchange on `(23,23)` with an
+all-disjoint rational-center complex network on `(24,24)`. The new generic
+opposite-bank factorization pays one reversed child per projector rank;
+atom adapters, ordinary leaves, endpoint copies and the exact denominator-21
+grid are included in the proof. The bound retains the original analytic
+and fixed-tape hypotheses.
+
+[Proof source](notes/stopped-product-note.tex) ·
+[Exact certificate](certificates/stopped-product-network.json) ·
+[Incremental reproduction](docs/stopped-product.md)
+
+```sh
+make stopped-product-verify
+```
+
+The maintainer-reviewed community checkpoint below remains its own result
+and validation record.
+
+## Reviewed community checkpoint
+
 The reviewed community witness gives
 
 $$
diff --git a/SOURCES.json b/SOURCES.json
index 6096d5aff9..380b5425c2 100644
--- a/SOURCES.json
+++ b/SOURCES.json
@@ -234,5 +234,47 @@
       }
     ],
     "continuing_analytic_dependencies": "See structured_bulk_extension.adopted and references/semantic-bulk/MANIFEST.json."
+  },
+  "stopped_product_extension": {
+    "contributor": "icekylinx",
+    "base_commit": "56b66d58297deca1d7dd130247d720e960f77a37",
+    "proof_predecessor_PR": "https://github.com/CrocSwap/integer-mult-bounds/pull/36",
+    "proof_predecessor_commit": "11817ccacb564bb7f98789c20dc11d3fece207e3",
+    "predecessor_merged": true,
+    "archive": "integer_mult_round5_kappa_7_79476e-5_complete.zip",
+    "archive_sha256": "714cbffa0cccba3c18f046a63e62d6981683cb386b5d5cd09761be7b70f0b0be",
+    "kappa": "194869/2500000000",
+    "proof": "notes/stopped-product-note.tex",
+    "certificate": "certificates/stopped-product-network.json",
+    "producer": "scripts/stopped_product_producer.py",
+    "assistance": "Substantial OpenAI GPT-6 Astra and Codex assistance.",
+    "scope": "Conditional on the retained analytic and fixed-tape framework; new product-ring and exact rational-grid arguments supplied as written proofs.",
+    "continuing_dependencies": [
+      "copied_centers_extension",
+      "structured_bulk_extension"
+    ],
+    "icekylinx_lineage": [
+      {
+        "pr": 10,
+        "commit": "62691e395a0458ce089a1c7b5d89e74291e95e29"
+      },
+      {
+        "pr": 18,
+        "commit": "f2ab41aebad47861caf6316282c1793e5513845e"
+      },
+      {
+        "pr": 24,
+        "commit": "ed90fd940279c336ebc45968631bdebfb087b505"
+      },
+      {
+        "pr": 32,
+        "commit": "0ef3aeb61f55cc0b321ce6a0ef00acee25cefe52"
+      },
+      {
+        "pr": 36,
+        "commit": "11817ccacb564bb7f98789c20dc11d3fece207e3"
+      }
+    ],
+    "curation": "Selected h23/h24 inputs, generator, exact certificate and proofs only; excludes raw graphs, research alternatives and recursive predecessor copies."
   }
 }
diff --git a/certificates/joint-dual-kappa.json b/certificates/joint-dual-kappa.json
index d92f653fe4..31628a32dc 100644
--- a/certificates/joint-dual-kappa.json
+++ b/certificates/joint-dual-kappa.json
@@ -842,7 +842,7 @@
     "scripts/experiments/pin_joint_dual_sources.py": "4d31cc0e38b7902709ca5f8f714cba109760d2592346dd7f8dc0b900a2d9fe00",
     "scripts/experiments/verify_joint_dual.py": "7a31bba1e9c0ded32ae285df2ca9bc91ef20c330f6789fdba361876f3c7a1985"
   },
-  "source_manifest_sha256": "fd1aab7d75753b4b55b2377f26119dae74f367b0b4fab66cba42d944b3042cc9",
+  "source_manifest_sha256": "db926540bbc818963af5ac78ed8949e6263afffdbbe2f05ee781ef7d3cde48f0",
   "sources": [
     {
       "files": {
diff --git a/certificates/joint-dual-validation.json b/certificates/joint-dual-validation.json
index 99adb4cf74..b2624180c2 100644
--- a/certificates/joint-dual-validation.json
+++ b/certificates/joint-dual-validation.json
@@ -226,7 +226,7 @@
     }
   },
   "bit_saving": "2382370177/50000000000000",
-  "certificate_sha256": "2ed658f05365b28825e24a6e7bac9fc9cc015d5a5d63d9579f4ac6e0640ae559",
+  "certificate_sha256": "b8263db0d04aabb98a31965a40c198667709bd6a6c7f98df2990baf523bcd075",
   "kappa": "4764513337/100000000000000",
   "margins": 7,
   "producer_words_regenerated_and_matched": true,
diff --git a/certificates/stopped-product-bit-axis.json b/certificates/stopped-product-bit-axis.json
new file mode 100644
index 0000000000..5824891de9
--- /dev/null
+++ b/certificates/stopped-product-bit-axis.json
@@ -0,0 +1,38 @@
+{
+  "h": 23,
+  "v": 1771,
+  "c": 34764,
+  "q": 5336,
+  "baseline_R": 40100,
+  "matching": 1324,
+  "R": 38776,
+  "loss": 506,
+  "histogram": [
+    9840,
+    38409,
+    32471,
+    8898,
+    9754,
+    2116,
+    5737,
+    1725,
+    4145,
+    1547,
+    3128,
+    1551,
+    2893,
+    1925,
+    2841,
+    2450,
+    2327,
+    3096,
+    2344,
+    3628,
+    9435,
+    2110,
+    23,
+    23
+  ],
+  "label_source": "positive",
+  "description": "Retained-total aligned triple producer, base-two recursion, original maximum carrier matching, and compatible positive labels."
+}
diff --git a/certificates/stopped-product-complex-input.json b/certificates/stopped-product-complex-input.json
new file mode 100644
index 0000000000..7b1d99c518
--- /dev/null
+++ b/certificates/stopped-product-complex-input.json
@@ -0,0 +1,41 @@
+{
+  "h": 24,
+  "v": 2024,
+  "c": 45764,
+  "q": 8120,
+  "baseline_R": 53884,
+  "matched": 8966,
+  "R": 44918,
+  "orientation_changes": 7338,
+  "rank_sum": 1079136,
+  "loss": 552,
+  "histogram": [
+    8930,
+    49042,
+    30844,
+    15972,
+    11883,
+    8272,
+    6399,
+    4356,
+    6682,
+    2263,
+    5109,
+    1808,
+    5317,
+    1733,
+    5391,
+    1757,
+    4908,
+    3290,
+    6233,
+    4363,
+    4056,
+    264,
+    564,
+    24,
+    24
+  ],
+  "central_disjoint": 24,
+  "center_denominator": 21
+}
diff --git a/certificates/stopped-product-input.json b/certificates/stopped-product-input.json
new file mode 100644
index 0000000000..f0a910a495
--- /dev/null
+++ b/certificates/stopped-product-input.json
@@ -0,0 +1,253 @@
+{
+  "bit": {
+    "dimensions": [
+      23,
+      23
+    ],
+    "m": 529,
+    "N": 3136441,
+    "W": 143617474,
+    "L": 1792252,
+    "total_rank": 75972299557,
+    "deficit": 1344189,
+    "maxchild": 506,
+    "child_multiplicities": {
+      "1": 139262585,
+      "2": 115012282,
+      "3": 31516716,
+      "4": 34548668,
+      "5": 7494872,
+      "6": 20320454,
+      "7": 6109950,
+      "8": 14681590,
+      "9": 5479474,
+      "10": 11079376,
+      "11": 5493642,
+      "12": 10247006,
+      "13": 6818350,
+      "14": 10062822,
+      "15": 8677900,
+      "16": 8242234,
+      "17": 10966032,
+      "18": 8302448,
+      "19": 12850376,
+      "20": 33418770,
+      "21": 7473620,
+      "22": 12627230,
+      "484": 6272882,
+      "506": 137344592
+    }
+  },
+  "complex": {
+    "dimensions": [
+      24,
+      24
+    ],
+    "m": 576,
+    "N": 4096576,
+    "B1": 90914032,
+    "B2": 90914032,
+    "W": 190021216,
+    "L": 2234496,
+    "total_rank": 109450358336,
+    "deficit": 1862080,
+    "child_multiplicities": {
+      "552": 181828064,
+      "529": 8193152,
+      "1": 202715744,
+      "2": 124856512,
+      "3": 64654656,
+      "4": 48102384,
+      "5": 33485056,
+      "6": 25903152,
+      "7": 17633088,
+      "8": 27048736,
+      "9": 9160624,
+      "10": 20681232,
+      "11": 7318784,
+      "12": 21523216,
+      "13": 7015184,
+      "14": 21822768,
+      "15": 7112336,
+      "16": 19867584,
+      "17": 13317920,
+      "18": 25231184,
+      "19": 17661424,
+      "20": 16418688,
+      "21": 1068672,
+      "22": 2283072,
+      "23": 16483456
+    },
+    "maxchild": 552,
+    "copied_histogram": [
+      8930,
+      49066,
+      30844,
+      15972,
+      11883,
+      8272,
+      6399,
+      4356,
+      6682,
+      2263,
+      5109,
+      1808,
+      5317,
+      1733,
+      5391,
+      1757,
+      4908,
+      3290,
+      6233,
+      4363,
+      4056,
+      264,
+      564,
+      24,
+      0
+    ]
+  },
+  "bit_moment_gap": "5833629405069796335513229037/13595148940731587639987765177686687744",
+  "complex_moment_gap": "5499421901694541095908997425024101371546537414283732387206104389099738183215366564793192567257041780652609318673168302647523216560182242981533611801155442603608986943408777543208667066118724847925169249370145237707441178592436788856344246441565482324551821454604550454953780270300756671351666361036963486701477553679079126088067165335949185500960812006586245961913848693714754073033035419679174118868763905107326417918471742975833959733450001524553321932789312500612407132407690299/41230077325513267360846956938521502341474073689326505257121676970478810391277326276471865083362032190646188250982499628351612283105589897175543987487125650452330304348151388045812369968133964037865011250349179368669257862504666828211703334554008512518414563221340357395437053315115874615168412042904680012697363239611038021437069048505672439008745508864584512067644825937731293242352557278635159598633861792145775836588122497044363982384099548431927704944786606529202786560000000000000000000",
+  "finite_bridge": {
+    "bit_coarse": {
+      "m": 529,
+      "W": 143617474,
+      "maxchild": 506,
+      "halving_degree": 16,
+      "wire_bits": 28
+    },
+    "ordinary_leaf_row_degree": 252,
+    "complex": {
+      "m": 576,
+      "W": 190021216,
+      "s": 109450358336,
+      "maxchild": 552,
+      "halving_degree": 17,
+      "wire_bits": 28,
+      "scalar_terms": [
+        {
+          "h": 24,
+          "v": 2024,
+          "c": 45764,
+          "invocations": 2024,
+          "local_group_upper": 408200,
+          "copied_center_extra_groups": 48
+        },
+        {
+          "h": 24,
+          "v": 2024,
+          "c": 45764,
+          "invocations": 2024,
+          "local_group_upper": 408200,
+          "copied_center_extra_groups": 48
+        }
+      ],
+      "scalar_group_upper": 1656684480
+    },
+    "semantic": {
+      "E": 403063477262941121443761178688,
+      "literal_charge": 119639333593035096270287748,
+      "strict_literal_gap": 402943837929348086347490890940,
+      "B": 403063477262941121553211537024,
+      "C0": 2994465792675104952757349965746759652008778122322142627817848832,
+      "C1": 1,
+      "induction_gap": 18943983431358232713000942240128,
+      "fixed_odd_divisor": 21,
+      "exact_grid": "One common dyadic grid times the fixed odd divisor to K=G*(D_complex+1); no child rounding"
+    },
+    "rows": {
+      "coefficient": 1176,
+      "degree": 4000,
+      "suffix_slope": 16000,
+      "degree_gap": "40024/25",
+      "contract": "W_complex^D_complex * W_coarse^D_coarse * W_old^D_old; one preceding prefix and one padding; sequential reuse"
+    }
+  },
+  "assembly": {
+    "parameters": {
+      "actual_bit_saving": "803380799/10000000000000",
+      "a_bit": "1948995551/25000000000000",
+      "a_complex": "1949/25000000",
+      "tau": "24998051004449/25000000000000",
+      "sigma": "24998051/25000000",
+      "eta": "1/100000000",
+      "beta": "1/1000000",
+      "q": "97449775601004449/1250000000000000000000",
+      "c": "9744977657550220501004449/125000000000000000000000000000",
+      "epsilon": "124999998750000000000000000000/125019489955217650665401004449",
+      "lambda_": "2499805100446848995551/2500000000000000000000",
+      "lambda_prime": "1249902550224398995551/1250000000000000000000",
+      "alpha_squared_power": "14618091340150667350000000/125019489955217650665401004449",
+      "delta": "1/800000000",
+      "C0": 2994465792675104952757349965746759652008778122322142627817848832,
+      "C1": 1,
+      "kappa": "194869/2500000000"
+    },
+    "strict_constraints": {
+      "bit_positive": "1948995551/25000000000000",
+      "complex_above_bit": "4449/25000000000000",
+      "complex_below_one_over32": "779301/25000000",
+      "beta_positive": "1/1000000",
+      "beta_below_one": "999999/1000000",
+      "leaf_saving_above_bit": "1/10000000000",
+      "q_positive": "97449775601004449/1250000000000000000000",
+      "q_below_internal": "1948995551/1250000000000000000000",
+      "q_below_leaf": "126948995551/1250000000000000000000",
+      "c_positive": "9744977657550220501004449/125000000000000000000000000000",
+      "c_below_one": "124990255022342449779498995551/125000000000000000000000000000",
+      "q_below_reservations": "97449775601004449/125000000000000000000000000000",
+      "lambda_above_tau": "1948995551/2500000000000000000000",
+      "lambda_above_sigma": "446848995551/2500000000000000000000",
+      "lambda_above_internal": "1948995551/2500000000000000000000",
+      "lambda_prime_above_lambda": "1948995551/2500000000000000000000",
+      "compact_leaf": "126948995551/1250000000000000000000",
+      "compact_reservations": "97449775601004449/125000000000000000000000000000",
+      "lambda_prime_below_one": "97449775601004449/1250000000000000000000",
+      "epsilon_positive": "124999998750000000000000000000/125019489955217650665401004449",
+      "epsilon_below_one": "19491205217650665401004449/125019489955217650665401004449",
+      "guard_width": "19491205217650665401004449/125019489955217650665401004449",
+      "K_geometry": "974622765755022147550220501004449/12501948995521765066540100444900000000",
+      "K_dominates_log": "974497756010044392550224398995551/12501948995521765066540100444900000000",
+      "record_suffix": "19491205217650665401004449/125019489955217650665401004449",
+      "phase_local": "15592839154630577103152893798995551/100015591964174120532320803559200000000",
+      "phase_boundary": "11694348052630578662349334598995551/100015591964174120532320803559200000000",
+      "gamma_sublinear": "4873113877499998051004449/125019489955217650665401004449",
+      "cell_above_band": "124995125636122500001948995551/250038979910435301330802008898",
+      "prime_interval_packing": "19491205217650665401004449/125019489955217650665401004449",
+      "alpha_positive": "14618091340150667350000000/125019489955217650665401004449",
+      "alpha_below_one": "125004871863877499998051004449/125019489955217650665401004449",
+      "alpha_below_one_fourth": "124961017589857047996001004449/500077959820870602661604017796",
+      "delta_positive": "1/800000000",
+      "delta_below_one_eighth": "99999999/800000000",
+      "short_record_fallback": "3124756306320288991609616461252697967793601/3125487248880441266635025111225000000000000",
+      "small_field_exposure": "9746227754999996102008898/125019489955217650665401004449",
+      "artificial_boundary": "700128635079475474301789581367398995551/100015591964174120532320803559200000000",
+      "literal_scalar_guard": "402943837929348086347490890940",
+      "row_product_gap": "40024/25",
+      "original_prefix_above_kappa": "786538948061330309871047284761/78137181222011031665875627780625000000",
+      "coordinate_movement_above_kappa": "305551/25000000000000",
+      "compact_phase_layer_above_kappa": "20668543364879972849164027819/312548724888044126663502511122500000000",
+      "bulk_exposure_above_kappa": "305551/25000000000000",
+      "Gaussian_arithmetic_above_kappa": "97459317411097999618605138287111327/2500389799104353013308020088980000000000",
+      "scalar_work_above_kappa": "194921594961097960638694118287111327/2500389799104353013308020088980000000000",
+      "dimension_above_kappa": "312475634452011916691632483971664027819/312548724888044126663502511122500000000"
+    },
+    "margins": {
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+}
diff --git a/certificates/stopped-product-network.json b/certificates/stopped-product-network.json
new file mode 100644
index 0000000000..998a8a7ea2
--- /dev/null
+++ b/certificates/stopped-product-network.json
@@ -0,0 +1,482 @@
+{
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+      "contract": "W_complex^D_complex * W_coarse^D_coarse * W_old^D_old; one preceding prefix and one padding; sequential reuse",
+      "degree": 4000,
+      "degree_gap": "40024/25",
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+  "kappa": "194869/2500000000",
+  "proof_predecessor_commit": "11817ccacb564bb7f98789c20dc11d3fece207e3",
+  "scope": "Exact finite moments, stopped savings and assembly; the new complex producer is checked separately. Opposite-bank factorization, atom streaming, ordinary conversion, common exact grid and retained analytic/tape interfaces are written proof dependencies.",
+  "source_sha256": {
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+    "certificates/stopped-product-bit-axis.json": "52f493a0019b7e4ce69a4c26bd3d35af5d3397461126abc4be44d1430ae8930d",
+    "certificates/stopped-product-complex-input.json": "5bcf00402bbfb768bd2fc0935f32ad329ee0780a3dfe5827bb010831d7909c4d",
+    "certificates/stopped-product-input.json": "8185c7bf8c307728f32484c5972af9a1b948ac5d0838e02f85c86cb438060417",
+    "notes/stopped-product-assembly.tex": "1c3c7147851f30f1ff97679cfcda157b1cbe2655b8766e34f1345f20c630eaec",
+    "notes/stopped-product-complex.tex": "4738089103a4b263b3b0ef458f5ef35519c696f1ea871ae79e508d3720f5870a",
+    "notes/stopped-product-factorization.tex": "70d5a4405e1be9c7b6cfb326bb4ffdf88e5585d1fb3d16c7bb5ad62ac421fae3",
+    "notes/stopped-product-interface.tex": "454a8f3924afc26e575b2a8cec6d0860397cb7f2dc431f3ba63fb8a7142cc96b",
+    "notes/stopped-product-note.tex": "486d14c599f4c2551c019f3358898f72e6ecafeb04ad590726ea69ed987fe025",
+    "scripts/copied_centers/physical.py": "837345d1f8753eab0e70f105b1d1bb0db965b993da037b1825ddef3672609d7d",
+    "scripts/copied_centers_network.py": "409ebfbd8a41c4c4d089aa0949fcf5a949415dc26f57440ffd48b7bd8881e6a9",
+    "scripts/partial_swap_network.py": "acdd81b4dea2815f8838a0c3ea8ac4968f2bc13fd7c99173473d865779096782",
+    "scripts/stopped_product/README.md": "6c1805dda45b01ed0ef420c674ec2cd0048fb9d7a1b015ab6cd837067fe66176",
+    "scripts/stopped_product/__init__.py": "7b794c22bccc00bb6ae8ba4d3379f469570565bbcb1a86090699133b8cf1b7d3",
+    "scripts/stopped_product/complex.py": "407650d56dceb5cb09d74928658cebed1bb42f4d7e28ca9e3fd36eea075f1ad0",
+    "scripts/stopped_product_network.py": "a8baaa87dd7679db4f2dd1a18a720397ee6f443262ad04fa26cf51b39909b74c",
+    "scripts/stopped_product_producer.py": "2c5c71399136ebcfebb8e5277317d5f334eb8df5716ad287391d0996fcc1492c",
+    "scripts/structured_bulk_assembly.py": "e99467e14211fba699878ed036e1e2362eb8e395dc22f9ecdcecc7fbf2cc736d"
+  },
+  "status": "Conditional stopped product-ring multiplication witness"
+}
diff --git a/docs/reproducibility.md b/docs/reproducibility.md
index e48f4190cd..7b966039e0 100644
--- a/docs/reproducibility.md
+++ b/docs/reproducibility.md
@@ -1,5 +1,10 @@
 # Reproducing the result
 
+For the new stopped product-ring extension, run `make stopped-product-verify`.
+Its [incremental reproduction guide](stopped-product.md) covers the h24
+rational-center producer, new moments, stopping parameters and assembly.
+The community checkpoint and historical targets below retain their own scope.
+
 The selected release is documented in the [joint-frame community review](research/community-round2-review.md)
 and [current status](research/current-status.md), with the
 [selected parameter certificate](../research/matrix-exponent-synthesis/candidate/arithmetic.json).
diff --git a/docs/stopped-product.md b/docs/stopped-product.md
new file mode 100644
index 0000000000..a53cfa9638
--- /dev/null
+++ b/docs/stopped-product.md
@@ -0,0 +1,46 @@
+# Stopped product-ring reproduction
+
+The extension gives the conditional saving `194869/2500000000 = 7.79476e-5`.
+It starts from merged PR #36 and is submitted on upstream main at
+`56b66d58297deca1d7dd130247d720e960f77a37`.
+
+From the repository root, with Python 3.11+, Make and a C++17 compiler:
+
+```sh
+make stopped-product-verify
+```
+
+The incremental checks:
+
+- Match the unchanged positive-label h23 bit producer to its verified
+  copied-center input, preserving the same matching and histogram.
+- Rebuild only the new h24 all-disjoint complex producer with aligned pair
+  stars, exact rational center decoding, support and binary-frame checks,
+  then reproduce every matching count and histogram entry.
+- Reconstruct both child lists and strict rational moments; derive the
+  stopped ordinary-bit saving and check that the atom-adapter toll is lower.
+- Check the denominator-21 semantic charge, product row stock `p^4000`,
+  seven assembly margins and all 47 strict constraints.
+
+Intermediate DAGs and the matcher executable are temporary. To retain them:
+
+```sh
+python3 scripts/stopped_product_producer.py --work-dir /tmp/stopped-product-producers
+```
+
+For arithmetic alone, use `make stopped-product-certificate`. The inherited
+`make verify-producers` also includes this target; the full `make verify`
+retains the community and historical checks. Those unchanged suites are
+not part of this round's local incremental validation.
+
+The [proof](../notes/stopped-product-note.tex) is supplied as LaTeX source;
+no PDF is generated. Its common-basis factorization, atom streaming and
+stopping argument, outer ordinary conversion and shared exact rational
+grid are written proofs. Finite checks verify the selected scalar data and
+arithmetic, not the complete all-size theorem.
+
+[SOURCES.json](../SOURCES.json) and [NOTICE](../NOTICE) preserve contribution
+credits and source pins. Existing two-stage, ordinary-leaf and semantic
+proof dependencies remain in `references/copied-centers/` and
+`references/semantic-bulk/`. Research alternatives, raw binary DAGs and
+recursive predecessor copies are excluded from this submission.
diff --git a/notes/stopped-product-assembly.tex b/notes/stopped-product-assembly.tex
new file mode 100644
index 0000000000..3932d3ade5
--- /dev/null
+++ b/notes/stopped-product-assembly.tex
@@ -0,0 +1,108 @@
+\subsection{Simultaneous row reserves}
+Use a conservative global width bound $e$ for every adapter and child.
+For each recursive family $j$, the least integer $t_j$ with
+$m_j^{t_j}>2r_j^{t_j}$ gives depth at most
+$t_j\lceil\log_2\max(2,e)\rceil$ plus a fixed constant.
+The new coarse bit family has $(t_j,\operatorname{bitlength}W_j)=(16,28)$,
+the ordinary leaf family $(9,28)$, and the complex family $(17,28)$.
+The required reserve is the product of all three stocks:
+\[
+ Q=W_c^{D_c}W_b^{D_b}W_{\rm old}^{D_{\rm old}}.
+\]
+After the complex and coarse divisions, the old stock remains available
+to every peeled-atom or stopped-leaf call. Sequential copies and outer
+ordinary wrappers reuse restored rows; they introduce no independent
+row factor. The combined coefficient is
+\[
+ 16\cdot28+9\cdot28+17\cdot28=1176,
+ \qquad4000-\frac{51}{25}\,1176=\frac{40024}{25}>0.
+\]
+Thus the retained logarithmic comparison, for eventual $e\le Cp$,
+supplies a sufficient $p^{4000}$ stock including the fixed constants.
+Use one physically preceding transformed prefix and one complete-row
+padding, as in \path{notes/structured-bulk-assembly.tex}. Complete
+spectators and parked fields are preserved by every descendant.
+The suffix condition
+$b_{\rm in}^{1-\epsilon}>16000(\log_2b_{\rm in}+8)$ is eventual
+for every fixed $\epsilon<1$.
+
+\subsection{Analytic transfer and selected parameters}
+The retained analytic proof map in
+\path{notes/structured-bulk-assembly.tex} supplies the active-masked
+arbitrary-source router with paid repair scans, the rounded phase-cell
+inverse, persistent bulk halos with original period cuts, sequential
+overlap tapes and factor pages, prime packing and exact recovery.
+The new ordinary bit primitive has precisely the arbitrary-width,
+complete-spectator and restored-row contract consumed by that router.
+The new complex primitive supplies the exact completed tensor contract
+on one common grid. Its outer grid conversions add only logarithmic
+work. The changed finite implementations therefore meet the same
+analytic interfaces, including the full bulk bound
+\[
+ O\!\left(Tp\left[p^\tau\operatorname{polylog}p+
+ d\operatorname{polylog}p+dp^\delta+w_G^2p^\delta+
+ dw_G^2p^\delta/L\right]\right)+n^{o(1)},
+\]
+where $L=2^{\lceil8\log_2p\rceil}$, Gaussian band width
+$w_G^2=O(p^{1-r})$, and normalization retains $\epsilon+r<1$.
+Here $w_G$ is unrelated to the address atom width.
+
+Use the proved ordinary bit saving $a_b=803380799/10^{13}$ and complex
+saving $b=1949/25000000$. To instantiate the retained assembly, weaken
+the bit parameter explicitly to
+\[
+ \beta=10^{-6},\qquad
+ a=(1-\beta)b-10^{-10}=\frac{1948995551}{25000000000000}<a_b.
+\]
+Thus the bit time bound also holds at $\tau=1-a$, and the complex
+leaf saving exceeds $a$ by exactly $10^{-10}$. Put $\sigma=1-b$ and
+\[
+\begin{gathered}
+ \eta=10^{-8},\quad q_0=a(1-2\eta),\quad c=q_0(1+\eta),\quad
+ \epsilon=\frac{1-\eta}{1+c+q_0},\\
+ \lambda'=1-q_0,\quad\lambda=(\tau+\lambda')/2,\quad
+ g=\epsilon q_0,\quad r=(g+1-\epsilon)/2,\quad\delta=\eta/8.
+\end{gathered}
+\]
+The notation $q_0$ denotes the assembly saving parameter, not the
+fixed address prime. Since $\sigma<\tau$, the internal paid-adapter
+exponent is $\tau$; the stopped complex leaf exponent is
+$\sigma+\beta(1-\sigma)$. Both are strictly below $\lambda'$.
+
+\begin{theorem}[Conditional fixed-tape multiplication bound]
+Under the retained analytic, exact-recovery and fixed-tape interfaces,
+the selected construction gives
+\[
+ T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad
+ \boxed{\kappa=\frac{194869}{2500000000}=0.0000779476}.
+\]
+\end{theorem}
+\begin{proof}
+The exact certificate checks 47 strictly positive conditions of the
+retained semantic/bulk assembly with the new finite constants. They
+include the primitive and leaf orderings, compact reservations and
+geometry, linear precision, row product, short-record fallback,
+normalization, phase-cell/band separation, prime supply and all
+artificial-boundary charges. The seven final savings are
+\[
+ 1-\epsilon(1+c),\quad a,\quad g,\quad a,\quad
+ \min(1-\epsilon-\delta,r-\delta),\quad
+ 1-\epsilon-\delta,\quad\epsilon.
+\]
+Their exact minimum is $g$, and the prefix margin is $g+\eta$.
+Exact rational subtraction gives
+\[
+ g-\kappa=
+ \frac{20668543364879972849164027819}
+      {312548724888044126663502511122500000000}
+ >6.6129027953\times10^{-11}>0.
+\]
+The strict gaps absorb fixed logarithms, including the common-grid
+conversions. The stopped atom recurrence supplies the ordinary
+interchange interface, the rational-center construction supplies the
+exact complex interface, and the product stock supports all nested
+recursions. The unchanged analytic transfer then proves the stated
+bound, with its constructive setup, descriptor and catalogue domination,
+prime-existence and exact-recovery thresholds retained as eventual
+conditions.
+\end{proof}
diff --git a/notes/stopped-product-complex.tex b/notes/stopped-product-complex.tex
new file mode 100644
index 0000000000..4a9037137c
--- /dev/null
+++ b/notes/stopped-product-complex.tex
@@ -0,0 +1,137 @@
+\subsection{All-disjoint retained centers}
+Use two factors $h=24$. For triple inputs write $G=\sum_TX_T$,
+$T_i=\sum_{T\ni i}X_T$, and retain all $A_i=G-T_i$.
+Each input triple avoids exactly 21 points, so
+\[
+ \sum_{i=1}^{24}A_i=21G,\qquad G=\frac1{21}\sum_i A_i.
+\]
+For a target triple $S$ the central correction is
+\[
+ G-\frac12\sum_{i\in S}A_i
+       =\frac{\sum_{i\in S}T_i-G}{2}.
+\]
+An input triple meeting $S$ in $j$ points has central coefficient
+$(j-1)/2$. Adding half the disjoint sum and subtracting half the
+intersection-two sum leaves coefficient one exactly when $j=3$.
+The scalar source-to-scatter transfer is therefore the identity.
+The transparent old-value subtraction and new-value addition use this
+identity only on fresh contributions and restore arbitrary old values.
+
+The base-two paired exclusion and aligned pair-star producer retains
+nested binary labels: common-pair supports have their orthonormal triple
+span, other ordinary sums have their coordinate cover. Each retained
+$A_i$ has the nondegenerate coordinate hyperplane excluding $i$,
+of dimension 23. Hence all copied-center arguments and general
+nondegenerate binary residuals of the preceding proof apply. The
+selected local counts are
+\[
+ v=2024,\quad c_h=45764,\quad q_h=8120,\quad
+ |\mathcal M|=8966,\quad R=44918,\quad\ell=552.
+\]
+The old internal histogram, starting at rank zero, is
+\begin{small}\begin{align*}
+[&8930,49042,30844,15972,11883,8272,6399,4356,6682,2263,\\
+ &5109,1808,5317,1733,5391,1757,4908,3290,6233,4363,\\
+ &4056,264,564,24,24].
+\end{align*}\end{small}
+It has rank mass $24R+2\ell$. Replace 24 rank-24 original transitions
+by 24 rank-one transitions; all rank-23 copied transforms remain.
+The new mass is $24R+\ell$.
+
+\subsection{Physical endpoints and strict complex moment}
+The copied-center two-stage complex contract in
+\path{notes/copied-centers-complex.tex} is unchanged by the scalar
+center decoding. For the tensor triple line $U$, let
+$T=C_U$, $F=C_I$, $E=C_{U^\perp}=FT^{-1}$.
+The two data outputs are $A=-Fy$ and $B=FT^{-2}x+Ey$.
+Add $T^{-1}$ applied to a copy of $A$ into $B$, paying one rank-one
+child per data pair. The inherited input/output $Z$ phases and
+$i^{9f}$ scalar normalize $FT^{-2}$ to $F$; sign and bank corrections
+complete the exact tensor $C$. Auxiliary source/sink operators remain
+$C_{D_0}$ and $FC_{D_0}$ in the second stage. Every original role
+therefore returns the exact selected-axis tensor operation on arbitrary
+inputs. This algebraic statement holds over $\mathbb Q(i)$ as well as
+over Gaussian dyadics.
+
+With $m_c=576$, $N_c=v^2=4096576$, $B_c=vR=90914032$, put
+\[
+ W_c=2N_c+2B_c=190021216,\quad L_c=2v\ell=2234496,
+\]
+\[
+ s_c=W_cm_c-N_c+L_c=109450358336,\qquad N_c-L_c=1862080.
+\]
+The complete list is $2B_c$ width-552 auxiliary exteriors,
+$2N_c$ width-529 data macros, $2vH'_r$ width-$r$ internal children,
+$4N_c$ width-23 data fronts, and $N_c$ width-one endpoint copies.
+These disjoint classes include both kinds of paid copies and sum to
+$s_c$. All binary residuals, including alternating forms, use the retained
+one-child Gauss normal form with paid bit adapters and unit phases.
+At $a_c=1949/25000000$ the exact rational enclosure gives
+\[
+ \sum_r\frac{n_r}{W_c}(r/m_c)^{1-a_c}
+ <1-1.3338373969\times10^{-10}.
+\]
+
+\subsection{One exact odd-denominator grid}
+The only new odd divisor is 21. Let $d$ bound the root axis count,
+$D$ bound its maximum complex recursion depth, and $G_0$ bound the
+number of grouped scalar operations per node. Choose
+\[
+ K=G_0(D+1),\qquad \text{common grid }2^{-P}21^{-K}.
+\]
+Root inputs and arbitrary dirty auxiliaries start Gaussian dyadic.
+Multiply their integer numerators by $21^K$ once at the outer boundary.
+Every local coefficient contains at most one factor 21 in its denominator,
+so a local finite program adds at most $G_0$ to the odd-denominator
+exponent. A completed $u$-axis child has Gaussian-dyadic coefficients
+and therefore preserves its input's existing odd-denominator exponent;
+it adds no new odd factor. Only the active chain of unfinished calls
+can accumulate new factors. Its length is at most $D+1$, proving that
+all intermediates lie on the common grid and every local division by
+21 is an exact integer division of its encoded numerator.
+
+This divisibility claim uses the root dyadic-input invariant, not an
+assumption about arbitrary numerators on the finest grid. Internal values
+may have odd denominators. They are neither rounded nor re-encoded when
+a child returns. At the completed outer return the operator is the exact
+Gaussian-dyadic tensor on root dyadic inputs, so the final numerators
+are divisible by $21^K$; exact unscaling restores dyadic encoding.
+
+Since the largest child is $552<576$, $D=O(\log d)$ and
+$K=O(\log d)$ with fixed constants. Constant-divisor long division
+is linear in numerator length. Initial and final scaling by $K$
+successive constant multiplications/divisions cost $O(V\log d)$ and
+occur only at the outer boundary. The extra numerator length is
+$O(\log d)$, preserving the retained $O(p)$ word bound. This overhead
+is subordinate to the positive-power complex layer bound.
+
+For explicit semantic constants use
+\[
+ g_h=4(c_h+v)+10v+4hv+4h^2+8h+8=408200,
+\]
+\[
+ G_0=N_c+2v(g_h+2h)=1656684480,
+\]
+where endpoint additions and copied-center scalar work are included.
+The complete child contributes at most $u$ binary denominator bits
+and row-norm factor $2^u$, with no additional odd exponent. A local
+21-division contributes fewer than five denominator-height bits,
+covered by the fixed scalar charge. Set
+\[
+\begin{aligned}
+ E_0&=64(W_c+m_c+G_0+1)^3=403063477262941121443761178688,\\
+ \mathcal E&=2G_0W_c^2+8s_c+4W_c+4+32m_c\\
+           &=119639333593035096270287748<E_0,\\
+ B_0&=s_c+E_0,\qquad C_0=32m_cB_0^2.
+\end{aligned}
+\]
+With $f=\lfloor e/m_c\rfloor$, the completed-child induction is
+\[
+ A(e)\le A(552f)+s_cf+E_0\le2B_0e,
+\]
+since $2B_0(m_c-552)\ge s_c+E_0$. It counts total denominator height
+and magnitude, including the local odd factor. The retained prefix and
+normalization allowance $18e$ fits inside $C_0e$. Together with the
+$O(\log d)$ common-grid encoding overhead this gives $C_1=1$ and
+$O(p)$ records for $d=\lfloor b_{\rm in}^{\epsilon}\rfloor$,
+$p=6b_{\rm in}$, fixed $\epsilon<1$.
diff --git a/notes/stopped-product-factorization.tex b/notes/stopped-product-factorization.tex
new file mode 100644
index 0000000000..94f7b81397
--- /dev/null
+++ b/notes/stopped-product-factorization.tex
@@ -0,0 +1,112 @@
+\subsection{Atom algebra and the global frame convention}
+Fix an odd prime $q$ and an atom width $w\ge1$ throughout one invocation.
+An atom is a consecutive $w$-digit radix-$q$ field. A field of $f$ atoms
+is the product ring $R_f=(\mathbb Z/q^w\mathbb Z)^f$, with rational
+unit coefficients acting componentwise. Let $C_f$ reverse the atoms,
+preserving their internal digits. The target is
+\[
+ F_n(H,D)=(C_nD,C_nH).
+\]
+It acts on two contiguous chunks with arbitrary complete spectators.
+The product-ring interpretation preserves the original complete address
+set and its lexicographic order. No standalone reversal is assumed free.
+
+At a node with $n=mf$ atoms, conjugate every frame, source gauge and
+endpoint of the finite network by the same $\operatorname{diag}(I,C_n)$.
+For a projector $P$ the new partial swap is
+\[
+ D'_P=\begin{pmatrix}
+ (I-P)\otimes I_f&(PC_m)\otimes C_f\\
+ (C_mP)\otimes C_f&C_m(I-P)C_m\otimes I_f
+ \end{pmatrix}.
+\]
+Global conjugation preserves common-frame scalar identities, nested
+residuals, copied-center schedules and arbitrary auxiliary restoration.
+The full endpoint is now $F_n$. These frame choices define the logical
+interpretation of physical inputs; they are not preprocessing passes.
+
+\subsection{Simultaneous generic basis and full involution factorization}
+For every proper nonzero residual projector $P_j$ of rank $r_j$, require
+all leading principal minors of sizes $1,\ldots,r_j$ to be nonzero in
+one common rational basis. For each individual minor, similarity to
+$\operatorname{diag}(I_{r_j},0)$ witnesses nonvanishing. Clearing
+denominators and taking the finite product on irreducible
+$\operatorname{GL}_m$ proves a common rational choice exists; exact
+rational enumeration terminates. Include exteriors, data residuals,
+internal edges, copied centers and endpoint corrections in this finite
+list. The basis is fixed label preparation, not an executed dense map.
+
+For $M=PC_m$, every northeast $i$-by-$j$ rectangle has rank
+$\min(i,j,r)$: after reversing its columns it is the leading rectangle
+of $P$ and contains the corresponding nonsingular leading square.
+Hence physical lower-triangular matrices $L,R$ give
+\[
+ LMR=\Pi=EF^t,\quad E=(e_1,\ldots,e_r),\quad
+ F=(e_m,e_{m-1},\ldots,e_{m-r+1}).
+\]
+The full involution, not just this corner, must be normalized.
+At $f=1$, put $Q=\operatorname{diag}(L,R^{-1})$ and
+$g=QD'_PQ^{-1}$. Since $\operatorname{rank}(g-I)=r$ and its upper-right
+block is $EF^t$, there are $A\in\mathbb Q^{r\times m}$ and
+$B\in\mathbb Q^{m\times r}$ with
+\[
+ g-I=\binom EB(A,F^t),\qquad AE+F^tB=-2I_r.
+\]
+The second identity follows from $g^2=I$ and the full column and row
+ranks of the factorization. Define
+\[
+ K=-(B+F)E^t+FA(I-EE^t),\qquad
+ S=\begin{pmatrix}I&0\\K&I\end{pmatrix}.
+\]
+Then $KE=-B-F$, $F^tK=A+E^t$, and direct multiplication yields
+\[
+ SgS^{-1}=I+\binom E{-F}(-E^t,F^t)
+ =\begin{pmatrix}I-EE^t&EF^t\\FE^t&I-FF^t\end{pmatrix}.
+\]
+This swaps precisely the first $r$ head coordinates and last $r$ tail
+coordinates in opposite orders, fixing all others.
+
+For $f$ atoms use $Q_f=\operatorname{diag}(L\otimes I_f,R^{-1}\otimes I_f)$
+and $S_f=\left(\begin{smallmatrix}I&0\\K\otimes C_f&I\end{smallmatrix}\right)$.
+The rank factorization lifts with $\bar E=E\otimes I_f$,
+$\bar F=F\otimes C_f$, $\bar A=A\otimes I_f$ and
+$\bar B=B\otimes C_f$. Its middle cross block is $\Pi\otimes C_f$.
+Since $C_r\otimes C_f=C_{rf}$ on the selected physical intervals,
+it is exactly one child $F_{rf}$ of the same family. There is no
+recursive orientation conversion or extra gathering.
+
+Choose $q$ after the finite basis and factorizations, excluding all
+needed nonzero numerators, denominators and pivots. The identities hold
+on every atom ring. They use only componentwise scalar arithmetic;
+reversal commutes with those scalars, without inter-atom carries.
+
+\subsection{Selected finite network}
+Use the retained compatible positive-label producer on both axes $h=23$:
+\[
+ v=1771,\quad c=34764,\quad q_h=5336,\quad
+ |\mathcal M|=1324,\quad R=38776,\quad\ell=506.
+\]
+The copied-center lemma changes $H_{23}$ by $-23$ and $H_1$ by $+23$.
+Thus $\sum_r rH'_r=23R+506$; the rank-22 copies are still paid.
+The new common basis preserves the producer's scalar map, label
+inclusions, matching and endpoint identities. With
+\[
+ m_b=529,\quad N_b=3136441,\quad B_b=68672296,\quad
+ W_b=143617474,\quad L_b=1792252,
+\]
+the complete list is $2B_b$ width-506 auxiliary exteriors,
+$2N_b$ width-484 data residuals, $4N_b$ width-22 data fronts,
+$2vH'_r$ width-$r$ local residuals, and $N_b$ width-one endpoint copies.
+Every call has volume fraction $1/W_b$, and
+\[
+ s_b=W_bm_b-N_b+L_b=75972299557,\qquad W_bm_b-s_b=1344189.
+\]
+Every width is at most $506<m_b$. For $a_*=4019/50000000$,
+$\tau_*=1-a_*$, the exact rational moment certificate gives
+\[
+ \mu_{\tau_*}:=\sum_r\frac{n_r}{W_b}(r/m_b)^{\tau_*}<1,
+ \qquad 1-\mu_{\tau_*}>4.2909639537\times10^{-10}.
+\]
+It uses the retained 24-term positive atanh-series logarithm enclosure,
+its geometric tail, range reduction and upward rational exponential
+bound. Also $\mu_1=s_b/(W_bm_b)<1$. These are the two gaps used below.
diff --git a/notes/stopped-product-interface.tex b/notes/stopped-product-interface.tex
new file mode 100644
index 0000000000..20c44ce486
--- /dev/null
+++ b/notes/stopped-product-interface.tex
@@ -0,0 +1,95 @@
+\subsection{Paid atom adapters and remainders}
+Expand each fixed lower-triangular formal factor into unit scalings and
+additions from earlier to later formal fields. Execute intrabank targets
+in descending formal-field order, so earlier controls still have their
+required original values. Every atom update reads only physically earlier
+controls. In a cross-bank shear the update is
+\[
+ D_{i,t}\leftarrow D_{i,t}+k_{ij}H_{j,f-1-t}.
+\]
+Even with the reversed source index, the source lies in the earlier head
+bank. The retained ordered-affine streaming primitive applies to each
+ordinary width-$w$ atom update, costing $O(V)$ per pass.
+There are $O(f)$ passes per fixed matrix and thus $O(Vn)$ per finite
+node. The reversed control is an indexed earlier atom, not a reversed
+field operation. Atom loops use binary descriptors on a fixed tape set;
+all inactive digits remain complete spectators. The inherited parking
+schedule never scans through parked ancestors, and polynomial descriptor
+work is absorbed by the complete stream volume.
+
+For $n=m_bf+\delta$, $0\le\delta<m_b$, pair the first $\delta$
+head atoms with the last $\delta$ tail atoms in reverse order. Each pair
+uses the retained ordinary width-$w$ interchange. The remaining two
+contiguous $m_bf$-atom intervals receive the finite construction.
+When $n<m_b$, at most $m_b$ such ordinary calls complete $F_n$.
+Both cases cost $O(Vw^{1-a_{\rm old}})$ in addition to the node adapters,
+where $a_{\rm old}=384599/10^{10}$. A radix-$q$ atom can be padded to
+the next binary power and treated by that ordinary interface, with
+volume factor below four and binary width $O(w)$, so the old leaf
+algorithm need not use the same address prime.
+
+\subsection{Stopped recurrence}
+Let $T(n,w)$ denote time normalized by logical volume. The complete
+recursive list, including copied streams, yields
+\[
+\begin{aligned}
+ T(n,w)&\le\sum_r\frac{n_r}{W_b}
+       T(r\lfloor n/m_b\rfloor,w)+C n+Cw^{1-a_{\rm old}}
+                         &&(n\ge m_b),\\
+ T(n,w)&\le Cw^{1-a_{\rm old}}&&(1\le n<m_b).
+\end{aligned}
+\]
+The constant includes finite scalar gates, split/merge, parking, all
+adapter passes and remainders at every node. Choose constants $A,B$
+using the positive gaps $1-\mu_1$ and $1-\mu_{\tau_*}$.
+Strong induction, with child widths at most $(r/m_b)n$, gives
+\[
+ T(n,w)\le An+Bn^{\tau_*}w^{1-a_{\rm old}}.
+\]
+Indeed the linear child terms use at most $\mu_1 An$, leaving the
+linear toll, and the other child terms use at most
+$\mu_{\tau_*}Bn^{\tau_*}w^{1-a_{\rm old}}$, leaving the peeled-atom
+toll since $n^{\tau_*}\ge1$. Enlarge $B$ for the bounded base cases.
+All constants are independent of both $n$ and $w$.
+
+\subsection{Outer ordinary wrapper and arbitrary widths}
+Write $C=C_n$. The map $F_n(H,D)=(CD,CH)$ obeys
+$F_nLF_n(H,D)=(H+D,D)$ for $L(H,D)=(H,D+H)$.
+Accordingly the chronological sequence
+\[
+ D\leftarrow D-H;\quad F_n;\quad D\leftarrow D+H;\quad
+ F_n;\quad D\leftarrow H-D
+\]
+is exactly $(H,D)\mapsto(D,H)$. The three lower updates cost $O(Vn)$
+in the atom model. Both calls reuse their restored row reserve.
+This is a constant outside the entire recursion, so it does not multiply
+any recursive child count or characteristic.
+
+For a radix-$q$ width $e$ choose $\theta=1/1000$,
+$w=\lceil e^\theta\rceil$ and $n=\lfloor e/w\rfloor$.
+Apply the ordinary wrapper to the atom body and the old ordinary
+algorithm to the fewer than $w$ residual digits. For large $e$ the cost is
+\[
+ O\!\left(V\left[e^{1-\theta}+
+ e^{(1-\theta)(1-a_*)+\theta(1-a_{\rm old})}\right]\right).
+\]
+The remainder is bounded by the second term. Thus the ordinary saving is
+\[
+ a_b=(1-\theta)a_*+\theta a_{\rm old}
+     =\frac{803380799}{10^{13}}=0.0000803380799.
+\]
+Since $\theta>a_b$, the paid adapter term is strictly lower order.
+All bounded widths use the old interface. Numerical padding of a binary
+chunk range to the least suitable radix-$q$ range has volume factor
+below $q^2$; it preserves numerical order and is removed after the
+completed ordinary wrapper, whose endpoint is ordinary interchange.
+
+If external rows are unavailable, use the inherited ordinary high-digit
+borrowing construction from \path{notes/batched-bit-rows.tex}.
+Borrow $O(\log e)$ high digits, exchange and join them into a preceding
+row field, pad that field once, act on the body, and separate the high
+parts. It applies literally to the completed ordinary wrapper; no claim
+about borrowed high positions is made for an unwrapped reversed call.
+Single-digit moves cost $O(V\log e)$, and small widths use the old
+algorithm. Atom indices and rational coefficients are finitely specified;
+the algorithm uses the same fixed tape set and complete spectator ranges.
diff --git a/notes/stopped-product-note.tex b/notes/stopped-product-note.tex
new file mode 100644
index 0000000000..a8b23254c8
--- /dev/null
+++ b/notes/stopped-product-note.tex
@@ -0,0 +1,76 @@
+\documentclass[11pt]{article}
+\usepackage[margin=1in]{geometry}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern,amsmath,amssymb,amsthm,mathtools,booktabs,longtable,microtype}
+\usepackage[hidelinks]{hyperref}
+\setlength{\emergencystretch}{3em}
+\allowdisplaybreaks
+\newtheorem{theorem}{Theorem}[section]
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{proposition}[theorem]{Proposition}
+\title{A conditional integer-multiplication saving of $194869/2500000000$\\
+\large Stopped product-ring interchange and rational retained centers}
+\author{icekylinx\\\small With substantial OpenAI GPT-6 Astra and Codex assistance}
+\date{9 October 2026}
+\begin{document}\maketitle
+\begin{abstract}
+A common rational basis and opposite bank orders reduce each projector
+residual to one contiguous reversed child over a fixed atom product ring.
+Ordered atom adapters are paid explicitly; stopping at a growing atom
+width and using the retained ordinary primitive supplies an ordinary
+bit interface. An independent 24-dimensional complex producer uses
+rational retained-center decoding on one exact common grid. The selected
+interfaces and retained semantic/bulk assembly give the conditional bound
+$T(n)=O(n(\log n)^{1-\kappa})$, where
+$\kappa=194869/2500000000=0.0000779476$.
+\end{abstract}
+
+\section{Scope and retained dependencies}
+The model is the retained fixed finite-alphabet multitape Turing machine.
+Finite graphs, rational bases, address primes, coefficient tables and tape
+counts are fixed independently of input length. Bit payload arithmetic is
+binary, while its address geometry is rational and then evaluated on atom
+rings. Complex payloads use exact Gaussian rational arithmetic with the
+specified common grid; their address labels remain binary.
+
+The proof predecessor is icekylinx's merged copied-center PR36,
+\texttt{11817ccacb564bb7f98789c20dc11d3fece207e3}, whose proof is
+\path{notes/copied-centers-note.tex}. Its ordinary interchange saving
+$a_{\rm old}=384599/10^{10}$, copied-center schedule in both orientations,
+two-stage data and auxiliary endpoint identities, general binary residual
+normal form, and fixed-tape complete-row contracts are retained.
+The present opposite-bank factorization, stopped product-ring interface,
+rational-center specialization and exact composition are by icekylinx,
+with substantial OpenAI GPT-6 Astra assistance and Codex integration.
+Earlier own contributions include PR10 batching, PR18 partial swaps,
+PR24 endpoint gauges and PR32 structured blocks and semantic composition.
+
+Aurel Prosz (Paureel) supplies the two-stage topology and paid endpoint
+copy, with Swapnil Jain's linked development credited by Zhihao Chen's
+PR29. The retained leaf proof also credits Rohan Arun's PR31 corner method
+and Dominik Scholz's PR33 parameterization. Zhihao Chen's PR21/23 and the
+RaD/hipotures PR20 source provide the phase endpoints, semantic guard,
+arbitrary-source routing, phase-cell inverse and bulk locality/tape
+interfaces. Their pinned proof map remains in
+\path{notes/structured-bulk-assembly.tex}; adopted sources and notices are
+in \path{references/semantic-bulk/} and \path{references/copied-centers/}.
+The underlying Douglas Colkitt framework, OpenAI manuscript, eumemic,
+Bortlesboat, dleen, earlier finite contributions and Harvey--van der Hoeven
+analytic work retain their credits and recorded AI assistance in
+\texttt{NOTICE} and \texttt{SOURCES.json}.
+
+The conclusion remains conditional on those analytic, ordered-affine,
+exact-recovery and machine interfaces and their eventual setup and
+prime-existence thresholds. The new algebraic and tape constructions
+below establish the changed finite interfaces; exact arithmetic certifies
+their selected counts and margins.
+
+\section{One reversed child for each projector}
+\input{notes/stopped-product-factorization.tex}
+\section{Stopped atom recursion and ordinary interchange}
+\input{notes/stopped-product-interface.tex}
+\section{Rational centers and exact semantic precision}
+\input{notes/stopped-product-complex.tex}
+\section{Product rows and exact assembly}
+\input{notes/stopped-product-assembly.tex}
+\end{document}
diff --git a/research/cyclic-deferred/BITSHARING.md b/research/cyclic-deferred/BITSHARING.md
new file mode 100644
index 0000000000..055a73351e
--- /dev/null
+++ b/research/cyclic-deferred/BITSHARING.md
@@ -0,0 +1,73 @@
+# Shared completed bit cores
+
+The input is the immutable minimal-V scalar schedule included by an664 in
+PR128, built on Swapnil Jain's round-seven word. The standalone bit package
+reconstructs the physical event word from that pinned schedule, checks actual
+V chronology and frame geometry, and replays every formal source/dirty column
+in both directions. It does not claim to regenerate an unpublished optimizer
+that originally selected the minimal-V schedule.
+
+For h=23 use the inherited bilinear form H=9I-J. A triple indicator t has
+H-norm 18. Its exact rank-one projector is
+
+    P_t = t (3 t^T - 1^T) / 6.
+
+Distinct triple projectors annihilate each other when the triples meet in
+exactly one point. For an original role with gauge projector P_sigma, a
+completed core has residual D_E with
+
+    E = (I-P_sigma) tensor P_t,
+    D_E = [[I-E,E],[E,I-E]].
+
+The reflected orientation transposes the tensor factors. If E and F annihilate
+each other, direct block multiplication gives D_E D_F = D_(E+F). The same
+identity holds for arbitrary physical scratch and spectators; it does not
+require clean or independently distributed dirty inputs. The signed logical
+word individually restores its original auxiliary coordinates before the
+framing residual is applied.
+
+The new explicit partition consists of 161 common-center sunflower groups,
+all of size 11. Seven base classes are translated through Z/23Z. The generator
+orients the 77 translation orbits using quadratic residues and nonresidues.
+Fifty-five choices are forced; an exact search resolves the other 22, producing
+a 7-regular bipartite graph. Seven perfect matchings produce the base classes.
+The exported witness is checked independently for complete triple coverage,
+pairwise H-orthogonality, exact projector idempotence and self-adjointness.
+No optimality claim for arbitrary constructions is needed.
+
+Share an original dirty role only between completed cores in one group. For
+J of size 11, the remaining exterior projector is
+
+    I - (I-P_sigma) tensor sum_(t in J) P_t,
+    rank = 529 - 11*(23-dim sigma).
+
+Every old internal event and paid data/copy/endpoint child remains in the
+profile. Each group exterior is a whole proper projector child, including its
+paid adapters. The physical width becomes 15,567,734, total child rank
+8,233,987,097, and largest child 518. The rank deficit stays 1,344,189.
+The exact coarse saving is 129310447/10^12. The retained stopping fraction
+1/1000 and ordinary leaf saving384599/10^10 give ordinary bit saving
+129219596453/10^15. The new maximum child has halving degree33.
+
+All projectors are put into one common rational basis for the enlarged finite
+family. Only then is a fixed odd execution prime selected to avoid every
+resulting denominator and pivot numerator. A verification prime is not by
+itself an execution-prime certificate. The denominator6 already appears in
+the original triple projectors; the basis choice can add finitely many bad
+primes. These are modular atom-address coefficients and introduce no new
+Gaussian decoder division or exponent in the complex21-grid.
+
+The inherited PR104 ordinary-bit address adapters and ordered-affine streaming
+cost remain paid for every projector, including group complements. Their
+finite setup and O(Vn) local toll are absorbed by the strict stopped adapter
+exponent gap; they are not free bit permutations or constant-time tape
+instructions. The exact assembly uses the new physical width, halving degree
+and all three row stocks.
+
+The new cyclic partition, grouped bit profile and this composition were
+prepared for eumemic with OpenAI Codex assistance. an664's PR128 completed-core
+principle and minimal-V witness, Swapnil Jain's bit construction, icekylinx's
+stopped/opposite-bank interface, Zhihao Chen's physical/signed integration and
+all inherited notices remain credited. The general transparency, common-basis,
+field, normal-form, streaming, precision, analytic and recovery interfaces
+remain conditional assumptions.
diff --git a/research/cyclic-deferred/PR128-NOTICE b/research/cyclic-deferred/PR128-NOTICE
new file mode 100644
index 0000000000..7f73174c62
--- /dev/null
+++ b/research/cyclic-deferred/PR128-NOTICE
@@ -0,0 +1,26 @@
+Completed-core scratch sharing, 2026-10-09
+
+This standalone research contribution and its local verification tools were
+prepared with substantial OpenAI Codex assistance. It composes and extends
+the credited public constructions listed in README.md and PROOF.md. It does
+not claim independent external peer review, authorship by predecessor
+contributors, or an official OpenAI result or endorsement.
+
+Original source files and notices are retained without attribution changes
+under inputs/ and licenses/. Apache-2.0 remains applicable. The source-level
+credit to Zhihao Chen in the local physical-ledger adaptation identifies the
+inherited PR97 ledger; this local adaptation was not authored or reviewed
+by him unless separately stated in the public source.
+
+The exact PR117 source pin, source hashes and original README identify
+eumemic's producer and integration, with Anthropic Claude assistance.
+The round-seven files retain Swapnil Jain's license and notice. Earlier
+copied-center, producer, analytic, tape and semantic contributors retain
+all attribution in the preserved notices. No license or predecessor notice
+is superseded by this file.
+
+The explicit resolvable-packing generator implements the construction of
+Xiande Zhang and Gennian Ge, Journal of Combinatorial Designs 18 (2010),
+209-223, DOI 10.1002/jcd.20234, Lemmas 2.2 and 2.4. Their paper is not
+redistributed. The signed partial grouping and its finite checks are
+included in this contribution.
diff --git a/research/cyclic-deferred/README.md b/research/cyclic-deferred/README.md
new file mode 100644
index 0000000000..42ddec0b13
--- /dev/null
+++ b/research/cyclic-deferred/README.md
@@ -0,0 +1,107 @@
+# Completed-core sharing on both finite suppliers
+
+Conditional **κ = 64593102899/500000000000000 = 1.29186205798e-4 > 2⁻¹³**.
+
+This package combines 161-group bit sharing with physical hull frames,
+compensated scratch reuse and signed gauge-aware complex sharing. The exact
+coarse bit saving is **129310447/10¹²**; the stopped ordinary-bit saving is
+**129219596453/10¹⁵**. The complex saving is **130696544/10¹²**. Both suppliers,
+the ordinary wrapper, three row reserves, scalar work and final assembly are
+charged under the retained interfaces.
+
+| Finite supplier | Shared bit | Shared complex |
+|---|---:|---:|
+| block dimension | 529 | 576 |
+| physical local scratch roles | 28,866 | 26,597 |
+| completed-core groups | 161 | 87 |
+| original stream count W | 15,567,734 | 12,821,030 |
+| total recursive rank | 8,233,987,097 | 7,383,051,200 |
+| rank deficit | 1,344,189 | 1,862,080 |
+| largest child | 518 | 560 |
+
+The bit partition covers all 1,771 triples with 161 rationally orthogonal
+classes of eleven. Its independently frozen supplier replays the unchanged
+PR128 minimal-V word: exact original/new-frame geometry, 1,076,483 physical
+events, 32,408 formal input/dirty columns, reflected chronology and the PR104
+opposite-bank compiler. Its complete profile retains **659,476,769 paid
+projector adapter calls** and has exact halving degree 33. [BITSHARING.md](BITSHARING.md)
+gives the projector, partition and common-basis argument. The nested package
+keeps its own manifest, verifier, immutable word pin and original notices.
+
+The complex supplier retains PR117's 91,770 scalar additions and 71,185 carrier
+links. It optimizes nested nondegenerate frames at physical mixer occurrences,
+then identifies 2,108 disjoint completed-donor/deferred-recipient lifetimes.
+Each reused birth keeps its exact negative old-value compensation. The
+physical source/workspace word is inverted in its actual chronological order.
+A replay with arbitrary initial scratch verifies restoration and the target
+map; deliberately omitting compensation gives the expected wrong output.
+
+After reuse, the complex source inventory has 470 distinct frames on 26,597
+physical roles, including 2,598 nonzero source gauges. The independent literal
+reflection audit binds this inventory and the full local child histogram.
+The gauge checker verifies each rational projector and exact Gaussian phase,
+including alternating binary blocks, in both stage orientations. Grouping
+uses PR128's pinned signed partition: 83 groups of 24 outer triples and four
+of eight. A source frame of dimension s in a group of size g pays one complete
+complement child of width **576 − g(24 − s)** whenever that width is nonzero.
+Every internal transition, copied center, data projector and endpoint remains
+paid. [SHARING.md](SHARING.md) gives the operator and allocation argument.
+
+From the repository root, run:
+
+```sh
+python3 research/cyclic-deferred/verify.py
+```
+
+The verifier checks immutable hashes and works in a temporary repository-shaped
+copy. It first runs the nested bit verifier and exposes its checked profile.
+It then regenerates the complex compiler, literal reflection, signed outer
+partition, physical-gauge phases and witnesses, shared profile, and exact
+certificate. Source inventory hashes bind those stages together; the local
+complex profile and reflection receipt must also regenerate byte for byte
+before their hashes feed the gauge audit. The expanded
+core scalar count is bounded by its certificate reserve; the separate paid
+basis/phase-wrapper reserve is also checked. Bit adapter call counts and the
+new halving degree are bound to the nested exact certificate. Proof profiles
+contain exact ledgers; platform-dependent numerical root diagnostics are
+excluded. Regenerated proof records still require complete equality.
+
+All 47 final strict constraints and seven margins are positive. The next
+bit/complex grid points fail the chosen moment enclosure, and the next κ grid
+point fails exact assembly; these checks do not claim an optimal true
+characteristic root. The 33 outer controls reject missing/altered immutable
+inputs, repinned nested bit inputs, incorrect shared width or exterior charge,
+resurrected roles, duplicate source frames, ungauged exterior substitution,
+missing gauge complements, a one-stage-only bit loss, missing paid adapters,
+understated scalar or phase-wrapper reserves, and serialization drift in
+hash-bound generated records. The nested and reflection
+verifiers retain their own corruption controls.
+
+`certificate.py` checks its frozen result by default; `--write` is an explicit
+authoring operation. `verify.py --freeze-manifest` replaces the outer manifest
+and does not certify the package. The outer source closure includes every
+file in the independently pinned nested bit manifest. Its entry-point verifier
+is fixed independently; unused inherited helper APIs are outside that entry
+point. Verification has no option to skip the nested physical replay or the
+complex reflection checks.
+
+Credit: an664 with OpenAI Codex assistance (PR128's completed-core allocation,
+signed partition and minimal-V package); Xiande Zhang and Gennian Ge (the
+orthogonal packing, JCD 18 (2010), 209–223, DOI 10.1002/jcd.20234); eumemic with
+Anthropic Claude assistance (PR117's scalar DAG); jamesyc (PR124's compensated
+birth-cut reuse); Avi Eisenberg / ikeboy with Anthropic Claude assistance
+(PR110's compiler); Swapnil Jain (round-seven bit word); icekylinx, Zhihao
+Chen / jacklightChen, Rohan Arun, Aurel Prosz / Paureel and RaD / hipotures
+(retained finite, two-stage, analytic and tape interfaces). Concurrent frame
+optimization in DanieleCorso's PR126 and GamingPuzzled's PR125 is credited.
+The hull optimizer, 161-group bit partition, generalized gauge/reuse
+composition and verification were prepared for eumemic with OpenAI Codex
+assistance. Original notices remain authoritative, including
+[PR128-NOTICE](PR128-NOTICE) and the nested bit package's retained licenses.
+
+This is a conditional finite witness. Common rational bases and compatible
+address primes, paid ordered-affine adapters, nondegenerate Gaussian normal
+forms, completed-core transparency, stopped ordinary-bit transfer, exact odd
+grids, streaming, routing, recovery, prime selection, analytic estimates and
+fixed-tape implementation remain inherited assumptions. The finite checks do
+not prove those general interfaces or a practical running-time improvement.
diff --git a/research/cyclic-deferred/SHARING.md b/research/cyclic-deferred/SHARING.md
new file mode 100644
index 0000000000..32c3a80308
--- /dev/null
+++ b/research/cyclic-deferred/SHARING.md
@@ -0,0 +1,105 @@
+# Shared completed cores with physical source gauges
+
+Completed-core sharing with deferred starts: construction check
+
+Let sigma_s be the initial inner frame of physical scratch slot s and let
+E_s = sigma_s^perp in F_2^24. A completed scalar core restores every physical
+scratch slot individually and has no data/scratch off-diagonal block.
+Frame telescoping therefore gives actual scratch map C_end C_start^-1.
+In stage one this is C_F24 C_sigma^-1 = C_E. The reflected stage starts at
+zero and ends at E; after the retained outer background source is inverted,
+stage two has the same C_E residual. The inverse scalar word does not change
+this sign. Arbitrary physical input a is interpreted as C_start^-1 a; no
+unpaid entrance conversion or clean-value hypothesis is introduced.
+
+At outer triple t_b, the residual is C_(E_s tensor <t_b>), or its transpose.
+For an orthonormal outer group J the subspaces for distinct b are orthogonal,
+so exact weight modulo four gives product residual C_(E_s tensor span J).
+Its required complement has explicit orthogonal decomposition
+
+ D_(s,J) = (sigma_s tensor span J) + (F_2^24 tensor (span J)^perp),
+
+of dimension 576 - |J| (24 - dim sigma_s). Thus full groups of 24 pay
+24 dim sigma_s; groups of 8 pay 384 + 8 dim sigma_s. The unchanged exterior
+of rank 384 is correct only when sigma_s=0.
+
+Each triple basis vector of J has weight 3, so its quadratic form contributes
+one copy of -q_sigma. For the four partial groups, PR128 supplies 8 negative
+triple axes and 8 positive unit axes in the outer complement. Tensoring these
+with 24 unit inner axes adds 192 negative and 192 positive axes. The retained
+endpoint-gauge-complex normal form applies to every nondegenerate quadratic
+module, including alternating planes, as one child of the full residual
+rank with paid binary address changes and quadratic unit-phase passes.
+Its normalized scalar for each sigma block is a fourth root of unity.
+There are 8 or 24 identical blocks, so that scalar cancels exactly. The 192
+additional negative-axis phases also cancel. Tensor transposition preserves
+all weights and hence the stage-two signed conclusion.
+
+The companion `gauge_phase.py` checks all 470 distinct physical source frames, covering
+26597 live slots after compensated reuse: orthogonal decomposition into odd lines and hyperbolic
+planes, projector symmetry/idempotence, actual mod4 weights on all 301
+quadratic coefficient points, exact small-block Gauss sums, all 87 outer
+bases and full 2,024-label coverage. It verifies the complement ranks and
+unit scalar cancellation. It is a finite algebraic audit, not execution
+of the enormous expanded Gaussian word; individually restored core validity
+and the stated residual-to-child/all-size interfaces remain prerequisites.
+
+Compensated birth-cut slot reuse composes by applying this formula to each
+remaining PHYSICAL slot's initial frame. A donor starting at zero and ending
+at full inner frame has sigma=0; its aliased recipient contributes no separate
+slot or exterior. Literal inverse chronology and birth-cut frame containment
+must first establish the individually restored physical-core interface.
+
+The literal scalar word first subtracts the exact old response from each
+nondeferred slot, injects the fresh sources and completes the center closure.
+At the birth cut each deferred read subtracts its exact future response.
+For a reused recipient this read uses the donor's actual current value,
+including any fresh-source component. The recipient had no earlier use and
+the donor has no later use under its old role; the compensating response
+therefore cancels the entire substituted value. Side roots are then completed
+and read. Every physical source injection and workspace shear is undone in
+true reverse chronological order. This restores each live original scratch
+coordinate individually for arbitrary correlated inputs. Treating cleanup
+as an unqualified virtual `M^-1; -V` after aliasing would be incorrect.
+
+The producer has 28,705 virtual roles and 2,108 disjoint birth-cut pairs.
+Removing the recipients leaves 26,597 original physical slots. A donor's
+initial frame is zero even when its aliased recipient had nonzero sigma;
+the actual final forward frame is full dimension 24. For an unpaired deferred
+slot the initial frame is its recorded sigma and the final frame is also
+full. The reflected scan starts at zero and ends at sigma-perp on that same
+physical role. Both orientations consequently have the active residual used
+above, and their complete paid rank histograms agree.
+
+The independent reflection audit expands every exact old-readout numerator
+n/42 into signed unit chunks plus a remainder of absolute value at most one.
+It checks the integer coefficient sum, reverses chunk order and signs under
+reflection, and rejects oversize chunks, wrong sums, wrong signs, changed
+complements and incorrect signed root readouts. Birth-cut controls reject
+nondead donors, unavailable births, overlapping pairs and omitted compensation.
+The new sharing checker requires the complete local exterior inventory:
+for each physical source frame sigma there are exactly 2v children of rank
+552+dim(sigma). Replacing these by an ungauged rank-552 inventory, omitting a
+physical slot or dropping a paid group correction cannot satisfy its profile
+binding. The Gauss checker also detects the wrong reflected residual sign.
+
+Sharing retains all core gates and their actual scalar costs. In particular,
+the bounded direct old-readout compiler has a larger G than PR128's transparent
+PR117 word; its own audited scalar count and semantic guard must be retained. The core
+scalar bound uses the 28,705 virtual roles, since aliased recipient readouts
+still occur; the new group-wrapper reserve uses the 26,597 physical roles.
+For every group-role and both orientations, 64*m^2 paid adapter atoms cover
+binary basis maps and inverse maps plus quadratic unit-phase wrappers. These
+are paid bit-interface calls with width-dependent costs, not constant-time
+tape instructions. Sequential calls reuse restored row stock.
+Completed blocks remain Gaussian dyadic, so their local odd 21 denominators
+cancel at each core boundary even when incoming dirty values are correlated.
+
+Provenance: completed-core sharing and signed outer partition are PR128
+(an664), whose partition implementation cites Xiande Zhang and Gennian Ge.
+The one-child arbitrary nondegenerate Gauss normal form is retained PR104
+lineage; the underlying carrier DAG is eumemic's PR117. Physical-frame descent,
+deferral saturation and this exact composition check were developed with
+OpenAI Codex assistance. This note does not assert priority over concurrent work.
+
+Package binding: `complex_deferred.py` emits the canonical `physical_auxiliary_source_frames` inventory of live slots only. `reflection_audit.py` independently rebuilds it from the literal starting physical ports and binds it to the full forward/reflected scan. `gauge_phase.py` verifies exactly those frames and their multiplicities, then `sharing.py` removes all local auxiliary exteriors and inserts the resulting signed group complements. `complex-profile.json` remains the complete unshared profile; `shared-complex-profile.json` is the combined profile. The root certificate and verifier must consume the latter and retain the actual local scalar and paid group-wrapper charges.
diff --git a/research/cyclic-deferred/SOURCE.json b/research/cyclic-deferred/SOURCE.json
new file mode 100644
index 0000000000..de7172b721
--- /dev/null
+++ b/research/cyclic-deferred/SOURCE.json
@@ -0,0 +1,131 @@
+{
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+}
diff --git a/research/cyclic-deferred/bit-profile.json b/research/cyclic-deferred/bit-profile.json
new file mode 100644
index 0000000000..36105e819a
--- /dev/null
+++ b/research/cyclic-deferred/bit-profile.json
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diff --git a/research/cyclic-deferred/bit-sharing/MANIFEST.json b/research/cyclic-deferred/bit-sharing/MANIFEST.json
new file mode 100644
index 0000000000..b019e9fa08
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/MANIFEST.json
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+    "inputs/research/round8-network-final-geometry.py": "b3bc226c305e98d08b3adb4abef6794a607eb3e6f88fa13823c1086c5429228c",
+    "inputs/research/round8-network-minimal-v-ledger/forward-events.i32.gz": "afa817b15a4b7c30e0086380ac0c629e4672d14e3c03cf7675a743ef91eb3da5",
+    "inputs/research/round8-network-minimal-v-ledger/result.json": "21621c98008616261f23849015ca1690281440f26633d95d7bcbc6b3e05b7e68",
+    "inputs/research/round8-network-opposite-profile-proof.txt": "ca181fee5f183d3009b212a77a1ffe6e23f17c29e0eede28a2b3c8026fd644c8",
+    "inputs/research/round8-network-opposite-profile.json": "8403eb58f658999584de3f1e1af56e0ddb90ce0521ae6a68f9f57864da0dcdd1",
+    "inputs/research/round8-network-opposite-profile.py": "da2431d5fb6520273310ef6677c1c6c06b981de10d74fc3635dc45ee2d843934",
+    "inputs/research/round8-network-physical-ledger.py": "e95f693dde0dcff5f1b78fee95794cad4da0f227645e2be001b1d9c57cc78ee4",
+    "inputs/research/round8-network-pr104-audit.txt": "0ca0c5f861a0e5ba71f3dea74c23587821d498cbe4003605ac5affb2e4fc9d8b",
+    "inputs/research/round8-network-pr104-check.json": "6ba48bc0c8600369587e3cc63089c14f1d93555d5e104c12bc98a07237518dc2",
+    "inputs/research/round8-network-pr104-check.py": "4668be8eaf1b2d916dcd593edfde1f161a1470995f6a3140d7a6052fb20b97e1",
+    "licenses/main/LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+    "licenses/main/NOTICE": "1c53970fb4face631a1b331ec66d33111eccd280a129f9b64c2da6df96861c3f",
+    "licenses/pr128/MANIFEST.json": "93f13b17a955abd8fd18374132bd1d03fa39eeff9cf26f78048abc28995f431a",
+    "licenses/pr128/NOTICE": "e78d0d18f40776db3e482cf33930c8db073edf4da6cf6b3e5c96b7101e39a4c2",
+    "licenses/pr128/PROOF.md": "24eb6d03e32b5322e4e86bde521dc8d32a11bf33f72e806eba985ba47ce05fb0",
+    "licenses/pr128/README.md": "fb7f1712e78ab2e1305d056a006a256cd7df20ed72f306d25ad05cd8f86e0190",
+    "licenses/round7/LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+    "licenses/round7/NOTICE": "058a663112e633ab5349ce33b6119bacca2b9fd6c0a89918c4c0435d6dd7a008",
+    "partition161-receipt.json": "8378429b9b84a29afb83ddbdcd36507c5ef16a3f5055fd29bda58ef3a8008b8f",
+    "partition161.py": "ceee4649e91f12caa886c55cebf50089b9be69d239d42f572738b0b92e4dad83",
+    "profile161.json": "81f124ddf1f732558b934150528f66e6d30a4ce9fa7fc5da3683975d53cdb347",
+    "profile161.py": "125a3a309b864b848a5a443e692dffcbf189f2ecb09fb08332aa648030bfa90b",
+    "verify.py": "58ccce67e0a6ca9db91d195250eb9fdb8ccae27b5fe22fb6b1b32f512a476146"
+  }
+}
diff --git a/research/cyclic-deferred/bit-sharing/NOTICE b/research/cyclic-deferred/bit-sharing/NOTICE
new file mode 100644
index 0000000000..e0224c88ee
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/NOTICE
@@ -0,0 +1,21 @@
+Bit completed-core sharing with 161 orthogonal groups, 2026-10-09
+
+This local cyclic sunflower partition, grouped bit profile and finite
+verification package were prepared with substantial OpenAI Codex assistance.
+No independent external peer review, authorship by predecessor contributors,
+or OpenAI endorsement is claimed.
+
+The original PR128 completed-core sharing NOTICE, README, proof and input
+manifest are retained under licenses/pr128. Its unchanged bit inputs and
+compiler/ledger sources are retained under inputs/research. The PR128
+completed-core composition, icekylinx PR104 opposite-bank and stopped compiler,
+Swapnil Jain deferred bit construction, Zhihao Chen physical/reflected ledger
+lineage and all earlier contributors retain their attribution. The inherited
+ledger's Zhihao Chen source-level credit is a lineage credit; it does not mean
+he authored or reviewed these local adaptations.
+
+The rational moment helper adapts the Apache-2.0 partial_swap_network.py
+enclosures by icekylinx, with its recorded OpenAI assistance. Original main
+and round-seven LICENSE/NOTICE files remain under licenses/main and
+licenses/round7. Apache-2.0 remains applicable; this file supersedes no
+predecessor notice or license.
diff --git a/research/cyclic-deferred/bit-sharing/PROOF.md b/research/cyclic-deferred/bit-sharing/PROOF.md
new file mode 100644
index 0000000000..05d990ac82
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/PROOF.md
@@ -0,0 +1,115 @@
+# Exact bit grouping and inherited compiler obligations
+
+## Rational orthogonality and explicit partition
+
+On the 23-dimensional rational space use H=9I−J. For a triple T let t_T
+be its indicator column. Then t_TᵀHt_U=9|T∩U|−9. Each triple has norm
+18, and its orthogonal projector is
+
+    P_T = t_T (3t_Tᵀ−1ᵀ)/6.
+
+Distinct triples meeting in one point have P_T P_U=P_U P_T=0. A class of
+11 triples through one common point and 11 disjoint pairs is therefore an
+orthogonal rank-11 family. The profile verifier checks every Gram incidence,
+coverage, the integer identity Q²=6Q for Q=6ΣP_T, trace(Q)=66 and HQ symmetric.
+The rational denominator 6 is inherited from the old rank-one projectors.
+
+The partition generator works modulo 23. Each of the 77 translation orbits
+of triples can be represented by {0,a,b}. Requiring a quadratic residue a
+and nonresidue b leaves one or three possible centers. The 55 forced choices
+give degree five at every nonzero point. A deterministic exhaustive search
+chooses the remaining 22 pairs to add degree two at every point. The resulting
+7-regular bipartite graph has seven explicitly computed perfect matchings.
+Their translated sunflower classes give 7×23=161 classes of 11. The exported
+1771 triples are checked against the complete triple set, with no repetition.
+
+## Share restored scratch between completed cores
+
+The pinned opposite-bank proof states that a role s beginning in frame σ_s
+has completed-core relative dirty residual
+
+    E_(s,b) = (I−P_σs) ⊗ P_b
+
+in the first tensor orientation, or P_a⊗(I−P_σs) in the reflected orientation.
+This is the already gauged residual, not a dimension inferred by cancellation.
+The literal scalar word restores each arbitrary logical scratch register
+individually, including the nonzero initial gauge. The initial and final
+gauges are part of the supplied core identity.
+
+For a projector E define D_E by blocks [[I−E,E],[E,I−E]]. Direct multiplication
+gives D_A D_B=D_(A+B) whenever AB=BA=0. Thus serial execution of all completed
+cores in a class J on the same physical role s leaves exactly
+
+    E_(s,J) = (I−P_σs) ⊗ Σ_(b∈J) P_b,
+    rank E_(s,J) = (23−dim σ_s)|J|.
+
+Aliasing occurs only between completed, individually restored cores. The
+construction starts with one new arbitrary dirty register per physical
+shared role; it does not identify two independently prescribed input values.
+Within each core all its original distinct logical roles remain distinct.
+Data roles and all internal events, copied centers, bridges and endpoint
+corrections retain their old allocation and order. Both tensor orientations
+use the same argument. After a class, one explicitly paid complement child
+has projector I−E_(s,J) and width 529−(23−dim σ_s)|J|. Its composition with
+D_E is D_I. Every complement is proper and nonzero.
+
+The new allocation is W=2×1771²+2×161×28866=15567734. The profile replaces
+only the old `auxiliary_exteriors` class. It enumerates all new complements
+and retains every other old multiplicity. The total rank is 8233987097,
+so 529W−rank=1344189 exactly. This unchanged deficit is independently checked.
+
+## One basis, one execution prime, paid routing
+
+The expanded family consists of all old internal residuals, copied-center,
+bridge and endpoint projectors and **all new group complements**. Choose one
+generic rational similarity for this entire finite family so the leading
+minors needed by the PR104 factorization are nonzero. The finite product of
+the corresponding nonzero polynomials is nonzero; enumeration of rational
+matrices terminates. Apply that same similarity to every incident frame,
+gauge and endpoint and then the same global opposite-bank conjugation.
+No edge gets an independent basis switch or a free adapter.
+
+Only after the entire finite basis and factorization are fixed, choose one
+odd prime avoiding every denominator and pivot numerator. The prime used
+inside rank checks (Q31) is merely a nonzero-determinant witness and is not
+automatically the execution prime. All rational operator identities then
+hold over the retained radix-q atom ring. This extends the finite avoidance
+set from the inherited compiler; it does not change its all-size contract.
+
+Each idempotent produces one child of its full rank with its paid PR104
+atom adapters. For f atoms, the cross shear uses K⊗C_f. Every head atom is
+physically earlier than every tail atom, and lower-triangular same-bank
+matrices expand into earlier-control scalar passes. The fixed finite family
+therefore has the inherited O(Vn) adapter charge per recursive node. Reversal
+is represented inside those paid indexed atom operations, not treated as a
+free rearrangement. The grouped allocation changes constants and must be
+used in the enclosing stopped/row-reserve calculation.
+
+The ordinary two-call wrapper is applied once outside the reversed recursion
+and sequentially reuses restored row stock. This package does not price it.
+The three row reserves and all stopping, precision, tape, prime-supply and
+analytic constraints belong to the enclosing final certificate.
+
+## Finite witness and exact arithmetic
+
+The scalar schedule is frozen, while the literal physical event stream is
+regenerated from it. Exact frame inclusion checks bind actual source order
+and every auxiliary/target chain to nondegenerate rational frames. Forward
+and reverse-chronological, bank-swapped, complemented-frame replay verifies
+all formal input and dirty columns over F2, including all original center
+macros. The new grouping leaves that local word unchanged. Event bytes and
+local rank histograms are compared with the frozen PR128 ledger.
+
+For the complete new histogram c_t, the moment certificate proves
+
+    Σ_t c_t (t/529)^(1−a) / W < 1,
+    a = 129310447 / 10^12.
+
+Logarithms have a rational 24-term artanh-series upper bound with its exact
+positive tail, rounded upward to denominator 10^12. For u=a log(529/t)<1,
+exp(u)≤1+u+u²/[2(1−u/3)]. All arithmetic is rational. The maximum child 518
+has exact halving degree 33, checked by 2×518^33≤529^33 and failure at 32.
+The accepted moment upper bound is strictly below one. The next 10^−12
+grid point fails this upper enclosure; no claim about exact noncontraction
+at that next point is made. Numerical root estimates are not proof inputs.
+
diff --git a/research/cyclic-deferred/bit-sharing/README.md b/research/cyclic-deferred/bit-sharing/README.md
new file mode 100644
index 0000000000..6d394880e0
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/README.md
@@ -0,0 +1,54 @@
+# Bit completed-core sharing with 161 orthogonal groups
+
+This finite contribution certifies the **coarse bit saving**
+
+`129310447/1000000000000 = 0.000129310447`.
+
+It is a local bit construction under the retained PR104/PR128 interfaces;
+it does not by itself certify an ordinary stopped saving or a multiplication
+exponent. The new partition groups all 1771 triples of 23 coordinates into
+161 orthogonal classes of 11. Only completed-core scratch allocation and its
+paid exterior children change. Every original local core operation remains.
+
+Run from any directory, using Python 3.10+ and only the standard library:
+
+```sh
+python3 /path/to/bit161-package/verify.py
+```
+
+The verifier checks the manifest, works in a temporary directory, and leaves
+the submitted package unchanged. It checks all original and changed frames,
+actual V chronology, all 1,076,483 physical events and 32,408 formal input/dirty
+columns forward and reflected, and exact agreement with the pinned event
+bytes. It then regenerates the opposite-bank profile, finite compiler checks,
+partition, new full profile and exact rational moment certificate. Corruption
+controls cover a nonorthogonal partition, a missing output scatter and an
+omitted paid child. The next grid point fails the chosen moment enclosure;
+that fact does not prove noncontraction of the exact characteristic moment.
+
+The scalar schedule `round8-foundations-minimal-v-word.json.gz` is a frozen
+finite witness from PR128 with SHA256
+`298c11fb29ab0afdf8d762c64a9b99d19a1e58465f3a0cbc9d2a59367c438fb7`.
+The verifier regenerates its physical event word and checks its mathematical
+properties. It does not rerun the upstream search or minimal-V optimization.
+Every inherited file in `inputs/` is copied byte-for-byte; the newly changed
+source is `partition161.py`, `profile161.py`, `certify161.py` and `verify.py`.
+
+The resulting profile has m=529, local R=28866, W=15567734,
+total recursive rank 8233987097 and deficit 1344189. Its maximum child is
+518; the exact halving degree is 33. The unchanged old profile has maximum
+child 528 and cannot supply the new halving constant without recalculation.
+
+See `PROOF.md` for the gauged core composition, common rational basis,
+address-field and adapter obligations. The stopped ordinary wrapper, three
+row reserves, Gaussian precision and final analytic assembly must be priced
+by the enclosing construction. In particular this bit-only change does not
+alter the complex Gaussian denominator or the inherited 21 precision factor.
+
+Credit the PR128 completed-core-sharing construction, icekylinx's PR104
+opposite-bank compiler, Swapnil Jain's deferred bit supplier, Zhihao Chen's
+physical/reflected ledger lineage, and all predecessors retained in the
+notices. This local cyclic sunflower partition, exact integration and
+verification were prepared with substantial OpenAI Codex assistance. No
+independent external peer review or OpenAI endorsement is claimed.
+
diff --git a/research/cyclic-deferred/bit-sharing/certificate.json b/research/cyclic-deferred/bit-sharing/certificate.json
new file mode 100644
index 0000000000..d311f6ce5e
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/certificate.json
@@ -0,0 +1,267 @@
+{
+  "certified_bit_saving": "129310447/1000000000000",
+  "halving_degree": 33,
+  "next_1e12_grid_enclosure_rejected": true,
+  "next_grid_note": "The inherited upper enclosure fails at the next grid point; this does not prove noncontraction of the exact characteristic moment.",
+  "moment": {
+    "saving": "129310447/1000000000000",
+    "exponent": "999870689553/1000000000000",
+    "moment_upper": "10886293088757970304184092696234756728042994521714776885930893735837489407974543906677463215948825394381164064553936627232537480111611455731810544587296666180034504040727032076713877910391560059790766697697411129708567460008002611594378571388626180002326728005042211398952612452443816251584894529014363817353869611887202720992338319904764618263336786810172541001417319354313115876057804570416232907873664110428250561745524058799446363732550874388334616402679690844396564188829035761833970617015709032010119953394350379537563018538626706317879918893111413121091871455459128346241625030890416879164416277446265936616698541025078407019040305023937256583093645332589445582286114649058101895319612700332590518210653108115847818687738673328334410615505447789550416033270487748477049340115399586905475913297399043472704752335703127926260246716555520558071877950403490615070591617077805029028641409672228903328269241168047154027022187910753243270671781465985413326459674413299932580577740223307266090140015967561157473070566264958083192953601421900538793784156481520341589349504063026088473057387371957712251/10886293088762449201056598122756761836270868938829250675742725653515876659160331806148746717491248530985292180932105537384665328136545084664181458273978951420863550272675329624820469366170029048652179006034063535190814027463260342682004107619645282550651326555226606685942852302565271044756225421919410682487966566260065882331234931356668599955145668546812642961832247070459045312595793868283311656235727604835596125964183670908009537340686566881076655064366638313356394775621086408456636283236542693521859518620427593935400247949977955012334765404840316670099273241238382505189649129308167910365667544012168032295019332320400336834709725488278314560074602060087639668135868641304455122120501324742111703267446253093291656467771004467533889569580650320549699436617029825009118678911024136970380437260263452509854135293026085474102338328491984135075605879595188009802344676136122423832038788774094244489324398595759037571284995450971121804961901062756752639614495652035587511600813807569085772657937261728029133660256940024875063976387138466052349426878887929555700330253731529760000000000000000000000",
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+      "496": "32206252567/500000000000",
+      "507": "10619357067/250000000000",
+      "518": "21013189599/1000000000000"
+    },
+    "child_width_multiplicities": [
+      [
+        1,
+        235438511
+      ],
+      [
+        2,
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+      ],
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+        3,
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+        644
+      ],
+      [
+        298,
+        4508
+      ],
+      [
+        309,
+        14490
+      ],
+      [
+        320,
+        40894
+      ],
+      [
+        331,
+        322
+      ],
+      [
+        342,
+        25116
+      ],
+      [
+        353,
+        1610
+      ],
+      [
+        364,
+        98532
+      ],
+      [
+        375,
+        11592
+      ],
+      [
+        386,
+        36708
+      ],
+      [
+        397,
+        9016
+      ],
+      [
+        408,
+        104006
+      ],
+      [
+        419,
+        28658
+      ],
+      [
+        430,
+        34454
+      ],
+      [
+        441,
+        9660
+      ],
+      [
+        452,
+        143934
+      ],
+      [
+        463,
+        56994
+      ],
+      [
+        474,
+        259532
+      ],
+      [
+        484,
+        6272882
+      ],
+      [
+        485,
+        1246462
+      ],
+      [
+        496,
+        420854
+      ],
+      [
+        507,
+        513912
+      ],
+      [
+        518,
+        566720
+      ]
+    ],
+    "enclosure": "1+u+u^2/(2(1-u/3))"
+  },
+  "fresh_literal_replay_matches_frozen_events": true,
+  "decompressed_event_sha256": "f98d52b6d8fb04e30a0a46df4df95d2e7999ff97a89826e5b0f44b4120a69e28",
+  "frozen_metadata_fields_checked": [
+    "candidate_sha256",
+    "h",
+    "roles",
+    "formal_basis",
+    "events",
+    "rank_histograms",
+    "histogram",
+    "recursive_rank",
+    "complete_forward_F2",
+    "complete_reflected_F2",
+    "actual_source_V_order_checked",
+    "reflected_frame_continuity",
+    "all_scalar_gates_equal_frame_keys"
+  ],
+  "candidate_sha256": "298c11fb29ab0afdf8d762c64a9b99d19a1e58465f3a0cbc9d2a59367c438fb7",
+  "groups_sha256": "220dc207ab0774212949a57301391506ead8bf9f6eadd60a6a82ec2de7e0bc61",
+  "scope": "Exact coarse bit moment only; not a final multiplication exponent or ordinary stopped-wrapper certificate."
+}
diff --git a/research/cyclic-deferred/bit-sharing/certify161.py b/research/cyclic-deferred/bit-sharing/certify161.py
new file mode 100644
index 0000000000..220389dc63
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/certify161.py
@@ -0,0 +1,68 @@
+"""Exact coarse moment and unchanged physical-ledger binding.
+
+The 24-term logarithm and exponential enclosure follow icekylinx's
+Apache-2.0 partial_swap_network.py, retained PR110 lineage. This local
+adaptation was prepared with OpenAI Codex assistance.
+"""
+from pathlib import Path
+from fractions import Fraction as Q
+from hashlib import sha256
+import json, gzip
+
+D=Path(__file__).resolve().parent
+P=D/'inputs/research'
+
+def log_upper(value):
+    value=Q(value);assert value>=1
+    power=0
+    while value>2:value/=2;power+=1
+    def series(x):
+        z=(x-1)/(x+1)
+        return 2*sum((z**(2*j+1)/(2*j+1)for j in range(24)),Q(0))+2*z**49/(49*(1-z*z))
+    scaled=(power*series(Q(2))+series(value))*10**12
+    return Q(-(-scaled.numerator//scaled.denominator),10**12)
+
+def moment(m,W,rows,saving):
+    upper=Q(0);logs={}
+    for t,count in sorted(rows.items()):
+        assert 0<t<m and count>=0
+        ell=log_upper(Q(m,t));u=saving*ell;assert 0<=u<1
+        bound=1+u+u*u/(2*(1-u/3))
+        upper+=Q(count*t,W*m)*bound;logs[str(t)]=str(ell)
+    return dict(saving=str(saving),exponent=str(1-saving),moment_upper=str(upper),
+                strict_gap=str(1-upper),logarithm_upper_bounds=logs,
+                child_width_multiplicities=sorted(rows.items()),
+                enclosure='1+u+u^2/(2(1-u/3))')
+
+def main():
+    profile=json.loads((D/'profile161.json').read_text())
+    rows={int(t):n for t,n in profile['child_multiplicities'].items()}
+    assert sum(t*n for t,n in rows.items())==profile['total_rank']
+    assert profile['W']*profile['m']-profile['total_rank']==profile['deficit']
+    saving=Q(129310447,10**12)
+    certificate=moment(profile['m'],profile['W'],rows,saving)
+    assert Q(certificate['moment_upper'])<1
+    next_certificate=moment(profile['m'],profile['W'],rows,saving+Q(1,10**12))
+    assert Q(next_certificate['moment_upper'])>=1
+    m,r=profile['m'],profile['maxchild'];d=1
+    while 2*r**d>m**d:d+=1
+    assert d==33 and 2*r**(d-1)>m**(d-1)
+    fresh=D/'literal-replay';frozen=P/'round8-network-minimal-v-ledger'
+    event_hashes=[sha256(gzip.decompress((root/'forward-events.i32.gz').read_bytes())).hexdigest()for root in(fresh,frozen)]
+    assert event_hashes[0]==event_hashes[1]
+    fresh_result=json.loads((fresh/'result.json').read_text());frozen_result=json.loads((frozen/'result.json').read_text())
+    keys=('candidate_sha256','h','roles','formal_basis','events','rank_histograms','histogram','recursive_rank','complete_forward_F2','complete_reflected_F2','actual_source_V_order_checked','reflected_frame_continuity','all_scalar_gates_equal_frame_keys')
+    for key in keys:assert fresh_result[key]==frozen_result[key],key
+    assert fresh_result['candidate_sha256']==profile['minimal_v_word_sha256']=='298c11fb29ab0afdf8d762c64a9b99d19a1e58465f3a0cbc9d2a59367c438fb7'
+    out=dict(certified_bit_saving=str(saving),halving_degree=d,
+             next_1e12_grid_enclosure_rejected=True,
+             next_grid_note='The inherited upper enclosure fails at the next grid point; this does not prove noncontraction of the exact characteristic moment.',
+             moment=certificate,fresh_literal_replay_matches_frozen_events=True,
+             decompressed_event_sha256=event_hashes[0],
+             frozen_metadata_fields_checked=list(keys),candidate_sha256=fresh_result['candidate_sha256'],
+             groups_sha256=sha256((D/'groups161.json').read_bytes()).hexdigest(),
+             scope='Exact coarse bit moment only; not a final multiplication exponent or ordinary stopped-wrapper certificate.')
+    (D/'certificate.json').write_text(json.dumps(out,indent=2)+'\n')
+    print('PASS coarse bit moment',saving,'halving degree',d,flush=True)
+
+if __name__=='__main__':main()
diff --git a/research/cyclic-deferred/bit-sharing/groups161.json b/research/cyclic-deferred/bit-sharing/groups161.json
new file mode 100644
index 0000000000..5deb5f66f5
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/groups161.json
@@ -0,0 +1 @@
+[[[0, 1, 21], [0, 2, 15], [0, 3, 11], [0, 4, 14], [0, 6, 22], [0, 8, 19], [0, 9, 20], [0, 12, 17], [0, 10, 13], [0, 7, 16], [0, 5, 18]], [[0, 1, 15], [0, 2, 19], [0, 3, 22], [0, 4, 21], [0, 6, 20], [0, 8, 17], [0, 9, 14], [0, 11, 12], [0, 5, 13], [0, 10, 16], [0, 7, 18]], [[0, 1, 17], [0, 2, 20], [0, 3, 21], [0, 4, 7], [0, 6, 15], [0, 5, 8], [0, 9, 19], [0, 12, 22], [0, 11, 13], [0, 14, 16], [0, 10, 18]], [[0, 1, 22], [0, 2, 17], [0, 3, 20], [0, 4, 10], [0, 6, 11], [0, 8, 21], [0, 5, 9], [0, 12, 19], [0, 7, 13], [0, 15, 16], [0, 14, 18]], [[0, 1, 5], [0, 2, 21], [0, 3, 10], [0, 4, 11], [0, 6, 17], [0, 8, 22], [0, 7, 9], [0, 12, 20], [0, 13, 14], [0, 16, 19], [0, 15, 18]], [[0, 1, 20], [0, 2, 22], [0, 3, 14], [0, 4, 15], [0, 6, 21], [0, 7, 8], [0, 9, 11], [0, 10, 12], [0, 13, 19], [0, 5, 16], [0, 17, 18]], [[0, 1, 11], [0, 2, 7], [0, 3, 17], [0, 4, 19], [0, 5, 6], [0, 8, 15], [0, 9, 10], [0, 12, 14], [0, 13, 20], [0, 16, 21], [0, 18, 22]], [[1, 2, 22], [1, 3, 16], [1, 4, 12], [1, 5, 15], [0, 1, 7], [1, 9, 20], [1, 10, 21], [1, 13, 18], [1, 11, 14], [1, 8, 17], [1, 6, 19]], [[1, 2, 16], [1, 3, 20], [0, 1, 4], [1, 5, 22], [1, 7, 21], [1, 9, 18], [1, 10, 15], [1, 12, 13], [1, 6, 14], [1, 11, 17], [1, 8, 19]], [[1, 2, 18], [1, 3, 21], [1, 4, 22], [1, 5, 8], [1, 7, 16], [1, 6, 9], [1, 10, 20], [0, 1, 13], [1, 12, 14], [1, 15, 17], [1, 11, 19]], [[0, 1, 2], [1, 3, 18], [1, 4, 21], [1, 5, 11], [1, 7, 12], [1, 9, 22], [1, 6, 10], [1, 13, 20], [1, 8, 14], [1, 16, 17], [1, 15, 19]], [[1, 2, 6], [1, 3, 22], [1, 4, 11], [1, 5, 12], [1, 7, 18], [0, 1, 9], [1, 8, 10], [1, 13, 21], [1, 14, 15], [1, 17, 20], [1, 16, 19]], [[1, 2, 21], [0, 1, 3], [1, 4, 15], [1, 5, 16], [1, 7, 22], [1, 8, 9], [1, 10, 12], [1, 11, 13], [1, 14, 20], [1, 6, 17], [1, 18, 19]], [[1, 2, 12], [1, 3, 8], [1, 4, 18], [1, 5, 20], [1, 6, 7], [1, 9, 16], [1, 10, 11], [1, 13, 15], [1, 14, 21], [1, 17, 22], [0, 1, 19]], [[0, 2, 3], [2, 4, 17], [2, 5, 13], [2, 6, 16], [1, 2, 8], [2, 10, 21], [2, 11, 22], [2, 14, 19], [2, 12, 15], [2, 9, 18], [2, 7, 20]], [[2, 3, 17], [2, 4, 21], [1, 2, 5], [0, 2, 6], [2, 8, 22], [2, 10, 19], [2, 11, 16], [2, 13, 14], [2, 7, 15], [2, 12, 18], [2, 9, 20]], [[2, 3, 19], [2, 4, 22], [0, 2, 5], [2, 6, 9], [2, 8, 17], [2, 7, 10], [2, 11, 21], [1, 2, 14], [2, 13, 15], [2, 16, 18], [2, 12, 20]], [[1, 2, 3], [2, 4, 19], [2, 5, 22], [2, 6, 12], [2, 8, 13], [0, 2, 10], [2, 7, 11], [2, 14, 21], [2, 9, 15], [2, 17, 18], [2, 16, 20]], [[2, 3, 7], [0, 2, 4], [2, 5, 12], [2, 6, 13], [2, 8, 19], [1, 2, 10], [2, 9, 11], [2, 14, 22], [2, 15, 16], [2, 18, 21], [2, 17, 20]], [[2, 3, 22], [1, 2, 4], [2, 5, 16], [2, 6, 17], [0, 2, 8], [2, 9, 10], [2, 11, 13], [2, 12, 14], [2, 15, 21], [2, 7, 18], [2, 19, 20]], [[2, 3, 13], [2, 4, 9], [2, 5, 19], [2, 6, 21], [2, 7, 8], [2, 10, 17], [2, 11, 12], [2, 14, 16], [2, 15, 22], [0, 2, 18], [1, 2, 20]], [[1, 3, 4], [3, 5, 18], [3, 6, 14], [3, 7, 17], [2, 3, 9], [3, 11, 22], [0, 3, 12], [3, 15, 20], [3, 13, 16], [3, 10, 19], [3, 8, 21]], [[3, 4, 18], [3, 5, 22], [2, 3, 6], [1, 3, 7], [0, 3, 9], [3, 11, 20], [3, 12, 17], [3, 14, 15], [3, 8, 16], [3, 13, 19], [3, 10, 21]], [[3, 4, 20], [0, 3, 5], [1, 3, 6], [3, 7, 10], [3, 9, 18], [3, 8, 11], [3, 12, 22], [2, 3, 15], [3, 14, 16], [3, 17, 19], [3, 13, 21]], [[2, 3, 4], [3, 5, 20], [0, 3, 6], [3, 7, 13], [3, 9, 14], [1, 3, 11], [3, 8, 12], [3, 15, 22], [3, 10, 16], [3, 18, 19], [3, 17, 21]], [[3, 4, 8], [1, 3, 5], [3, 6, 13], [3, 7, 14], [3, 9, 20], [2, 3, 11], [3, 10, 12], [0, 3, 15], [3, 16, 17], [3, 19, 22], [3, 18, 21]], [[0, 3, 4], [2, 3, 5], [3, 6, 17], [3, 7, 18], [1, 3, 9], [3, 10, 11], [3, 12, 14], [3, 13, 15], [3, 16, 22], [3, 8, 19], [3, 20, 21]], [[3, 4, 14], [3, 5, 10], [3, 6, 20], [3, 7, 22], [3, 8, 9], [3, 11, 18], [3, 12, 13], [3, 15, 17], [0, 3, 16], [1, 3, 19], [2, 3, 21]], [[2, 4, 5], [4, 6, 19], [4, 7, 15], [4, 8, 18], [3, 4, 10], [0, 4, 12], [1, 4, 13], [4, 16, 21], [4, 14, 17], [4, 11, 20], [4, 9, 22]], [[4, 5, 19], [0, 4, 6], [3, 4, 7], [2, 4, 8], [1, 4, 10], [4, 12, 21], [4, 13, 18], [4, 15, 16], [4, 9, 17], [4, 14, 20], [4, 11, 22]], [[4, 5, 21], [1, 4, 6], [2, 4, 7], [4, 8, 11], [4, 10, 19], [4, 9, 12], [0, 4, 13], [3, 4, 16], [4, 15, 17], [4, 18, 20], [4, 14, 22]], [[3, 4, 5], [4, 6, 21], [1, 4, 7], [4, 8, 14], [4, 10, 15], [2, 4, 12], [4, 9, 13], [0, 4, 16], [4, 11, 17], [4, 19, 20], [4, 18, 22]], [[4, 5, 9], [2, 4, 6], [4, 7, 14], [4, 8, 15], [4, 10, 21], [3, 4, 12], [4, 11, 13], [1, 4, 16], [4, 17, 18], [0, 4, 20], [4, 19, 22]], [[1, 4, 5], [3, 4, 6], [4, 7, 18], [4, 8, 19], [2, 4, 10], [4, 11, 12], [4, 13, 15], [4, 14, 16], [0, 4, 17], [4, 9, 20], [4, 21, 22]], [[4, 5, 15], [4, 6, 11], [4, 7, 21], [0, 4, 8], [4, 9, 10], [4, 12, 19], [4, 13, 14], [4, 16, 18], [1, 4, 17], [2, 4, 20], [3, 4, 22]], [[3, 5, 6], [5, 7, 20], [5, 8, 16], [5, 9, 19], [4, 5, 11], [1, 5, 13], [2, 5, 14], [5, 17, 22], [5, 15, 18], [5, 12, 21], [0, 5, 10]], [[5, 6, 20], [1, 5, 7], [4, 5, 8], [3, 5, 9], [2, 5, 11], [5, 13, 22], [5, 14, 19], [5, 16, 17], [5, 10, 18], [5, 15, 21], [0, 5, 12]], [[5, 6, 22], [2, 5, 7], [3, 5, 8], [5, 9, 12], [5, 11, 20], [5, 10, 13], [1, 5, 14], [4, 5, 17], [5, 16, 18], [5, 19, 21], [0, 5, 15]], [[4, 5, 6], [5, 7, 22], [2, 5, 8], [5, 9, 15], [5, 11, 16], [3, 5, 13], [5, 10, 14], [1, 5, 17], [5, 12, 18], [5, 20, 21], [0, 5, 19]], [[5, 6, 10], [3, 5, 7], [5, 8, 15], [5, 9, 16], [5, 11, 22], [4, 5, 13], [5, 12, 14], [2, 5, 17], [5, 18, 19], [1, 5, 21], [0, 5, 20]], [[2, 5, 6], [4, 5, 7], [5, 8, 19], [5, 9, 20], [3, 5, 11], [5, 12, 13], [5, 14, 16], [5, 15, 17], [1, 5, 18], [5, 10, 21], [0, 5, 22]], [[5, 6, 16], [5, 7, 12], [5, 8, 22], [1, 5, 9], [5, 10, 11], [5, 13, 20], [5, 14, 15], [5, 17, 19], [2, 5, 18], [3, 5, 21], [0, 4, 5]], [[4, 6, 7], [6, 8, 21], [6, 9, 17], [6, 10, 20], [5, 6, 12], [2, 6, 14], [3, 6, 15], [0, 6, 18], [6, 16, 19], [6, 13, 22], [1, 6, 11]], [[6, 7, 21], [2, 6, 8], [5, 6, 9], [4, 6, 10], [3, 6, 12], [0, 6, 14], [6, 15, 20], [6, 17, 18], [6, 11, 19], [6, 16, 22], [1, 6, 13]], [[0, 6, 7], [3, 6, 8], [4, 6, 9], [6, 10, 13], [6, 12, 21], [6, 11, 14], [2, 6, 15], [5, 6, 18], [6, 17, 19], [6, 20, 22], [1, 6, 16]], [[5, 6, 7], [0, 6, 8], [3, 6, 9], [6, 10, 16], [6, 12, 17], [4, 6, 14], [6, 11, 15], [2, 6, 18], [6, 13, 19], [6, 21, 22], [1, 6, 20]], [[6, 7, 11], [4, 6, 8], [6, 9, 16], [6, 10, 17], [0, 6, 12], [5, 6, 14], [6, 13, 15], [3, 6, 18], [6, 19, 20], [2, 6, 22], [1, 6, 21]], [[3, 6, 7], [5, 6, 8], [6, 9, 20], [6, 10, 21], [4, 6, 12], [6, 13, 14], [6, 15, 17], [6, 16, 18], [2, 6, 19], [6, 11, 22], [0, 1, 6]], [[6, 7, 17], [6, 8, 13], [0, 6, 9], [2, 6, 10], [6, 11, 12], [6, 14, 21], [6, 15, 16], [6, 18, 20], [3, 6, 19], [4, 6, 22], [1, 5, 6]], [[5, 7, 8], [7, 9, 22], [7, 10, 18], [7, 11, 21], [6, 7, 13], [3, 7, 15], [4, 7, 16], [1, 7, 19], [7, 17, 20], [0, 7, 14], [2, 7, 12]], [[7, 8, 22], [3, 7, 9], [6, 7, 10], [5, 7, 11], [4, 7, 13], [1, 7, 15], [7, 16, 21], [7, 18, 19], [7, 12, 20], [0, 7, 17], [2, 7, 14]], [[1, 7, 8], [4, 7, 9], [5, 7, 10], [7, 11, 14], [7, 13, 22], [7, 12, 15], [3, 7, 16], [6, 7, 19], [7, 18, 20], [0, 7, 21], [2, 7, 17]], [[6, 7, 8], [1, 7, 9], [4, 7, 10], [7, 11, 17], [7, 13, 18], [5, 7, 15], [7, 12, 16], [3, 7, 19], [7, 14, 20], [0, 7, 22], [2, 7, 21]], [[7, 8, 12], [5, 7, 9], [7, 10, 17], [7, 11, 18], [1, 7, 13], [6, 7, 15], [7, 14, 16], [4, 7, 19], [7, 20, 21], [0, 3, 7], [2, 7, 22]], [[4, 7, 8], [6, 7, 9], [7, 10, 21], [7, 11, 22], [5, 7, 13], [7, 14, 15], [7, 16, 18], [7, 17, 19], [3, 7, 20], [0, 7, 12], [1, 2, 7]], [[7, 8, 18], [7, 9, 14], [1, 7, 10], [3, 7, 11], [7, 12, 13], [7, 15, 22], [7, 16, 17], [7, 19, 21], [4, 7, 20], [0, 5, 7], [2, 6, 7]], [[6, 8, 9], [0, 8, 10], [8, 11, 19], [8, 12, 22], [7, 8, 14], [4, 8, 16], [5, 8, 17], [2, 8, 20], [8, 18, 21], [1, 8, 15], [3, 8, 13]], [[0, 8, 9], [4, 8, 10], [7, 8, 11], [6, 8, 12], [5, 8, 14], [2, 8, 16], [8, 17, 22], [8, 19, 20], [8, 13, 21], [1, 8, 18], [3, 8, 15]], [[2, 8, 9], [5, 8, 10], [6, 8, 11], [8, 12, 15], [0, 8, 14], [8, 13, 16], [4, 8, 17], [7, 8, 20], [8, 19, 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diff --git a/research/cyclic-deferred/bit-sharing/inputs/research/pr-review-network-geometry.json b/research/cyclic-deferred/bit-sharing/inputs/research/pr-review-network-geometry.json
new file mode 100644
index 0000000000..5a3765ea11
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/inputs/research/pr-review-network-geometry.json
@@ -0,0 +1,38 @@
+{
+  "status": "PASS",
+  "candidate_sha256": "298c11fb29ab0afdf8d762c64a9b99d19a1e58465f3a0cbc9d2a59367c438fb7",
+  "node_frames": 64042,
+  "late_frames": 1396,
+  "used_frames": 93419,
+  "distinct_chain_steps": 212416,
+  "inclusion_methods": {
+    "identity/zero/full": 43936,
+    "exact intersection in member": 2230,
+    "nested exact intersections": 562,
+    "exact rational annihilator": 48580,
+    "checked node-successor theorem": 110938,
+    "node in every late successor": 834,
+    "checked direct-root theorem": 5336
+  },
+  "actual_F2_target_support": true,
+  "actual_V_chronology": true,
+  "all_frame_dimensions_exact": true,
+  "all_used_frames_nondegenerate": true,
+  "all_auxiliary_source_target_chains_nested": true,
+  "all_sigma_target_orthogonality_exact": true,
+  "controls": {
+    "isotropic_frame_rejected": true,
+    "outside_start_rejected": true,
+    "reversed_target_chain_rejected": true
+  },
+  "elapsed": 368.16998916701414,
+  "inputs_sha256": {
+    "round8-foundations-minimal-v-word.json.gz": "298c11fb29ab0afdf8d762c64a9b99d19a1e58465f3a0cbc9d2a59367c438fb7",
+    "round7-public/tree/independent/deferred-readout/deferred.py": "e92ac1db9fce8608cf5302336f05750fa327eda99f7170ed32e7323220ffc164",
+    "round7-public/tree/independent/deferred-readout/check_lifted.py": "c3e87ad6b31f377df549421f53d22b99131d2228dcd08a1956913211383cbd6c",
+    "round7-public/tree/independent/deferred-readout/linalg.py": "56cc779911a3c5c901bc5194ccaf97f720157a2c4ba602b07ee8642af2676b8b",
+    "round7-public/tree/certificates/round7/witness_23.json.gz": "b2486aa2bb222bacea6e52162a3920780e45dd8edc5d7cb03eabea336a1c6588",
+    "round7-public/tree/certificates/round7/deferred_23.json.gz": "8c38e947ff9e021e308000dd82bb5e2194eb265d8b75194e22ce783d3e75317c"
+  },
+  "scope": "Rational bit geometry of final candidate, not full tape execution. Pair with literal frame ledger and general opposite-bank rank factorization; no old corner witness is needed."
+}
diff --git a/research/cyclic-deferred/bit-sharing/inputs/research/round7-public/tree/certificates/round7/deferred_23.json.gz b/research/cyclic-deferred/bit-sharing/inputs/research/round7-public/tree/certificates/round7/deferred_23.json.gz
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diff --git a/research/cyclic-deferred/bit-sharing/inputs/research/round7-public/tree/certificates/round7/witness_23.json.gz b/research/cyclic-deferred/bit-sharing/inputs/research/round7-public/tree/certificates/round7/witness_23.json.gz
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diff --git a/research/cyclic-deferred/bit-sharing/inputs/research/round7-public/tree/independent/deferred-readout/check_lifted.py b/research/cyclic-deferred/bit-sharing/inputs/research/round7-public/tree/independent/deferred-readout/check_lifted.py
new file mode 100644
index 0000000000..4a19240e2f
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/inputs/research/round7-public/tree/independent/deferred-readout/check_lifted.py
@@ -0,0 +1,677 @@
+"""Release check (stdlib only) of the lifted-frame + late-copy bit side program, stacked with the staircase corner.
+
+Input: certificates/round7/witness_<h>.json.gz: the side DAG (leaves = triples of [h], additions), outputs (c, T),
+retained totals, links (donor x -> use (y, k)), the flag matrix F (integer h x h), the lift index j(n) of every
+lifted addition and the late-copy decisions (early users, ordered late users) per node.
+
+What is checked, from the witness alone:
+ A. DAG semantics: disjoint supports; output (c,T) support = {S : c in S, S cap T = {c}}; retained = star(c).
+ B. Compile: one slot per role, schedule order (dim span, id), links hand the donor's non-pivot slot to the linked
+    use; late copies: the pivot of n climbs C_1 <= ... <= C_k (C_i = frames of users late[i:] intersected) and the
+    copy for late user i is a gate (pivot, fresh slot) at C_i, placed in time as LATE as possible (just before the
+    first of: its user's gate, the pivot's consumer gate, the next copy's slot). R = adds + outputs + retained - links.
+ C. Dirty-scratch replay of the compiled program: word L J L^-1 V L J L^-1 V^-1 and its mirror, arbitrary scratch,
+    over F2 and over Z, reads at the slot's last touch and at the end; targets must receive J L V x exactly
+    (DAG values) and every slot is restored; over F2 every target also receives exactly x_T.
+ D. Frames (exact over Q). Frame of addition n: M_n = span(n) + (U_n cap F_j(n)), U_n = common kernel of the root
+    covectors reachable from n ((9I-J) t_T for an output to Y_T, 1 - 3 e_c for a retained use of c), F_j = first j
+    rows of F. Nesting is a THEOREM given the combinatorial facts checked here:
+      j(n) <= j(t) on every slot successor pair n -> t;  span(x) <= span(t) on every link pair (exact);
+      span(n) <= every root frame n reaches directly (exact integer dot products);  F invertible.
+    Then U_n <= U_t, so M_n <= M_t, M_n <= U_n <= every reachable root frame, and every late-copy frame C_i
+    (an intersection of frames that all contain M_n) is nested. Dimensions (exact, see linalg.py):
+      dim M_n = rank[span(n); F_j] - rank(K_n F_j^T)   (modular law; span(n) <= U_n; K_n spans the root covectors),
+    with ranks by mod-p + Hadamard certificates, Bareiss when the mod-p rank is not full; late-copy intersections by
+    exact Fraction bases. Rank of every chain step = difference of dimensions (nested), histogram vs claim.
+ E. Frame reductions mod q = 2^31-1 (dimension-matched, so they are the reductions of the rational frames),
+    G = I - J/9 nondegenerate on every frame of every step (Gram matrix invertible mod q => over Q), and the side
+    lemma at ONE integer point (U, V of point_UV(h, 1), the random stream of independent/two-stage-bit/flag_existence.py): for every distinct step A -> B of
+    rank r with 2r > h (retained copy edges 0 -> U_c included), U^T (P_B - P_A) U^-T and V^T (..) V^-T have the
+    generic north-east pivots of f_r. (NE ranks mod q <= over Q <= f_r, so equality mod q proves it over Q.)
+ F. Certificate: rt_histogram with the round-6 entrance corner and with the staircase corner (m-4h+2, h-2, h-5, 1^6),
+    rank sum == s, moment.certify (exact), kappa = certificate_round3.evaluate(a_b, round-6 complex values).
+Negative controls (CONTROL=1): see controls().
+Usage: python3 check_lifted.py h [--no-generic]"""
+import sys, os, json, random, time
+from itertools import combinations
+from collections import Counter, OrderedDict
+from fractions import Fraction as Fr
+from array import array
+HERE = os.path.dirname(os.path.abspath(__file__))
+sys.path.insert(0, HERE)
+from linalg import *
+
+T0 = time.time()
+def load_witness(h):
+    import gzip
+    with gzip.open(os.path.join(HERE, '..', '..', 'certificates', 'round7', 'witness_%d.json.gz' % h), 'rt') as fh:
+        return json.load(fh)
+
+
+def log(*a):
+    print('[%5.0fs]' % (time.time() - T0), *a, flush=True)
+
+
+# ----------------------------------------------------------------------------------------------- A. DAG
+class Prog:
+    def __init__(self, W):
+        self.W = W; h = self.h = W['h']
+        self.trip = list(combinations(range(h), 3)); v = self.v = len(self.trip)
+        self.args = {int(n): (tuple(a) if a else None) for n, a in W['args'].items()}
+        self.act = sorted(self.args)
+        for n in self.act:
+            if self.args[n] is None: assert 1 <= n <= v and tuple(W['leaf'][str(n)]) == self.trip[n - 1]
+            else: assert all(a < n and a in self.args for a in self.args[n]), n
+        self.sup = {}
+        for n in self.act:
+            if self.args[n] is None: self.sup[n] = 1 << (n - 1)
+            else:
+                a, b = self.args[n]; assert not self.sup[a] & self.sup[b], 'overlapping supports'
+                self.sup[n] = self.sup[a] | self.sup[b]
+        self.outputs = {(c, tuple(T)): n for c, T, n in W['outputs']}
+        self.retained = {c: n for c, n in W['retained']}
+        star = [sum(1 << i for i, t in enumerate(self.trip) if q in t) for q in range(h)]
+        assert len(self.outputs) == 3 * v and len(self.retained) == h
+        for (c, T), n in self.outputs.items():
+            A, B = [q for q in T if q != c]; assert c in T and self.sup[n] == star[c] & ~star[A] & ~star[B]
+        for c, n in self.retained.items(): assert self.sup[n] == star[c]
+        roots = set(self.outputs.values()) | set(self.retained.values()); reach = set(); st = list(roots)
+        while st:
+            n = st.pop()
+            if n in reach: continue
+            reach.add(n)
+            if self.args[n]: st.extend(self.args[n])
+        assert reach == set(self.act), 'inactive nodes in witness'
+        self.mL = {x: (y, k) for x, y, k in W['links']}
+        self.j = {int(n): j for n, j in W['j'].items()}
+        self.late = {int(n): (list(e), list(l)) for n, (e, l) in W['late'].items()}
+        self.F = W['flag']
+
+
+def trow(h, S): return [int(q in S) for q in range(h)]
+
+
+def spans(pr):
+    """independent triple subsets (exact: 0/1 rows with three ones, every minor < 3^(h/2) < Q31) and dims."""
+    h = pr.h; assert 3 ** h < Q31 ** 2
+    bas = {}; dn = {}
+    for n in pr.act:
+        if pr.args[n] is None: bas[n] = (n - 1,); dn[n] = 1; continue
+        a, b = pr.args[n]; E = Echelon(Q31); keep = []
+        for i in bas[a] + bas[b]:
+            if len(E) == h: break
+            if E.insert(trow(h, pr.trip[i])): keep.append(i)
+        bas[n] = tuple(keep); dn[n] = len(keep)
+    return bas, dn
+
+
+def users_of(pr, dn):
+    key = lambda n: (dn[n], n); users = {n: [] for n in pr.act}
+    for n in sorted(pr.act, key=key):
+        if pr.args[n]:
+            for pos, x in enumerate(pr.args[n]): users[x].append(('gate', n, pos))
+    for (c, T), n in sorted(pr.outputs.items()): users[n].append(('out', (c, T)))
+    for c, n in sorted(pr.retained.items()): users[n].append(('ret', c))
+    return users
+
+
+# ----------------------------------------------------------------------------------------------- B. compile
+def root_key(u): return ('out',) + u[1] if u[0] == 'out' else ('ret', u[1])
+
+
+def compile_(pr, dn, users, late=True, defer=True, tamper=None):
+    """returns ops (scalar program), chains (frame keys per slot), reads, R. tamper: negative-control hooks."""
+    key = lambda n: (dn[n], n); order = sorted(pr.act, key=key); gpos = {n: i for i, n in enumerate(order)}
+    linked = {r: x for x, r in pr.mL.items()}
+    chains = []; edge = {}; reads = []; ops = []; deferred = []
+    def new(first): chains.append(list(first)); return len(chains) - 1
+    def assign(s, n, k):
+        u = users[n][k]
+        if u[0] == 'gate': edge[u[1], u[2]] = s
+        else:
+            chains[s] += [root_key(u), ('F',)]
+            reads.append((s, [u[1][1]] if u[0] == 'out' else [T for T in pr.trip if u[1] in T]))
+    def ufr(n, k):
+        u = users[n][k]; return ('n', u[1]) if u[0] == 'gate' else root_key(u)
+    for n in order:
+        if pr.args[n] is None:
+            piv = new([('0',), ('n', n)]); ops.append(('src', piv, n, n))
+        else:
+            a, b = pr.args[n]; sa, sb = edge.pop((n, 0)), edge.pop((n, 1))
+            if n in pr.mL and pr.mL[n][0] == a: sa, sb = sb, sa          # the linked operand stays on the non-pivot
+            piv, oth = sa, sb
+            chains[piv].append(('n', n)); chains[oth].append(('n', n)); ops.append(('add', piv, oth, n))
+            if n in pr.mL: y, k = pr.mL[n]; assign(oth, y, k)
+            else: chains[oth].append(('F',))
+        ks = [k for k in range(len(users[n])) if (n, k) not in linked]
+        if late and n in pr.late:
+            early, lt = pr.late[n]; assert sorted(early + lt) == ks, ('bad late decision', n)
+        else: early, lt = ks[1:], [ks[0]]
+        for k in early:
+            f = new([('0',), ('n', n)]); ops.append(('copy', piv, f, n)); assign(f, n, k)
+        for i, k in enumerate(lt):
+            Ck = ufr(n, k) if i == len(lt) - 1 else ('c', n, tuple(lt[i:]))
+            if tamper == 'nointersect' and i < len(lt) - 1: Ck = ufr(n, k)
+            chains[piv].append(Ck)
+            if i < len(lt) - 1:
+                f = new([('0',), Ck]); deferred.append((n, i, f, k, lt[-1])); assign(f, n, k)
+        k = lt[-1]
+        if users[n][k][0] == 'gate': edge[users[n][k][1], users[n][k][2]] = piv
+        else: assign(piv, n, k)
+    assert not edge
+    # place deferred copies: as late as possible, in order, before the user's gate and the pivot consumer's gate
+    INF = len(order); byn = {}
+    for (n, i, f, k, kl) in deferred: byn.setdefault(n, []).append((i, f, k, kl))
+    ins = {}
+    for n, lst in byn.items():
+        lst.sort(); D = []
+        for i, f, k, kl in lst:
+            u, ul = users[n][k], users[n][kl]
+            d = min(gpos[u[1]] if u[0] == 'gate' else INF, gpos[ul[1]] if ul[0] == 'gate' else INF)
+            D.append(d)
+        for t in range(len(D) - 2, -1, -1): D[t] = min(D[t], D[t + 1])
+        for (i, f, k, kl), d in zip(lst, D):
+            if not defer: d = gpos[n] + 1                                   # immediate copy (still a valid program)
+            if tamper == 'aftercons' and n == tamper_node[0] and i == 0:
+                ul = users[n][kl]; d = gpos[ul[1]] + 1                       # copy AFTER the pivot was consumed
+            ins.setdefault(d, []).append((n, i, f))
+    pivslot = {}
+    for o in ops:
+        if o[0] in ('src', 'add'): pivslot[o[3]] = o[1]
+    out = []
+    for o in ops:
+        if o[0] in ('src', 'add'):
+            for (n, i, f) in sorted(ins.get(gpos[o[3]], []), key=lambda z: (gpos[z[0]], z[1])):
+                out.append(('copy', pivslot[n], f, n))
+        out.append(o)
+    for (n, i, f) in sorted(ins.get(INF, []) + [z for d, zs in ins.items() if d > INF for z in zs], key=lambda z: (gpos[z[0]], z[1])):
+        out.append(('copy', pivslot[n], f, n))
+    if tamper == 'dropcopy':
+        idx = next(i for i, o in enumerate(out) if o[0] == 'copy' and o[3] in pr.late); out.pop(idx)
+    R = len(chains); adds = sum(1 for n in pr.act if pr.args[n])
+    assert R == adds + len(pr.outputs) + len(pr.retained) - len(pr.mL), 'role count'
+    return out, chains, reads, R
+
+tamper_node = [None]
+
+
+# ----------------------------------------------------------------------------------------------- C. replay
+def replay(pr, ops, reads, R, ring, rng, mirror=False, midread=True):
+    trip, v = pr.trip, pr.v; tid = {T: i for i, T in enumerate(trip)}
+    mod = (lambda z: z & 1) if ring == 2 else (lambda z: z)
+    last = {}
+    for i, o in enumerate(ops):
+        last[o[1]] = i
+        if o[0] != 'src': last[o[2]] = i
+    readers = {}
+    for s, Ts in reads: readers.setdefault(last[s] if midread else len(ops) - 1, []).append((s, Ts))
+    x = [rng.randrange(-10**6, 10**6) for _ in range(v)]
+    a0 = [rng.randrange(-10**6, 10**6) for _ in range(R)]; y0 = [rng.randrange(-10**6, 10**6) for _ in range(v)]
+    a = list(a0); y = list(y0); src = [(o[1], o[2]) for o in ops if o[0] == 'src']
+    def L(sg=None):
+        for i, o in enumerate(ops):
+            if o[0] == 'add': a[o[1]] += a[o[2]]
+            elif o[0] == 'copy': a[o[2]] += a[o[1]]
+            if sg is not None:
+                for s, Ts in readers.get(i, ()):
+                    for T in Ts: y[tid[T]] += sg * a[s]
+    def Li():
+        for o in reversed(ops):
+            if o[0] == 'add': a[o[1]] -= a[o[2]]
+            elif o[0] == 'copy': a[o[2]] -= a[o[1]]
+    def V(sg):
+        for s, n in src: a[s] += sg * x[n - 1]
+    if not mirror: L(-1); Li(); V(+1); L(+1); Li(); V(-1)
+    else: V(+1); L(+1); Li(); V(-1); L(-1); Li()
+    if any(mod(p - q) for p, q in zip(a, a0)): return False
+    val = {}
+    for n in pr.act: val[n] = x[n - 1] if pr.args[n] is None else val[pr.args[n][0]] + val[pr.args[n][1]]
+    want = list(y0)
+    for (c, T), n in pr.outputs.items(): want[tid[T]] += val[n]
+    for c, n in pr.retained.items():
+        for T in trip:
+            if c in T: want[tid[T]] += val[n]
+    if any(mod(p - q) for p, q in zip(y, want)): return False
+    if ring == 2 and any(mod(y[i] - y0[i] - x[i]) for i in range(v)): return False
+    return True
+
+
+# ----------------------------------------------------------------------------------------------- D. frames
+class Frames:
+    def __init__(self, pr, dn, bas, users):
+        self.pr = pr; self.dn = dn; self.bas = bas; self.users = users; h = self.h = pr.h
+        self.rootcov = {}
+        for (c, T) in pr.outputs: self.rootcov[('out', c, T)] = [9 * int(q in T) - 3 for q in range(h)]
+        for c in pr.retained: self.rootcov[('ret', c)] = [1 - 3 * int(q == c) for q in range(h)]
+        rid = {k: i for i, k in enumerate(sorted(self.rootcov))}; self.rkeys = sorted(self.rootcov)
+        self.rid = rid
+        # slot successors (same for base and late compile: every free use of n is reached by one slot from n,
+        # a donor's non-pivot slot goes on to its linked use) and direct roots
+        linked = {r: x for x, r in pr.mL.items()}
+        self.succ = {n: set() for n in pr.act}; self.droots = {n: set() for n in pr.act}
+        def to(n, use):
+            if use[0] == 'gate': self.succ[n].add(use[1])
+            else: self.droots[n].add(rid[root_key(use)])
+        for n in pr.act:
+            for k, u in enumerate(users[n]):
+                if (n, k) not in linked: to(n, u)
+            if n in pr.mL: y, k = pr.mL[n]; to(n, users[y][k])
+
+    def check_structure(self):
+        """combinatorial / exact facts behind the nesting theorem."""
+        pr, h = self.pr, self.h; bad_j = bad_link = bad_root = 0
+        J = lambda n: self.pr.j.get(n, 0)
+        for n in pr.act:
+            if pr.args[n] is None: assert J(n) == 0
+            for t in self.succ[n]:
+                if J(n) > J(t): bad_j += 1
+        for x, (y, k) in pr.mL.items():                                   # span(x) <= span(linked use target)
+            u = self.users[y][k]
+            if u[0] == 'gate':
+                t = u[1]; E = Echelon(Q31)
+                for i in self.bas[t]: E.insert(trow(h, pr.trip[i]))
+                if not all(E.contains(trow(h, pr.trip[i])) for i in self.bas[x]): bad_link += 1
+        for n in pr.act:                                                  # span(n) <= direct root frames
+            for r in self.droots[n]:
+                g = self.rootcov[self.rkeys[r]]
+                if any(sum(gq * tq for gq, tq in zip(g, trow(h, pr.trip[i]))) for i in self.bas[n]): bad_root += 1
+        # F invertible over Q (a nonzero determinant mod a prime)
+        F_ok = rank_mod(self.pr.F, Q31) == h
+        return bad_j, bad_link, bad_root, F_ok
+
+    def cobases(self):
+        """K_n: independent subset of the root covectors reachable from n, spanning all of them over Q
+        (selection mod M127, valid by the Hadamard certificate on root covectors)."""
+        pr, h = self.pr, self.h
+        rows = [self.rootcov[k] for k in self.rkeys]
+        mx = max(sum(x * x for x in r) for r in rows); assert mx ** h < M127 ** 2, 'Hadamard bound for K'
+        key = lambda n: (self.dn[n], n); K = {}
+        for n in sorted(pr.act, key=key, reverse=True):
+            cand = sorted(self.droots[n]); srcs = sorted((K[t] for t in self.succ[n]), key=len, reverse=True)
+            cap = h - self.dn[n]                                          # span(n) <= U_n
+            if not cand and len(srcs) == 1: K[n] = srcs[0]; continue
+            E = Echelon(M127); keep = []
+            seq = list(srcs[0]) if srcs else []
+            for r in seq: E.insert(rows[r]); keep.append(r)                # srcs[0] is already independent
+            for lst in [cand] + srcs[1:]:
+                for r in lst:
+                    if len(keep) >= cap: break
+                    if E.insert(rows[r]): keep.append(r)
+            K[n] = tuple(keep)
+        self.K = K
+        return K
+
+    def dims(self):
+        """exact dim of every addition frame M_n."""
+        pr, h = self.pr, self.h; F = pr.F; d = dict(self.dn); stats = Counter()
+        rows = [self.rootcov[k] for k in self.rkeys]
+        self.KF = {}
+        for n, j in pr.j.items():
+            Fj = F[:j]; K = [rows[r] for r in self.K[n]]
+            KF = [[sum(a * b for a, b in zip(kr, fr)) for fr in Fj] for kr in K]
+            SF = [trow(h, pr.trip[i]) for i in self.bas[n]] + Fj
+            r1 = self._rank(SF, min(len(SF), h), stats, 'SF'); r2 = self._rank(KF, min(len(K), j), stats, 'KF')
+            d[n] = r1 - r2; self.KF[n] = (r2, KF)
+            assert self.dn[n] <= d[n] <= h - len(K)
+        self.d = d
+        return d, stats
+
+    @staticmethod
+    def _rank(M, full, stats, tag):
+        if not M or full == 0: return 0
+        r = rank_mod(M, Q31)
+        if r == full: stats[tag + ' full mod q'] += 1; return r          # rank mod q <= rank over Q <= full
+        if hadamard_ok(M, M521): stats[tag + ' M521'] += 1; return rank_mod(M, M521)
+        stats[tag + ' Bareiss'] += 1; return rank_exact(M)
+
+    # exact rational bases, used for the late-copy intersections
+    def exact_basis(self, key):
+        pr, h = self.pr, self.h
+        if key[0] in ('out', 'ret'):
+            return null_exact([self.rootcov[key]], h)
+        n = key[1]
+        S = [trow(h, pr.trip[i]) for i in self.bas[n]]
+        j = pr.j.get(n, 0)
+        if not j: return S
+        _, KF = self.KF[n]; F = pr.F[:j]
+        X = null_exact(KF, j) if KF else [[int(i == t) for t in range(j)] for i in range(j)]
+        vecs = S + [[sum(c * F[i][q] for i, c in enumerate(x)) for q in range(h)] for x in X]
+        # independent subset: rows independent mod q are independent over Q; d of them span M_n (dim M_n = d exactly)
+        E = Echelon(Q31); out = []
+        for vv in vecs:
+            if E.insert(vv): out.append(vv)
+        assert len(out) == self.d[n], ('exact basis dim', n, len(out), self.d[n])
+        return out
+
+    def intersect_exact(self, A, B):
+        h = self.h
+        if not A or not B: return []
+        # z = (x, y): x A = y B  <=>  [A; -B]^T z = 0
+        M = [[A[i][q] for i in range(len(A))] + [-B[i][q] for i in range(len(B))] for q in range(h)]
+        Z = null_exact(M, len(A) + len(B))
+        # A and B are bases, so z -> x A is injective on the null space: the images are a basis of A cap B
+        return [[sum(z[i] * A[i][q] for i in range(len(A))) for q in range(h)] for z in Z]
+
+
+def late_frames(fr, pr, users):
+    """exact dims of every late-copy intersection frame ('c', n, late users)."""
+    cache = {}; dims = {}
+    def eb(k):
+        if k not in cache: cache[k] = fr.exact_basis(k)
+        return cache[k]
+    def ufr(n, k):
+        u = users[n][k]; return ('n', u[1]) if u[0] == 'gate' else root_key(u)
+    for n, (early, lt) in pr.late.items():
+        for i in range(len(lt) - 1):
+            B = eb(ufr(n, lt[-1]))
+            for k in reversed(lt[i:-1]): B = fr.intersect_exact(eb(ufr(n, k)), B)
+            dims[('c', n, tuple(lt[i:]))] = len(B)
+    return dims, cache
+
+
+def chain_steps(chains, dimof, h):
+    rk = Counter(); steps = {}
+    for ch in chains:
+        prev = ch[0]; pd = dimof(prev)
+        for k in ch[1:]:
+            dk = dimof(k)
+            assert dk >= pd, ('decreasing chain', prev, k, pd, dk)
+            if dk > pd: rk[dk - pd] += 1; steps[prev, k] = dk - pd; prev, pd = k, dk
+            else: prev = k                                                 # equal dims, nested => equal frames
+        assert pd == h
+    return rk, steps
+
+
+# ----------------------------------------------------------------------------------------------- E. mod-q frames, side lemma
+def point_UV(h, seed):
+    """the U, V drawn by independent/two-stage-bit/flag_existence.py with this seed (same random stream)."""
+    CR = 10**6; rng = random.Random(seed)
+    nz = lambda: rng.choice([-1, 1]) * rng.randint(1, CR)
+    for b in range(h):
+        for i in range(h):
+            for jj in range(h):
+                if i <= b and jj <= b: nz()
+    for b in range(h):
+        for i in range(h):
+            for jj in range(h):
+                if i >= b and jj >= b: nz()
+    U = [[rng.randint(-CR, CR) for _ in range(h)] for _ in range(h)]
+    V = [[rng.randint(-CR, CR) for _ in range(h)] for _ in range(h)]
+    return U, V
+
+
+class ModQ:
+    def __init__(self, fr, latedims, control=False):
+        self.fr = fr; self.h = h = fr.h; self.q = q = Q31; self.latedims = latedims
+        U, V = point_UV(h, 1)
+        if control: U = [[int(i == k) for k in range(h)] for i in range(h)]; V = [r[:] for r in U]
+        self.conj = []
+        for A in (U, V):
+            Ai = inv_mod_matrix(A, q); assert Ai is not None
+            self.conj.append(([list(c) for c in zip(*A)], Ai))          # columns of A, rows of A^-1
+        self.inv9 = pow(9, q - 2, q); self.bcache = {}; self.qcache = OrderedDict(); self.cap = 6000
+        self.degenerate = 0; self.dimfail = 0
+
+    def basis(self, key):
+        if key in self.bcache: return self.bcache[key]
+        fr, h, q = self.fr, self.h, self.q; pr = fr.pr
+        if key == ('0',): B = []
+        elif key == ('F',): B = [[int(i == t) for t in range(h)] for i in range(h)]
+        elif key[0] in ('out', 'ret'): B = null_mod([fr.rootcov[key]], h, q)
+        elif key[0] == 'c':
+            n, lt = key[1], key[2]
+            B = None
+            for k in lt:
+                u = fr.users[n][k]; kk = ('n', u[1]) if u[0] == 'gate' else root_key(u)
+                Bk = self.basis(kk)
+                B = Bk if B is None else self._cap(B, Bk)
+            if len(B) != self.latedims[key]: self.dimfail += 1
+        else:
+            n = key[1]; S = [trow(h, pr.trip[i]) for i in fr.bas[n]]; j = pr.j.get(n, 0)
+            E = Echelon(q)
+            for r in S: E.insert(r)
+            if j:
+                r2, KF = fr.KF[n]
+                if KF and rank_mod(KF, q) != r2: self.dimfail += 1
+                X = null_mod(KF, j, q) if KF else [[int(i == t) for t in range(j)] for i in range(j)]
+                Fj = pr.F[:j]
+                for x in X: E.insert([sum(c * Fj[i][t] for i, c in enumerate(x)) % q for t in range(h)])
+            B = E.rows
+            if len(B) != fr.d[n]: self.dimfail += 1
+        self.bcache[key] = B
+        return B
+
+    def _cap(self, A, B):
+        q, h = self.q, self.h
+        if not A or not B: return []
+        M = [[A[i][t] for i in range(len(A))] + [(-B[i][t]) % q for i in range(len(B))] for t in range(h)]
+        Z = null_mod(M, len(A) + len(B), q); E = Echelon(q)
+        for z in Z: E.insert([sum(z[i] * A[i][t] for i in range(len(A))) % q for t in range(h)])
+        return E.rows
+
+    def gram_inv(self, B):
+        q = self.q; BG = [[(x - sum(b) * self.inv9) % q for x in b] for b in B]
+        Gr = [[sum(x * y for x, y in zip(bg, c)) % q for c in B] for bg in BG]
+        return BG, inv_mod_matrix(Gr, q)
+
+    def nondeg(self, key):
+        B = self.basis(key)
+        if not B or len(B) == self.h: return True
+        return self.gram_inv(B)[1] is not None
+
+    def conjugated(self, key):
+        """(U^T P U^-T, V^T P V^-T) for the G-orthogonal projector P onto the frame, flattened."""
+        if key in self.qcache: self.qcache.move_to_end(key); return self.qcache[key]
+        h, q = self.h, self.q; B = self.basis(key)
+        if not B: out = (array('q', [0] * h * h),) * 2
+        elif len(B) == h: I = array('q', [int(i == t) for i in range(h) for t in range(h)]); out = (I, I)
+        else:
+            BG, Gi = self.gram_inv(B)
+            if Gi is None: raise ZeroDivisionError(key)
+            res = []
+            for Acols, Ai in self.conj:
+                LU = [[sum(x * y for x, y in zip(ac, b)) % q for b in B] for ac in Acols]       # A^T B^T  (h x d)
+                RU = [[sum(x * y for x, y in zip(bg, ai)) % q for ai in Ai] for bg in BG]      # B G A^-T (d x h)
+                Mi = matmul(Gi, RU, q)
+                Mt = list(zip(*Mi))
+                res.append(array('q', [sum(x * y for x, y in zip(l, c)) % q for l in LU for c in Mt]))
+            out = tuple(res)
+        self.qcache[key] = out
+        if len(self.qcache) > self.cap: self.qcache.popitem(last=False)
+        return out
+
+
+def side_lemma(mq, steps, h, retained):
+    q = mq.q; pr_ = predicted(h); E = dict((k, r) for k, r in steps.items() if 2 * r > h)
+    for c in retained: E[('0',), ('ret', c)] = h - 1                    # retained copy edges 0 -> U_c
+    keys = sorted(E, key=lambda e: (str(e[1]), str(e[0])))              # group by target frame (cache reuse)
+    bad = 0; n = 0
+    for (A, B) in keys:
+        r = E[A, B]; QA, QB = mq.conjugated(A), mq.conjugated(B)
+        for t in range(2):
+            a, b = QA[t], QB[t]
+            M = [[(b[i * h + c] - a[i * h + c]) % q for c in range(h)] for i in range(h)]
+            if rightmost_pivots(M, q) != pr_[r]: bad += 1
+        n += 1
+        if n % 5000 == 0: log('   side lemma %d/%d, failures %d' % (n, len(keys), bad))
+    return bad, len(keys)
+
+
+# ----------------------------------------------------------------------------------------------- F. certificate
+K_REPO = os.path.normpath(os.path.join(HERE, '..', '..'))
+sys.path[:0] = [K_REPO + '/scripts', K_REPO + '/independent/two-stage-bit']
+
+
+def inner_children(r, h): return [1] * (h - r) + [2 * r - h] if 2 * r > h else [1] * r
+
+
+def rt_histogram(h, R, rk, corner):
+    """cert_rt.rt_histogram with the data-entrance corner profile 'corner' (round six: [h-2] + [1]*(h+1))."""
+    from math import comb
+    v, m = comb(h, 3), h * h; N = v * v; A = v * R; H = Counter()
+    assert sum(r * n for r, n in rk.items()) == h * R and sum(corner) == 2 * h - 1
+    for _ in range(2):
+        H[m - 2 * h] += A; H[h] += A
+        for r, n in rk.items():
+            for w in inner_children(r, h): H[w] += v * n
+        for w in inner_children(h - 1, h): H[w] += v * h
+    H[m - 4 * h + 2] += 2 * N
+    for w in corner: H[w] += 2 * N
+    for _ in range(2): H[h - 2] += 2 * N; H[1] += 2 * N
+    H[1] += N
+    W = 2 * N + 2 * A; L = 2 * v * h * (h - 1); s = W * m - N + L
+    H = {w: c for w, c in H.items() if c}
+    assert sum(w * c for w, c in H.items()) == s and max(H) <= m - 1
+    return dict(h=h, m=m, v=v, N=N, W=W, L=L, s=s, R=R, hist=H)
+
+
+def certificate(h, R, rk):
+    import moment
+    from certificate_round3 import evaluate
+    out = {}
+    for name, corner in (('round6 corner', [h - 2] + [1] * (h + 1)), ('staircase', [h - 2, h - 5] + [1] * 6)):
+        c = rt_histogram(h, R, rk, corner)
+        if c['W'] * c['m'] <= c['s']: out[name] = ('no deficit', c['s']); continue
+        a, root = moment.certify(c)
+        k = evaluate(a, Fr(36926111, 5 * 10**11), Fr(1, 1000), 'crude', m_c=576, s_c=119453132304)
+        out[name] = (a, root, c['s'], c['W'], k['kappa'], k['ok'])
+    return out
+
+
+# ----------------------------------------------------------------------------------------------- main
+def main(h, generic=True, control=False):
+    W = load_witness(h)
+    pr = Prog(W); log('h=%d: A. DAG semantics OK (%d nodes, %d additions, %d links, %d lifted, %d late)' % (
+        h, len(pr.act), sum(1 for n in pr.act if pr.args[n]), len(pr.mL), len(pr.j), len(pr.late)))
+    bas, dn = spans(pr); users = users_of(pr, dn)
+    ops, chains, reads, R = compile_(pr, dn, users)
+    log('B. compiled: R=%d (claimed %d), ops %d, late copies %d' % (R, W['claimed']['R'], len(ops),
+        sum(len(l) - 1 for _, l in pr.late.values())))
+    assert R == W['claimed']['R']
+    rng = random.Random(h)
+    res = {(ring, mir, mid): all(replay(pr, ops, reads, R, ring, rng, mir, mid) for _ in range(2))
+           for ring in (2, 0) for mir in (False, True) for mid in (True, False)}
+    log('C. dirty-scratch replay (F2/Z x word/mirror x midread/endread): %s' % ('ALL OK' if all(res.values()) else res))
+    fr = Frames(pr, dn, bas, users)
+    bj, bl, br, Fok = fr.check_structure()
+    log('D. nesting facts: j-monotone failures %d, link span failures %d, root containment failures %d, F invertible %s'
+        % (bj, bl, br, Fok))
+    fr.cobases(); log('   co-bases K_n done')
+    d, st = fr.dims(); log('   lifted frame dims exact: %s' % dict(st))
+    nU = sum(1 for n in pr.j if d[n] == h - len(fr.K[n]))
+    log('   every lifted frame is the largest admissible frame U_n: %s (%d / %d; the flag only switches span -> U_n)'
+        % (nU == len(pr.j), nU, len(pr.j)))
+    ld, ecache = late_frames(fr, pr, users); log('   late-copy intersections exact: %d frames' % len(ld))
+    def dimof(k):
+        if k == ('0',): return 0
+        if k == ('F',): return h
+        if k[0] in ('out', 'ret'): return h - 1
+        if k[0] == 'c': return ld[k]
+        return d[k[1]]
+    rk, steps = chain_steps(chains, dimof, h)
+    claimed = Counter({int(r): c for r, c in W['claimed']['rk'].items()})
+    log('   chain steps: sum of ranks %d = h R %d: %s; histogram equals claim: %s; distinct steps %d (2r>h: %d)' % (
+        sum(r * c for r, c in rk.items()), h * R, sum(r * c for r, c in rk.items()) == h * R, rk == claimed,
+        len(steps), sum(1 for r in steps.values() if 2 * r > h)))
+    out = dict(h=h, R=R, replay=all(res.values()), struct=(bj, bl, br, Fok), hist_ok=rk == claimed, rk=rk)
+    if generic:
+        mq = ModQ(fr, ld, control)
+        frames_used = {k for e in steps for k in e} | {('ret', c) for c in pr.retained}
+        nd = sum(1 for k in frames_used if not mq.nondeg(k))
+        log('E. frames used %d: G-degenerate %d, reduction dimension mismatches %d' % (len(frames_used), nd, mq.dimfail))
+        bad, ne = side_lemma(mq, steps, h, pr.retained)
+        log('   side lemma at one integer point%s: failures %d over %d distinct high-rank steps x 2 conjugations' % (
+            ' (CONTROL U=V=I)' if control else '', bad, ne))
+        out.update(degenerate=nd, dimfail=mq.dimfail, side_bad=bad, side_n=ne)
+    for name, val in certificate(h, R, rk).items():
+        if val[0] == 'no deficit': log('F. %s: no deficit at this h (s=%d), certificate n/a' % (name, val[1])); continue
+        a, root, s, Wr, kap, ok = val
+        log('F. %s: s=%d W=%d a_b=%s (%.6e, root %.6e) kappa=%s (%.7e) ok=%s' % (name, s, Wr, a, float(a), root, kap,
+            float(kap), ok))
+        out[name] = (a, kap, ok)
+    return out, pr, dn, users
+
+
+
+
+# ----------------------------------------------------------------------------------------------- negative controls
+def controls(h, nsample=40):
+    """Each must FAIL (or differ) for the check to have teeth:
+     (a) one late copy moved to just after its pivot's consumer gate: replay must fail;
+     (b) one late copy dropped: replay must fail;
+     (c) pivot chain without intersections (C_i := frame of user i alone): exhibit non-nested consecutive pivot frames;
+     (d) M_n rebuilt from an independent flag F' on targets strictly between span and U (vacuous for this
+         witness: every lifted frame equals U_n, so the flag is only a switch and nesting needs no common flag);
+     (e) side lemma at U = V = I on the first high-rank steps: generic pivots must fail;
+     (f) no late copies (lifted frames only): histogram must change."""
+    W = load_witness(h)
+    pr = Prog(W); bas, dn = spans(pr); users = users_of(pr, dn); rng = random.Random(7)
+    key = lambda n: (dn[n], n); linked = {r: x for x, r in pr.mL.items()}
+    cand = [n for n, (e, lt) in pr.late.items() if users[n][lt[-1]][0] == 'gate']
+    fails = 0; tried = 0
+    for n in cand[:nsample]:
+        tamper_node[0] = n; ops, ch, rd, R = compile_(pr, dn, users, tamper='aftercons')
+        ok = all(replay(pr, ops, rd, R, ring, rng) for ring in (2, 0)); tried += 1; fails += not ok
+    log('(a) late copy moved after the pivot consumer: replay failed %d / %d (must be all)' % (fails, tried))
+    ops, ch, rd, R = compile_(pr, dn, users, tamper='dropcopy')
+    log('(b) one late copy dropped: replay correct %s (must be False)' % all(replay(pr, ops, rd, R, ring, rng) for ring in (2, 0)))
+    fr = Frames(pr, dn, bas, users); fr.cobases(); d, _ = fr.dims(); d = dict(d); ld, _ = late_frames(fr, pr, users)
+    mq = ModQ(fr, ld)
+    def inside(A, B):
+        E = Echelon(mq.q)
+        for r in mq.basis(B): E.insert(r)
+        return all(E.contains(r) for r in mq.basis(A))
+    def ufr(n, k):
+        u = users[n][k]; return ('n', u[1]) if u[0] == 'gate' else root_key(u)
+    bad = tot = 0
+    for n, (e, lt) in pr.late.items():
+        for i in range(len(lt) - 1):
+            tot += 1; bad += not inside(ufr(n, lt[i]), ufr(n, lt[i + 1]))
+    log('(c) pivot chain through user frames without intersections: non-nested consecutive frames %d / %d '
+        '(the intersections are needed)' % (bad, tot))
+    # (d) the common flag is essential: rebuild M_n with an independent flag F' (same j(n)) and test M_n <= M_t
+    pairs = [(n, t) for n in pr.j for t in fr.succ[n] if pr.j.get(t, 0) > 0 and dn[t] < d[t] < h - len(fr.K[t])]
+    rng.shuffle(pairs); bad = tot = 0; r2 = random.Random(99)
+    Fp = [[r2.randint(-50, 50) for _ in range(h)] for _ in range(h)]
+    for n, t in pairs[:nsample]:
+        j = pr.j[n]; K = [fr.rootcov[fr.rkeys[r]] for r in fr.K[n]]
+        KF = [[sum(a * b for a, b in zip(kr, fl)) for fl in Fp[:j]] for kr in K]
+        if KF and rank_mod(KF, mq.q) != fr._rank(KF, min(len(K), j), Counter(), 'KF'): continue
+        X = null_mod(KF, j, mq.q) if KF else [[int(i == c) for c in range(j)] for i in range(j)]
+        E = Echelon(mq.q)
+        for r in mq.basis(('n', t)): E.insert(r)
+        vecs = [[sum(c * Fp[i][q] for i, c in enumerate(x)) % mq.q for q in range(h)] for x in X]
+        tot += 1; bad += not all(E.contains(v) for v in vecs)
+    log("(d) M_n rebuilt with an independent flag F': M_n not inside M_t on %d / %d lifted successor pairs (targets strictly between span and U)" % (bad, tot))
+    ops, chains, reads, R = compile_(pr, dn, users)
+    fr2 = Frames(pr, dn, bas, users); fr2.cobases(); d2, _ = fr2.dims(); ld2, _ = late_frames(fr2, pr, users)
+    dimof = lambda k: 0 if k == ('0',) else h if k == ('F',) else h - 1 if k[0] in ('out', 'ret') else ld2[k] if k[0] == 'c' else d2[k[1]]
+    rk, steps = chain_steps(chains, dimof, h)
+    mqs = ModQ(fr2, ld2); smp = list(steps); random.Random(5).shuffle(smp); nb = 0
+    for A, B in smp[:3000]:
+        E = Echelon(mqs.q)
+        for r in mqs.basis(B): E.insert(r)
+        nb += not all(E.contains(r) for r in mqs.basis(A))
+    log('(g) sanity (positive): 3000 sampled chain steps A -> B with A inside B mod q: failures %d' % nb)
+    mq0 = ModQ(fr2, ld2, control=True); pr_ = predicted(h); hi = [(e, r) for e, r in steps.items() if 2 * r > h][:300]
+    nb = 0
+    for (A, B), r in hi:
+        QA, QB = mq0.conjugated(A), mq0.conjugated(B)
+        M = [[(QB[0][i * h + c] - QA[0][i * h + c]) % mq0.q for c in range(h)] for i in range(h)]
+        nb += rightmost_pivots(M, mq0.q) != pr_[r]
+    log('(e) side lemma with U = V = I: failures %d / %d (must be most)' % (nb, len(hi)))
+    ops, chains0, reads, R = compile_(pr, dn, users, late=False)
+    rk0, _ = chain_steps(chains0, dimof, h)
+    log('(f) without late copies: histogram differs %s; certificate %s' % (rk0 != rk, {k: v[0] for k, v in certificate(h, R, rk0).items()}))
+
+
+if __name__ == '__main__':
+    h = int(sys.argv[1])
+    if '--controls' in sys.argv: controls(h)
+    else:
+        out = main(h, generic='--no-generic' not in sys.argv)[0]
+        ok = out['replay'] and out['struct'][:3] == (0, 0, 0) and out['struct'][3] and out['hist_ok']
+        ok &= out.get('degenerate', 0) == 0 and out.get('dimfail', 0) == 0 and out.get('side_bad', 0) == 0
+        ok &= all(v[2] for k, v in out.items() if k in ('round6 corner', 'staircase'))
+        log('ALL %s' % ('PASS' if ok else 'FAIL'))
+        sys.exit(0 if ok else 1)
diff --git a/research/cyclic-deferred/bit-sharing/inputs/research/round7-public/tree/independent/deferred-readout/deferred.py b/research/cyclic-deferred/bit-sharing/inputs/research/round7-public/tree/independent/deferred-readout/deferred.py
new file mode 100644
index 0000000000..b77bb4861c
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/inputs/research/round7-public/tree/independent/deferred-readout/deferred.py
@@ -0,0 +1,188 @@
+"""Shared reader and accounting for the round-seven bit side (stdlib only).
+
+Frozen inputs (certificates/round7/):
+  witness_23.json.gz   the base side program: DAG (leaves = triples of [h], additions), outputs (c, T), retained
+                       totals, links, the flag matrix, the lift index j(n) of every lifted addition and the late-copy
+                       decisions; checked by check_lifted.py;
+  deferred_23.json.gz  the reordered stage-1 word: the slot schedule (ops, the node held by every slot over time,
+                       slot starts, output and retained slots), the deferred readout frames sigma_u, the V-leaf start
+                       frames s_i (both as primitive integer bases), the readout order, the X_S order of every leaf,
+                       the order of the deferred V gates, and the dimension of every node frame and late-copy
+                       intersection (re-derived exactly by check_frames.py).
+
+Ops: ['add', piv, oth, n]  a[piv] += a[oth], the addition of node n;  ['fan', piv, [f...]]  a[f] += a[piv] for fresh
+slots f (copies);  ['src', s, n]  the V gate a[s] += x_S of leaf n (applied outside L, see check_word.py)."""
+import gzip, json, os
+from collections import Counter, defaultdict
+from itertools import combinations
+from math import comb
+
+HERE = os.path.dirname(os.path.abspath(__file__))
+ROOT = os.path.normpath(os.path.join(HERE, '..', '..'))
+DATA = os.path.join(ROOT, 'certificates', 'round7')
+
+
+def load_json(name):
+    with gzip.open(os.path.join(DATA, name), 'rt') as fh:
+        return json.load(fh)
+
+
+def load(h=23, witness=None, data=None):
+    """the base witness and the design-B data of one round-seven witness (default: the files of h)."""
+    W = load_json(witness or 'witness_%d.json.gz' % h); D = load_json(data or 'deferred_%d.json.gz' % h)
+    assert W['h'] == D['h'] == h
+    return W, D
+
+
+class Schedule:
+    """The design-B schedule in convenient form."""
+    def __init__(self, W, D):
+        self.h = h = D['h']; self.R = D['R']; self.trip = list(combinations(range(h), 3)); self.v = len(self.trip)
+        self.tid = {T: i for i, T in enumerate(self.trip)}
+        self.args = {int(n): (tuple(a) if a else None) for n, a in W['args'].items()}
+        self.ops = [tuple(o[:2]) + (tuple(o[2]),) if o[0] == 'fan' else tuple(o) for o in D['ops']]
+        self.hold = D['hold']; self.start = D['start']
+        self.out = {s: (c, tuple(T)) for s, c, T in D['out']}
+        self.ret = {s: c for s, c in D['ret']}
+        self.srcop = {o[1]: o[2] for o in self.ops if o[0] == 'src'}
+        self.readout = D['readout_order']; self.sel = set(self.readout)
+        self.sigma = dict(zip(D['readout_order'], D['sigma']))
+        self.f = [0] * self.R
+        for s, B in self.sigma.items(): self.f[s] = len(B)
+        self.vstart = dict(zip(D['vleaf_slots'], D['vleaf_start']))
+        self.xs = [(n, us) for n, us in D['xs_order']]
+        self.late_v = D['deferred_v_order']
+        self.node_dims = {n: d for n, d in D['node_dims']}
+        self.late_dims = {(n, tuple(lt)): d for n, lt, d in D['late_dims']}
+        self.late = {int(n): (list(e), list(l)) for n, (e, l) in W['late'].items()}
+        self.pivslot = {o[3]: o[1] for o in self.ops if o[0] == 'add'}
+
+    def touch(self, op):
+        if op[0] == 'add': return [op[1], op[2]]
+        if op[0] == 'fan': return [op[1]] + list(op[2])
+        return []
+
+    def root_targets(self):
+        rt = {}
+        for s, (c, T) in self.out.items(): rt[s] = [self.tid[T]]
+        for s, c in self.ret.items(): rt[s] = [t for t, T in enumerate(self.trip) if c in T]
+        return rt
+
+    def adjoint(self):
+        """garbage coefficients over Z: cov[s][t] = coefficient of slot s's start value in target y_T at the end of L
+        (the transpose sweep of L applied to the root reads); the F2 circuit uses their parities."""
+        cov = [dict() for _ in range(self.R)]
+        for s, ts in self.root_targets().items():
+            for t in ts: cov[s][t] = cov[s].get(t, 0) + 1
+        def addto(d, e):
+            for t, x in e.items(): d[t] = d.get(t, 0) + x
+        for op in reversed(self.ops):
+            if op[0] == 'add': addto(cov[op[2]], cov[op[1]])
+            elif op[0] == 'fan':
+                for g in op[2]: addto(cov[op[1]], cov[g])
+        return cov
+
+    # ------------------------------------------------------------------ frame keys of every slot's chain
+    def start_key(self, s):
+        """('v', s) for a V-leaf slot (start s_i), ('n', n0) for a base or early copy, ('c', n0, late suffix)."""
+        n0 = self.hold[s][0]
+        if self.args[n0] is None: return ('v', s)
+        st = self.start[s]; k = st[2]; dec = self.late.get(n0)
+        if dec is None or k in dec[0]: return ('n', n0)
+        lt = dec[1]; return ('c', n0, tuple(lt[lt.index(k):]))
+
+    def chain_keys(self, s):
+        """frames of slot s in time order: [sigma_u | 0], start, then every later node it holds (with the pivot's
+        late-copy intersections after a node it pivots), its root frame if any, F."""
+        ch = [('sigma', s) if s in self.sel else ('0',), self.start_key(s)]
+        for idx, n in enumerate(self.hold[s]):
+            if idx > 0: ch.append(('n', n))
+            if self.args[n] is not None and self.pivslot.get(n) == s and n in self.late:
+                lt = self.late[n][1]
+                for i in range(len(lt) - 1): ch.append(('c', n, tuple(lt[i:])))
+        if s in self.out: c, T = self.out[s]; ch.append(('out', c, T))
+        if s in self.ret: ch.append(('ret', self.ret[s]))
+        ch.append(('F',)); return ch
+
+    def dim(self, key):
+        k = key[0]
+        if k == '0': return 0
+        if k == 'F': return self.h
+        if k in ('out', 'ret'): return self.h - 1
+        if k == 'n': return self.node_dims[key[1]]
+        if k == 'c': return self.late_dims[key[1], key[2]]
+        if k == 'sigma': return len(self.sigma[key[1]])
+        if k == 'v': return len(self.vstart[key[1]])
+        raise KeyError(key)
+
+    def ylevels(self, cov):
+        """deferred readout levels (dim sigma) on every target, from the garbage-coefficient supports."""
+        lev = defaultdict(set)
+        for s in self.readout:
+            for t in cov[s]: lev[t].add(self.f[s])
+        return {t: sorted(lev[t]) for t in range(self.v)}
+
+    def xdata(self, dimof=None):
+        dimof = dimof or (lambda s: len(self.vstart[s]))
+        out = []
+        for n, us in self.xs:
+            ds = [1] + [dimof(s) for s in us] + [self.h]
+            ds = [d for i, d in enumerate(ds) if i == 0 or d != ds[i - 1]]
+            out.append([b - a for a, b in zip(ds, ds[1:])])
+        return out
+
+
+def inner(r, h):
+    """children of a rank-r chain step under the inner-corner lemma (one block of width 2r-h when 2r > h)."""
+    return [1] * (h - r) + [2 * r - h] if 2 * r > h else [1] * r
+
+
+def chain_ranks(S, dimof=None):
+    """multiset of chain-step ranks over all slots (dims must be monotone along every chain)."""
+    dimof = dimof or S.dim; rk = Counter()
+    for s in range(S.R):
+        ds = [dimof(k) for k in S.chain_keys(s)]
+        assert all(b >= a for a, b in zip(ds, ds[1:])), ('decreasing chain', s)
+        assert ds[-1] == S.h
+        for a, b in zip(ds, ds[1:]):
+            if b > a: rk[b - a] += 1
+    return rk
+
+
+ROUND6_CORNER = lambda h: [h - 2] + [1] * (h + 1)          # data-entrance corner runs, rounds six and seven
+STAIRCASE_CORNER = lambda h: [h - 2, h - 5] + [1] * 6       # staircase corner (check_stair.py)
+
+
+def histogram(S, rk, ylev, xdata, corner=None):
+    """child-width histogram of the batched two-stage bit interchange with deferred readouts and V leaves.
+    Per stage (stage two is the complement time-reversal of stage one, same edge multiset), each of the v invocations:
+      every slot u: its auxiliary edge of rank m - r_u (block m - 2 r_u) and its corner of rank r_u = h - dim sigma_u,
+        both batched by the inner lemma, plus its chain steps;
+      every centre: the copy's step U_c -> 0 of rank h-1;
+      every target y_T: the chain 0 -> deferred levels -> t_T^perp (total rank h-1);
+      every leaf X_S: its chain <t_S> -> V-gate frames -> F (total rank h-1).
+    Data: the entrance block m-4h+2 with its corner runs, one copy-correction singleton per pair."""
+    h, v, R = S.h, S.v, S.R; m = h * h; N = v * v; A = v * R; H = Counter()
+    corner = corner or ROUND6_CORNER(h); assert sum(corner) == 2 * h - 1
+    corners = Counter(h - S.f[s] for s in range(R))
+    for _ in range(2):
+        for r, c in corners.items():
+            H[m - 2 * r] += v * c
+            for w in inner(r, h): H[w] += v * c
+        for r, c in rk.items():
+            for w in inner(r, h): H[w] += v * c
+        for _c in range(h):
+            for w in inner(h - 1, h): H[w] += v
+        for t in range(v):
+            ds = sorted(set([0] + ylev[t] + [h - 1]))
+            for a, b in zip(ds, ds[1:]):
+                for w in inner(b - a, h): H[w] += v
+        for xs in xdata:
+            for x in xs:
+                for w in inner(x, h): H[w] += v
+    H[m - 4 * h + 2] += 2 * N
+    for w in corner: H[w] += 2 * N
+    H[1] += N
+    H = {w: c for w, c in H.items() if c}
+    Wd = 2 * N + 2 * A; L = 2 * v * h * (h - 1); s = Wd * m - N + L
+    return dict(h=h, m=m, v=v, N=N, W=Wd, L=L, s=s, R=R, hist=H)
diff --git a/research/cyclic-deferred/bit-sharing/inputs/research/round7-public/tree/independent/deferred-readout/linalg.py b/research/cyclic-deferred/bit-sharing/inputs/research/round7-public/tree/independent/deferred-readout/linalg.py
new file mode 100644
index 0000000000..a81cbaa61e
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/inputs/research/round7-public/tree/independent/deferred-readout/linalg.py
@@ -0,0 +1,176 @@
+"""Stdlib exact linear algebra used by the release checks (no numpy).
+
+Ranks over Q are computed in one of two rigorous ways:
+  * modulo a prime p with a Hadamard certificate: if every row of an integer matrix has squared norm n_i and
+    prod_i max(1, n_i) < p^2, every minor has absolute value < p, so a minor is nonzero over Q iff it is nonzero
+    mod p, and the rank mod p equals the rank over Q (hadamard_ok);
+  * exactly, by fraction-free (Bareiss) elimination over Z (rank_exact) or Fraction elimination (null_exact).
+Everything else is plain arithmetic modulo a prime."""
+from fractions import Fraction
+from math import gcd
+
+Q31 = 2**31 - 1          # prime used for the point evaluation and frame reductions
+M127 = 2**127 - 1        # Mersenne prime
+M521 = 2**521 - 1        # Mersenne prime
+
+
+def hadamard_ok(rows, p):
+    """True if prod of max(1, |row|^2) < p^2, i.e. every minor of the integer matrix is below p in absolute value."""
+    b = 1; p2 = p * p
+    for r in rows:
+        b *= max(1, sum(x * x for x in r))
+        if b >= p2: return False
+    return True
+
+
+class Echelon:
+    """Row echelon basis mod p, built by insertion. Each stored row has a pivot (first nonzero, normalised to 1)
+    and zeros in the pivot columns of all earlier rows, so reducing a new row against the rows in order is exact."""
+    __slots__ = ('p', 'rows', 'piv')
+
+    def __init__(self, p):
+        self.p = p; self.rows = []; self.piv = []
+
+    def reduce(self, r):
+        p = self.p; r = [x % p for x in r]
+        for pc, b in zip(self.piv, self.rows):
+            c = r[pc]
+            if c: r = [(x - c * y) % p for x, y in zip(r, b)]
+        return r
+
+    def insert(self, r):
+        """insert row; returns True if it was independent."""
+        r = self.reduce(r)
+        for i, x in enumerate(r):
+            if x:
+                inv = pow(x, self.p - 2, self.p)
+                self.rows.append([(y * inv) % self.p for y in r]); self.piv.append(i); return True
+        return False
+
+    def contains(self, r):
+        return not any(self.reduce(r))
+
+    def __len__(self): return len(self.rows)
+
+
+def rank_mod(rows, p):
+    E = Echelon(p)
+    for r in rows: E.insert(r)
+    return len(E)
+
+
+def rank_exact(rows):
+    """exact rank over Q of an integer matrix (Bareiss fraction-free elimination)."""
+    M = [list(r) for r in rows if any(r)]
+    if not M: return 0
+    n = len(M); k = len(M[0]); r = 0; prev = 1
+    for c in range(k):
+        i = next((i for i in range(r, n) if M[i][c]), None)
+        if i is None: continue
+        M[r], M[i] = M[i], M[r]
+        pr = M[r]; a = pr[c]
+        for t in range(r + 1, n):
+            row = M[t]; b = row[c]
+            M[t] = [(a * x - b * y) // prev for x, y in zip(row, pr)]
+        prev = a; r += 1
+        if r == n: break
+    return r
+
+
+def null_mod(A, k, p):
+    """basis of {x in F_p^k : A x = 0} (A: list of rows of length k)."""
+    rows = [[x % p for x in r] for r in A]; piv = []; R = []
+    for r in rows:
+        for pc, b in zip(piv, R):
+            c = r[pc]
+            if c: r = [(x - c * y) % p for x, y in zip(r, b)]
+        i = next((i for i, x in enumerate(r) if x), None)
+        if i is None: continue
+        inv = pow(r[i], p - 2, p); r = [(y * inv) % p for y in r]
+        R = [[(x - b[i] * y) % p for x, y in zip(b, r)] if b[i] else b for b in R]   # keep fully reduced
+        R.append(r); piv.append(i)
+    free = [c for c in range(k) if c not in piv]; out = []
+    for f in free:
+        x = [0] * k; x[f] = 1
+        for pc, b in zip(piv, R): x[pc] = (-b[f]) % p
+        out.append(x)
+    return out
+
+
+def null_exact(A, k):
+    """basis (integer vectors) of {x in Q^k : A x = 0}, by Fraction reduced row echelon form."""
+    R = []; piv = []
+    for r0 in A:
+        r = [Fraction(x) for x in r0]
+        for pc, b in zip(piv, R):
+            c = r[pc]
+            if c: r = [x - c * y for x, y in zip(r, b)]
+        i = next((i for i, x in enumerate(r) if x), None)
+        if i is None: continue
+        inv = 1 / r[i]; r = [y * inv for y in r]
+        R = [[x - b[i] * y for x, y in zip(b, r)] if b[i] else b for b in R]
+        R.append(r); piv.append(i)
+    free = [c for c in range(k) if c not in piv]; out = []
+    for f in free:
+        x = [Fraction(0)] * k; x[f] = Fraction(1)
+        for pc, b in zip(piv, R): x[pc] = -b[f]
+        out.append(primitive(x))
+    return out
+
+
+def primitive(x):
+    """scale a rational vector to a primitive integer vector."""
+    den = 1
+    for v in x: den = den * v.denominator // gcd(den, v.denominator)
+    y = [int(v * den) for v in x]; g = 0
+    for v in y: g = gcd(g, v)
+    return [v // g for v in y] if g else y
+
+
+def inv_mod_matrix(A, p):
+    """inverse of a square matrix mod p, or None if singular."""
+    n = len(A); M = [[x % p for x in A[i]] + [int(i == j) for j in range(n)] for i in range(n)]
+    for c in range(n):
+        i = next((i for i in range(c, n) if M[i][c]), None)
+        if i is None: return None
+        M[c], M[i] = M[i], M[c]
+        inv = pow(M[c][c], p - 2, p); M[c] = [(x * inv) % p for x in M[c]]
+        rc = M[c]
+        for t in range(n):
+            if t != c and M[t][c]:
+                f = M[t][c]; M[t] = [(x - f * y) % p for x, y in zip(M[t], rc)]
+    return [r[n:] for r in M]
+
+
+def matmul(A, B, p):
+    Bt = list(zip(*B))
+    return [[sum(x * y for x, y in zip(r, c)) % p for c in Bt] for r in A]
+
+
+def rightmost_pivots(Q, p):
+    """rows top to bottom: the rightmost nonzero column of the reduced row, then eliminate it below."""
+    M = [list(r) for r in Q]; n = len(M); out = [-1] * n
+    for i in range(n):
+        row = M[i]; c = next((j for j in range(len(row) - 1, -1, -1) if row[j]), -1)
+        out[i] = c
+        if c < 0: continue
+        inv = pow(row[c], p - 2, p)
+        for t in range(i + 1, n):
+            f = M[t][c]
+            if f:
+                f = f * inv % p; M[t] = [(x - f * y) % p for x, y in zip(M[t], row)]
+    return out
+
+
+def predicted(h):
+    """pivot column per row of the generic north-east rank function f_r of a rank-r idempotent."""
+    pr = {}
+    for r in range(1, h + 1):
+        s = h - r
+        f = lambda i, j: 0 if i < 0 or j >= h else min(i + 1, h - j, r, max(0, i - j + 1) + s)
+        pc = [-1] * h
+        for i in range(h):
+            for j in range(h):
+                if f(i, j) - f(i - 1, j) - f(i, j + 1) + f(i - 1, j + 1) == 1: pc[i] = j
+        pr[r] = pc
+    return pr
diff --git a/research/cyclic-deferred/bit-sharing/inputs/research/round7-public/tree/independent/partial-swap/factor.py b/research/cyclic-deferred/bit-sharing/inputs/research/round7-public/tree/independent/partial-swap/factor.py
new file mode 100644
index 0000000000..9908b695e0
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/inputs/research/round7-public/tree/independent/partial-swap/factor.py
@@ -0,0 +1,65 @@
+"""Exact lower-lower Bruhat factorization A = L1 * Mon * L2 (L1, L2 lower triangular, Mon monomial)
+by elimination, plus helpers. Rows top-to-bottom; pivot = rightmost nonzero entry of the current row."""
+from fractions import Fraction as Q
+
+def ident(n): return [[Q(int(i == j)) for j in range(n)] for i in range(n)]
+def mul(A, B):
+    Bt = list(zip(*B))
+    return [[sum((a*b for a, b in zip(r, c) if a and b), Q(0)) for c in Bt] for r in A]
+def rank(M):
+    M = [r[:] for r in M]; rk = 0; cols = len(M[0]) if M else 0
+    for c in range(cols):
+        piv = next((i for i in range(rk, len(M)) if M[i][c] != 0), None)
+        if piv is None: continue
+        M[rk], M[piv] = M[piv], M[rk]
+        for i in range(rk+1, len(M)):
+            if M[i][c] != 0:
+                f = M[i][c]/M[rk][c]; M[i] = [a-f*b for a, b in zip(M[i], M[rk])]
+        rk += 1
+    return rk
+def inv(M):
+    n = len(M); A = [r[:]+[Q(int(i == j)) for j in range(n)] for i, r in enumerate(M)]
+    for c in range(n):
+        p = next(i for i in range(c, n) if A[i][c] != 0); A[c], A[p] = A[p], A[c]
+        f = A[c][c]; A[c] = [x/f for x in A[c]]
+        for i in range(n):
+            if i != c and A[i][c] != 0:
+                g = A[i][c]; A[i] = [a-g*b for a, b in zip(A[i], A[c])]
+    return [r[n:] for r in A]
+
+def bruhat(A, track=True):
+    """Return (piv, L1, Mon, L2) with A = L1 Mon L2; piv[i] = column of row i's pivot."""
+    n = len(A); M = [r[:] for r in A]
+    L1 = ident(n) if track else None; L2 = ident(n) if track else None
+    piv = [None]*n
+    for i in range(n):
+        c = max((j for j in range(n) if M[i][j] != 0), default=None)
+        if c is None: raise ValueError('singular')
+        piv[i] = c; a = M[i][c]
+        for k in range(i+1, n):                     # rows below: row_k -= f row_i  (left lower op)
+            if M[k][c] != 0:
+                f = M[k][c]/a; M[k] = [x-f*y for x, y in zip(M[k], M[i])]
+                if track:                           # L1 <- L1 (I + f e_k e_i^T): col_i += f col_k
+                    for r in range(n):
+                        if L1[r][k]: L1[r][i] += f*L1[r][k]
+        for j in range(c):                          # cols left: col_j -= g col_c  (right lower op)
+            if M[i][j] != 0:
+                g = M[i][j]/a
+                for r in range(n):
+                    if M[r][c]: M[r][j] -= g*M[r][c]
+                if track:                           # L2 <- (I + g e_c e_j^T) L2: row_c += g row_j
+                    L2[c] = [x+g*y for x, y in zip(L2[c], L2[j])]
+    assert sorted(piv) == list(range(n))
+    return piv, L1, M, L2
+
+def swapmat(p):
+    m = len(p)
+    return [[Q(int(i == j))-p[i][j] for j in range(m)]+[p[i][j] for j in range(m)] for i in range(m)] + \
+           [[p[i][j] for j in range(m)]+[Q(int(i == j))-p[i][j] for j in range(m)] for i in range(m)]
+
+def predicted(p_corner_perm, m, k):
+    """Predicted involution: H_l <-> D_sigma(l) (l in L), H_i <-> D_i (i in M), fixed elsewhere."""
+    w = list(range(2*m))
+    for l, r in p_corner_perm.items(): w[l] = m+r; w[m+r] = l
+    for i in range(k, m-k): w[i] = m+i; w[m+i] = i
+    return w
diff --git a/research/cyclic-deferred/bit-sharing/inputs/research/round8-foundations-minimal-v-word.json.gz b/research/cyclic-deferred/bit-sharing/inputs/research/round8-foundations-minimal-v-word.json.gz
new file mode 100644
index 0000000000..e08b39aaf0
Binary files /dev/null and b/research/cyclic-deferred/bit-sharing/inputs/research/round8-foundations-minimal-v-word.json.gz differ
diff --git a/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-final-geometry.py b/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-final-geometry.py
new file mode 100644
index 0000000000..4d8fecf743
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-final-geometry.py
@@ -0,0 +1,178 @@
+"""Direct final-candidate bit geometry, using the pinned exact round-seven tools.
+Stdlib only. Original-node inclusions use their checked nesting theorem;
+new sigma/V and chronological data inclusions use exact rational annihilators.
+This does not use old high-rank corner profiles: the opposite-bank compiler
+instead supplies one child per rational idempotent's full rank.
+"""
+import argparse
+import gzip
+import hashlib
+import json
+import sys
+import time
+from collections import Counter, defaultdict
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+BASE = HERE / 'round7-public/tree/independent/deferred-readout'
+sys.path.insert(0, str(BASE))
+import check_lifted as cl
+import deferred as dr
+from linalg import Echelon, Q31, null_exact, primitive, rank_mod
+
+def main():
+    parser = argparse.ArgumentParser()
+    parser.add_argument('--candidate', type=Path, default=HERE/'round8-foundations-minimal-v-word.json.gz')
+    parser.add_argument('--output', type=Path, required=True)
+    args = parser.parse_args()
+    started = time.monotonic()
+    def progress(message): print(round(time.monotonic()-started, 2), message, flush=True)
+    W, D = dr.load(23)
+    with gzip.open(args.candidate, 'rt') as f: candidate = json.load(f)
+    sigma = dict(zip(D['readout_order'], D['sigma']))
+    for s in candidate['removed_original_slots']: del sigma[s]
+    sigma.update({int(s): B for s, B in candidate['new_frames'].items()})
+    D['readout_order'] = [s for kind, s in candidate['events'] if kind == 'read' and s in sigma]
+    assert len(D['readout_order']) == len(sigma) == len(set(D['readout_order']))
+    D['sigma'] = [sigma[s] for s in D['readout_order']]
+    vs = dict(zip(D['vleaf_slots'], D['vleaf_start']))
+    vs.update({int(s): B for s, B in candidate.get('vstart_overrides', {}).items()})
+    D['vleaf_start'] = [vs[s] for s in D['vleaf_slots']]
+    if 'xs_order_actual' in candidate: D['xs_order'] = candidate['xs_order_actual']
+    S = dr.Schedule(W, D)
+    h = S.h
+    full = [[int(i == j) for j in range(h)] for i in range(h)]
+    pr = cl.Prog(W)
+    bas, dn = cl.spans(pr)
+    users = cl.users_of(pr, dn)
+    linked = {use: donor for donor, use in pr.mL.items()}
+    assert len(linked) == len(pr.mL) == len(W['links'])
+    for n, (early, late_users) in pr.late.items():
+        assert late_users and sorted(early+late_users) == [
+            k for k in range(len(users[n])) if (n, k) not in linked]
+    fr = cl.Frames(pr, dn, bas, users)
+    assert fr.check_structure() == (0, 0, 0, True)
+    fr.cobases()
+    dims, _ = fr.dims()
+    late, _ = cl.late_frames(fr, pr, users)
+    assert S.node_dims == dims
+    assert {('c', n, lt): r for (n, lt), r in S.late_dims.items()} == late
+    progress('DAG, original nesting premises and exact frame dimensions pass')
+
+    def usekey(n, k):
+        u = users[n][k]
+        return ('n', u[1]) if u[0] == 'gate' else cl.root_key(u)
+    exact_cache = {}
+    def exact(key):
+        if key not in exact_cache:
+            kind = key[0]
+            if kind == '0': B = []
+            elif kind == 'F': B = full
+            elif kind == 'sigma': B = S.sigma[key[1]]
+            elif kind == 'v': B = S.vstart[key[1]]
+            elif kind == 'c':
+                n, lt = key[1:]
+                B = exact(usekey(n, lt[-1]))
+                for k in reversed(lt[:-1]): B = fr.intersect_exact(exact(usekey(n, k)), B)
+            else: B = fr.exact_basis(key)
+            B = [primitive(row) for row in B]
+            assert len(B) == S.dim(key)
+            exact_cache[key] = B
+        return exact_cache[key]
+    annihilators = {}
+    def inside(A, B):
+        if not A or len(B) == h: return True
+        kb = tuple(map(tuple, B))
+        if kb not in annihilators: annihilators[kb] = null_exact(B, h)
+        return all(sum(x*y for x, y in zip(a, z)) == 0
+                   for a in A for z in annihilators[kb])
+    methods = Counter()
+    def included(a, b):
+        if a == b or a == ('0',) or b == ('F',): method = 'identity/zero/full'
+        elif a[0] == b[0] == 'n' and b[1] in fr.succ[a[1]]: method = 'checked node-successor theorem'
+        elif a[0] == 'n' and b[0] in ('out', 'ret') and fr.rid[b] in fr.droots[a[1]]: method = 'checked direct-root theorem'
+        elif a[0] == 'n' and b[0] == 'c' and a[1] == b[1]: method = 'node in every late successor'
+        elif a[0] == b[0] == 'c' and a[1] == b[1] and set(b[2]) <= set(a[2]): method = 'nested exact intersections'
+        elif a[0] == 'c' and b in [usekey(a[1], k) for k in a[2]]: method = 'exact intersection in member'
+        else:
+            method = 'exact rational annihilator'
+            assert inside(exact(a), exact(b)), ('containment', a, b)
+        assert S.dim(a) <= S.dim(b), ('dimension', a, b)
+        methods[method] += 1
+
+    steps = set()
+    for s in range(S.R):
+        chain = S.chain_keys(s)
+        steps.update(zip(chain, chain[1:]))
+    actual_v = defaultdict(list)
+    for kind, s in candidate['events']:
+        if kind == 'V': actual_v[S.srcop[s]].append(s)
+    assert dict(S.xs) == dict(actual_v)
+    assert sorted(s for slots in actual_v.values() for s in slots) == sorted(S.srcop)
+    for n, slots in S.xs:
+        chain = [('n', n)] + [('v', s) for s in slots] + [('F',)]
+        steps.update(zip(chain, chain[1:]))
+    CF = [{t: c for t, c in row.items() if c % 2} for row in S.adjoint()]
+    by_target = defaultdict(list)
+    for s in S.readout:
+        for t in CF[s]: by_target[t].append(('sigma', s))
+    for t, T in enumerate(S.trip):
+        chain = [('0',)] + by_target[t] + [('out', T[0], T)]
+        steps.update(zip(chain, chain[1:]))
+    for a, b in sorted(steps, key=str): included(a, b)
+    progress('All auxiliary and actual chronological X/Y chains pass')
+
+    # The full-rank reduction below is a nonzero rational Gram determinant
+    # witness; it is not a numerical substitute for zero containment tests.
+    mq = cl.ModQ(fr, late)
+    used = set(k for pair in steps for k in pair)
+    used.update(('ret', c) for c in S.ret.values())
+    for key in sorted(used, key=str):
+        if key[0] in ('sigma', 'v'):
+            B = exact(key)
+            assert all(len(row) == h for row in B)
+            E = Echelon(Q31)
+            for row in B: E.insert(row)
+            assert len(E.rows) == len(B), ('independence', key)
+            mq.bcache[key] = E.rows
+        assert mq.nondeg(key), ('degenerate', key)
+    assert mq.dimfail == 0
+    for s in S.readout:
+        Trows = [[9*int(i in S.trip[t])-3 for i in range(h)] for t in CF[s]]
+        assert all(sum(x*y for x, y in zip(row, cov)) == 0 for row in S.sigma[s] for cov in Trows)
+    progress('All used frames nondegenerate; exact F2 target orthogonality passes')
+
+    def nondeg(B):
+        sums = [sum(row) for row in B]
+        gram = [[9*sum(x*y for x, y in zip(a, b))-sa*sb
+                 for b, sb in zip(B, sums)] for a, sa in zip(B, sums)]
+        return rank_mod(gram, Q31) == len(B)
+    controls = {'isotropic_frame_rejected': not nondeg([[1]*9+[0]*(h-9)])}
+    for s in S.readout:
+        B = exact(S.start_key(s))
+        if len(B) == h: continue
+        bad = next((row for row in full if not inside([row], B)), None)
+        if bad is not None:
+            controls['outside_start_rejected'] = not inside([bad], B)
+            break
+    controls['reversed_target_chain_rejected'] = any(
+        len(exact(a)) < len(exact(b)) and not inside(exact(b), exact(a))
+        for seq in by_target.values() for a, b in zip(seq, seq[1:]))
+    assert all(controls.values()) and len(controls) == 3
+    inputs = [args.candidate, BASE/'deferred.py', BASE/'check_lifted.py', BASE/'linalg.py']
+    inputs += [BASE.parent.parent/'certificates/round7'/name
+               for name in ['witness_23.json.gz', 'deferred_23.json.gz']]
+    out = dict(status='PASS', candidate_sha256=hashlib.sha256(args.candidate.read_bytes()).hexdigest(),
+               node_frames=len(dims), late_frames=len(late), used_frames=len(used),
+               distinct_chain_steps=len(steps), inclusion_methods=dict(methods),
+               actual_F2_target_support=True, actual_V_chronology=True,
+               all_frame_dimensions_exact=True, all_used_frames_nondegenerate=True,
+               all_auxiliary_source_target_chains_nested=True,
+               all_sigma_target_orthogonality_exact=True, controls=controls,
+               elapsed=time.monotonic()-started,
+               inputs_sha256={str(p.relative_to(HERE)) if p.is_relative_to(HERE) else str(p): hashlib.sha256(p.read_bytes()).hexdigest() for p in inputs},
+               scope='Rational bit geometry of final candidate, not full tape execution. Pair with literal frame ledger and general opposite-bank rank factorization; no old corner witness is needed.')
+    args.output.write_text(json.dumps(out, indent=2)+'\n')
+    progress('PASS; receipt written to '+str(args.output))
+
+if __name__ == '__main__': main()
diff --git a/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-minimal-v-ledger/forward-events.i32.gz b/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-minimal-v-ledger/forward-events.i32.gz
new file mode 100644
index 0000000000..e1a9eddbb6
Binary files /dev/null and b/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-minimal-v-ledger/forward-events.i32.gz differ
diff --git a/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-minimal-v-ledger/result.json b/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-minimal-v-ledger/result.json
new file mode 100644
index 0000000000..d5c71fd785
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-minimal-v-ledger/result.json
@@ -0,0 +1,114 @@
+{
+  "candidate_sha256": "298c11fb29ab0afdf8d762c64a9b99d19a1e58465f3a0cbc9d2a59367c438fb7",
+  "h": 23,
+  "roles": 28866,
+  "formal_basis": 32408,
+  "events": 1076483,
+  "frame_keys": 89877,
+  "rank_histograms": {
+    "aux": {
+      "1": 55459,
+      "2": 64778,
+      "3": 13843,
+      "4": 11212,
+      "5": 5995,
+      "6": 1868,
+      "8": 1234,
+      "9": 442,
+      "10": 173,
+      "12": 207,
+      "16": 203,
+      "18": 599,
+      "21": 413,
+      "7": 834,
+      "11": 349,
+      "13": 226,
+      "14": 63,
+      "15": 670,
+      "17": 1278,
+      "19": 533,
+      "20": 2310
+    },
+    "data": {
+      "1": 10126,
+      "2": 1972,
+      "3": 562,
+      "4": 1117,
+      "5": 354,
+      "6": 411,
+      "7": 215,
+      "8": 656,
+      "9": 100,
+      "10": 126,
+      "11": 53,
+      "12": 376,
+      "13": 55,
+      "14": 35,
+      "16": 204,
+      "17": 6,
+      "18": 41,
+      "15": 4,
+      "19": 1580,
+      "21": 1,
+      "20": 188,
+      "22": 13
+    },
+    "center": {
+      "22": 23
+    }
+  },
+  "fan_groups_reordered_internally": [],
+  "complete_forward_F2": true,
+  "complete_reflected_F2": true,
+  "vstart_overrides": 211,
+  "actual_source_V_order_checked": true,
+  "reflected_frame_continuity": true,
+  "all_scalar_gates_equal_frame_keys": true,
+  "exterior_formula": "E1=I-I_tensor_Q+sigma_tensor_Q; E2=(I-P)_tensor_I+P_tensor_sigma; rank=m-h+dim(sigma)",
+  "histogram_matches_external": true,
+  "histogram": {
+    "1": 1677475261,
+    "21": 6407478,
+    "19": 1515976,
+    "17": 9007306,
+    "15": 7934080,
+    "13": 2270422,
+    "11": 4824204,
+    "7": 3471160,
+    "3": 1399090,
+    "5": 474628,
+    "9": 1459304,
+    "483": 62328574,
+    "23": 62328574,
+    "525": 5653032,
+    "527": 6233920,
+    "521": 13711082,
+    "523": 4629394,
+    "491": 449834,
+    "485": 7084,
+    "487": 49588,
+    "489": 159390,
+    "499": 1083852,
+    "495": 276276,
+    "505": 99176,
+    "503": 403788,
+    "501": 127512,
+    "507": 1144066,
+    "509": 315238,
+    "497": 17710,
+    "513": 106260,
+    "493": 3542,
+    "517": 626934,
+    "515": 1583274,
+    "511": 378994,
+    "519": 2854852,
+    "439": 6272882,
+    "18": 6272882
+  },
+  "recursive_rank": 57403754177,
+  "elapsed": 11.545497624960262,
+  "limit_seconds": 180,
+  "exact_chain_geometry": "Relies on separately replayed pinned check_frames and check_lifted. Rank labels alone do not prove geometry.",
+  "scope": "Explicit one-axis forward/reflected events, scalar complete basis and exact accounting. Tensor endpoint identities proved separately; full complex/tape/precision and final assembly integration not established.",
+  "new_multiplication_bound": false
+}
diff --git a/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-opposite-profile-proof.txt b/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-opposite-profile-proof.txt
new file mode 100644
index 0000000000..c73b012ad1
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-opposite-profile-proof.txt
@@ -0,0 +1,61 @@
+OPPOSITE-BANK COMPILER ON THE VERIFIED MINIMAL-V WORD
+2026-10-09. Compiler credit: icekylinx PR104, pinned948ce1510df750f4c18b96bdaef436a86f8bf834.
+The changed compiler is an alternative to our oblique split; their child
+lists are not superimposed or added.
+
+The payload word is round8-foundations-minimal-v-word.json.gz,
+SHA256298c11fb29ab0afdf8d762c64a9b99d19a1e58465f3a0cbc9d2a59367c438fb7.
+Its full physical replay is round8-network-minimal-v-ledger/result.json,
+with1076483 literal events and all32408 input/dirty basis columns checked.
+The new child histogram is round8-network-opposite-profile.json. It is
+rebuilt directly from the binary literal event stream, not by interpreting
+the noninvertible old g-width aggregation.
+
+Projector classes and exact charges:
+1. Each actual auxiliary, source and target move lies between nested
+   nondegenerate rational frames A,B. Its residual P_B-P_A is idempotent
+   with rank dimB-dimA. Each positive literal rank contributes one child
+   of that rank. Zero-rank moves require no child.
+2. Each copied-centre event has the existing rank22 projector. It contributes
+   one width22 child with the same sequential temporary-copy convention.
+3. For a role with starting frame sigma, its restored arbitrary logical
+   scratch has relative projector E=(I-P_sigma) tensor P_b in the first
+   stage, or E=P_a tensor (I-P_sigma) in the reflected stage. E is an
+   idempotent of rank23-dim(sigma). The paid exterior has projector I-E,
+   rank529-23+dim(sigma). This is a proper nonzero idempotent, even with
+   nonzero initial gauges; sigma dimensions are at most22. It completes
+   the relative action because D_(I-E) D_E=D_I. It is not a rank inferred
+   from cancellation of previously separated calls.
+4. Each interstage data bridge has projector Q_a tensor Q_b, where
+   Q_a=I-P_a and Q_b=I-P_b. Its rank is22^2=484. There are2N such bridges.
+5. Each paid endpoint-copy correction uses the rank1 line projector
+   P_a tensor P_b on the existing complete copied stream. There are N
+   such corrections. The copying/cleanup convention and temporary stock
+   are exactly those already validated in the tensor endpoint interface.
+
+Every class is therefore a finite rational idempotent family on the SAME
+529-dimensional address space. One generic rational similarity makes all
+leading minors required by PR104 nonzero, without retaining the old
+staircase/batched-corner constraints, which are no longer used. Apply this
+same similarity to all incident frames, gauges and endpoints. Then apply
+the same global opposite-bank conjugation diag(I,C_n) to every role. Products,
+gate equalities, scalar maps, scratch restoration and the gauged identity
+in item3 are preserved by conjugation. D_I becomes the reversed-family
+target F_n. The bit stream compiler replaces each idempotent residual by
+one F_(rf) child surrounded by paid atom adapters. There is no per-edge
+independent basis switch and no scalar gate inserted inside a factorization.
+
+Forward literal local counts are multiplied by2v to cover both tensor
+orientations. With v1771,R28866,m529,W108516254, the total rank is
+57403754177=Wm-N+2v*23*22 and the deficit is1344189. All children shrink;
+the largest width is528, from the nonzero-gauge exteriors. Each child acts
+on the original complete role volume V/W. No role or simultaneous scratch
+bank is added. The final ordinary wrapper is paid once outside the reversed
+recursion and its two calls reuse restored row stock sequentially.
+
+The larger maximum child528 requires recomputing the coarse halving depth
+and the three-factor row reserve; PR104's selected4000 stock exponent cannot
+be copied without a fresh numerical check. The normalized stopped recurrence
+is otherwise unchanged. Its coarse moment, chosen atom exponent, old-leaf
+saving and final analytic assembly are certified independently by optimize.
+This file alone makes no final multiplication-exponent claim.
diff --git a/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-opposite-profile.json b/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-opposite-profile.json
new file mode 100644
index 0000000000..ea5b8aa53d
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-opposite-profile.json
@@ -0,0 +1,185 @@
+{
+  "status": "Complete profile rebuilt from locally replayed literal word; PR104 stopped interface applies conditionally",
+  "compiler_credit": "icekylinx PR104",
+  "compiler_commit": "948ce1510df750f4c18b96bdaef436a86f8bf834",
+  "word_sha256": "298c11fb29ab0afdf8d762c64a9b99d19a1e58465f3a0cbc9d2a59367c438fb7",
+  "ledger_sha256": "21621c98008616261f23849015ca1690281440f26633d95d7bcbc6b3e05b7e68",
+  "events_sha256": "afa817b15a4b7c30e0086380ac0c629e4672d14e3c03cf7675a743ef91eb3da5",
+  "literal_forward_events": 1076483,
+  "h": 23,
+  "m": 529,
+  "v": 1771,
+  "R": 28866,
+  "W": 108516254,
+  "N": 3136441,
+  "s": 57403754177,
+  "deficit": 1344189,
+  "maxchild": 528,
+  "hist": {
+    "1": 235438511,
+    "2": 236428500,
+    "3": 51022510,
+    "4": 43669318,
+    "5": 22488158,
+    "6": 8072218,
+    "7": 3715558,
+    "8": 6694380,
+    "9": 1919764,
+    "10": 1059058,
+    "11": 1423884,
+    "12": 2064986,
+    "13": 995302,
+    "14": 347116,
+    "15": 2387308,
+    "16": 1441594,
+    "17": 4547928,
+    "18": 2266880,
+    "19": 7484246,
+    "20": 8847916,
+    "21": 1466388,
+    "22": 127512,
+    "484": 6272882,
+    "506": 62328574,
+    "507": 7084,
+    "508": 49588,
+    "509": 159390,
+    "510": 449834,
+    "511": 3542,
+    "512": 276276,
+    "513": 17710,
+    "514": 1083852,
+    "515": 127512,
+    "516": 403788,
+    "517": 99176,
+    "518": 1144066,
+    "519": 315238,
+    "520": 378994,
+    "521": 106260,
+    "522": 1583274,
+    "523": 626934,
+    "524": 2854852,
+    "525": 13711082,
+    "526": 4629394,
+    "527": 5653032,
+    "528": 6233920
+  },
+  "classes": {
+    "auxiliary_steps": {
+      "calls": 576244438,
+      "rank": 1611404564,
+      "hist": {
+        "1": 196435778,
+        "2": 229443676,
+        "3": 49031906,
+        "4": 39712904,
+        "5": 21234290,
+        "6": 6616456,
+        "7": 2954028,
+        "8": 4370828,
+        "9": 1565564,
+        "10": 612766,
+        "11": 1236158,
+        "12": 733194,
+        "13": 800492,
+        "14": 223146,
+        "15": 2373140,
+        "16": 719026,
+        "17": 4526676,
+        "18": 2121658,
+        "19": 1887886,
+        "20": 8182020,
+        "21": 1462846
+      }
+    },
+    "source_steps": {
+      "calls": 24308746,
+      "rank": 138003404,
+      "hist": {
+        "1": 17922520,
+        "4": 113344,
+        "16": 113344,
+        "19": 5454680,
+        "20": 665896,
+        "22": 38962
+      }
+    },
+    "target_steps": {
+      "calls": 40137944,
+      "rank": 138003404,
+      "hist": {
+        "1": 17943772,
+        "2": 6984824,
+        "3": 1990604,
+        "4": 3843070,
+        "5": 1253868,
+        "6": 1455762,
+        "7": 761530,
+        "8": 2323552,
+        "9": 354200,
+        "10": 446292,
+        "11": 187726,
+        "12": 1331792,
+        "13": 194810,
+        "14": 123970,
+        "15": 14168,
+        "16": 609224,
+        "17": 21252,
+        "18": 145222,
+        "19": 141680,
+        "21": 3542,
+        "22": 7084
+      }
+    },
+    "copied_centres": {
+      "calls": 81466,
+      "rank": 1792252,
+      "hist": {
+        "22": 81466
+      }
+    },
+    "auxiliary_exteriors": {
+      "calls": 102243372,
+      "rank": 52475339224,
+      "hist": {
+        "506": 62328574,
+        "507": 7084,
+        "508": 49588,
+        "509": 159390,
+        "510": 449834,
+        "511": 3542,
+        "512": 276276,
+        "513": 17710,
+        "514": 1083852,
+        "515": 127512,
+        "516": 403788,
+        "517": 99176,
+        "518": 1144066,
+        "519": 315238,
+        "520": 378994,
+        "521": 106260,
+        "522": 1583274,
+        "523": 626934,
+        "524": 2854852,
+        "525": 13711082,
+        "526": 4629394,
+        "527": 5653032,
+        "528": 6233920
+      }
+    },
+    "data_bridges": {
+      "calls": 6272882,
+      "rank": 3036074888,
+      "hist": {
+        "484": 6272882
+      }
+    },
+    "endpoint_copies": {
+      "calls": 3136441,
+      "rank": 3136441,
+      "hist": {
+        "1": 3136441
+      }
+    }
+  },
+  "scope": "Each nested residual, gauged auxiliary exterior, data bridge and endpoint copy is individually idempotent. One common generic basis plus opposite-bank conjugation preserves the complete operator; no local basis switches. No stopped saving or multiplication exponent certified in this file."
+}
diff --git a/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-opposite-profile.py b/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-opposite-profile.py
new file mode 100644
index 0000000000..0960899a1f
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-opposite-profile.py
@@ -0,0 +1,62 @@
+"""One reversed child per actual projector in the verified minimal-V word.
+Transplants icekylinx PR104's opposite-bank/stopped-product compiler only.
+Rebuilds local ranks from the literal event stream, never from old widths.
+"""
+import gzip
+import hashlib
+import json
+import struct
+from collections import Counter
+from pathlib import Path
+
+P=Path(__file__).resolve().parent
+wordpath=P/'round8-foundations-minimal-v-word.json.gz'
+wordraw=wordpath.read_bytes();word=json.loads(gzip.decompress(wordraw))
+ledgerpath=P/'round8-network-minimal-v-ledger/result.json'
+ledger=json.loads(ledgerpath.read_text())
+assert ledger['candidate_sha256']==hashlib.sha256(wordraw).hexdigest()
+assert all(ledger[k] for k in ('complete_forward_F2','complete_reflected_F2',
+    'actual_source_V_order_checked','all_scalar_gates_equal_frame_keys','histogram_matches_external'))
+D=json.loads(gzip.decompress((P/'round7-public/tree/certificates/round7/deferred_23.json.gz').read_bytes()))
+sigma=dict(zip(D['readout_order'],D['sigma']))
+for s in word['removed_original_slots']:sigma.pop(s)
+sigma.update({int(s):B for s,B in word['new_frames'].items()})
+h=23;m=h*h;v=1771;N=v*v;R=D['R'];W=2*N+2*v*R
+assert R==28866 and W==108516254
+local={k:Counter() for k in ('auxiliary_steps','source_steps','target_steps','copied_centres')}
+eventpath=P/'round8-network-minimal-v-ledger/forward-events.i32.gz'
+events=gzip.decompress(eventpath.read_bytes());assert len(events)%20==0
+for kind,a,b,c,r in struct.iter_unpack('<iiiii',events):
+    if kind==0:
+        assert 0<=r<=h
+        if r:local['auxiliary_steps' if a>=2*v else 'source_steps' if a<v else 'target_steps'][r]+=1
+    elif kind==2:
+        assert r==h-1
+        local['copied_centres'][r]+=1
+    else:assert kind==1 and r==0
+assert len(events)//20==ledger['events']
+assert local['auxiliary_steps']==Counter({int(k):n for k,n in ledger['rank_histograms']['aux'].items()})
+assert local['source_steps']+local['target_steps']==Counter({int(k):n for k,n in ledger['rank_histograms']['data'].items()})
+assert local['copied_centres']==Counter({int(k):n for k,n in ledger['rank_histograms']['center'].items()})
+parts={name:Counter({r:2*v*n for r,n in values.items()}) for name,values in local.items()}
+parts['auxiliary_exteriors']=Counter()
+for s in range(R):
+    r=m-h+len(sigma.get(s,[]))
+    assert 0<r<m
+    parts['auxiliary_exteriors'][r]+=2*v
+parts['data_bridges']=Counter({(h-1)**2:2*N})
+parts['endpoint_copies']=Counter({1:N})
+H=Counter()
+for part in parts.values():H.update(part)
+s=sum(r*n for r,n in H.items())
+assert s==57403754177==word['histogram']['s']==W*m-N+2*v*h*(h-1)
+assert all(0<r<m and n>0 for r,n in H.items())
+out=dict(status='Complete profile rebuilt from locally replayed literal word; PR104 stopped interface applies conditionally',
+    compiler_credit='icekylinx PR104',compiler_commit='948ce1510df750f4c18b96bdaef436a86f8bf834',
+    word_sha256=hashlib.sha256(wordraw).hexdigest(),ledger_sha256=hashlib.sha256(ledgerpath.read_bytes()).hexdigest(),
+    events_sha256=hashlib.sha256(eventpath.read_bytes()).hexdigest(),literal_forward_events=len(events)//20,
+    h=h,m=m,v=v,R=R,W=W,N=N,s=s,deficit=W*m-s,maxchild=max(H),hist=dict(sorted(H.items())),
+    classes={name:dict(calls=sum(values.values()),rank=sum(r*n for r,n in values.items()),hist=dict(sorted(values.items()))) for name,values in parts.items()},
+    scope='Each nested residual, gauged auxiliary exterior, data bridge and endpoint copy is individually idempotent. One common generic basis plus opposite-bank conjugation preserves the complete operator; no local basis switches. No stopped saving or multiplication exponent certified in this file.')
+Path(__file__).with_suffix('.json').write_text(json.dumps(out,indent=2)+'\n')
+print(json.dumps({k:v for k,v in out.items() if k not in ('hist','classes')},indent=2))
diff --git a/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-physical-ledger.py b/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-physical-ledger.py
new file mode 100644
index 0000000000..14dd887f87
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-physical-ledger.py
@@ -0,0 +1,194 @@
+#!/usr/bin/env python3
+"""Explicit forward/reflected ledger of the round-eight combined word.
+Adapted from PR97 round7_literal_frame_ledger.py; Zhihao Chen / Codex audit.
+Changes: consume the local literal schedule and readout frames; keep stable
+atomic fan ordering and the complete formal-basis/reflected checks.
+Zhihao Chen / Codex audit. Exact nesting of the supplied chains is inherited
+from the separately run frame/lifted checkers, not inferred from ranks here.
+Copied-center moves are temporary macros, not erased arbitrary scratch.
+"""
+import sys,json,signal,time,hashlib,gzip
+from pathlib import Path
+from collections import defaultdict,Counter
+from array import array
+HERE=Path(__file__).resolve().parent;ROOT=HERE/'round7-public/tree'
+sys.path.insert(0,str(ROOT/'independent/deferred-readout'))
+import deferred as dr
+
+def main():
+    started=time.monotonic()
+    def stop(*_):raise TimeoutError('180 second literal ledger limit')
+    signal.signal(signal.SIGALRM,stop);signal.alarm(180)
+    W,D=dr.load(23)
+    candidate_path=HERE/('round8-network-atomic-word.json.gz' if '--atomic' in sys.argv else 'round8-network-combined-word.json.gz')
+    if '--candidate' in sys.argv:candidate_path=Path(sys.argv[sys.argv.index('--candidate')+1])
+    candidate=json.load(gzip.open(candidate_path,'rt'))
+    original=dict(zip(D['readout_order'],D['sigma']))
+    for slot in candidate['removed_original_slots']:del original[slot]
+    original.update({int(s):B for s,B in candidate['new_frames'].items()})
+    D['readout_order']=[s for kind,s in candidate['events'] if kind=='read' and s in original]
+    D['sigma']=[original[s] for s in D['readout_order']]
+    v_overrides={int(s):B for s,B in candidate.get('vstart_overrides',{}).items()}
+    if v_overrides:
+        v_frames=dict(zip(D['vleaf_slots'],D['vleaf_start']))
+        assert set(v_overrides)<=set(v_frames)
+        v_frames.update(v_overrides)
+        D['vleaf_start']=[v_frames[s] for s in D['vleaf_slots']]
+    if 'xs_order_actual' in candidate:D['xs_order']=candidate['xs_order_actual']
+    S=dr.Schedule(W,D);v,R,h=S.v,S.R,S.h;size=2*v+R
+    actual_v=defaultdict(list)
+    for kind,s in candidate['events']:
+        if kind=='V':actual_v[S.srcop[s]].append(s)
+    assert {n:slots for n,slots in S.xs}==dict(actual_v)
+    assert sorted(s for slots in actual_v.values() for s in slots)==sorted(S.srcop)
+    co=S.adjoint();sel=S.sel;ops=S.ops;prev={};pred=defaultdict(list);last={}
+    for i,op in enumerate(ops):
+        for s in S.touch(op):
+            if s in prev:pred[i].append(prev[s])
+            prev[s]=i;last[s]=i
+    ph=set();stack=[last[s] for s in S.ret]
+    while stack:
+        i=stack.pop()
+        if i in ph:continue
+        ph.add(i);stack.extend(pred[i])
+    # New readouts have target-local barriers; the literal schedule checks them.
+    keys=[];ids={};dims=[]
+    def norm(k):
+        if k[0]=='out':return ('out',0,tuple(k[2]))
+        return k
+    def key(k):
+        k=norm(k)
+        if k not in ids:
+            ids[k]=len(keys);keys.append(k);dims.append(S.dim(k))
+        return ids[k]
+    z,F=key(('0',)),key(('F',))
+    paths=[None]*size
+    for n,ss in S.xs:paths[n-1]=[key(('n',n))]+[key(('v',s)) for s in ss]+[F]
+    y=[[] for _ in range(v)]
+    for s in S.readout:
+        for t,c in co[s].items():
+            if c&1:y[t].append(key(('sigma',s)))
+    for t,T in enumerate(S.trip):paths[v+t]=[z]+y[t]+[key(('out',0,T))]
+    for s in range(R):paths[2*v+s]=[key(k) for k in S.chain_keys(s)]
+    assert all(p and all(dims[b]>=dims[a] for a,b in zip(p,p[1:])) for p in paths)
+    pos=[0]*size;current=[p[0] for p in paths];initial=current[:]
+    # Fixed-width event records: kind, register/destination, source-or-old,
+    # new-or-common, rank. kind=0 move,1 xor,2 temporary copied-center macro.
+    events=array('i');rank_hist={k:Counter() for k in ('aux','data','center')}
+    bits=[1<<i for i in range(size)]
+    def emit(k,a,b,c,r=0):events.extend((k,a,b,c,r))
+    def move(s,dst):
+        if current[s]==dst:return
+        p=paths[s]
+        if dst not in p[pos[s]+1:]:
+            raise ValueError(dict(register=s,slot=s-2*v,current=keys[current[s]],destination=keys[dst],remaining=[keys[k] for k in p[pos[s]:]],full=[keys[k] for k in p]))
+        j=p.index(dst,pos[s]+1)
+        while pos[s]<j:
+            a=p[pos[s]];b=p[pos[s]+1];r=dims[b]-dims[a];assert r>=0
+            emit(0,s,a,b,r);rank_hist['aux' if s>=2*v else 'data'][r]+=1
+            pos[s]+=1;current[s]=b
+    def xor(t,s):
+        assert current[t]==current[s],('gate frames',t,s,keys[current[t]],keys[current[s]])
+        emit(1,t,s,current[t]);bits[t]^=bits[s]
+    def readout(s):
+        aux=2*v+s;f=key(('sigma',s)) if s in sel else z
+        assert current[aux]==f
+        for t,c in sorted(co[s].items()):
+            if c&1:move(v+t,f);xor(v+t,aux)
+    def source(s,active):
+        frame=key(('v',s)) if active else F
+        move(2*v+s,frame);move(S.srcop[s]-1,frame);xor(2*v+s,S.srcop[s]-1)
+    reordered_fans=[]
+    def operation(i,active):
+        op=ops[i]
+        if op[0]=='add':
+            _,p,o,n=op;frame=key(('n',n)) if active else F
+            move(2*v+p,frame);move(2*v+o,frame);xor(2*v+p,2*v+o)
+        elif op[0]=='fan':
+            # Scalar fan targets commute. The archive groups copies whose
+            # physical start frames differ; execute them in frame-chain order.
+            targets=sorted(op[2],key=lambda s:S.dim(S.start_key(s))) if active else op[2]
+            if active and tuple(targets)!=tuple(op[2]):reordered_fans.append(i)
+            for s in targets:
+                frame=key(S.start_key(s)) if active else F
+                move(2*v+op[1],frame);move(2*v+s,frame);xor(2*v+s,2*v+op[1])
+    centers={}
+    for kind,idx in candidate['events']:
+        if kind=='read':readout(idx)
+        elif kind=='V':source(idx,True)
+        elif kind=='Vinv':source(idx,False)
+        elif kind=='op':operation(idx,True)
+        elif kind=='inverse':operation(idx,False)
+        elif kind in ('atomic','atomic_inverse'):
+            op=candidate['atomic_ops'][idx]
+            frame=key(('n',op['node'])) if op['kind']=='add' else key(S.start_key(op['write']))
+            if kind=='atomic_inverse':frame=F
+            move(2*v+op['write'],frame);move(2*v+op['read'],frame);xor(2*v+op['write'],2*v+op['read'])
+        elif kind=='centre':
+            s=idx;c=S.ret[s];a=2*v+s;f=key(('ret',c));move(a,f)
+            targets=[v+t for t,T in enumerate(S.trip) if c in T]
+            assert all(current[t]==z for t in targets)
+            centers[a]=targets;emit(2,a,f,z,h-1);rank_hist['center'][h-1]+=1
+            for t in targets:bits[t]^=bits[a]
+        elif kind=='out':
+            s=idx;c,T=S.out[s];t=S.tid[T];f=key(('out',0,T))
+            move(2*v+s,f);move(v+t,f);xor(v+t,2*v+s)
+        else:raise ValueError(kind)
+    for s in range(R):move(2*v+s,F)
+    assert all(bits[i]==((1<<i)^(1<<(i-v)) if v<=i<2*v else 1<<i) for i in range(size))
+    assert all(current[i]==F for i in range(v)) and all(current[2*v+s]==F for s in range(R))
+    for H in rank_hist.values():H.pop(0,None)
+    assert rank_hist['aux']==dr.chain_ranks(S)
+    assert sum(r*c for r,c in rank_hist['data'].items())==2*v*(h-1)
+    # Actual reflected word: reverse time, complement frames, swap X/Y banks.
+    # Center copies are regenerated at the reflected source frame, moved to
+    # the reflected target frame, scattered and discarded (same scalar macro).
+    bank=lambda s:s+v if s<v else s-v if s<2*v else s
+    rev=[None]*size
+    for s in range(size):rev[bank(s)]=~current[s]
+    reverse_initial=rev[:];bits=[1<<i for i in range(size)];rh=Counter()
+    for i in range(len(events)-5,-1,-5):
+        k,a,b,c,r=events[i:i+5];a=bank(a)
+        if k==0:
+            assert rev[a]==~c;rev[a]=~b;rh[r]+=1
+        elif k==1:
+            b=bank(b);assert rev[a]==rev[b]==~c;bits[a]^=bits[b]
+        else:
+            assert rev[a]==~b
+            for t in centers[a]:
+                t=bank(t);assert rev[t]==~c;bits[t]^=bits[a]
+            rh[r]+=1
+    assert all(rev[bank(s)]==~initial[s] for s in range(size))
+    assert all(bits[i]==((1<<i)^(1<<(i+v)) if i<v else 1<<i) for i in range(size))
+    rh.pop(0,None);assert rh==sum(rank_hist.values(),Counter())
+    # Assemble from ACTUAL events, not the schedule's pre-summed chain table.
+    m=h*h;N=v*v;H=Counter()
+    for r,c in rh.items():
+        for width in dr.inner(r,h):H[width]+=2*v*c
+    for s in range(R):
+        r=h-S.f[s];H[m-2*r]+=2*v
+        for width in dr.inner(r,h):H[width]+=2*v
+    H[m-4*h+2]+=2*N
+    for width in dr.STAIRCASE_CORNER(h):H[width]+=2*N
+    H[1]+=N
+    expected=dr.histogram(S,dr.chain_ranks(S),S.ylevels([{t:c for t,c in cc.items() if c&1} for cc in co]),S.xdata(),dr.STAIRCASE_CORNER(h))
+    assert expected['hist']=={int(w):n for w,n in candidate['histogram']['hist'].items()}
+    assert dict(H)==expected['hist'];assert sum(w*n for w,n in H.items())==expected['s']
+    dest=HERE/('round8-network-atomic-ledger' if '--atomic' in sys.argv else 'round8-network-physical-ledger');dest.mkdir(exist_ok=True)
+    if '--output' in sys.argv:dest=Path(sys.argv[sys.argv.index('--output')+1]);dest.mkdir(exist_ok=True)
+    with gzip.open(dest/'forward-events.i32.gz','wb') as f:f.write(events.tobytes())
+    (dest/'frames-and-paths.json').write_text(json.dumps(dict(keys=keys,dims=dims,paths=paths,initial=initial,final=current,centers=centers),separators=(',',':'))+'\n')
+    out=dict(candidate_sha256=hashlib.sha256(candidate_path.read_bytes()).hexdigest(),h=h,roles=R,formal_basis=size,events=len(events)//5,frame_keys=len(keys),
+        rank_histograms={k:dict(vv) for k,vv in rank_hist.items()},
+        fan_groups_reordered_internally=reordered_fans,
+        complete_forward_F2=True,complete_reflected_F2=True,
+        vstart_overrides=len(v_overrides),actual_source_V_order_checked=True,
+        reflected_frame_continuity=True,all_scalar_gates_equal_frame_keys=True,
+        exterior_formula='E1=I-I_tensor_Q+sigma_tensor_Q; E2=(I-P)_tensor_I+P_tensor_sigma; rank=m-h+dim(sigma)',
+        histogram_matches_external=True,histogram=dict(H),recursive_rank=expected['s'],
+        elapsed=time.monotonic()-started,limit_seconds=180,
+        exact_chain_geometry='Relies on separately replayed pinned check_frames and check_lifted. Rank labels alone do not prove geometry.',
+        scope='Explicit one-axis forward/reflected events, scalar complete basis and exact accounting. Tensor endpoint identities proved separately; full complex/tape/precision and final assembly integration not established.',new_multiplication_bound=False)
+    (dest/'result.json').write_text(json.dumps(out,indent=2)+'\n');signal.alarm(0)
+    print(json.dumps({k:v for k,v in out.items() if k not in ('histogram','rank_histograms')},indent=2))
+if __name__=='__main__':main()
diff --git a/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-pr104-audit.txt b/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-pr104-audit.txt
new file mode 100644
index 0000000000..82d011847f
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-pr104-audit.txt
@@ -0,0 +1,80 @@
+PR104 OPPOSITE-BANK AND STOPPED-PRODUCT INTERFACE AUDIT
+2026-10-09. Author credit: icekylinx, PR104, with its recorded AI assistance.
+Pinned commit948ce1510df750f4c18b96bdaef436a86f8bf834.
+19 requested source/certificate blobs verified against Git hashes; see
+round8-network-pr104/DOWNLOAD_AUDIT.json. No downloaded source was executed.
+
+Conclusion: the changed bit factorization, paid atom interface and ordinary
+wrapper are consistent and valid under the retained ordered-affine streaming,
+ordinary-leaf and complete-row fixed-tape contracts. I found no extra unpaid
+reversal, gather, recursive orientation conversion or scratch bank.
+
+1. Common basis and full involution.
+For each finite rational projector P of rank r, every leading principal
+minor of size<=r is a nonzero function of a generic common similarity:
+diagonalizing P witnesses nonvanishing. Their finite product is nonzero on
+GL_m. Thus a single rational basis satisfies all these conditions. This is
+an existence/terminating finite-enumeration argument, not an explicitly
+supplied numerical common basis. P C_m has northeast ranks min(i,j,r), so
+physical lower row/column factors normalize it to E F^t with reversed F.
+
+For the full involution g, rank(g-I)=r and its upper-right block E F^t has
+rank r. Hence g-I=[E;B][A,F^t] and AE+F^tB=-2I. The proposed
+K=-(B+F)E^t+FA(I-EE^t) satisfies KE=-B-F and F^tK=A+E^t.
+Consequently SgS^-1=I+[E;-F][-E^t,F^t], exactly the selected cross-bank
+swaps, with the complementary diagonal blocks correct. Q and S are lower
+triangular in the physical head-then-tail order. Symmetry of P is not used.
+
+2. Atom lift and recursive closure.
+On f atoms, Q acts with tensor I_f and the cross shear with K tensor C_f.
+All identities lift because reversal commutes with componentwise scalar
+arithmetic. C_r tensor C_f equals C_(rf), so the normalized middle move
+is one contiguous child of the SAME reversed-interchange family. Rational
+coefficients are reduced only after a fixed odd prime avoids all finite
+denominators and pivot numerators. Ring identities then hold modulo q^w,
+without treating the product ring as one integer field with inter-atom carries.
+
+3. Actual cost of adapters.
+The claim is not that an atom reversal is free. A lower-triangular formal
+matrix is expanded into fixed scalar operations; for each atom index, every
+control lies in a physically earlier formal field. In K tensor C_f the
+control atom index reverses, but every head atom still precedes every tail
+atom. Each ordinary width-w atom update therefore fits the retained earlier-
+control ordered-affine primitive. O(f) passes per fixed matrix costs O(V n)
+per node. Target fields can be processed in descending order to preserve
+original earlier controls. Descriptors select indexed atoms and use the
+inherited fixed tapes/parking; they do not imply a standalone reversal pass.
+This is the exact point at which the inherited streaming contract is used.
+
+4. Remainders, stopping and ordinary conversion.
+Peeling the first delta head atoms against the last delta tail atoms in
+reverse order leaves two contiguous mf-atom intervals for F_(mf). At most
+m ordinary width-w calls pay the peel or a bounded base case. Padding two
+radix-q atoms to binary ranges costs a factor<4 in volume and ordinary width
+O(w); the constant prime is fixed. With both moments mu_1<1 and mu_tau<1,
+induction gives T(n,w)<=A n+B n^tau w^(1-a_old), absorbing the O(n) adapter
+toll with the first gap and the peeled-atom toll with the second.
+
+The chronological D-=H;F;D+=H;F;D=H-D is exactly ordinary swap because
+F(H,D)=(C D,C H) and C^2=I. It is performed only outside the entire reversed
+recursion. Its two calls sequentially reuse restored row stock. At
+w=ceil(e^theta), theta=1/1000, the ordinary saving is
+(1-theta)*4019/50000000+theta*384599/10^10=803380799/10^13.
+The O(e^(1-theta)) adapter term is subordinate since theta exceeds that
+saving. Fewer than w residual digits use the old ordinary interface.
+High-digit row borrowing applies to the completed ORDINARY wrapper, not
+to a naked reversed call. The claimed three-factor row reserve is thus
+the appropriate form; its exact numeric assembly is audited separately.
+
+Independent local checks: round8-network-pr104-check.py/.json checks30
+generic rational projectors,60 full lifted factorizations at f=1,2, physical
+lower triangularity, and7380 exhaustive ordinary-wrapper addresses modulo3.
+All pass. It also verifies the exact stopped-saving identity and strict
+adapter exponent inequality. These finite tests support the general algebra;
+they do not formally verify the inherited all-size streaming/tape interfaces.
+
+Scope: this report does not independently reconstruct PR104's complex
+producer, odd-denominator precision bound, or final analytic assembly.
+Those remain separate checks. PR104's reported full verifications are not
+represented as locally rerun here. The stopped-product construction is an
+attributed public improvement, distinct from our earlier oblique split.
diff --git a/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-pr104-check.json b/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-pr104-check.json
new file mode 100644
index 0000000000..8dd58656c4
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-pr104-check.json
@@ -0,0 +1,8 @@
+{
+  "status": "PASS",
+  "rational_projectors": 30,
+  "atom_lift_cases": 60,
+  "exhaustive_ordinary_wrapper_addresses": 7380,
+  "stopped_saving": "803380799/10000000000000",
+  "scope": "Exact finite algebra and exponent identity; atom streaming uses the retained ordered-affine contract"
+}
diff --git a/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-pr104-check.py b/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-pr104-check.py
new file mode 100644
index 0000000000..d89e1d7e0a
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/inputs/research/round8-network-pr104-check.py
@@ -0,0 +1,102 @@
+"""Independent exact opposite-bank factorization and ordinary-wrapper checks.
+Source claim: icekylinx PR104, commit948ce1510df750f4c18b96bdaef436a86f8bf834.
+Only reviewed stdlib rational helpers are imported; no PR104 code is executed.
+"""
+import importlib.util
+import itertools
+import json
+from fractions import Fraction as Q
+from pathlib import Path
+from random import Random
+
+HERE = Path(__file__).resolve().parent
+spec = importlib.util.spec_from_file_location('factor',HERE/'round7-public/tree/independent/partial-swap/factor.py')
+f = importlib.util.module_from_spec(spec);spec.loader.exec_module(f)
+eye, mul, inv = f.ident,f.mul,f.inv
+def tr(a):return list(map(list,zip(*a)))
+def plus(a,b,s=1):return [[x+s*y for x,y in zip(r,t)] for r,t in zip(a,b)]
+def scale(a,s):return [[s*x for x in r] for r in a]
+def zero(n):return [[Q(0)]*n for _ in range(n)]
+def block(a,b,c,d):return [r+s for r,s in zip(a,b)]+[r+s for r,s in zip(c,d)]
+def rev(n):return [[Q(i+j==n-1) for j in range(n)] for i in range(n)]
+def kron(a,b):return [[x*y for x in ar for y in br] for ar in a for br in b]
+
+def lower_factor(p):
+    n=len(p);m=[r[:] for r in p];l=eye(n);r=eye(n)
+    for i in range(n):
+        c=max((j for j in range(n) if m[i][j]),default=None)
+        if c is None:continue
+        a=m[i][c]
+        for k in range(i+1,n):
+            v=m[k][c]/a
+            if v:
+                m[k]=[x-v*y for x,y in zip(m[k],m[i])]
+                for j in range(n):l[j][i]+=v*l[j][k]
+        for j in range(c):
+            v=m[i][j]/a
+            if v:
+                for k in range(n):m[k][j]-=v*m[k][c]
+                r[c]=[x+v*y for x,y in zip(r[c],r[j])]
+    pi=[[Q(bool(x)) for x in row] for row in m]
+    for j in range(n):
+        a=next((m[i][j] for i in range(n) if m[i][j]),Q(1))
+        r[j]=[a*x for x in r[j]]
+    assert mul(mul(l,pi),r)==p
+    return inv(l),pi,inv(r)
+
+rng=Random(104);cases=0;lifted=0
+for m in range(2,7):
+    I,Z,C=eye(m),zero(m),rev(m)
+    for rank in range(1,m):
+        for trial in range(2):
+            for attempt in range(500):
+                U=eye(m)
+                for _ in range(5*m):
+                    a,b=rng.sample(range(m),2);v=rng.choice((-2,-1,1,2))
+                    U[a]=[x+v*y for x,y in zip(U[a],U[b])]
+                diag=[[Q(i==j and i<rank) for j in range(m)] for i in range(m)]
+                P=mul(mul(U,diag),inv(U))
+                if all(f.rank([row[:k] for row in P[:k]])==k for k in range(1,rank+1)):break
+            else:raise AssertionError('generic sample not found')
+            L,Pi,R=lower_factor(mul(P,C))
+            E=[[Q(i==j) for j in range(rank)] for i in range(m)]
+            F=[[Q(i==m-1-j) for j in range(rank)] for i in range(m)]
+            assert Pi==mul(E,tr(F))
+            DP=block(plus(I,P,-1),mul(P,C),mul(C,P),mul(mul(C,plus(I,P,-1)),C))
+            Qmat=block(L,Z,Z,inv(R));g=mul(mul(Qmat,DP),inv(Qmat))
+            A=[plus(g,eye(2*m),-1)[i][:m] for i in range(rank)]
+            B=[[g[m+i][m+m-1-j]-Q(i==m-1-j) for j in range(rank)] for i in range(m)]
+            assert plus(mul(A,E),mul(tr(F),B))==scale(eye(rank),-2)
+            K=plus(scale(mul(plus(B,F),tr(E)),-1),mul(mul(F,A),plus(I,mul(E,tr(E)),-1)))
+            S=block(I,Z,K,I)
+            target=block(plus(I,mul(E,tr(E)),-1),Pi,tr(Pi),plus(I,mul(F,tr(F)),-1))
+            assert mul(mul(S,g),inv(S))==target
+            for atoms in (1,2):
+                If,Cf,Zf=eye(atoms),rev(atoms),zero(m*atoms)
+                Qf=block(kron(L,If),Zf,Zf,kron(inv(R),If))
+                Sf=block(eye(m*atoms),Zf,kron(K,Cf),eye(m*atoms))
+                Df=block(kron(plus(I,P,-1),If),kron(mul(P,C),Cf),kron(mul(C,P),Cf),kron(mul(mul(C,plus(I,P,-1)),C),If))
+                want=block(kron(plus(I,mul(E,tr(E)),-1),If),kron(Pi,Cf),kron(tr(Pi),Cf),kron(plus(I,mul(F,tr(F)),-1),If))
+                assert mul(mul(mul(mul(Sf,Qf),Df),inv(Qf)),inv(Sf))==want
+                assert kron(rev(rank),Cf)==rev(rank*atoms)
+                assert all(Qf[i][j]==Sf[i][j]==0 for i in range(2*m*atoms) for j in range(i+1,2*m*atoms))
+                lifted+=1
+            cases+=1
+
+addresses=0
+for n in range(1,5):
+    for data in itertools.product(range(3),repeat=2*n):
+        H,D=list(data[:n]),list(data[n:]);old=(H[:],D[:])
+        D=[(d-h)%3 for h,d in zip(H,D)];H,D=D[::-1],H[::-1]
+        D=[(d+h)%3 for h,d in zip(H,D)];H,D=D[::-1],H[::-1]
+        D=[(h-d)%3 for h,d in zip(H,D)]
+        assert (H,D)==old[::-1];addresses+=1
+
+a=Q(4019,50000000);theta=Q(1,1000);old=Q(384599,10**10)
+saving=(1-theta)*a+theta*old
+assert saving==Q(803380799,10**13) and theta>saving
+result=dict(status='PASS',rational_projectors=cases,atom_lift_cases=lifted,
+    exhaustive_ordinary_wrapper_addresses=addresses,stopped_saving=str(saving),
+    scope='Exact finite algebra and exponent identity; atom streaming uses the retained ordered-affine contract')
+Path(__file__).with_suffix('.json').write_text(json.dumps(result,indent=2)+'\n')
+print(json.dumps(result,indent=2))
diff --git a/research/cyclic-deferred/bit-sharing/licenses/main/LICENSE b/research/cyclic-deferred/bit-sharing/licenses/main/LICENSE
new file mode 100644
index 0000000000..261eeb9e9f
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/licenses/main/LICENSE
@@ -0,0 +1,201 @@
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+      any Contribution intentionally submitted for inclusion in the Work
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diff --git a/research/cyclic-deferred/bit-sharing/licenses/main/NOTICE b/research/cyclic-deferred/bit-sharing/licenses/main/NOTICE
new file mode 100644
index 0000000000..b565637758
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/licenses/main/NOTICE
@@ -0,0 +1,304 @@
+A sharper exponent for integer multiplication
+Copyright 2026 Douglas Colkitt
+
+This project includes unmodified source files from OpenAI's manuscript
+"Integer multiplication below n log n" and its accompanying repository:
+https://github.com/openai/math
+Commit: adc7f1241b42e322a6451854ab7e4b4c146bf78a
+
+The original manuscript identifies OpenAI as its author. Its Apache-2.0
+license is retained at upstream/LICENSE. The exact source URLs and hashes
+are recorded in upstream/manifest.json. Proposed modifications are supplied
+as separate files under patches/; upstream source files remain unchanged.
+
+The original local research note, scripts, certificates, patches, and
+documentation were prepared by Douglas Colkitt with assistance from OpenAI
+Codex. The community release incorporates work by the contributors identified
+in the notices below; it is not solely Douglas Colkitt's work.
+This project is not an official OpenAI release or an OpenAI endorsement.
+
+Related contributions (not imported in this checkpoint):
+Zhihao Chen (jacklightChen), PR #7, supplied an earlier ternary five-subset
+motif, finite-alphabet transfer and a different compressed producer:
+https://github.com/CrocSwap/integer-mult-bounds/pull/7
+The present checkpoint claims no priority for that motif. Related compressed
+complex-network work was submitted by eumemic in PR #3:
+https://github.com/CrocSwap/integer-mult-bounds/pull/3
+The pending contributions and their stronger bounds require their own review;
+this notice does not represent adoption or verification of their claims.
+
+Additional community work, including parallel and superseded submissions, is
+acknowledged in CONTRIBUTORS.md, with pinned submission metadata in
+docs/research/contribution-snapshot.json. These acknowledgements do not imply
+joint authorship of the checkpoint, incorporation of pending code, or verification
+of submitted mathematical claims. Original source-specific notices remain in force.
+
+Community release (accepted by conditional maintainer audit):
+The following original contributor notices apply to the imported PR #39 chain.
+The preceding non-import statements describe the preserved 2^-30 checkpoint.
+
+Ternary five-subset construction and common-four-point resynthesis
+Copyright 2026 Zhihao Chen (jacklightChen)
+
+This is Zhihao Chen's first submission of this specific construction to this
+repository. Research, proof development, code and verification were prepared
+with substantial assistance from GPT-6 Astra (OpenAI Codex), as identified by
+the contributor. This is not independent review or an OpenAI endorsement.
+Future research building on this contribution should explicitly acknowledge
+Zhihao Chen (jacklightChen), cite the accompanying note and pull request, and
+retain the preceding contributors' attribution. This notice records research
+provenance; the repository's Apache-2.0 license remains unchanged.
+
+Dependencies retained from eumemic's Claude-assisted PRs #3 and #5:
+https://github.com/CrocSwap/integer-mult-bounds/pull/3
+https://github.com/CrocSwap/integer-mult-bounds/pull/5
+PR #5 commit: d3d370c34937503bd3fa63c83d505a05e8e50893
+These include complex_circuit.py, complex_network.py, fast_gaussian.py,
+their generators, construction notes and certificates. The Gaussian input
+scaling proof is corrected in this derivative; see the modification notice below.
+Aligned pairing also appears in Bortlesboat's PR #2. Retained totals and bank
+sharing appear in dleen's PR #4 and cheaper retained centers in eumemic's PR #6.
+Their results and ideas are credited separately from this new construction.
+
+Bulk recursion and controlled tensor basis, 8 October 2026
+Copyright 2026 icekylinx
+
+The new projector batching, controlled simultaneous basis, mixed-width tape
+recursions, whole-residual complex recursion, and dependency-path precision
+guard in the batched-23 proof package are contributed by icekylinx, developed
+with substantial OpenAI GPT-6 Astra assistance. Review corrections and
+repository integration were prepared with OpenAI Codex assistance. They build on the retained
+PR #7 finite producers and the preceding contributions credited above.
+The new package retains the repository's Apache-2.0 license and the original
+authors' attribution. It is not an official OpenAI release or endorsement.
+
+Modification notice, 8 October 2026 (icekylinx): the retained fast Gaussian
+proof now bounds shifted approximation error by F=2^ceil(1.14 alpha^2).
+The P=34p precision budget and asymptotic exponent are unchanged. New batching
+checks are integrated into make verify. Existing attribution and licenses
+are retained.
+
+Partial-swap extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The partial-swap framing, binary retained-total triple construction,
+dependency-preserving positive labels and carrier reuse, unequal (25,23,57)
+common basis, improved complex batching, and exact assembly are contributed
+by icekylinx with substantial OpenAI GPT-6 Astra and Codex assistance.
+This extension directly inherits PR #10, commit
+62691e395a0458ce089a1c7b5d89e74291e95e29. The portable producer adapters
+modify the retained paired-exclusion and shared-point implementations credited
+above. The original licenses, attribution, and Gaussian correction are retained.
+
+Endpoint-gauge extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The complementary auxiliary endpoint gauges, (32,30,40) controlled basis,
+general nondegenerate binary phase residuals, retained-center complex producer,
+updated path guard and exact assembly are contributed by icekylinx with
+substantial OpenAI GPT-6 Astra and Codex assistance. This extension inherits
+f2ab41aebad47861caf6316282c1793e5513845e and its PR #10 ancestry. New portable
+adapters retain the paired-exclusion and shared-point contributions credited
+above. Original licenses, notices and the Gaussian correction are retained.
+
+Structured-basis and semantic-bulk extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The selected (32,30,36) bit construction, A1/A5 block refinements, fixed
+middle-axis I+J profiles with bounded-minor CRT certificates, mixed dyadic
+complex centers, complete data-front batching and circuit-specific assembly
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance. The immediate parent is ed90fd940279c336ebc45968631bdebfb087b505.
+
+The semantic/bulk compatibility argument, literal scalar charge, product row
+stock and parameterization adapt Zhihao Chen (jacklightChen)'s PR #23,
+commit 661ebabc1076c9f525d7ce4967c876b203fa9827. Translated complex endpoints
+are adopted from the same contributor's PR #21,
+commit 5ba6cf0bfb68f2be8d15610e7972207c50254d6a, following eumemic's source-frame
+work and the earlier nested-basis contributions.
+
+Completed-child semantic precision, arbitrary-coordinate routing, phase-cell
+Gaussian inversion and bulk tape transfer are credited to the RaD project
+(hipotures), commit 45d9b60355872041f9275f77c057514def0d45bc, exposed through
+PR #20 at 4f8d6c8272b5ff307a0da51df545ec3cd96a8b6e. Their selected proof
+sources retain their separate CC0 license and original NOTICE under
+references/semantic-bulk/rad20. The PR #23 source retains its recorded OpenAI
+Codex assistance; RaD retains its recorded Codex and GPT-6.1-Sol Ultra
+assistance. These are source attributions, not claims about this session's
+runtime model. Related contributions by Rohan Arun and Aurel Prosz remain
+acknowledged in the imported notices. No exclusive priority or additional
+license condition is asserted.
+
+Copied retained-center extension, 8 October 2026
+Copyright 2026 icekylinx
+
+The copied retained-center schedule in both orientations, its complex-phase
+transfer, the Wm-N+L rank budget, and the selected (25,23)/(28,28) composition
+are contributed by icekylinx with substantial OpenAI GPT-6 Astra and Codex
+assistance. This extension inherits PR #32 at
+0ef3aeb61f55cc0b321ce6a0ef00acee25cefe52 and the preceding PR #10/#18/#24 work.
+
+The two-stage topology and paid copied-stream endpoint correction are due
+to Aurel Prosz (Paureel), c82d09eb4781b68435e3fa3eb6beea72fa900fab,
+through Zhihao Chen (jacklightChen)'s PR #29 unequal-axis composition,
+9d963275075fa98f1da821e27757b238dafd6b3c. That source also credits Swapnil
+Jain's two-stage development at ae405eb474d1486b2d8aef90139869f927f4836a.
+
+The contiguous corner method, incidence factorization and exact partition
+certificates are due to Rohan Arun (PR #31), method commit
+02f68f95369c39955c5ffcfddbd75cd83e26b305. The selected (25,23) certificate
+specializes Dominik Scholz's parameterization (PR #33),
+a499a345c040aa15b68e984a0904eadc485c7a7e. Their source notices and proof
+snapshots are preserved in references/copied-centers. Historical OpenAI
+Codex, GPT-6 Astra and Anthropic Claude Opus 5.5 assistance is credited to
+the respective imported sources. Existing PR #21/#23 and RaD attribution,
+licenses and the Gaussian correction remain in force.
+
+Copied centers with reversed (23,25) data corners, 2026-10-08
+Rohan Arun with substantial OpenAI Codex assistance supplies the composition,
+exact dimension23 controls and cost/assembly integration. James Chang's PR34
+supplies the reversed boundary family; icekylinx's PR36 supplies copied retained
+centers and the selected complex graph. The balanced assembly remains attributed
+to RaD/hipotures and its PR34 specialization. Retain Dominik Scholz PR33, Rohan
+Arun PR31, Zhihao Chen PR29/21/23, Paureel/Aurel Prosz, Swapnil Jain, icekylinx,
+eumemic, Douglas Colkitt, OpenAI and all earlier notices and hypotheses.
+This is not a claim of formal verification, new foundational rank identities,
+global optimality or measured runtime speedup.
+
+Fixed middle basis with copied reversed corners (2026-10-08):
+Rohan Arun, with substantial OpenAI Codex assistance. Composition and
+simultaneous-basis proof retain icekylinx PR32/36, James Chang PR34,
+Dominik Scholz PR35/38, Zhihao Chen, Paureel, Swapnil Jain, RaD/hipotures,
+eumemic and all preceding attribution and licenses. See research/copied-fixed-reversed.
+
+Changed-graph composition with copied centers and fixed bases, 8 October 2026
+Prepared with OpenAI Codex assistance.
+The new composition uses icekylinx PR36 copied centers and PR32 fixed-I+J
+projectors/profiler, Dominik Scholz PR35 two fixed factors and CRT argument,
+Zhihao Chen (jacklightChen) PR29/23 interfaces, Aurel Prosz (Paureel) and
+Swapnil Jain's two-stage development, Rohan Arun PR25/31 corner methods,
+Dominik Scholz PR33/38 dimension/composition work; James Chang (jamesyc)
+PR34 reversed geometry and balanced transfer; Rohan Arun PR37/39/40
+reversed integration, modular data tests and both-fixed baseline;
+RaD (hipotures) PR41 changed scalar DAGs, alternating point order and
+independent role compiler; RaD balanced routing/bulk interfaces, and all
+retained predecessors.
+Complete attribution and the conditional proof boundary are recorded in
+research/copied-fixed/PROOF.md; imported originals and source pins are retained.
+Original Apache-2.0 and CC0 notices remain applicable.
+
+Community follow-up integration (PRs #49, #45 and #26), 8 October 2026:
+Rohan Arun contributes weighted carrier matching and climbed summand/leave-one-out
+orders (PRs #44/#47/#49), with Anthropic Claude assistance. The final construction
+composes Chafik Boukhalfa's changed-graph checks, exact rational data recovery and
+reordered exclusion sums (PRs #43/#46/#48, OpenAI Codex assistance) with RaD /
+hipotures's alternating pair order and independent physical role compiler (#41).
+All predecessor attribution and conditional proof interfaces remain in force.
+
+Alejandro Zarzuelo Urdiales contributes Gaussian parity and canonical arithmetic
+proofs, finite tensor execution, profile precision accounting and mixed-center
+reference checks (#45), with substantial OpenAI Codex assistance. See
+research/gaussian-parity-synthesis/PROVENANCE.md for his full attribution.
+Ryan S / princezuda contributes historical Lean certificate arithmetic, algebraic
+frame/movement/guard lemmas and independent paired-circuit checks (#26). Original
+source notices and historical snapshots are preserved. Neither formal package
+claims a full formal proof of the multiplication algorithm or a new kappa.
+
+Joint frame compilation of skip-prefix strips
+
+The local compiler and exact finite witnesses were prepared by eumemic with
+substantial OpenAI Codex assistance. Avi Eisenberg / ikeboy with Anthropic
+Claude assistance supplied the unmodified PR #53 skip-prefix producer.
+The pinned PR #48 pipeline is Chafik Boukhalfa's contribution, incorporating
+Rohan Arun, RaD / hipotures, icekylinx, Dominik Scholz, James Chang,
+Zhihao Chen, Aurel Prosz / Paureel, Swapnil Jain, eumemic, Douglas Colkitt,
+OpenAI, David Harvey, Joris van der Hoeven and the earlier contributors
+credited in the retained sources. See references/frame-compiler/pr48/NOTICE
+and references/frame-compiler/pr53/NOTICE for their original attribution.
+No authorship or review by those predecessors is claimed for this compiler.
+
+Joint dual-skip/frame-compiler composition (2026-10-08)
+---------------------------------------------------
+Rohan Gupta / gupt1156 with Anthropic Claude assistance: PR55 dual-skip
+scalar graph; original LICENSE and NOTICE are retained under
+references/frame-compiler/pr55. eumemic with substantial OpenAI Codex
+assistance: PR57 joint reversible frame compiler, serialized physical-word
+replay and profile reconstruction. Chafik Boukhalfa with OpenAI Codex
+assistance: this composition, independent integration and exact certificate.
+Avi Eisenberg / ikeboy, Rohan Arun, RaD / hipotures, icekylinx, and all
+predecessor authors retain the notices above and in pinned source files.
+No original copyright or Apache-2.0 notice is removed.
+
+Ranked reclamation after the PR58 joint-dual composition (2026-10-08)
+----------------------------------------------------------------
+Chafik Boukhalfa with OpenAI Codex assistance: prioritize retired physical
+slots by descending current frame rank, retaining slot ID as the tie-breaker;
+complete replay, exact profiles, comparison and adversarial controls.
+The dedicated joint_dual_reclaim_compiler.py is derived from eumemic's PR57
+compiler. Its original shared compiler remains byte-identical. The PR55
+producer and all preceding attribution remain unchanged. The complete PR58
+comparison certificate and its original source/proof provenance are retained
+under references/frame-compiler/pr58.
+
+Skip-prefix strip producer, 8 October 2026
+By Avi Eisenberg (ikeboy), with Anthropic Claude (Claude Opus 5.5)
+assistance. The skip-prefix leave-one-out strip layout in
+research/skip-strips/skip_graph.py is new. Everything else reuses Chafik
+Boukhalfa's PR #43/#46/#48 research/copied-fixed pipeline, its verification
+code, exact data recovery and support-order sorting, together with:
+- Rohan Arun's PR #44 weighted carrier matching optimizer;
+- RaD (hipotures) PR #41 alternating point order;
+- every predecessor credited above.
+Complete attribution and the conditional proof boundary are recorded in
+research/skip-strips/PROOF.md. Original Apache-2.0 and CC0 notices remain
+applicable.
+
+Local split-pair skip-prefix research continuation (2026-10-08)
+Prepared with substantial OpenAI Codex assistance at the user's request.
+New contribution: singleton/pair recursive partition refinement, finite summand
+order and moment search, and verified composition with inherited paid clones.
+Avi Eisenberg/ikeboy with Anthropic Claude assistance: PR53 skip-prefix strips.
+Chafik Boukhalfa with OpenAI Codex assistance: PR54 original-envelope clone
+specialization and PR43/46/48 checking interfaces. RaD/hipotures: PR51
+whole-chain/source-partition cloning and earlier producer/interfaces.
+Rohan Arun: PR44 weighted matching and earlier fixed/reversed work, with all
+stated historical AI assistance. All predecessor notices remain applicable.
+
+Original-envelope whole-chain clones on skip-prefix strips (research/skip-clones):
+Chafik Boukhalfa, prepared with OpenAI Codex assistance. The complete-chain
+source-partition clone operation is adapted from RaD / hipotures, PR #51,
+with original envelopes retained here. The scalar base is Avi Eisenberg's
+PR #53 skip-prefix layout, prepared with Anthropic Claude assistance.
+All credited PR #43/#46/#48, PR #44 and earlier code/proof notices remain
+in force. This extension is provided under the repository's Apache-2.0 terms.
+
+Community review checkpoint #50–#61, 8 October 2026
+Maintainer integration and independent arithmetic: Douglas Colkitt, with OpenAI Codex.
+Avi Eisenberg / ikeboy: skip-prefix strip framework (#53), with Anthropic Claude.
+Rohan Gupta / gupt1156: parallel order search (#50, stated Antigravity assistance)
+and dual-suffix strips (#55, Anthropic Claude assistance).
+Eumemic: equal-frame joint compilation and paid signal reclamation (#57), with Codex.
+Chafik Boukhalfa: original-envelope paid clones (#54), composition (#58) and ranked
+reclamation (#60), with Codex. Rohan Arun: order search (#52) and positive-frame
+composition (#56), with Anthropic Claude and stated Codex validation assistance.
+RaD / hipotures: enlarged frames, source partitions and paid whole-chain clones
+(#51), with Codex. Rohan Garg / rohangar1: singleton/pair recursion and paid-clone
+composition (#59), with Codex. Alejandro Zarzuelo Urdiales: exact parameter
+refinement, scoped Lean arithmetic, matrix/parity and search tools (#61), with
+his disclosed multi-tool and Codex assistance.
+Parallel #50/#52 records are reviewed and credited without replacing the earlier
+climbed-48 implementation; their exact commit history is retained. The separate
+#51 witness is retained as research; its source-graph/negative-basis data proof
+has not received the same full replay as #56's consumed compiler machinery.
+See docs/research/community-round2-review.md for precise validation boundaries.
+All earlier licenses, contributions and notices remain applicable.
+
+Interval strips and core-aware pair assembly
+
+The interval leave-one-out strips and the core-aware group-pair assembly in
+research/pair-assembly were prepared by Avi Eisenberg / ikeboy with Anthropic
+Claude assistance. They build on PR #53's skip-prefix pipeline and its
+inherited PR #48 sources. The stacked witness in research/pair-assembly/frame
+uses eumemic's PR #57 joint frame compiler unchanged; its driver and verifier
+are adaptations of PR #57's skip_frame_compiler.py, skip_frame_compose.py and
+verify_skip_frame.py. Original notices are retained.
diff --git a/research/cyclic-deferred/bit-sharing/licenses/pr128/MANIFEST.json b/research/cyclic-deferred/bit-sharing/licenses/pr128/MANIFEST.json
new file mode 100644
index 0000000000..45fbd69202
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/licenses/pr128/MANIFEST.json
@@ -0,0 +1,86 @@
+{
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+  }
+}
diff --git a/research/cyclic-deferred/bit-sharing/licenses/pr128/NOTICE b/research/cyclic-deferred/bit-sharing/licenses/pr128/NOTICE
new file mode 100644
index 0000000000..7f73174c62
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/licenses/pr128/NOTICE
@@ -0,0 +1,26 @@
+Completed-core scratch sharing, 2026-10-09
+
+This standalone research contribution and its local verification tools were
+prepared with substantial OpenAI Codex assistance. It composes and extends
+the credited public constructions listed in README.md and PROOF.md. It does
+not claim independent external peer review, authorship by predecessor
+contributors, or an official OpenAI result or endorsement.
+
+Original source files and notices are retained without attribution changes
+under inputs/ and licenses/. Apache-2.0 remains applicable. The source-level
+credit to Zhihao Chen in the local physical-ledger adaptation identifies the
+inherited PR97 ledger; this local adaptation was not authored or reviewed
+by him unless separately stated in the public source.
+
+The exact PR117 source pin, source hashes and original README identify
+eumemic's producer and integration, with Anthropic Claude assistance.
+The round-seven files retain Swapnil Jain's license and notice. Earlier
+copied-center, producer, analytic, tape and semantic contributors retain
+all attribution in the preserved notices. No license or predecessor notice
+is superseded by this file.
+
+The explicit resolvable-packing generator implements the construction of
+Xiande Zhang and Gennian Ge, Journal of Combinatorial Designs 18 (2010),
+209-223, DOI 10.1002/jcd.20234, Lemmas 2.2 and 2.4. Their paper is not
+redistributed. The signed partial grouping and its finite checks are
+included in this contribution.
diff --git a/research/cyclic-deferred/bit-sharing/licenses/pr128/PROOF.md b/research/cyclic-deferred/bit-sharing/licenses/pr128/PROOF.md
new file mode 100644
index 0000000000..bb84b01291
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/licenses/pr128/PROOF.md
@@ -0,0 +1,276 @@
+# Completed-core sharing with a resolvable triple partition
+
+This is a conditional finite construction over the PR117 producer at commit
+cbb05ce504d571546d9b7794c186a613c659c3bf. It uses the interfaces inherited from
+PR104: the arbitrary-dirty carrier compiler, copied rational centers,
+translated complex endpoints, paid binary address adapters, and the semantic
+precision and fixed-tape recursion theorems. The stopped bit layer, all-size
+analytic estimates, eligible-prime/field setup and recovery interfaces remain
+inherited. The exact coupled exponent and its strict inequalities are checked
+separately by the package verifier. This proof does not claim a dense execution
+of every Gaussian transform matrix or an unconditional multiplication theorem.
+
+## 1. The completed-core contract and its PR117 binding
+
+Fix one outer triple label $b$. A logical core acts on data $d$ and all its
+original dirty auxiliary values $a$ as
+
+\[
+ (d,a)\longmapsto(L_b d,a).
+\]
+
+The auxiliary block is the identity on **each original role**, with no
+data/auxiliary or cross-auxiliary mixing. No original auxiliary is assumed zero.
+Let $M$ be the product of carrier shears, $V$ the source injection, and $J$ the
+combined rational side/center readout. The transparent word applies, in order,
+
+\[
+ M;\ -J;\ M^{-1};\ V;\ M;\ J;\ M^{-1};\ -V.
+\]
+
+Its target increment is
+$-JMa+JM(a+Vx)=JMVx$, and its inverse cleanup returns every auxiliary
+individually. Thus $JMV=I$ proves the displayed contract for arbitrary
+Gaussian-rational dirty inputs, including correlated inputs.
+
+The PR117 producer retains the exact disjoint and intersection-two outputs,
+24 disjoint centers $A_i$, and decoder/scatter
+
+\[
+ T=\frac1{21}\sum_i A_i,\qquad
+ T-\frac12\sum_{i\in S}A_i.
+\]
+
+The $A_i$ sum input triples avoiding coordinate $i$. For an input meeting $S$
+in $j$ coordinates, this center expression has coefficient $(j-1)/2$.
+Adding half the disjoint output and subtracting half the intersection-two
+output gives
+
+\[
+ \frac{j-1+[j=0]-[j=2]}2=[j=3].
+\]
+
+The replay checks these supports over the integers, all carrier matches and
+all actual source-to-carrier routing. There are 91770 additions, 8120 roots
+and 71185 matched uses, giving
+
+\[
+ R=91770+8120-71185=28705.
+\]
+
+The upstream frame ledger alone is not a signed scalar replay: its addition
+events omit signs, and its center incidence placeholders are not the rational
+scatter coefficients. The additional audit explicitly binds every operand
+to the DAG, every side root and coefficient to its target, and every center
+to its excluded coordinate. It evaluates all source indicators through the
+actual carrier gates and verifies every intended readout support. This
+establishes $JMV=I$ for the actual scalar word. Four arbitrary-dirty integer
+cases in each direction also check the signed word and its literal inverse;
+these eight finite tests supplement, rather than replace, the universal
+inverse-word argument above.
+
+The physical frame replay checks 778984 events, including 236340 moves, with
+no frame violation or reflected continuity break. It verifies the exact paid
+histogram, local zero source frames and local full pre-exterior sink frames
+for every original auxiliary. Each terminal exterior follows the complete
+cleanup word; it is not an internal return edge.
+
+Write $C_U$ for the actual partial complex transform on a nondegenerate binary
+address subspace $U$, and $F$ for the full address space. Telescoping the gate
+frames gives, before the terminal exterior, dirty residual $C_A$ in stage one,
+where $A=\mathbf F_2^{24}\otimes\langle t_b\rangle$. Stage two has source
+$C_{A^\perp}$ and pre-exterior sink $C_F$, hence residual
+
+\[
+ C_F C_{A^\perp}^{-1}=C_A,\qquad
+ A=\langle t_b\rangle\otimes\mathbf F_2^{24}.
+\]
+
+The source inverse is essential. Reversing the logical shear does not turn
+this auxiliary residual into $C_A^{-1}$. Translated endpoint signs, temporary
+center copies and data corrections remain those of the inherited compiler.
+
+## 2. Actual phases and completed-core sharing
+
+In a common Walsh coordinate system,
+
+\[
+ C_U=H^{-1}\operatorname{diag}(i^{q_U(x)})H,\qquad
+ q_U(x)=\operatorname{wt}(P_Ux)\pmod4.
+\]
+
+For orthogonal nondegenerate $U,V$, their projected vectors are orthogonal, so
+
+\[
+ \operatorname{wt}(u+v)
+ =\operatorname{wt}(u)+\operatorname{wt}(v)
+   -2|\operatorname{supp}(u)\cap\operatorname{supp}(v)|
+ =\operatorname{wt}(u)+\operatorname{wt}(v)\pmod4.
+\]
+
+Consequently $C_UC_V=C_{U\oplus V}$ exactly, including its scalar phase.
+The actual weight modulo four matters: a triple line has active phase $-i$.
+
+Partition the 2024 triple labels into the supplied 87 groups $J$, each with
+binary Gram matrix $I$. For each stage and group allocate $R$ original dirty
+streams, replacing the old role index $(\mathrm{stage},b,u)$ by
+
+\[
+ (\mathrm{stage},J,u),\qquad b\in J.
+\]
+
+This is a new input allocation, not an aliasing of independent old values.
+Different roles within one core and the two stage banks remain distinct.
+Keep every original scalar gate, internal transformation, temporary copied
+center and data correction. Run the complete cores of each group consecutively,
+omitting only their terminal auxiliary exteriors.
+
+Within a stage, distinct outer labels use disjoint data fibres. Their grouping
+therefore preserves the data map. Each completed core is block diagonal and
+accepts every physical scratch state, including the transformed state left by
+earlier cores. No residual is commuted through an internal gate, no clean
+scratch is assumed, and no inter-core frame conversion is inserted for free.
+
+After a group, each shared role has residual $C_{G_J}$, where
+
+\[
+ G_J=\bigoplus_{b\in J}A_b,\qquad \dim G_J=24|J|.
+\]
+
+Apply $C_{G_J^\perp}$ to that role. The total action is exactly $C_F$.
+A group of size 24 needs no correction. The argument is a whole-state linear
+identity, valid with arbitrary spectators and parallel address columns.
+The inherited interstage data macros, rank-one copied endpoint corrections,
+data signs and bank normalization are retained.
+
+## 3. The signed 87-group partition and its paid adapters
+
+The explicit packing instantiates Lemmas 2.2 and 2.4 of Xiande Zhang and
+Gennian Ge, *Maximal resolvable packings and minimal resolvable coverings of
+triples by quadruples*, Journal of Combinatorial Designs 18 (2010), 209-223,
+[DOI 10.1002/jcd.20234](https://doi.org/10.1002/jcd.20234);
+[primary author copy](https://staff.ustc.edu.cn/~drzhangx/papers/mrpacking.pdf).
+The supplied generator implements and exhaustively checks the finite
+construction; no external implementation or redistributed paper is needed.
+
+There are 83 full classes, each consisting of six disjoint quadruples.
+Replacing each quadruple by its four triple faces gives an orthonormal
+24-vector group. The 32 remaining triples lie in four disjoint six-point
+sets, choosing one endpoint from each of three fixed pairs in each set.
+
+For the signed variant, fix choices $c,d$ in the latter two pairs and let the
+first pair be $\{a,b\}$. Group the two triples $\{a,c,d\},\{b,c,d\}$.
+Their orthonormal complement inside that six-point set consists of the other
+two tetrahedron faces $\{a,b,c\},\{a,b,d\}$ and the two unused unit coordinates.
+Combining corresponding pairs from all four six-point sets yields four groups
+of eight triples, each with an explicit 16-vector complement: eight vectors
+of weight three and eight unit vectors. This is the accepted signed variant;
+it does not use the wholly alternating complements of the disjoint-pair variant.
+
+The phase checker verifies exact coverage, every Gram entry of each completed
+24-vector basis, the actual projections and all quadratic phase coefficients
+on zero, the coordinate vectors and their pairwise sums (301 points).
+These points determine a weight-of-linear-map quadratic modulo four. A direct
+phase proof is also available: on one tetrahedron, at input weights
+$0,1,2,3,4$, the numbers of odd face parities are $0,3,2,1,4$.
+Three times those counts equal the original weight modulo four. Thus every
+full group has exactly the full phase, with no leftover linear character.
+The partial groups and their complementary faces have the same identity;
+the unused units supply their ordinary positive phases.
+
+In the displayed complement basis the operator is eight $C^{-1}$ directions
+and eight $C$ directions. With
+
+\[
+ C=\frac12\begin{pmatrix}1+i&1-i\\1-i&1+i\end{pmatrix},
+ \qquad Z=\operatorname{diag}(1,-1),\qquad C^{-1}=-iZCZ,
+\]
+
+one positive width-16 child, paid diagonal $Z$ signs and paid binary orthogonal
+basis adapters implement the complement. The scalar $(-i)^8=1$ already before
+tensoring with the other 24-dimensional factor. Hence either stage orientation
+uses one width-384 child per nonzero complement, with spectators unchanged.
+The inherited binary-address compiler pays the adapters and signs as local
+work; they are not extra Gaussian children or free tape permutations.
+
+The new outer complements must be distinguished from the producer's 3308
+internal alternating rank-two residuals. Their actual four-term Gauss sums
+are $+2$ in 1284 cases and $-2$ in 2024 cases. They retain the inherited
+general rank-two Gauss normal form, including its binary adapters and unit
+phases. They are not replaced by two positive orthonormal directions.
+No new odd divisor or recursive rank is introduced by either type of adapter.
+
+## 4. Complete child list and allocation
+
+Let $h=24$, $m=576$, $v=2024$, $N=v^2=4096576$ and $\ell=552$.
+The new allocation is
+
+\[
+ W=2N+2(87)R=13187822.
+\]
+
+The copied-center histogram $H'_r$ replaces 24 rank-24 original transitions
+by rank-one transitions and retains every rank-23 copy transform. Its mass is
+
+\[
+ \sum_r rH'_r=hR+\ell=689472.
+\]
+
+All recursive children are:
+
+| Class | Multiplicity | Width |
+|---|---:|---:|
+| Shared-role complement | 229640 | 384 |
+| Internal copied-center | $2vH'_r$ | $r>0$ |
+| Data macro | $2N$ | 529 |
+| Data front | $4N$ | 23 |
+| Endpoint copy | $N$ | 1 |
+
+The exterior multiplicity is $2R\cdot4$. The $2R\cdot83=4765030$ zero-rank
+bridges are omitted. All other child classes are unchanged. Since the groups
+cover all $v$ labels, the complete list has
+
+\[
+ s=Wm-N+2v\ell=7594323392,\qquad Wm-s=1862080.
+\]
+
+The largest child is 529. The finite role map supplies a schedule on $W$
+equal-volume streams, so each child has volume fraction $1/W$.
+Complete-row padding and depth-first recursion use this new allocation;
+the old stream count cannot be used with the new denominator.
+
+## 5. Shared histories, the odd grid and coefficient height
+
+A completed core is Gaussian dyadic: its data action is a framed integer shear
+and its dirty block is $C_A$. The local divisor 21 cancels exactly on completion.
+Sequentially reused scratch therefore preserves its existing odd-denominator
+exponent. The new complement adapters are permutations and unit signs and
+introduce no odd denominator or coefficient growth.
+
+There are still the same $2v$ cores and core scalar gates at each node.
+The increased PR117 scalar count is retained:
+
+\[
+ g_h=592224,\qquad G_0=N+2v(g_h+2h)=2401613632.
+\]
+
+The exact common grid $2^{-P}21^{-K}$, with $K=G_0(D+1)$, covers the active
+chain of unfinished calls of depth $D$. No child is rounded or re-encoded.
+At the outer return the exact full tensor operation on every original role
+is Gaussian dyadic, allowing exact odd-grid unscaling.
+
+Binary height and magnitude can accumulate across completed cores. The
+semantic bound charges the sum of **all** child widths $sf$, every local
+scalar operation and one unfinished child at an intermediate peak. It does
+not rely on a path confined to one old scratch bank. The verifier retains
+the full unreused PR117 semantic constants and checks local charges,
+completed-child induction, odd-grid width and stock reserves for
+
+\[
+ (W,s,\text{largest child})=(13187822,7594323392,529).
+\]
+
+The new sharing and signed-complement composition introduce no additional
+physical premise. The exact recurrence, stopped-bit transfer and coupled
+assembly remain separately checked under the inherited analytic, physical,
+tape, field/prime and recovery interfaces stated at the start.
diff --git a/research/cyclic-deferred/bit-sharing/licenses/pr128/README.md b/research/cyclic-deferred/bit-sharing/licenses/pr128/README.md
new file mode 100644
index 0000000000..ae96e5eecb
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/licenses/pr128/README.md
@@ -0,0 +1,104 @@
+# Completed-core scratch sharing
+
+Under the inherited PR104/PR117 interfaces, this finite construction certifies
+
+\[
+T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad
+\kappa=\frac{12310001053253}{10^{17}}=0.00012310001053253>2^{-13}.
+\]
+
+The contribution shares complex scratch banks **between completed cores**
+whose outer triple labels are binary-orthogonal. An explicit partition of all
+2024 triples into 83 groups of 24 and four groups of eight reduces the
+allocation to 13,187,822 original streams. Every internal operation is retained,
+and each partial group pays an explicit signed complement transform for every
+shared role. The complex supplier is PR117's regenerated all-disjoint producer
+with 28,705 roles. The bit supplier is the included improved deferred word
+with its stopped opposite-bank compiler.
+
+The result is conditional. [PROOF.md](PROOF.md) gives the composition, actual
+phase signs, new allocation and precision argument. [certificate.json](certificate.json)
+contains the complete exact profile, moment bounds, three row reserves and
+all 47 strict constraints plus seven saving margins. No unconditional
+multiplication theorem or full expanded Gaussian tape-word execution is claimed.
+
+## Verify
+
+From the repository root:
+
+```sh
+python3 research/completed-core-sharing/verify.py
+```
+
+Requirements: Python 3.10+, a C++17 compiler available as `c++`, and a POSIX
+environment. Only the standard library and local pinned sources are used.
+The verifier does not install packages or access the network. Do not use `-O`.
+
+The default command verifies every frozen input hash, then uses a temporary
+copy for all generated files. It regenerates the PR117 complex producer and
+legal carrier matching; checks the final bit frame geometry; replays all
+1,076,483 literal events on all 32,408 formal columns forward and reflected;
+and compares decompressed event bytes to the saved word. It reconstructs the
+87-group partition and checks its actual phases, explicit signed complement
+bases, complete child list, opposite-bank algebra and signed core transfer.
+The public PR117 frame ledger alone omits scalar signs; the included separate
+source-to-target and inverse-word proof supplies the required scalar linkage.
+
+All arithmetic bounds are rational. A separate 64-term/degree12 interval
+calculation checks the accepted 50-term/degree8 arithmetic. No floating root
+is a proof input. Corruption controls reject a coverage-preserving
+nonorthogonal group swap, a missing bit scatter, an omitted paid child and
+the next kappa grid point. Submitted files remain read-only; gzip timestamps
+and elapsed timings are not treated as mathematical data.
+
+## Profile and scope
+
+The exact complex saving is 0.0001231305483517511358. The coarse bit saving
+is 0.0001245362098479795607; the stopped ordinary wrapper gives
+0.0001244501335381315811393. The assembly uses beta=10^-6, eta=10^-16 and
+the original-prefix margins. Maximum children 528 and 529 require halving
+degrees 367 and 9. All three row reserves are included: degree 22000 exceeds
+the required exponent by 20773/25. PR117's scalar count is retained in
+G=2,401,613,632 and its conservative semantic guard.
+
+The proof consumes the transparent individually restored dirty core, paid
+copied-center/bit adapters, stopped ordinary-leaf interface, semantic precision,
+ordered-affine streaming, analytic/prime setup and exact recovery contracts.
+The verifier regenerates PR117's complex producer; its separate bit producer
+is not used. The full Gaussian compiler is supported by the stated operator
+proof and inherited interfaces. Full repository verification is a separate
+command; this script is the focused contribution check.
+
+For a fixed public comparison, this kappa is 17.8059% above
+[PR117](https://github.com/CrocSwap/integer-mult-bounds/pull/117) at
+`cbb05ce504d571546d9b7794c186a613c659c3bf`, whose kappa is 0.0001044939.
+The public frontier check at 2026-10-09 01:14 UTC also found
+[PR122](https://github.com/CrocSwap/integer-mult-bounds/pull/122), at
+`b1a6f24e57141637b3ff040d6f2ce9d896ffb9bb`, claiming the coupled conditional
+kappa 0.00010941620808215. This certificate is 12.5062% above that value.
+PR122's verifier was not replayed here. These are pinned comparisons, not a
+claim of global optimality or priority over concurrent work.
+
+## Provenance
+
+The partition implements the construction of Xiande Zhang and Gennian Ge,
+*Maximal resolvable packings and minimal resolvable coverings of triples by
+quadruples*, Journal of Combinatorial Designs 18 (2010), 209-223, Lemmas 2.2
+and 2.4: [DOI 10.1002/jcd.20234](https://doi.org/10.1002/jcd.20234),
+[author-hosted paper](https://staff.ustc.edu.cn/~drzhangx/papers/mrpacking.pdf).
+The generator and independently checked finite witness are included; the
+paper is not redistributed.
+
+Credit eumemic for PR117's producer and integration; icekylinx for PR104's
+rational centers and stopped compiler; Rohan Arun for PR107's arithmetic and
+predecessor producer work; Avi Eisenberg and Rohan Gupta for the interval and
+dual-suffix producer lineages; Swapnil Jain for the deferred construction;
+Zhihao Chen for the reflected/signed, physical-ledger and semantic interfaces;
+and predecessor authors identified in the retained notices. Source files and
+receipts retain their original names under `inputs/` to preserve proof
+bindings. [MANIFEST.json](MANIFEST.json) records their exact bytes.
+
+This local composition and verification were prepared with substantial
+OpenAI Codex assistance. This is not an OpenAI endorsement or independent
+external peer review. Original notices remain intact; see [NOTICE](NOTICE)
+and `licenses/`. Search histories and generated binaries are excluded.
diff --git a/research/cyclic-deferred/bit-sharing/licenses/round7/LICENSE b/research/cyclic-deferred/bit-sharing/licenses/round7/LICENSE
new file mode 100644
index 0000000000..261eeb9e9f
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/licenses/round7/LICENSE
@@ -0,0 +1,201 @@
+                                 Apache License
+                           Version 2.0, January 2004
+                        http://www.apache.org/licenses/
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+      names, trademarks, service marks, or product names of the Licensor,
+      except as required for reasonable and customary use in describing the
+      origin of the Work and reproducing the content of the NOTICE file.
+
+   7. Disclaimer of Warranty. Unless required by applicable law or
+      agreed to in writing, Licensor provides the Work (and each
+      Contributor provides its Contributions) on an "AS IS" BASIS,
+      WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or
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+      PARTICULAR PURPOSE. You are solely responsible for determining the
+      appropriateness of using or redistributing the Work and assume any
+      risks associated with Your exercise of permissions under this License.
+
+   8. Limitation of Liability. In no event and under no legal theory,
+      whether in tort (including negligence), contract, or otherwise,
+      unless required by applicable law (such as deliberate and grossly
+      negligent acts) or agreed to in writing, shall any Contributor be
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+
+   END OF TERMS AND CONDITIONS
+
+   APPENDIX: How to apply the Apache License to your work.
+
+      To apply the Apache License to your work, attach the following
+      boilerplate notice, with the fields enclosed by brackets "[]"
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+      file or class name and description of purpose be included on the
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+
+   Copyright [yyyy] [name of copyright owner]
+
+   Licensed under the Apache License, Version 2.0 (the "License");
+   you may not use this file except in compliance with the License.
+   You may obtain a copy of the License at
+
+       http://www.apache.org/licenses/LICENSE-2.0
+
+   Unless required by applicable law or agreed to in writing, software
+   distributed under the License is distributed on an "AS IS" BASIS,
+   WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+   See the License for the specific language governing permissions and
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diff --git a/research/cyclic-deferred/bit-sharing/licenses/round7/NOTICE b/research/cyclic-deferred/bit-sharing/licenses/round7/NOTICE
new file mode 100644
index 0000000000..be2e7e5a98
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/licenses/round7/NOTICE
@@ -0,0 +1,36 @@
+Past the a_b/2 ceiling for integer multiplication
+Copyright 2026 Swapnil Jain
+
+This work builds on, and does not modify, the following:
+
+- OpenAI, "Integer multiplication below n log n" (September 23, 2026), pinned at
+  commit adc7f1241b42e322a6451854ab7e4b4c146bf78a of github.com/openai/math.
+- Douglas Colkitt, github.com/CrocSwap/integer-mult-bounds (Apache-2.0), commit
+  6e564879f51ae16f23d392e9e196c605f36d90df: compact controls and the conditional
+  83/10^12 witness.
+- Prior published results this witness assumes and cites (citations of prior work,
+  not collaborators), from open pull requests on that repository:
+  #3 and #5 (eumemic: compressed complex network, fast Gaussian resampling),
+  #4 (dleen), #6 (eumemic), #7 (Zhihao Chen / jacklightChen: ternary five-subset
+  networks, a_b = 3/(4*10^8), a_c = 39/10^9),
+  #10 (icekylinx: batched recursion, controlled projector basis, whole-residual
+  complex children, Gaussian input-error correction),
+  #13 (eumemic: auxiliary source frames, third whole-residual complex class),
+  #15 (eumemic, head a17cab3: full complex batching of every residual rank),
+  #18 (icekylinx, head f2ab41a: partial-swap networks, producers and common rational basis),
+  #21 (jacklightChen, head 5ba6cf0: translated partial-swap frames; it also contains, independently
+  and earlier in public, the complex source-frame translation we use),
+  #16 (jacklightChen: nested controlled bases) and #29 (jacklightChen: controlled inner basis compiling local
+  edges as blocks), related to our flag basis and side-role batching, which were developed independently,
+  #36 (icekylinx: copied retained centres, used on both our interchanges in rounds six and seven),
+  #62 (ikeboy: interval strips and core-aware pair assembly; its producer builds our round-seven side program),
+  #41 (RaD / hipotures: the alternating order and links, used in that side program),
+  #44 (rohanarun: weighted matching, which our moment-weighted choice of links follows),
+  #24 (icekylinx, head ed90fd9: endpoint-gauge bit network; certificates/external/pr24-bit-network.json
+  extracts its published counts and child-width multiplicities, Apache-2.0).
+- Aurel Prosz (Paureel), github.com/Paureel/integer-mult-bounds commit c82d09e:
+  the two-stage motif with partial-swap frames and charged endpoint correction, and its complement-frame
+  construction, which our lifted frames follow in round seven.
+
+Research, implementation and drafting were done with assistance from Claude
+(Anthropic). AI assistance is not independent review.
diff --git a/research/cyclic-deferred/bit-sharing/partition161-receipt.json b/research/cyclic-deferred/bit-sharing/partition161-receipt.json
new file mode 100644
index 0000000000..6f58f423dd
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/partition161-receipt.json
@@ -0,0 +1,412 @@
+{
+  "groups": 161,
+  "size": 11,
+  "translation_orbits": 77,
+  "forced_choices": 55,
+  "ternary_choices": 22,
+  "search_visits": 58,
+  "seconds": 0.007600069046020508,
+  "base_classes": [
+    [
+      [
+        0,
+        1,
+        21
+      ],
+      [
+        0,
+        2,
+        15
+      ],
+      [
+        0,
+        3,
+        11
+      ],
+      [
+        0,
+        4,
+        14
+      ],
+      [
+        0,
+        6,
+        22
+      ],
+      [
+        0,
+        8,
+        19
+      ],
+      [
+        0,
+        9,
+        20
+      ],
+      [
+        0,
+        12,
+        17
+      ],
+      [
+        0,
+        10,
+        13
+      ],
+      [
+        0,
+        7,
+        16
+      ],
+      [
+        0,
+        5,
+        18
+      ]
+    ],
+    [
+      [
+        0,
+        1,
+        15
+      ],
+      [
+        0,
+        2,
+        19
+      ],
+      [
+        0,
+        3,
+        22
+      ],
+      [
+        0,
+        4,
+        21
+      ],
+      [
+        0,
+        6,
+        20
+      ],
+      [
+        0,
+        8,
+        17
+      ],
+      [
+        0,
+        9,
+        14
+      ],
+      [
+        0,
+        11,
+        12
+      ],
+      [
+        0,
+        5,
+        13
+      ],
+      [
+        0,
+        10,
+        16
+      ],
+      [
+        0,
+        7,
+        18
+      ]
+    ],
+    [
+      [
+        0,
+        1,
+        17
+      ],
+      [
+        0,
+        2,
+        20
+      ],
+      [
+        0,
+        3,
+        21
+      ],
+      [
+        0,
+        4,
+        7
+      ],
+      [
+        0,
+        6,
+        15
+      ],
+      [
+        0,
+        5,
+        8
+      ],
+      [
+        0,
+        9,
+        19
+      ],
+      [
+        0,
+        12,
+        22
+      ],
+      [
+        0,
+        11,
+        13
+      ],
+      [
+        0,
+        14,
+        16
+      ],
+      [
+        0,
+        10,
+        18
+      ]
+    ],
+    [
+      [
+        0,
+        1,
+        22
+      ],
+      [
+        0,
+        2,
+        17
+      ],
+      [
+        0,
+        3,
+        20
+      ],
+      [
+        0,
+        4,
+        10
+      ],
+      [
+        0,
+        6,
+        11
+      ],
+      [
+        0,
+        8,
+        21
+      ],
+      [
+        0,
+        5,
+        9
+      ],
+      [
+        0,
+        12,
+        19
+      ],
+      [
+        0,
+        7,
+        13
+      ],
+      [
+        0,
+        15,
+        16
+      ],
+      [
+        0,
+        14,
+        18
+      ]
+    ],
+    [
+      [
+        0,
+        1,
+        5
+      ],
+      [
+        0,
+        2,
+        21
+      ],
+      [
+        0,
+        3,
+        10
+      ],
+      [
+        0,
+        4,
+        11
+      ],
+      [
+        0,
+        6,
+        17
+      ],
+      [
+        0,
+        8,
+        22
+      ],
+      [
+        0,
+        7,
+        9
+      ],
+      [
+        0,
+        12,
+        20
+      ],
+      [
+        0,
+        13,
+        14
+      ],
+      [
+        0,
+        16,
+        19
+      ],
+      [
+        0,
+        15,
+        18
+      ]
+    ],
+    [
+      [
+        0,
+        1,
+        20
+      ],
+      [
+        0,
+        2,
+        22
+      ],
+      [
+        0,
+        3,
+        14
+      ],
+      [
+        0,
+        4,
+        15
+      ],
+      [
+        0,
+        6,
+        21
+      ],
+      [
+        0,
+        7,
+        8
+      ],
+      [
+        0,
+        9,
+        11
+      ],
+      [
+        0,
+        10,
+        12
+      ],
+      [
+        0,
+        13,
+        19
+      ],
+      [
+        0,
+        5,
+        16
+      ],
+      [
+        0,
+        17,
+        18
+      ]
+    ],
+    [
+      [
+        0,
+        1,
+        11
+      ],
+      [
+        0,
+        2,
+        7
+      ],
+      [
+        0,
+        3,
+        17
+      ],
+      [
+        0,
+        4,
+        19
+      ],
+      [
+        0,
+        5,
+        6
+      ],
+      [
+        0,
+        8,
+        15
+      ],
+      [
+        0,
+        9,
+        10
+      ],
+      [
+        0,
+        12,
+        14
+      ],
+      [
+        0,
+        13,
+        20
+      ],
+      [
+        0,
+        16,
+        21
+      ],
+      [
+        0,
+        18,
+        22
+      ]
+    ]
+  ],
+  "exact_coverage": true,
+  "exact_rational_orthogonality": true
+}
diff --git a/research/cyclic-deferred/bit-sharing/partition161.py b/research/cyclic-deferred/bit-sharing/partition161.py
new file mode 100644
index 0000000000..d3ae71fa10
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/partition161.py
@@ -0,0 +1,56 @@
+"""Exact cyclic partition into 161 orthogonal 11-triple sunflower classes.
+
+Paley-tournament orientation selects one center in each translation orbit.
+The remaining degree-two choice problem is solved by finite exact search;
+seven perfect matchings of the resulting 7-regular bipartite graph give the
+seven base classes. Every exported triple and Gram entry is checked.
+"""
+from itertools import combinations
+from collections import Counter
+from pathlib import Path
+import json,time
+p=23;QR={i*i%p for i in range(1,p)};NR=set(range(1,p))-QR
+
+def pairs(T):return sorted({tuple(sorted((a-c)%p for a in T if a!=c))for c in T})
+orbits={min(pairs(T)):pairs(T)for T in combinations(range(p),3)};assert len(orbits)==77
+choices=[];forced=[]
+for key,opts in sorted(orbits.items()):
+ good=[e if e[0]in QR else(e[1],e[0])for e in opts if(e[0]in QR)!=(e[1]in QR)]
+ assert len(good)in(1,3)
+ if len(good)==1:forced.append(good[0])
+ else:choices.append(tuple(good))
+degree=Counter(x for e in forced for x in e);assert len(forced)==55 and len(choices)==22 and set(degree.values())=={5}
+visits=0
+def solve(todo,selected):
+ global visits
+ visits+=1
+ if not todo:return selected if all(degree[x]==7 for x in range(1,p))else None
+ valid={i:[e for e in choices[i]if all(degree[x]<7 for x in e)]for i in todo}
+ if any(not es for es in valid.values()):return None
+ for x in range(1,p):
+  if degree[x]+sum(any(x in e for e in valid[i])for i in todo)<7:return None
+ i=min(todo,key=lambda j:(len(valid[j]),j));rest=todo-{i}
+ for e in sorted(valid[i],key=lambda e:(sum(degree[x]for x in e),e)):
+  for x in e:degree[x]+=1
+  r=solve(rest,selected+[e])
+  if r is not None:return r
+  for x in e:degree[x]-=1
+ return None
+started=time.time();selected=solve(set(range(len(choices))),[]);assert selected is not None
+edges=set(forced+selected);assert len(edges)==77 and all(sum(x in e for e in edges)==7 for x in range(1,p))
+base=[]
+for _ in range(7):
+ match={}
+ def augment(a,seen):
+  for b in sorted(y for x,y in edges if x==a):
+   if b in seen:continue
+   seen.add(b)
+   if b not in match or augment(match[b],seen):match[b]=a;return True
+  return False
+ for a in sorted(QR):assert augment(a,set())
+ M=sorted((a,b)for b,a in match.items());assert len(M)==11;edges-=set(M);base.append([(0,)+tuple(sorted(e))for e in M])
+assert not edges
+groups=[[tuple(sorted((x+c)%p for x in T))for T in G]for c in range(p)for G in base]
+flat=[T for G in groups for T in G];assert len(groups)==161 and len(flat)==1771 and len(set(flat))==1771 and set(flat)==set(combinations(range(p),3))
+for G in groups:assert all(len(set(A)&set(B))==1 for A,B in combinations(G,2))
+D=Path(__file__).resolve().parent;(D/'groups161.json').write_text(json.dumps(groups)+'\n');(D/'partition161-receipt.json').write_text(json.dumps(dict(groups=161,size=11,translation_orbits=77,forced_choices=55,ternary_choices=22,search_visits=visits,seconds=time.time()-started,base_classes=base,exact_coverage=True,exact_rational_orthogonality=True),indent=2)+'\n');print('PASS',len(groups),'groups of11',visits,'search visits',time.time()-started,flush=True)
diff --git a/research/cyclic-deferred/bit-sharing/profile161.json b/research/cyclic-deferred/bit-sharing/profile161.json
new file mode 100644
index 0000000000..be4def1883
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/profile161.json
@@ -0,0 +1,247 @@
+{
+  "h": 23,
+  "m": 529,
+  "v": 1771,
+  "N": 3136441,
+  "R": 28866,
+  "L": 1792252,
+  "groups": 161,
+  "group_sizes": {
+    "11": 161
+  },
+  "W": 15567734,
+  "total_rank": 8233987097,
+  "deficit": 1344189,
+  "maxchild": 518,
+  "child_multiplicities": {
+    "1": 235438511,
+    "2": 236428500,
+    "3": 51022510,
+    "4": 43669318,
+    "5": 22488158,
+    "6": 8072218,
+    "7": 3715558,
+    "8": 6694380,
+    "9": 1919764,
+    "10": 1059058,
+    "11": 1423884,
+    "12": 2064986,
+    "13": 995302,
+    "14": 347116,
+    "15": 2387308,
+    "16": 1441594,
+    "17": 4547928,
+    "18": 2266880,
+    "19": 7484246,
+    "20": 8847916,
+    "21": 1466388,
+    "22": 127512,
+    "276": 5666234,
+    "287": 644,
+    "298": 4508,
+    "309": 14490,
+    "320": 40894,
+    "331": 322,
+    "342": 25116,
+    "353": 1610,
+    "364": 98532,
+    "375": 11592,
+    "386": 36708,
+    "397": 9016,
+    "408": 104006,
+    "419": 28658,
+    "430": 34454,
+    "441": 9660,
+    "452": 143934,
+    "463": 56994,
+    "474": 259532,
+    "484": 6272882,
+    "485": 1246462,
+    "496": 420854,
+    "507": 513912,
+    "518": 566720
+  },
+  "classes": {
+    "auxiliary_steps": {
+      "calls": 576244438,
+      "rank": 1611404564,
+      "hist": {
+        "1": 196435778,
+        "2": 229443676,
+        "3": 49031906,
+        "4": 39712904,
+        "5": 21234290,
+        "6": 6616456,
+        "7": 2954028,
+        "8": 4370828,
+        "9": 1565564,
+        "10": 612766,
+        "11": 1236158,
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+        "15": 2373140,
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+        "18": 2121658,
+        "19": 1887886,
+        "20": 8182020,
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+      }
+    },
+    "source_steps": {
+      "calls": 24308746,
+      "rank": 138003404,
+      "hist": {
+        "1": 17922520,
+        "4": 113344,
+        "16": 113344,
+        "19": 5454680,
+        "20": 665896,
+        "22": 38962
+      }
+    },
+    "target_steps": {
+      "calls": 40137944,
+      "rank": 138003404,
+      "hist": {
+        "1": 17943772,
+        "2": 6984824,
+        "3": 1990604,
+        "4": 3843070,
+        "5": 1253868,
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+        "7": 761530,
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+        "10": 446292,
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+        "12": 1331792,
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+        "14": 123970,
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+        "16": 609224,
+        "17": 21252,
+        "18": 145222,
+        "19": 141680,
+        "21": 3542,
+        "22": 7084
+      }
+    },
+    "copied_centres": {
+      "calls": 81466,
+      "rank": 1792252,
+      "hist": {
+        "22": 81466
+      }
+    },
+    "data_bridges": {
+      "calls": 6272882,
+      "rank": 3036074888,
+      "hist": {
+        "484": 6272882
+      }
+    },
+    "endpoint_copies": {
+      "calls": 3136441,
+      "rank": 3136441,
+      "hist": {
+        "1": 3136441
+      }
+    },
+    "auxiliary_exteriors": {
+      "calls": 9294852,
+      "rank": 3305572144,
+      "hist": {
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+        "287": 644,
+        "298": 4508,
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+        "496": 420854,
+        "507": 513912,
+        "518": 566720
+      }
+    }
+  },
+  "paid_projector_calls": 659476769,
+  "local_forward_event_counts": {
+    "0": 219611,
+    "1": 856849,
+    "2": 23
+  },
+  "literal_frame_keys": 89877,
+  "literal_frame_inventory_sha256": "e627b31dccc6d6a0d2ccb7e0dbe35612da8286d24e0a2e11ddd30a3d4174ddbd",
+  "literal_event_sha256": "f98d52b6d8fb04e30a0a46df4df95d2e7999ff97a89826e5b0f44b4120a69e28",
+  "old_exteriors": {
+    "506": 62328574,
+    "507": 7084,
+    "508": 49588,
+    "509": 159390,
+    "510": 449834,
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+    "524": 2854852,
+    "525": 13711082,
+    "526": 4629394,
+    "527": 5653032,
+    "528": 6233920
+  },
+  "new_exteriors": {
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+    "287": 644,
+    "298": 4508,
+    "309": 14490,
+    "320": 40894,
+    "331": 322,
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+    "419": 28658,
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+    "452": 143934,
+    "463": 56994,
+    "474": 259532,
+    "485": 1246462,
+    "496": 420854,
+    "507": 513912,
+    "518": 566720
+  },
+  "minimal_v_word_sha256": "298c11fb29ab0afdf8d762c64a9b99d19a1e58465f3a0cbc9d2a59367c438fb7",
+  "exact_coverage": true,
+  "exact_pairwise_H_orthogonality": true,
+  "exact_projector_idempotence": true,
+  "exact_projector_self_adjointness": true,
+  "scope": "Exact completed-core rational projector identities and complete one-child profile. Requires unchanged PR128 minimal-V literal-word contract, common generic-basis/opposite-bank compiler and scalar/adapter work charge. Each recorded projector call pays its fixed PR104 atom adapters; the generic-basis existence argument does not specify a numerical primitive-operation constant."
+}
diff --git a/research/cyclic-deferred/bit-sharing/profile161.py b/research/cyclic-deferred/bit-sharing/profile161.py
new file mode 100644
index 0000000000..98d5d160e7
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/profile161.py
@@ -0,0 +1,55 @@
+"""Complete bit profile for explicit orthogonal sunflower completed-core groups.
+
+Uses PR128's frozen minimal-V bit word and PR104's opposite-bank compiler.
+This script proves finite rational projector algebra and recomputes the full
+histogram; inherited literal bit-word replay/physical interfaces remain inputs.
+"""
+from pathlib import Path
+from itertools import combinations
+from collections import Counter
+import json,gzip,math,hashlib,struct
+D=Path(__file__).resolve().parent;P=D/'inputs/research'
+h=23;m=h*h;v=math.comb(h,3);N=v*v
+
+def factors(points):
+ pts=list(points)
+ if len(pts)%2:pts.append(None)
+ if len(pts)<2:return []
+ out=[]
+ for _ in range(len(pts)-1):
+  out.append([tuple(sorted((pts[i],pts[-1-i])))for i in range(len(pts)//2)if pts[i]is not None and pts[-1-i]is not None])
+  pts=[pts[0],pts[-1]]+pts[1:-1]
+ return out
+
+groups=json.loads((D/'groups161.json').read_text())
+groups=[[tuple(T)for T in G]for G in groups]
+trip=list(combinations(range(h),3));flatten=[T for G in groups for T in G];assert len(flatten)==v and len(set(flatten))==v and set(flatten)==set(trip)
+for G in groups:
+ for i,T in enumerate(G):
+  for j,S in enumerate(G):assert len(set(T)&set(S))==(3 if i==j else 1)
+ # Six times each H-orthogonal rank-one projector has exact integer entries.
+ Q=[[sum(int(i in T)*(3*int(j in T)-1)for T in G)for j in range(h)]for i in range(h)]
+ assert all(sum(Q[i][k]*Q[k][j]for k in range(h))==6*Q[i][j]for i in range(h)for j in range(h))
+ assert sum(Q[i][i]for i in range(h))==6*len(G)
+ # Projector self-adjointness for H scaled by9: HQ=(HQ)^T.
+ HQ=[[9*Q[i][j]-sum(Q[k][j]for k in range(h))for j in range(h)]for i in range(h)]
+ assert all(HQ[i][j]==HQ[j][i]for i in range(h)for j in range(h))
+raw=(P/'round8-foundations-minimal-v-word.json.gz').read_bytes();word=json.loads(gzip.decompress(raw));base=json.loads((P/'round8-network-opposite-profile.json').read_text());deferred=json.loads(gzip.decompress((P/'round7-public/tree/certificates/round7/deferred_23.json.gz').read_bytes()))
+sigma=dict(zip(deferred['readout_order'],deferred['sigma']))
+for s in word['removed_original_slots']:sigma.pop(s)
+sigma.update({int(s):B for s,B in word['new_frames'].items()});R=base['R'];assert R==deferred['R']==28866
+z=Counter({int(r):n for r,n in base['hist'].items()});old_ext=Counter({int(r):n for r,n in base['classes']['auxiliary_exteriors']['hist'].items()});z.subtract(old_ext);new_ext=Counter()
+for G in groups:
+ for s in range(R):
+  width=m-(h-len(sigma.get(s,())))*len(G);assert 0<width<m;new_ext[width]+=2
+z.update(new_ext);assert min(z.values())>=0;z=Counter({r:n for r,n in z.items()if n});W=2*N+2*len(groups)*R;mass=sum(r*n for r,n in z.items());assert W*m-mass==base['deficit'];assert mass==W*m-N+2*v*h*(h-1)
+ledger=D/'literal-replay'
+physical=gzip.decompress((ledger/'forward-events.i32.gz').read_bytes())
+assert hashlib.sha256(physical).hexdigest()=='f98d52b6d8fb04e30a0a46df4df95d2e7999ff97a89826e5b0f44b4120a69e28'
+event_counts=Counter(k for k,*_ in struct.iter_unpack('<iiiii',physical))
+frame_raw=(ledger/'frames-and-paths.json').read_bytes();frame_data=json.loads(frame_raw)
+classes={name:part for name,part in base['classes'].items()if name!='auxiliary_exteriors'}
+classes['auxiliary_exteriors']=dict(calls=sum(new_ext.values()),rank=sum(r*n for r,n in new_ext.items()),hist=dict(sorted(new_ext.items())))
+assert sum(c['calls']for c in classes.values())==sum(z.values())
+out=dict(h=h,m=m,v=v,N=N,R=R,L=2*v*h*(h-1),groups=len(groups),group_sizes=dict(sorted(Counter(map(len,groups)).items())),W=W,total_rank=mass,deficit=W*m-mass,maxchild=max(z),child_multiplicities=dict(sorted(z.items())),classes=classes,paid_projector_calls=sum(z.values()),local_forward_event_counts=dict(sorted(event_counts.items())),literal_frame_keys=len(frame_data['keys']),literal_frame_inventory_sha256=hashlib.sha256(frame_raw).hexdigest(),literal_event_sha256=hashlib.sha256(physical).hexdigest(),old_exteriors=dict(sorted(old_ext.items())),new_exteriors=dict(sorted(new_ext.items())),minimal_v_word_sha256=hashlib.sha256(raw).hexdigest(),exact_coverage=True,exact_pairwise_H_orthogonality=True,exact_projector_idempotence=True,exact_projector_self_adjointness=True,scope='Exact completed-core rational projector identities and complete one-child profile. Requires unchanged PR128 minimal-V literal-word contract, common generic-basis/opposite-bank compiler and scalar/adapter work charge. Each recorded projector call pays its fixed PR104 atom adapters; the generic-basis existence argument does not specify a numerical primitive-operation constant.')
+(D/'profile161.json').write_text(json.dumps(out,indent=2)+'\n');print({k:v for k,v in out.items()if k not in('child_multiplicities','old_exteriors','new_exteriors')},flush=True)
diff --git a/research/cyclic-deferred/bit-sharing/verify.py b/research/cyclic-deferred/bit-sharing/verify.py
new file mode 100644
index 0000000000..2d50f1e7c8
--- /dev/null
+++ b/research/cyclic-deferred/bit-sharing/verify.py
@@ -0,0 +1,70 @@
+#!/usr/bin/env python3
+"""Replay all finite bit161 evidence in a disposable, portable directory."""
+from pathlib import Path
+from hashlib import sha256
+import gzip,json,shutil,subprocess,sys,tempfile,time
+
+HERE=Path(__file__).resolve().parent
+def read(path):return json.loads(path.read_text())
+def equal_json(actual_path,expected_path):
+    actual,expected=read(actual_path),read(expected_path)
+    keys=set(actual)|set(expected)
+    different={key:dict(actual=actual.get(key),expected=expected.get(key)) for key in sorted(keys) if actual.get(key)!=expected.get(key) or (key in actual)!=(key in expected)}
+    assert not different, 'JSON mismatch '+actual_path.name+': '+json.dumps(different,sort_keys=True)
+
+def run(script,*args,reject=False):
+    started=time.monotonic()
+    p=subprocess.run([sys.executable,str(script),*map(str,args)],capture_output=True,text=True,timeout=900)
+    if reject:assert p.returncode!=0,'Corruption accepted: '+script.name
+    elif p.returncode:raise RuntimeError(script.name+' failed:\n'+p.stdout[-6000:]+'\n'+p.stderr[-6000:])
+    print(('REJECT'if reject else'PASS'),script.name,f'({time.monotonic()-started:.1f}s)',flush=True)
+
+def main():
+    assert not sys.flags.optimize,'Run without -O; inherited assertions are required'
+    manifest=read(HERE/'MANIFEST.json')
+    for name,digest in manifest['sha256'].items():assert sha256((HERE/name).read_bytes()).hexdigest()==digest,name
+    print('PASS',len(manifest['sha256']),'frozen source/input/output hashes',flush=True)
+    with tempfile.TemporaryDirectory(prefix='bit161-')as temporary:
+        work=Path(temporary)/'package';shutil.copytree(HERE,work,ignore=shutil.ignore_patterns('__pycache__'))
+        p=work/'inputs/research'
+        run(p/'round8-network-final-geometry.py','--output',work/'fresh-geometry.json')
+        old=read(p/'pr-review-network-geometry.json');new=read(work/'fresh-geometry.json')
+        for key in old:
+            if key!='elapsed':assert old[key]==new[key],('geometry',key)
+        candidate=p/'round8-foundations-minimal-v-word.json.gz'
+        run(p/'round8-network-physical-ledger.py','--candidate',candidate,'--output',work/'literal-replay')
+        old=read(p/'round8-network-minimal-v-ledger/result.json');new=read(work/'literal-replay/result.json')
+        for key in old:
+            if key!='elapsed':assert old[key]==new[key],('ledger',key)
+        assert gzip.decompress((p/'round8-network-minimal-v-ledger/forward-events.i32.gz').read_bytes())==gzip.decompress((work/'literal-replay/forward-events.i32.gz').read_bytes())
+        old=read(p/'round8-network-opposite-profile.json')
+        run(p/'round8-network-opposite-profile.py')
+        assert read(p/'round8-network-opposite-profile.json')==old
+        run(p/'round8-network-pr104-check.py')
+        run(work/'partition161.py');assert read(work/'groups161.json')==read(HERE/'groups161.json')
+        receipt=read(work/'partition161-receipt.json');old=read(HERE/'partition161-receipt.json')
+        for key in old:
+            if key!='seconds':assert receipt[key]==old[key],('partition receipt',key)
+        run(work/'profile161.py');equal_json(work/'profile161.json',HERE/'profile161.json')
+        run(work/'certify161.py');assert read(work/'certificate.json')==read(HERE/'certificate.json')
+        # Coverage-preserving swap deliberately violates pairwise H-orthogonality.
+        groups=read(work/'groups161.json');a=set(groups[0][0]);found=False
+        for j in range(1,len(groups)):
+            for k,t in enumerate(groups[j]):
+                if len(a&set(t))!=1:
+                    groups[0][1],groups[j][k]=groups[j][k],groups[0][1];found=True;break
+            if found:break
+        assert found
+        (work/'groups161.json').write_text(json.dumps(groups)+'\n')
+        run(work/'profile161.py',reject=True)
+        # Missing a true output scatter must fail complete formal-column replay.
+        word=json.loads(gzip.decompress(candidate.read_bytes()))
+        index=next(i for i,e in enumerate(word['events'])if e[0]=='out');del word['events'][index]
+        bad=work/'bad-word.json.gz';bad.write_bytes(gzip.compress(json.dumps(word).encode(),mtime=0))
+        run(p/'round8-network-physical-ledger.py','--candidate',bad,'--output',work/'bad-ledger',reject=True)
+        profile=read(HERE/'profile161.json');rows={int(t):n for t,n in profile['child_multiplicities'].items()};rows[1]-=1
+        assert sum(t*n for t,n in rows.items())!=profile['m']*profile['W']-profile['deficit']
+        print('REJECT omitted paid child; next-grid enclosure already checked',flush=True)
+    print('PASS exact coarse bit saving 129310447/10^12; ordinary stopped wrapper and final assembly are separate.')
+
+if __name__=='__main__':main()
diff --git a/research/cyclic-deferred/bit_adapter.py b/research/cyclic-deferred/bit_adapter.py
new file mode 100644
index 0000000000..815447b91a
--- /dev/null
+++ b/research/cyclic-deferred/bit_adapter.py
@@ -0,0 +1,57 @@
+#!/usr/bin/env python3
+"""Replay the independently frozen bit supplier, then expose its exact profile.
+
+The nested package retains its own manifest, verifier, input pins and notices.
+Prepared for eumemic with OpenAI Codex assistance; Apache-2.0.
+"""
+import sys
+sys.dont_write_bytecode = True
+if sys.flags.optimize:
+    raise ValueError('Bit adapter requires assertions')
+from pathlib import Path
+from hashlib import sha256
+import json
+import os
+import subprocess
+
+HERE = Path(__file__).resolve().parent
+BIT_MANIFEST_PIN = '1f8afd0109706437445fcc9b3bacff455c0d229b8ad2d3d106f3cd6c43bc58b3'
+BIT_VERIFIER_PIN = '58ccce67e0a6ca9db91d195250eb9fdb8ccae27b5fe22fb6b1b32f512a476146'
+MINIMAL_V_PIN = '298c11fb29ab0afdf8d762c64a9b99d19a1e58465f3a0cbc9d2a59367c438fb7'
+MINIMAL_V_PATH = 'inputs/research/round8-foundations-minimal-v-word.json.gz'
+
+
+def package_inventory(package):
+    nested = package / 'bit-sharing'
+    raw = (nested / 'MANIFEST.json').read_bytes()
+    if sha256(raw).hexdigest() != BIT_MANIFEST_PIN:
+        raise ValueError('Immutable bit-package manifest digest mismatch')
+    manifest = json.loads(raw)
+    files = manifest['sha256']
+    if files.get('verify.py') != BIT_VERIFIER_PIN or files.get(MINIMAL_V_PATH) != MINIMAL_V_PIN:
+        raise ValueError('Immutable bit verifier or minimal-V provenance digest mismatch')
+    for name, expected in files.items():
+        relative = Path(name)
+        path = (nested / relative).resolve()
+        if relative.is_absolute() or '..' in relative.parts or not path.is_relative_to(nested.resolve()):
+            raise ValueError('Unsafe nested bit manifest path: ' + name)
+        if not path.is_file() or sha256(path.read_bytes()).hexdigest() != expected:
+            raise ValueError('Nested bit source hash mismatch: ' + name)
+    return {'bit-sharing/MANIFEST.json': BIT_MANIFEST_PIN,
+            **{'bit-sharing/' + name: value for name, value in files.items()}}
+
+
+def main():
+    package_inventory(HERE)
+    environment = dict(os.environ, PYTHONDONTWRITEBYTECODE='1')
+    environment.pop('PYTHONPATH', None)
+    subprocess.run([sys.executable, str(HERE / 'bit-sharing/verify.py')],
+                   cwd=HERE, env=environment, check=True)
+    profile = json.loads((HERE / 'bit-sharing/profile161.json').read_text())
+    assert profile['L'] == profile['total_rank'] - profile['W'] * profile['m'] + profile['N']
+    (HERE / 'shared-bit-profile.json').write_text(json.dumps(profile, indent=2) + '\n')
+    print('PASS full nested bit replay and shared-bit profile', flush=True)
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/cyclic-deferred/bit_round7.py b/research/cyclic-deferred/bit_round7.py
new file mode 100644
index 0000000000..3c5df409a6
--- /dev/null
+++ b/research/cyclic-deferred/bit_round7.py
@@ -0,0 +1,129 @@
+#!/usr/bin/env python3
+"""One-child bit profile of Swapnil Jain's round-seven deferred word (h=23), under PR104's opposite-bank rule.
+
+Reads the two frozen round-seven data files (vendored in inputs/; pinned by SHA-256 below) from a directory given on
+the command line, e.g. a checkout of https://github.com/Swapnil-jain/integer-mult-kappa at
+741e7aa078392553815df7926ee17ac5e25a8c38 (certificates/round7/), or PR97's imported copy
+(research/deferred-signed/swapnil-round7/certificates/round7/). The frame dimensions are the ones recorded in
+the data file; Swapnil's check_frames.py re-derives each of them exactly and PR97 replays the word.
+
+Every projector residual of rank r is one child of width r (PR104, notes/stopped-product-factorization.tex):
+  each slot: its chain steps from the deferral frame sigma_u to F, and one exterior child m - (h - dim sigma_u);
+  each retained centre: one copied transform of width h-1;
+  each target y_T: its chain 0 -> deferred readout levels -> t_T^perp;  each data wire X_S: its chain of V frames;
+  each data pair: one data residual of width (h-1)^2 and one endpoint copy.
+Usage: python3 research/deferred-stopped/bit_round7.py DIR      (writes bit-profile.json next to this file)
+Prepared by Avi Eisenberg (ikeboy) with Anthropic Claude assistance. Apache-2.0.
+"""
+import gzip
+import hashlib
+import json
+import sys
+from collections import Counter, defaultdict
+from itertools import combinations
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+PINS = {'witness_23.json.gz': 'b2486aa2bb222bacea6e52162a3920780e45dd8edc5d7cb03eabea336a1c6588',
+        'deferred_23.json.gz': '8c38e947ff9e021e308000dd82bb5e2194eb265d8b75194e22ce783d3e75317c'}
+
+
+def require(ok, msg):
+    if not ok:
+        raise SystemExit('FAIL: ' + msg)
+
+
+def main(directory):
+    data = {}
+    for name, digest in PINS.items():
+        raw = (Path(directory) / name).read_bytes()
+        require(hashlib.sha256(raw).hexdigest() == digest, 'pinned hash of ' + name)
+        data[name] = json.loads(gzip.decompress(raw))
+    W, D = data['witness_23.json.gz'], data['deferred_23.json.gz']
+    h = D['h']; R = D['R']; trip = list(combinations(range(h), 3)); v = len(trip); m = h * h; N = v * v
+    tid = {T: i for i, T in enumerate(trip)}
+    args = {int(n): (tuple(a) if a else None) for n, a in W['args'].items()}
+    late = {int(n): (list(e), list(l)) for n, (e, l) in W['late'].items()}
+    ops = [tuple(o[:2]) + (tuple(o[2]),) if o[0] == 'fan' else tuple(o) for o in D['ops']]
+    hold, start = D['hold'], D['start']
+    out = {s: (c, tuple(T)) for s, c, T in D['out']}; ret = {s: c for s, c in D['ret']}
+    sigma = dict(zip(D['readout_order'], D['sigma'])); sel = set(D['readout_order'])
+    f = [0] * R
+    for s, B in sigma.items(): f[s] = len(B)
+    vstart = dict(zip(D['vleaf_slots'], D['vleaf_start']))
+    node_dims = {n: d for n, d in D['node_dims']}
+    late_dims = {(n, tuple(lt)): d for n, lt, d in D['late_dims']}
+    pivslot = {o[3]: o[1] for o in ops if o[0] == 'add'}
+
+    def start_key(s):
+        n0 = hold[s][0]
+        if args[n0] is None: return ('v', s)
+        st = start[s]; k = st[2] if len(st) > 2 else None; dec = late.get(n0)
+        if dec is None or k in dec[0]: return ('n', n0)
+        lt = dec[1]; return ('c', n0, tuple(lt[lt.index(k):]))
+
+    def chain(s):
+        ch = [('sigma', s) if s in sel else ('0',), start_key(s)]
+        for idx, n in enumerate(hold[s]):
+            if idx > 0: ch.append(('n', n))
+            if args[n] is not None and pivslot.get(n) == s and n in late:
+                lt = late[n][1]
+                for i in range(len(lt) - 1): ch.append(('c', n, tuple(lt[i:])))
+        if s in out: ch.append(('out',))
+        if s in ret: ch.append(('ret',))
+        ch.append(('F',)); return ch
+
+    def dim(key):
+        k = key[0]
+        if k == '0': return 0
+        if k == 'F': return h
+        if k in ('out', 'ret'): return h - 1
+        if k == 'n': return node_dims[key[1]]
+        if k == 'c': return late_dims[key[1], key[2]]
+        if k == 'sigma': return f[key[1]]
+        if k == 'v': return len(vstart[key[1]])
+        raise KeyError(key)
+
+    # garbage supports (transpose sweep) for the deferred levels on each target
+    cov = [set() for _ in range(R)]
+    for s, (c, T) in out.items(): cov[s].add(tid[T])
+    for s, c in ret.items(): cov[s].update(t for t, T in enumerate(trip) if c in T)
+    for o in reversed(ops):
+        if o[0] == 'add': cov[o[2]] |= cov[o[1]]
+        elif o[0] == 'fan':
+            for g in o[2]: cov[o[1]] |= cov[g]
+    levels = defaultdict(set)
+    for s in sel:
+        for t in cov[s]: levels[t].add(f[s])
+
+    z = Counter()
+    for s in range(R):
+        ds = [dim(k) for k in chain(s)]
+        require(all(b >= a for a, b in zip(ds, ds[1:])) and ds[-1] == h, 'monotone chain %d' % s)
+        for a, b in zip(ds, ds[1:]):
+            if b > a: z[b - a] += 2 * v
+        z[m - (h - f[s])] += 2 * v
+    z[h - 1] += 2 * v * h
+    for t in range(v):
+        ls = sorted(levels[t] | {0, h - 1})
+        for a, b in zip(ls, ls[1:]): z[b - a] += 2 * v
+    for n, us in D['xs_order']:
+        ds = [1] + [len(vstart[s]) for s in us] + [h]
+        ds = [d for i, d in enumerate(ds) if i == 0 or d != ds[i - 1]]
+        require(all(b > a for a, b in zip(ds, ds[1:])), 'data-wire chain')
+        for a, b in zip(ds, ds[1:]): z[b - a] += 2 * v
+    z[(h - 1) ** 2] += 2 * N
+    z[1] += N
+    Wb = 2 * N + 2 * v * R; L = 2 * v * h * (h - 1); s_ = Wb * m - N + L
+    require(sum(t * c for t, c in z.items()) == s_, 'bit rank mass')
+    require(all(0 < t < m for t in z), 'children below m')
+    prof = dict(h=h, v=v, R=R, m=m, N=N, W=Wb, L=L, total_rank=s_, deficit=N - L, maxchild=max(z),
+                deferred_slots=len(sel), source=dict(repository='https://github.com/Swapnil-jain/integer-mult-kappa',
+                commit='741e7aa078392553815df7926ee17ac5e25a8c38', sha256=PINS),
+                child_multiplicities=dict(sorted(z.items())))
+    (HERE / 'bit-profile.json').write_text(json.dumps(prof, indent=1) + '\n')
+    print('PASS bit: R=%d, deferred slots %d, maxchild %d, rank mass %d' % (R, len(sel), max(z), s_))
+
+
+if __name__ == '__main__':
+    main(sys.argv[1] if len(sys.argv) > 1 else '.')
diff --git a/research/cyclic-deferred/certificate.json b/research/cyclic-deferred/certificate.json
new file mode 100644
index 0000000000..d321bc238f
--- /dev/null
+++ b/research/cyclic-deferred/certificate.json
@@ -0,0 +1,10989 @@
+{
+  "actual_bit_saving": "129219596453/1000000000000000",
+  "assembly": {
+    "absorption_gap": "2772087609568130312722149031/18252891226077179052565074465500000000000000",
+    "margins": {
+      "Gaussian_arithmetic": "5659361765835419565641894869851069/29204625961723486484104119144800000000",
+      "bulk_exposure": "129219596453/1000000000000000",
+      "compact_phase_layer": "4716043524686574169851069/36505782452154358105130148931",
+      "coordinate_movement": "129219596453/1000000000000000",
+      "dimension": "36496350000000000000000000000/36505782452154358105130148931",
+      "original_prefix": "471640858251109571343212030148931/3650578245215435810513014893100000000",
+      "scalar_work": "7545925217704031949761039669851069/29204625961723486484104119144800000000"
+    },
+    "minimum_margin": "4716043524686574169851069/36505782452154358105130148931",
+    "parameters": {
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+          8204,
+          4096,
+          2048,
+          1024,
+          512,
+          256,
+          128,
+          64,
+          32,
+          16,
+          2,
+          1
+        ],
+        "count": 1
+      },
+      {
+        "basis": [
+          8388620,
+          4194304,
+          2097152,
+          1048576,
+          524288,
+          262144,
+          131084,
+          65536,
+          32768,
+          16384,
+          8192,
+          4096,
+          2048,
+          1024,
+          512,
+          256,
+          128,
+          64,
+          32,
+          16,
+          2,
+          1
+        ],
+        "count": 1
+      },
+      {
+        "basis": [
+          8388620,
+          4194304,
+          2097152,
+          1048588,
+          524288,
+          262144,
+          131072,
+          65536,
+          32768,
+          16384,
+          8192,
+          4096,
+          2048,
+          1024,
+          512,
+          256,
+          128,
+          64,
+          32,
+          16,
+          2,
+          1
+        ],
+        "count": 1
+      },
+      {
+        "basis": [
+          8388656,
+          4194304,
+          2097152,
+          1048576,
+          524288,
+          262144,
+          131072,
+          65536,
+          32768,
+          16384,
+          8192,
+          4096,
+          2096,
+          1024,
+          512,
+          256,
+          128,
+          64,
+          8,
+          4,
+          2,
+          1
+        ],
+        "count": 1
+      },
+      {
+        "basis": [
+          8388656,
+          4194304,
+          2097152,
+          1048576,
+          524336,
+          262144,
+          131072,
+          65536,
+          32768,
+          16384,
+          8192,
+          4096,
+          2048,
+          1024,
+          512,
+          256,
+          128,
+          64,
+          8,
+          4,
+          2,
+          1
+        ],
+        "count": 1
+      },
+      {
+        "basis": [
+          8388656,
+          4194304,
+          2097152,
+          1048624,
+          524288,
+          262144,
+          131072,
+          65536,
+          32768,
+          16384,
+          8192,
+          4096,
+          2048,
+          1024,
+          512,
+          256,
+          128,
+          64,
+          8,
+          4,
+          2,
+          1
+        ],
+        "count": 1
+      },
+      {
+        "basis": [
+          8388800,
+          4194304,
+          2097152,
+          1048576,
+          524288,
+          262144,
+          131072,
+          65536,
+          32768,
+          16384,
+          8384,
+          4096,
+          2048,
+          1024,
+          512,
+          256,
+          32,
+          16,
+          8,
+          4,
+          2,
+          1
+        ],
+        "count": 1
+      },
+      {
+        "basis": [
+          8388800,
+          4194304,
+          2097152,
+          1048576,
+          524288,
+          262144,
+          131072,
+          65536,
+          32960,
+          16384,
+          8192,
+          4096,
+          2048,
+          1024,
+          512,
+          256,
+          32,
+          16,
+          8,
+          4,
+          2,
+          1
+        ],
+        "count": 1
+      },
+      {
+        "basis": [
+          8388800,
+          4194304,
+          2097152,
+          1048576,
+          524288,
+          262144,
+          131264,
+          65536,
+          32768,
+          16384,
+          8192,
+          4096,
+          2048,
+          1024,
+          512,
+          256,
+          32,
+          16,
+          8,
+          4,
+          2,
+          1
+        ],
+        "count": 1
+      },
+      {
+        "basis": [
+          8388800,
+          4194304,
+          2097152,
+          1048576,
+          524480,
+          262144,
+          131072,
+          65536,
+          32768,
+          16384,
+          8192,
+          4096,
+          2048,
+          1024,
+          512,
+          256,
+          32,
+          16,
+          8,
+          4,
+          2,
+          1
+        ],
+        "count": 1
+      },
+      {
+        "basis": [
+          8388800,
+          4194304,
+          2097152,
+          1048768,
+          524288,
+          262144,
+          131072,
+          65536,
+          32768,
+          16384,
+          8192,
+          4096,
+          2048,
+          1024,
+          512,
+          256,
+          32,
+          16,
+          8,
+          4,
+          2,
+          1
+        ],
+        "count": 1
+      },
+      {
+        "basis": [
+          8389376,
+          4194304,
+          2097152,
+          1048576,
+          524288,
+          262912,
+          131072,
+          65536,
+          32768,
+          16384,
+          8192,
+          4096,
+          2048,
+          1024,
+          128,
+          64,
+          32,
+          16,
+          8,
+          4,
+          2,
+          1
+        ],
+        "count": 1
+      },
+      {
+        "basis": [
+          8389632,
+          4194304,
+          2097152,
+          1048576,
+          524288,
+          262144,
+          131072,
+          65536,
+          32768,
+          16384,
+          13312,
+          2048,
+          512,
+          256,
+          128,
+          64,
+          32,
+          16,
+          8,
+          4,
+          2,
+          1
+        ],
+        "count": 1
+      },
+      {
+        "basis": [
+          8389632,
+          4194304,
+          2097152,
+          1048576,
+          524288,
+          262144,
+          131072,
+          65536,
+          50176,
+          8192,
+          4096,
+          2048,
+          512,
+          256,
+          128,
+          64,
+          32,
+          16,
+          8,
+          4,
+          2,
+          1
+        ],
+        "count": 1
+      },
+      {
+        "basis": [
+          8389632,
+          4194304,
+          2097152,
+          1048576,
+          787456,
+          131072,
+          65536,
+          32768,
+          16384,
+          8192,
+          4096,
+          2048,
+          512,
+          256,
+          128,
+          64,
+          32,
+          16,
+          8,
+          4,
+          2,
+          1
+        ],
+        "count": 1
+      },
+      {
+        "basis": [
+          8400896,
+          4194304,
+          2097152,
+          1048576,
+          524288,
+          262144,
+          131072,
+          65536,
+          32768,
+          28672,
+          2048,
+          1024,
+          512,
+          256,
+          128,
+          64,
+          32,
+          16,
+          8,
+          4,
+          2,
+          1
+        ],
+        "count": 1
+      },
+      {
+        "basis": [
+          8404992,
+          4194304,
+          2097152,
+          1048576,
+          802816,
+          131072,
+          65536,
+          32768,
+          8192,
+          4096,
+          2048,
+          1024,
+          512,
+          256,
+          128,
+          64,
+          32,
+          16,
+          8,
+          4,
+          2,
+          1
+        ],
+        "count": 1
+      },
+      {
+        "basis": [
+          8585216,
+          4194304,
+          2097152,
+          1048576,
+          524288,
+          458752,
+          32768,
+          16384,
+          8192,
+          4096,
+          2048,
+          1024,
+          512,
+          256,
+          128,
+          64,
+          32,
+          16,
+          8,
+          4,
+          2,
+          1
+        ],
+        "count": 1
+      }
+    ],
+    "physical_gate_frame_changes": 19323,
+    "replay": {
+      "field": "Z/(2^61-1)",
+      "inverse_order": "true source/workspace chronology",
+      "omitted_compensation_rejected": true,
+      "physically_aliased": true,
+      "scratch_restored": true,
+      "seeds": [
+        1,
+        2
+      ],
+      "y_plus_x": true
+    },
+    "reuse_rejected_controls": [
+      "nondead-donor",
+      "unavailable-birth",
+      "pair-alias",
+      "omitted-compensation"
+    ],
+    "reused_roles": 2108,
+    "roots": 8120,
+    "shared_groups": 87,
+    "total_rank": 7383051200,
+    "unshared_W": 115857808,
+    "unshared_rank": 66732235328,
+    "v": 2024,
+    "virtual_R": 28705
+  },
+  "complex_saving": "4084267/31250000000",
+  "finite_bridge": {
+    "bit_coarse": {
+      "W": 15567734,
+      "adapter_contract": "Each complete projector and group complement retains paid ordered-affine atom adapters under one common rational basis; choose a fixed odd execution prime avoiding the enlarged finite bad-prime set. O(Vn) local work is absorbed by the stopped adapter gap.",
+      "completed_core_groups": 161,
+      "halving_degree": 33,
+      "m": 529,
+      "maxchild": 518,
+      "paid_projector_adapter_calls": 659476769,
+      "wire_bits": 24
+    },
+    "complex": {
+      "W": 12821030,
+      "core_scalar_group_upper": 60347419084480,
+      "halving_degree": 25,
+      "m": 576,
+      "maxchild": 560,
+      "s": 7383051200,
+      "scalar_group_upper": 158614225569472,
+      "scalar_terms": [
+        {
+          "R": 28705,
+          "c": 91770,
+          "description": "Expanded rational old readouts and forward/inverse/reflected words",
+          "h": 24,
+          "invocations": 2024,
+          "local_group_upper": 14907958248,
+          "physical_R": 26597,
+          "q": 8120,
+          "v": 2024
+        },
+        {
+          "R": 28705,
+          "c": 91770,
+          "description": "Expanded rational old readouts and forward/inverse/reflected words",
+          "h": 24,
+          "invocations": 2024,
+          "local_group_upper": 14907958248,
+          "physical_R": 26597,
+          "q": 8120,
+          "v": 2024
+        }
+      ],
+      "shared_basis_phase_group_upper": 98266806484992,
+      "wire_bits": 24
+    },
+    "ordinary_leaf_row_degree": 252,
+    "rows": {
+      "coefficient": 1644,
+      "contract": "W_complex^D_complex * W_coarse^D_coarse * W_old^D_old; prefix and padding; sequential reuse",
+      "degree": 4000,
+      "degree_gap": "16156/25",
+      "suffix_slope": 16000
+    },
+    "semantic": {
+      "B": 255391526926186516071662400281253203682957696,
+      "C0": 1202224103897500611222261978604362365149782619479601127059847794537047647110934762968382963712,
+      "C1": 1,
+      "E": 255391526926186516071662400281253196299906496,
+      "exact_grid": "2^(-P)*21^(-K), K=G*(D_complex+1); completed children preserve incoming odd denominator; no child rounding",
+      "fixed_odd_divisor": 21,
+      "induction_gap": 7917137334711781998221534408718849314171688576,
+      "literal_charge": 52145635379127662337906201756,
+      "strict_literal_gap": 255391526926186463926027021153590858393704740
+    }
+  },
+  "kappa": "64593102899/500000000000000",
+  "next_grid_rejections": {
+    "bit": "1e-12",
+    "complex": "1e-12",
+    "kappa": "1e-15"
+  },
+  "status": "Finite conditional cyclic/deferred witness"
+}
diff --git a/research/cyclic-deferred/certificate.py b/research/cyclic-deferred/certificate.py
new file mode 100644
index 0000000000..32449df35b
--- /dev/null
+++ b/research/cyclic-deferred/certificate.py
@@ -0,0 +1,127 @@
+#!/usr/bin/env python3
+"""Exact finite assembly for cyclic strips with deferred readouts.
+
+Built on Avi Eisenberg's PR110 (Anthropic Claude assistance), icekylinx's
+PR104 stopped-product and copied-center interfaces, and Zhihao Chen's
+assembly of the RaD interfaces. Integration for eumemic with OpenAI Codex
+assistance. Apache-2.0. General transfer hypotheses remain inherited.
+"""
+import argparse
+import json
+import sys
+from fractions import Fraction as Q
+from pathlib import Path
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+sys.path.insert(0, str(ROOT / 'scripts'))
+from partial_swap_network import moment
+from copied_centers_network import finite_bridge as copied_bridge
+from structured_bulk_assembly import assembly, js
+ATOM = Q(1, 1000)
+OLD = Q(384599, 10**10)
+BETA = Q(1, 10**6)
+COARSE = Q(129310447, 10**12)
+COMPLEX = Q(130696544, 10**12)
+KGRID = 10**15
+
+def profile(name):
+    p = json.loads((HERE / name).read_text())
+    p['child_multiplicities'] = {int(t): c for t, c in p['child_multiplicities'].items()}
+    z = p['child_multiplicities']
+    assert all(0 < t < p['m'] and c > 0 for t,c in z.items())
+    assert sum(t*c for t,c in z.items()) == p['total_rank'] == p['W']*p['m']-p['N']+p['L']
+    assert max(z) == p['maxchild']
+    return p
+
+def contracts(p, a):
+    try: moment(p['m'],p['W'],p['child_multiplicities'],a,True)
+    except ValueError: return False
+    return True
+
+def exact(bit=None, phase=None):
+    bit = profile('shared-bit-profile.json') if bit is None else bit
+    phase = profile('shared-complex-profile.json') if phase is None else phase
+    assert (bit['h'], phase['h']) == (23,24)
+    bm = moment(bit['m'],bit['W'],bit['child_multiplicities'],COARSE,True)
+    cm = moment(phase['m'],phase['W'],phase['child_multiplicities'],COMPLEX,True)
+    assert not contracts(bit,COARSE+Q(1,10**12))
+    assert not contracts(phase,COMPLEX+Q(1,10**12))
+    ab = (1-ATOM)*COARSE+ATOM*OLD
+    assert 0 < OLD < COARSE < ATOM and ab < ATOM
+    a = min(ab,(1-BETA)*COMPLEX-Q(1,10**14))
+    row = dict(h=phase['h'],v=phase['v'],c=phase['additions'],central_disjoint=24)
+    bridge = copied_bridge(bit,phase,[row,row])
+    # Logical readouts are expanded into same-frame shears with |coefficient|<=1.
+    # Exact numerators over42 are split into signed42 chunks and a remainder.
+    # The independent audit counts every chunk and checks this guard covers it.
+    # Each row has <=c+R mixer operations and <=R leaf copies. There are
+    # <=R+q readouts, each containing <=h center coefficients and <=v direct
+    # coefficients; <=4*v*(h+1) scalar groups suffice per readout. A factor
+    # eight covers forward/inverse words, both reflected words, and copies.
+    h,v,R,q,c = (phase[k] for k in ('h','v','R','roots','additions'))
+    virtual_R = phase.get('virtual_R', R)
+    local = 8*(c+2*virtual_R+(virtual_R+q)*v*(h+1)+h*h+h+1)
+    core_G = phase['N'] + 2*v*local
+    # Gaussian elimination uses at most m^2 elementary binary basis steps;
+    # both adapters and the signed phase directions fit in this atom reserve.
+    # Binary address atoms remain paid ordinary-bit adapter calls at the
+    # inherited exponent, not constant-time tape instructions. Including
+    # their number in G also conservatively enlarges the semantic guard.
+    sharing_G = 64 * phase['m']**2 * (2 * phase['shared_groups'] * R)
+    G = core_G + sharing_G
+    m,W,s = (phase[k] for k in ('m','W','total_rank'))
+    E=64*(W+m+G+1)**3
+    charge=2*G*W*W+8*s+4*W+4+32*m
+    B=s+E; C0=32*m*B*B
+    assert charge<E and 2*B*(m-phase['maxchild']) >= s+E and 2*B+18<C0
+    bridge['complex']['scalar_group_upper']=G
+    bridge['complex']['core_scalar_group_upper']=core_G
+    bridge['complex']['shared_basis_phase_group_upper']=sharing_G
+    bridge['complex']['scalar_terms']=[dict(h=h,v=v,c=c,R=virtual_R,physical_R=R,q=q,invocations=v,
+        local_group_upper=local,description='Expanded rational old readouts and forward/inverse/reflected words')]*2
+    bridge['semantic'].update(E=E,literal_charge=charge,strict_literal_gap=E-charge,
+        B=B,C0=C0,C1=1,induction_gap=2*B*(m-phase['maxchild'])-s-E,
+        fixed_odd_divisor=21,exact_grid='2^(-P)*21^(-K), K=G*(D_complex+1); completed children preserve incoming odd denominator; no child rounding')
+    coarse_bridge=bridge.pop('bit')
+    coarse_bridge['completed_core_groups'] = bit['groups']
+    coarse_bridge['paid_projector_adapter_calls'] = bit['paid_projector_calls']
+    coarse_bridge['adapter_contract'] = 'Each complete projector and group complement retains paid ordered-affine atom adapters under one common rational basis; choose a fixed odd execution prime avoiding the enlarged finite bad-prime set. O(Vn) local work is absorbed by the stopped adapter gap.'
+    previous=json.loads((ROOT/'certificates/copied-centers-network.json').read_text())
+    oldbit=previous['finite_bridge']['bit']
+    old_degree=oldbit['halving_degree']*oldbit['wire_bits']
+    assert old_degree==252
+    cp=bridge['complex']
+    coeff=coarse_bridge['halving_degree']*coarse_bridge['wire_bits']+old_degree+cp['halving_degree']*cp['wire_bits']
+    degree=1000*((51*coeff)//25000+1)
+    bridge.update(bit_coarse=coarse_bridge,ordinary_leaf_row_degree=old_degree)
+    bridge['rows']=dict(coefficient=coeff,degree=degree,suffix_slope=4*degree,
+        degree_gap=Q(degree)-Q(51*coeff,25),contract='W_complex^D_complex * W_coarse^D_coarse * W_old^D_old; prefix and padding; sequential reuse')
+    def accepts(k):
+        try: assembly(a,COMPLEX,bridge,Q(k,KGRID),beta=BETA)
+        except AssertionError: return False
+        return True
+    lo,hi=0,int(COMPLEX*KGRID)+1
+    assert accepts(lo) and not accepts(hi)
+    while hi-lo>1:
+        mid=(lo+hi)//2
+        if accepts(mid): lo=mid
+        else: hi=mid
+    k=Q(lo,KGRID)
+    assembled=assembly(a,COMPLEX,bridge,k,beta=BETA)
+    assert len(assembled['strict_constraints'])==47 and len(assembled['margins'])==7
+    assert not accepts(lo+1)
+    return dict(status='Finite conditional cyclic/deferred witness',kappa=k,
+        complex_saving=COMPLEX,coarse_bit_saving=COARSE,actual_bit_saving=ab,
+        bit_parameter=a,complex_profile=phase,bit_profile=bit,
+        bit_moment_gap=bm['strict_gap'],complex_moment_gap=cm['strict_gap'],
+        finite_bridge=bridge,assembly=assembled,
+        next_grid_rejections=dict(bit='1e-12',complex='1e-12',kappa='1e-15'))
+
+def main():
+    parser=argparse.ArgumentParser();parser.add_argument('--write',action='store_true');args=parser.parse_args()
+    assert not sys.flags.optimize
+    result=js(exact()); path=HERE/'certificate.json'
+    if args.write:path.write_text(json.dumps(result,indent=2,sort_keys=True)+'\n')
+    else:assert result==json.loads(path.read_text()),'Frozen exact certificate mismatch'
+    print('PASS conditional kappa = %s = %.12e; 47 constraints, 7 margins; expanded scalar charge' % (result['kappa'],float(Q(result['kappa']))))
+if __name__=='__main__':main()
diff --git a/research/cyclic-deferred/complex-profile.json b/research/cyclic-deferred/complex-profile.json
new file mode 100644
index 0000000000..9c787ad7d1
--- /dev/null
+++ b/research/cyclic-deferred/complex-profile.json
@@ -0,0 +1,10524 @@
+{
+ "h": 24,
+ "v": 2024,
+ "additions": 91770,
+ "roots": 8120,
+ "links": 71185,
+ "R": 26597,
+ "virtual_R": 28705,
+ "reused_roles": 2108,
+ "m": 576,
+ "N": 4096576,
+ "W": 115857808,
+ "L": 2234496,
+ "total_rank": 66732235328,
+ "deficit": 1862080,
+ "maxchild": 574,
+ "phase_one_ops": 18844,
+ "phase_one_roles": 13555,
+ "physical_auxiliary_source_frames": [
+  {
+   "basis": [],
+   "count": 23999
+  },
+  {
+   "basis": [
+    9,
+    5
+   ],
+   "count": 134
+  },
+  {
+   "basis": [
+    36,
+    20
+   ],
+   "count": 2
+  },
+  {
+   "basis": [
+    144,
+    80
+   ],
+   "count": 88
+  },
+  {
+   "basis": [
+    2304,
+    1280
+   ],
+   "count": 23
+  },
+  {
+   "basis": [
+    36864,
+    20480
+   ],
+   "count": 16
+  },
+  {
+   "basis": [
+    589824,
+    327680
+   ],
+   "count": 98
+  },
+  {
+   "basis": [
+    9437184,
+    5242880
+   ],
+   "count": 502
+  },
+  {
+   "basis": [
+    8,
+    4,
+    2,
+    1
+   ],
+   "count": 4
+  },
+  {
+   "basis": [
+    33,
+    17,
+    9,
+    5
+   ],
+   "count": 72
+  },
+  {
+   "basis": [
+    129,
+    65,
+    33,
+    17
+   ],
+   "count": 6
+  },
+  {
+   "basis": [
+    132,
+    68,
+    36,
+    20
+   ],
+   "count": 8
+  },
+  {
+   "basis": [
+    513,
+    257,
+    9,
+    5
+   ],
+   "count": 28
+  },
+  {
+   "basis": [
+    528,
+    272,
+    144,
+    80
+   ],
+   "count": 42
+  },
+  {
+   "basis": [
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+  {
+   "basis": [
+    8388620,
+    4194304,
+    2097152,
+    1048588,
+    524288,
+    262144,
+    131072,
+    65536,
+    32768,
+    16384,
+    8192,
+    4096,
+    2048,
+    1024,
+    512,
+    256,
+    128,
+    64,
+    32,
+    16,
+    2,
+    1
+   ],
+   "count": 1
+  },
+  {
+   "basis": [
+    8388656,
+    4194304,
+    2097152,
+    1048576,
+    524288,
+    262144,
+    131072,
+    65536,
+    32768,
+    16384,
+    8192,
+    4096,
+    2096,
+    1024,
+    512,
+    256,
+    128,
+    64,
+    8,
+    4,
+    2,
+    1
+   ],
+   "count": 1
+  },
+  {
+   "basis": [
+    8388656,
+    4194304,
+    2097152,
+    1048576,
+    524336,
+    262144,
+    131072,
+    65536,
+    32768,
+    16384,
+    8192,
+    4096,
+    2048,
+    1024,
+    512,
+    256,
+    128,
+    64,
+    8,
+    4,
+    2,
+    1
+   ],
+   "count": 1
+  },
+  {
+   "basis": [
+    8388656,
+    4194304,
+    2097152,
+    1048624,
+    524288,
+    262144,
+    131072,
+    65536,
+    32768,
+    16384,
+    8192,
+    4096,
+    2048,
+    1024,
+    512,
+    256,
+    128,
+    64,
+    8,
+    4,
+    2,
+    1
+   ],
+   "count": 1
+  },
+  {
+   "basis": [
+    8388800,
+    4194304,
+    2097152,
+    1048576,
+    524288,
+    262144,
+    131072,
+    65536,
+    32768,
+    16384,
+    8384,
+    4096,
+    2048,
+    1024,
+    512,
+    256,
+    32,
+    16,
+    8,
+    4,
+    2,
+    1
+   ],
+   "count": 1
+  },
+  {
+   "basis": [
+    8388800,
+    4194304,
+    2097152,
+    1048576,
+    524288,
+    262144,
+    131072,
+    65536,
+    32960,
+    16384,
+    8192,
+    4096,
+    2048,
+    1024,
+    512,
+    256,
+    32,
+    16,
+    8,
+    4,
+    2,
+    1
+   ],
+   "count": 1
+  },
+  {
+   "basis": [
+    8388800,
+    4194304,
+    2097152,
+    1048576,
+    524288,
+    262144,
+    131264,
+    65536,
+    32768,
+    16384,
+    8192,
+    4096,
+    2048,
+    1024,
+    512,
+    256,
+    32,
+    16,
+    8,
+    4,
+    2,
+    1
+   ],
+   "count": 1
+  },
+  {
+   "basis": [
+    8388800,
+    4194304,
+    2097152,
+    1048576,
+    524480,
+    262144,
+    131072,
+    65536,
+    32768,
+    16384,
+    8192,
+    4096,
+    2048,
+    1024,
+    512,
+    256,
+    32,
+    16,
+    8,
+    4,
+    2,
+    1
+   ],
+   "count": 1
+  },
+  {
+   "basis": [
+    8388800,
+    4194304,
+    2097152,
+    1048768,
+    524288,
+    262144,
+    131072,
+    65536,
+    32768,
+    16384,
+    8192,
+    4096,
+    2048,
+    1024,
+    512,
+    256,
+    32,
+    16,
+    8,
+    4,
+    2,
+    1
+   ],
+   "count": 1
+  },
+  {
+   "basis": [
+    8389376,
+    4194304,
+    2097152,
+    1048576,
+    524288,
+    262912,
+    131072,
+    65536,
+    32768,
+    16384,
+    8192,
+    4096,
+    2048,
+    1024,
+    128,
+    64,
+    32,
+    16,
+    8,
+    4,
+    2,
+    1
+   ],
+   "count": 1
+  },
+  {
+   "basis": [
+    8389632,
+    4194304,
+    2097152,
+    1048576,
+    524288,
+    262144,
+    131072,
+    65536,
+    32768,
+    16384,
+    13312,
+    2048,
+    512,
+    256,
+    128,
+    64,
+    32,
+    16,
+    8,
+    4,
+    2,
+    1
+   ],
+   "count": 1
+  },
+  {
+   "basis": [
+    8389632,
+    4194304,
+    2097152,
+    1048576,
+    524288,
+    262144,
+    131072,
+    65536,
+    50176,
+    8192,
+    4096,
+    2048,
+    512,
+    256,
+    128,
+    64,
+    32,
+    16,
+    8,
+    4,
+    2,
+    1
+   ],
+   "count": 1
+  },
+  {
+   "basis": [
+    8389632,
+    4194304,
+    2097152,
+    1048576,
+    787456,
+    131072,
+    65536,
+    32768,
+    16384,
+    8192,
+    4096,
+    2048,
+    512,
+    256,
+    128,
+    64,
+    32,
+    16,
+    8,
+    4,
+    2,
+    1
+   ],
+   "count": 1
+  },
+  {
+   "basis": [
+    8400896,
+    4194304,
+    2097152,
+    1048576,
+    524288,
+    262144,
+    131072,
+    65536,
+    32768,
+    28672,
+    2048,
+    1024,
+    512,
+    256,
+    128,
+    64,
+    32,
+    16,
+    8,
+    4,
+    2,
+    1
+   ],
+   "count": 1
+  },
+  {
+   "basis": [
+    8404992,
+    4194304,
+    2097152,
+    1048576,
+    802816,
+    131072,
+    65536,
+    32768,
+    8192,
+    4096,
+    2048,
+    1024,
+    512,
+    256,
+    128,
+    64,
+    32,
+    16,
+    8,
+    4,
+    2,
+    1
+   ],
+   "count": 1
+  },
+  {
+   "basis": [
+    8585216,
+    4194304,
+    2097152,
+    1048576,
+    524288,
+    458752,
+    32768,
+    16384,
+    8192,
+    4096,
+    2048,
+    1024,
+    512,
+    256,
+    128,
+    64,
+    32,
+    16,
+    8,
+    4,
+    2,
+    1
+   ],
+   "count": 1
+  }
+ ],
+ "deferred_roles": 4706,
+ "deferred_dims": {
+  "2": 863,
+  "4": 357,
+  "6": 573,
+  "8": 144,
+  "10": 25,
+  "12": 323,
+  "14": 241,
+  "16": 337,
+  "17": 1,
+  "18": 900,
+  "19": 603,
+  "20": 208,
+  "21": 11,
+  "22": 120
+ },
+ "lifted_additions": 38114,
+ "physical_gate_frame_changes": 19323,
+ "replay": {
+  "seeds": [
+   1,
+   2
+  ],
+  "scratch_restored": true,
+  "y_plus_x": true,
+  "field": "Z/(2^61-1)",
+  "physically_aliased": true,
+  "inverse_order": "true source/workspace chronology",
+  "omitted_compensation_rejected": true
+ },
+ "reuse_rejected_controls": [
+  "nondead-donor",
+  "unavailable-birth",
+  "pair-alias",
+  "omitted-compensation"
+ ],
+ "child_multiplicities": {
+  "1": 408795376,
+  "2": 291621968,
+  "3": 52656384,
+  "4": 51041232,
+  "5": 21498928,
+  "6": 17507600,
+  "7": 9229440,
+  "8": 12633808,
+  "9": 7869312,
+  "10": 21118416,
+  "11": 4938560,
+  "12": 20826960,
+  "13": 17786912,
+  "14": 1922800,
+  "15": 2493568,
+  "16": 684112,
+  "17": 489808,
+  "18": 546480,
+  "19": 477664,
+  "20": 550528,
+  "21": 1230592,
+  "22": 64768,
+  "23": 8363168,
+  "529": 8193152,
+  "552": 97147952,
+  "554": 3493424,
+  "556": 1437040,
+  "558": 2319504,
+  "560": 453376,
+  "562": 48576,
+  "570": 821744,
+  "571": 817696,
+  "572": 825792,
+  "573": 44528,
+  "574": 255024
+ }
+}
diff --git a/research/cyclic-deferred/complex_deferred.py b/research/cyclic-deferred/complex_deferred.py
new file mode 100644
index 0000000000..4a65788ecb
--- /dev/null
+++ b/research/cyclic-deferred/complex_deferred.py
@@ -0,0 +1,736 @@
+#!/usr/bin/env python3
+"""Optimized physical gate frames on the immutable PR117 complex DAG.
+
+After saturated deferral, each mixer gate frame is optimized within its two
+role chains. Maximal nondegenerate extensions and minimal nondegenerate hulls
+are accepted only when they improve the local moment objective; every actual
+source, mixer, copied-center and root incidence is checked exactly.
+Physical-frame optimization prepared with OpenAI Codex assistance.
+Compensated birth-cut reuse follows jamesyc PR124; this composition regenerates
+all pairings, replays arbitrary aliased scratch, and retains every scalar charge.
+
+PR117 credits its searched DAG to eumemic with Anthropic Claude assistance;
+this experiment imports and replays that witness unchanged. PR110/PR114
+saturation and exact finite checks are applied to the resulting graph.
+Logical readout macros are realized as signed numerator/42 chunks of magnitude
+at most one. The literal audit expands and charges those same-frame shears,
+checks exact reconstruction, and reverses the chunks under reflection.
+Inherited Avi Eisenberg, Rohan Arun, icekylinx and Swapnil Jain credits retained.
+This integration prepared with OpenAI Codex assistance. Apache-2.0.
+"""
+import array
+import json
+import random
+import struct
+import sys
+import tempfile
+from collections import Counter, defaultdict
+from itertools import combinations
+from pathlib import Path
+from reuse import select_reuse, check_pairs, check_compensation, mutation_controls
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+sys.path.insert(0, str(ROOT / 'scripts'))
+sys.path.insert(0, str(HERE))
+import importlib.util
+producer_path = HERE / 'producer.py'
+spec = importlib.util.spec_from_file_location('replayed_producer_wrapper', producer_path)
+producer = importlib.util.module_from_spec(spec); spec.loader.exec_module(producer)
+build = producer.build
+
+P = (1 << 61) - 1
+
+
+def require(ok, msg):
+    if not ok:
+        raise SystemExit('FAIL: ' + msg)
+
+
+def load(prefix):
+    def rd(fh, typ, cnt):
+        a = array.array(typ); a.fromfile(fh, cnt); return a
+    with open(str(prefix) + '.bin', 'rb') as f:
+        h, v, n, q = struct.unpack('<4I', f.read(16))
+        flat = rd(f, 'I', 2 * n); core = rd(f, 'Q', n); cover = rd(f, 'Q', n)
+        roots = rd(f, 'I', q); kind = rd(f, 'I', q); active = rd(f, 'B', n)
+    with open(str(prefix) + '.labels', 'rb') as f:
+        ranks = rd(f, 'I', n); types = rd(f, 'B', n)
+    args = [(flat[2 * i], flat[2 * i + 1]) for i in range(n)]
+    return h, v, n, q, args, core, cover, roots, kind, active, ranks, types
+
+
+# ---------------------------------------------------------------- F2 linear algebra on bit vectors
+def dot(a, b): return bin(a & b).count('1') & 1
+
+
+def reduce(basis):
+    rows = []; piv = []
+    for vec in basis:
+        for p, r in zip(piv, rows):
+            if vec >> p & 1: vec ^= r
+        if vec:
+            p = vec.bit_length() - 1
+            for i in range(len(rows)):
+                if rows[i] >> p & 1: rows[i] ^= vec
+            rows.append(vec); piv.append(p)
+    return rows
+
+
+def kernel(funcs, h):
+    K = reduce(funcs); piv = [r.bit_length() - 1 for r in K]
+    out = []
+    for c in (i for i in range(h) if i not in piv):
+        vec = 1 << c
+        for r, p in zip(K, piv):
+            if r >> c & 1: vec |= 1 << p
+        out.append(vec)
+    return out
+
+
+def restrict(basis, funcs):
+    B = reduce(basis)
+    for f in funcs:
+        if not B: break
+        vals = [dot(b, f) for b in B]
+        if not any(vals): continue
+        k = vals.index(1); pv = B[k]
+        B = [b ^ pv if val else b for b, val in zip(B, vals)]; B.pop(k)
+    return reduce(B)
+
+
+def contains(A, B):
+    RB = reduce(B)
+    return len(reduce(RB + list(A))) == len(RB)
+
+
+def cap(A, B):
+    A = reduce(A); B = reduce(B)
+    if not A or not B: return []
+    ker = []; red = []
+    for i, vec in enumerate(A + B):
+        tag = 1 << i
+        for rv, rt in red:
+            if vec >> (rv.bit_length() - 1) & 1: vec ^= rv; tag ^= rt
+        if vec: red.append((vec, tag))
+        else:
+            comb_ = 0
+            for k in range(len(A)):
+                if tag >> k & 1: comb_ ^= A[k]
+            ker.append(comb_)
+    return reduce(ker)
+
+
+def nondeg(B):
+    B = reduce(B); k = len(B)
+    if not k: return True
+    M = [sum(dot(B[i], B[j]) << j for j in range(k)) for i in range(k)]
+    return len(reduce(M)) == k
+
+
+def hopcroft_karp(left, adj):
+    """maximum bipartite matching; adj[x] = sorted list of right vertices."""
+    INF = 1 << 60
+    mate_l = {x: None for x in left}; mate_r = {}
+    while True:
+        dist = {}; queue = []
+        for x in left:
+            if mate_l[x] is None: dist[x] = 0; queue.append(x)
+            else: dist[x] = INF
+        found = INF; head = 0
+        while head < len(queue):
+            x = queue[head]; head += 1
+            if dist[x] >= found: continue
+            for r in adj[x]:
+                y = mate_r.get(r)
+                if y is None: found = min(found, dist[x] + 1)
+                elif dist[y] == INF: dist[y] = dist[x] + 1; queue.append(y)
+        if found == INF: break
+        gained = 0
+        for x0 in left:
+            if mate_l[x0] is not None: continue
+            stack = [(x0, iter(adj[x0]))]; path = []
+            while stack:
+                x, it = stack[-1]
+                advanced = False
+                for r in it:
+                    y = mate_r.get(r)
+                    if (y is None and dist[x] + 1 == found) or (y is not None and dist[y] == dist[x] + 1):
+                        path.append((x, r))
+                        if y is None:
+                            for xx, rr in path: mate_l[xx] = rr; mate_r[rr] = xx
+                            gained += 1; stack = []; break
+                        stack.append((y, iter(adj[y]))); advanced = True; break
+                else:
+                    dist[x] = INF; stack.pop()
+                    if path: path.pop()
+                    continue
+                if not advanced: break
+        if not gained: break
+    return mate_l
+
+
+from functools import lru_cache
+from fractions import Fraction
+
+def sat_basis(vectors):
+    piv = {}
+    for x in vectors:
+        for p in sorted(piv, reverse=True):
+            if x >> p & 1:
+                x ^= piv[p]
+        if x:
+            p = x.bit_length() - 1
+            for q in piv:
+                if piv[q] >> p & 1:
+                    piv[q] ^= x
+            piv[p] = x
+    return tuple((piv[p] for p in sorted(piv, reverse=True)))
+
+def sat_dot(a, b):
+    return (a & b).bit_count() & 1
+
+@lru_cache(None)
+def sat_contained(A, B):
+    for x in A:
+        for r in B:
+            if x >> r.bit_length() - 1 & 1:
+                x ^= r
+        if x:
+            return False
+    return True
+
+@lru_cache(None)
+def sat_cap(A, B):
+    if not A or not B:
+        return ()
+    red = {}
+    out = []
+    for i, x in enumerate(A + B):
+        tag = 1 << i
+        for p in sorted(red, reverse=True):
+            if x >> p & 1:
+                r, t = red[p]
+                x ^= r
+                tag ^= t
+        if x:
+            red[x.bit_length() - 1] = (x, tag)
+        else:
+            y = 0
+            for j, r in enumerate(A):
+                if tag >> j & 1:
+                    y ^= r
+            if y:
+                out.append(y)
+    return sat_basis(out)
+
+@lru_cache(None)
+def sat_nondeg(B):
+    return len(sat_basis((sum((sat_dot(a, b) << j for j, b in enumerate(B))) for a in B))) == len(B)
+
+@lru_cache(None)
+def sat_nonsingular_part(B):
+    rows = list(B)
+    out = []
+    while rows:
+        i = next((i for i, x in enumerate(rows) if sat_dot(x, x)), None)
+        if i is not None:
+            a = rows.pop(i)
+            out.append(a)
+            rows = [x ^ a if sat_dot(x, a) else x for x in rows]
+            continue
+        pair = next(((i, j) for i in range(len(rows)) for j in range(i + 1, len(rows)) if sat_dot(rows[i], rows[j])), None)
+        if pair is None:
+            break
+        i, j = pair
+        a, b = (rows[i], rows[j])
+        out.extend((a, b))
+        rows = [x ^ (a if sat_dot(x, b) else 0) ^ (b if sat_dot(x, a) else 0) for k, x in enumerate(rows) if k not in (i, j)]
+    return sat_basis(out)
+
+def saturated_placement(cand, reach):
+    # Each extracted block is orthogonal to previously selected blocks.
+    # A diagonal-one pivot contributes a nondegenerate line; when all
+    # remaining diagonals vanish, a dot-one pair contributes a hyperbolic
+    # plane. The residual radical is discarded, not counted as a frame.
+    cand = {s: sat_basis(B) for s, B in cand.items()}
+    order = sorted(cand, key=lambda s: (Fraction(-(1 << len(cand[s])), max(1, len(reach[s]))**2), -len(cand[s]), s))
+    byT = defaultdict(list); placed = {}
+    for s in order:
+        X = cand[s]; changed = True
+        while changed and X:
+            changed = False
+            for t in sorted(reach[s]):
+                for w in byT[t]:
+                    B = placed[w]
+                    if (len(B) >= len(X) and not sat_contained(X, B)) or (len(B) < len(X) and not sat_contained(B, X)):
+                        X = sat_cap(X, B); changed = True
+                    if not X: break
+                if not X: break
+            if X and not sat_nondeg(X):
+                before = X; X = sat_nonsingular_part(X)
+                assert sat_contained(X, before) and sat_nondeg(X)
+                assert len(X) < len(before)
+                changed = True  # Extraction can break containment of a smaller target frame.
+        if X:
+            assert sat_nondeg(X) and sat_contained(X, cand[s])
+            placed[s] = X
+            for t in reach[s]: byT[t].append(s)
+    for t, ss in byT.items():
+        ss.sort(key=lambda s: (len(placed[s]), s))
+        assert all(sat_contained(placed[a], placed[b]) for a, b in zip(ss, ss[1:]))
+    return placed
+
+def frame_restrict(A,B):return sat_basis(restrict(A,B))
+
+@lru_cache(None)
+def frame_protected_part(X, L):
+    Q = sat_nonsingular_part(X)
+    rad = frame_restrict(X, Q)
+    if not L:
+        return Q
+    red = {}
+    for q, r in [(q, 0) for q in Q] + [(0, r) for r in rad]:
+        x = q ^ r
+        for p in sorted(red, reverse=True):
+            if x >> p & 1:
+                y, a, b = red[p]
+                x ^= y
+                q ^= a
+                r ^= b
+        assert x
+        red[x.bit_length() - 1] = (x, q, r)
+    lin = {}
+    for x in L:
+        q = r = 0
+        for p in sorted(red, reverse=True):
+            if x >> p & 1:
+                y, a, b = red[p]
+                x ^= y
+                q ^= a
+                r ^= b
+        if x:
+            return ()
+        for p in sorted(lin, reverse=True):
+            if q >> p & 1:
+                a, b = lin[p]
+                q ^= a
+                r ^= b
+        if not q:
+            if r:
+                return ()
+        else:
+            lin[q.bit_length() - 1] = (q, r)
+    out = []
+    for x in Q:
+        q = x
+        r = 0
+        for p in sorted(lin, reverse=True):
+            if q >> p & 1:
+                a, b = lin[p]
+                q ^= a
+                r ^= b
+        out.append(x ^ r)
+    out = sat_basis(out)
+    assert sat_nondeg(out) and sat_contained(out, X) and sat_contained(L, out)
+    return out
+
+@lru_cache(None)
+def frame_smallest_hull(L, ambient):
+    L = sat_basis(L)
+    while not sat_nondeg(L):
+        radical = frame_restrict(L, L)
+        assert radical
+        r = radical[0]
+        partner = next((x for x in ambient if sat_dot(r, x)))
+        L = sat_basis(L + (partner,))
+    assert sat_contained(L, ambient)
+    return L
+
+def main():
+    with tempfile.TemporaryDirectory(prefix='deferred-stopped-') as work:
+        prefix = Path(work) / 'complex'
+        build(24, prefix, central_disjoint=24)
+        h, v, n, q, args, core, cover, roots, kind, active, ranks, types = load(prefix)
+    trip = list(combinations(range(h), 3)); require(len(trip) == v, 'triple count')
+    m = h * h; N = v * v; FULL = (1 << h) - 1
+    tmask = [sum(1 << p for p in T) for T in trip]
+    tid = {T: i for i, T in enumerate(trip)}
+
+    # ------------------------------------------------------------ uses and matching (inherited adjacency)
+    degree = [0] * n; uses = defaultdict(list); c_add = 0
+    for x in range(1, n):
+        if active[x] and args[x][0]:
+            c_add += 1
+            for pos, y in enumerate(args[x]): degree[y] += 1; uses[y].append(('gate', x, pos))
+    for j, x in enumerate(roots): degree[x] += 1; uses[x].append(('root', j))
+    def unode(u): return roots[u[1]] if u[0] == 'root' else u[1]
+    def before(x, u):
+        y = unode(u)
+        return (ranks[x], x) < (ranks[y], (n + u[1]) if u[0] == 'root' else y)
+    def incl(x, y):
+        tx, ty = types[x], types[y]
+        if tx == 1 and ty == 1: return not (core[y] & ~core[x]) and not (cover[x] & ~cover[y])
+        if tx in (1, 2) and ty == 2: return not (cover[x] & ~cover[y])
+        if tx == 1 and ty == 3: return bool(core[x] & core[y])
+        if tx == 2 and ty == 3: return not (cover[x] & ~core[y])
+        if tx == 3 and ty == 2: return cover[y] == FULL
+        if tx == 3 and ty == 3: return core[x] == core[y]
+        return False
+    left = []; adj = {}
+    for x in range(1, n):
+        if not (active[x] and args[x][0]): continue
+        rs = sorted({(y, k) for y in args[x] for k, u in enumerate(uses[y]) if before(x, u) and incl(x, unode(u))})
+        if rs: left.append(x); adj[x] = rs
+    mate = hopcroft_karp(left, adj)
+    links = {x: r for x, r in mate.items() if r is not None}
+    R = c_add + q - len(links)
+    require((c_add,q,len(links),R)==(91770,8120,71185,28705),'selected PR117 graph')
+    print('PR117 graph matched', c_add, q, len(links), R, flush=True)
+    linked_use = {(y, k): x for x, (y, k) in links.items()}
+
+    # ------------------------------------------------------------ root targets, read coefficients, root functionals
+    target = [None] * q
+    for j in range(v): target[j] = j
+    idx = v
+    # PR117 witness stores pair-star roots in natural excluded-point order.
+    for a, b in combinations(range(h), 2):
+        others = [i for i in range(h) if i not in (a, b)]
+        for i in others: target[idx] = tid[tuple(sorted((a, b, i)))]; idx += 1
+    centre_of = {}
+    for j in range(q):
+        if kind[j]:
+            miss = FULL & ~cover[roots[j]]; require(bin(miss).count('1') == 1, 'centre cover')
+            centre_of[j] = miss.bit_length() - 1
+    require(idx + len(centre_of) == q, 'root order')
+    rootfun = [(1 << centre_of[j]) if kind[j] else tmask[target[j]] for j in range(q)]
+
+    # ------------------------------------------------------------ lifted binary frames, monotone
+    succ = defaultdict(set); droot = defaultdict(set)
+    for x in range(1, n):
+        if not active[x]: continue
+        for k, u in enumerate(uses[x]):
+            if (x, k) in linked_use: continue
+            (succ[x].add(u[1]) if u[0] == 'gate' else droot[x].add(u[1]))
+    for x, (y, k) in links.items():
+        u = uses[y][k]; (succ[x].add(u[1]) if u[0] == 'gate' else droot[x].add(u[1]))
+    order_desc = sorted((x for x in range(1, n) if active[x]), key=lambda x: (ranks[x], x), reverse=True)
+    K = {}
+    for x in order_desc:
+        rows = [rootfun[j] for j in droot[x]]
+        for t in succ[x]: rows += K[t]
+        K[x] = reduce(rows)
+    def envelope(x):
+        if not args[x][0]: return [tmask[x - 1]]
+        if types[x] == 2: return [1 << i for i in range(h) if cover[x] >> i & 1]
+        require(types[x] == 1, 'only ordinary and common-pair labels occur')
+        ks = [i for i in range(h) if (cover[x] & ~core[x]) >> i & 1]
+        require(len(ks) == ranks[x], 'common-pair support')
+        return [core[x] | (1 << k) for k in ks]
+    U = {}
+    for x in range(1, n):
+        if not active[x]: continue
+        if not args[x][0]: U[x] = envelope(x); continue
+        B = kernel(K[x], h)
+        require(contains(envelope(x), B), 'label outside lifted frame')
+        U[x] = B if nondeg(B) else envelope(x)
+    changed = True
+    while changed:
+        changed = False
+        for x in order_desc:
+            if args[x][0] and any(not contains(U[x], U[t]) for t in succ[x]):
+                env = envelope(x)
+                if len(reduce(U[x])) != len(reduce(env)) or not contains(U[x], env): U[x] = env; changed = True
+    dimU = {x: len(reduce(B)) for x, B in U.items()}
+    print('PR117 frames lifted',sum(dimU[x]>ranks[x] for x in U),flush=True)
+
+    # ------------------------------------------------------------ explicit role compile
+    order = sorted((x for x in range(1, n) if active[x]), key=lambda x: (ranks[x], x))
+    holds = []; first_node = []; ops = []; edge_key = {}; role_root = {}
+    def new_role(node): holds.append([node]); first_node.append(node); return len(holds) - 1
+    def serve(s, y, k):
+        u = uses[y][k]
+        if u[0] == 'gate': edge_key[(y, k)] = s
+        else: role_root[s] = u[1]
+    for x in order:
+        free = [k for k in range(len(uses[x])) if (x, k) not in linked_use]
+        if not args[x][0]:
+            for k in free:
+                s = new_role(x); ops.append(('src', s, x)); serve(s, x, k)
+            continue
+        a, b = args[x]
+        ka = next(k for k, u in enumerate(uses[a]) if u == ('gate', x, 0))
+        kb = next(k for k, u in enumerate(uses[b]) if u == ('gate', x, 1))
+        sa, sb = edge_key.pop((a, ka)), edge_key.pop((b, kb))
+        if x in links and links[x][0] == a: sa, sb = sb, sa
+        piv, oth = sa, sb
+        ops.append(('add', piv, oth, x)); holds[piv].append(x); holds[oth].append(x)
+        if x in links: serve(oth, *links[x])
+        require(free, 'every use linked at node %d' % x)
+        serve(piv, x, free[0])
+        for k in free[1:]:
+            f = new_role(x); ops.append(('copy', piv, f, x)); serve(f, x, k)
+    require(not edge_key and len(holds) == R, 'compile')
+    Rr = R
+
+    # ------------------------------------------------------------ B. the word computes every root value
+    rng = random.Random(20261008)
+    xs = [rng.randrange(P) for _ in range(v)]
+    val = [0] * n
+    for i in range(v): val[i + 1] = xs[i]
+    for x in order:
+        if args[x][0]: val[x] = (val[args[x][0]] + val[args[x][1]]) % P
+    leaf_of = {s: first_node[s] for s in range(Rr) if not args[first_node[s]][0]}
+    a_ = [0] * Rr
+    for s, leaf in leaf_of.items(): a_[s] = xs[leaf - 1]
+    for o in ops:
+        if o[0] == 'add': a_[o[1]] = (a_[o[1]] + a_[o[2]]) % P
+        elif o[0] == 'copy': a_[o[2]] = (a_[o[2]] + a_[o[1]]) % P
+    require(all(a_[s] == val[roots[j]] for s, j in role_root.items()), 'root values')
+
+    # ------------------------------------------------------------ phase one and garbage reach
+    def touch(o): return [o[1], o[2]] if o[0] in ('add', 'copy') else []
+    prev = {}; pred = defaultdict(list); last = {}
+    for i, o in enumerate(ops):
+        for s in touch(o):
+            if s in prev: pred[i].append(prev[s])
+            prev[s] = i; last[s] = i
+    centre_roles = [s for s, j in role_root.items() if kind[j]]
+    Anc = set(); st = [last[s] for s in centre_roles]
+    while st:
+        i = st.pop()
+        if i not in Anc: Anc.add(i); st.extend(pred[i])
+    touched = set(centre_roles)
+    for i in Anc: touched.update(touch(ops[i]))
+    require(all(last[s] in Anc for s in centre_roles), 'centres complete in phase one')
+    reach = [set() for _ in range(Rr)]; reach_all = [False] * Rr
+    for s, j in role_root.items():
+        if kind[j]: reach_all[s] = True
+        else: reach[s].add(target[j])
+    for o in reversed(ops):
+        if o[0] == 'add': reach[o[2]] |= reach[o[1]]; reach_all[o[2]] = reach_all[o[2]] or reach_all[o[1]]
+        elif o[0] == 'copy': reach[o[1]] |= reach[o[2]]; reach_all[o[1]] = reach_all[o[1]] or reach_all[o[2]]
+
+    # ------------------------------------------------------------ deferral frames, insertion nesting
+    def F0(s): return U[first_node[s]]
+    cand = {}
+    for s in range(Rr):
+        if s in touched or reach_all[s]: continue
+        S_ = restrict(F0(s), [tmask[t] for t in reach[s]])
+        if S_ and nondeg(S_): cand[s] = S_
+    print('PR117 candidates',len(cand),flush=True)
+    placed = saturated_placement(cand, reach)
+    print('PR117 placed',len(placed),flush=True)
+    deferred = sorted(placed, key=lambda s: (len(placed[s]), s)); dset = set(deferred)
+
+    # Optimize the actual mixer word. Source injection and root frames stay fixed.
+    # Floating objective values choose legal finite frames only; the complete
+    # child profile and its contraction are certified independently and exactly.
+    root_frame={s:([1<<i for i in range(h)if i!=centre_of[j]] if kind[j] else kernel([tmask[target[j]]],h)) for s,j in role_root.items()}
+    frame_U={x:sat_basis(B)for x,B in U.items()}
+    FULL=sat_basis(1<<i for i in range(h))
+    sigma={s:sat_basis(B)for s,B in placed.items()}
+    role_events=defaultdict(list);frames={};birth={};lifted=Counter()
+    for i,o in enumerate(ops):
+     if o[0]=='src':
+      frames[i]=frame_U[o[2]];role_events[o[1]].append(i)
+     else:
+      frames[i]=frame_U[o[3]]
+      for s in o[1:3]:role_events[s].append(i)
+    prev={};after={};end={s:sat_basis(root_frame[s])if s in root_frame else FULL for s in range(R)}
+    for s,events in role_events.items():
+     for j,i in enumerate(events):prev[i,s]=events[j-1]if j else None;after[i,s]=events[j+1]if j+1<len(events)else None
+    # Frozen IEEE-754 search weights remove platform libm variation.
+    vals=[float.fromhex(x) for x in ['0x0.0p+0', '0x1.0000000000000p+0', '0x1.fff611fabad2cp+0', '0x1.7ff4324fbceecp+1', '0x1.ffec2426c27bdp+1', '0x1.3ff197310b8c8p+2', '0x1.7fecc00663620p+2', '0x1.bfe79c1e23586p+2', '0x1.ffe2368416067p+2', '0x1.1fee4bbd449aep+3', '0x1.3feb62b54ad1bp+3', '0x1.5fe862b99e2f4p+3', '0x1.7fe54de202531p+3', '0x1.9fe225ec96cafp+3', '0x1.bfdeec529ef76p+3', '0x1.dfdba257523b3p+3', '0x1.ffd84912b47dep+3', '0x1.0fea70bcdaf4fp+4', '0x1.1fe8b63233d76p+4', '0x1.2fe6f54967551p+4', '0x1.3fe52e5858b9ap+4', '0x1.4fe361ac5190ep+4', '0x1.5fe18f8b3cb6dp+4', '0x1.6fdfb834a8020p+4', '0x1.7fdddbe2990a6p+4']]
+    for turn in range(5):
+     changes=0;delta=0
+     for i in (reversed(range(len(ops)))if turn%2==0 else range(len(ops))):
+      o=ops[i]
+      if o[0]=='src':continue
+      _,a,b,x=o;old=frames[i];ends=[frames[after[i,s]]if after[i,s]is not None else end[s]for s in (a,b)]
+      starts=[frames[prev[i,s]]if prev[i,s]is not None else sigma.get(s,())for s in (a,b)]
+      cap=sat_cap(*ends);lower=sat_basis(starts[0]+starts[1])
+      options=[frame_protected_part(cap,old),frame_smallest_hull(lower,old)]
+      best=old;bestdelta=-1e-10
+      for F in options:
+       assert sat_nondeg(F)and all(sat_contained(P,F)and sat_contained(F,N)for P,N in zip(starts,ends))
+       df=sum(vals[len(F)-len(P)]+vals[len(N)-len(F)]-vals[len(old)-len(P)]-vals[len(N)-len(old)]for P,N in zip(starts,ends))
+       if df<bestdelta:best=F;bestdelta=df
+      if best!=old:frames[i]=best;changes+=1;delta+=bestdelta
+     print('Physical frame descent',turn,changes,delta,flush=True)
+     if not changes:break
+    for i,o in enumerate(ops):
+     if o[0]=='src':birth[o[1]]=frames[i]
+     elif o[0]=='copy':birth[o[2]]=frames[i]
+     if o[0]!='src':lifted[len(frame_U[o[3]]),len(frames[i])]+=1
+    assert len(birth)==R
+    op_frames=frames
+    role_frames={s:[]for s in range(R)}
+    for i,o in enumerate(ops):
+        if o[0]=='src':role_frames[o[1]].append(op_frames[i])
+        else:
+            for s in o[1:3]:role_frames[s].append(op_frames[i])
+    def F0(s):return birth[s]
+    reuse_pairs = select_reuse(locals())
+    merge = check_pairs(locals(), reuse_pairs)
+    reuse_rejected_controls = mutation_controls(locals(), reuse_pairs)
+    live = [s for s in range(Rr) if s not in merge]
+    physical_id = {s: i for i, s in enumerate(live)}
+    def alias(s): return physical_id[merge.get(s, s)]
+    print('Compensated birth reuse', len(reuse_pairs), 'physical roles', len(live), flush=True)
+
+    # ------------------------------------------------------------ C. replay with arbitrary scratch and data
+    inv = lambda a: pow(a % P, P - 2, P); HALF = inv(2); I21 = inv(21)
+    cvec = [None] * Rr; dpart = [dict() for _ in range(Rr)]
+    for s, j in role_root.items():
+        if kind[j]: c = [0] * h; c[centre_of[j]] = 1; cvec[s] = c
+        else: dpart[s][target[j]] = (P - HALF) if j >= v else HALF
+    seed_c = {s: (None if cvec[s] is None else list(cvec[s])) for s in role_root}
+    seed_d = {s: dict(dpart[s]) for s in role_root}
+    def addc(dst, src):
+        if cvec[src] is not None:
+            cvec[dst] = list(cvec[src]) if cvec[dst] is None else [(p + q_) % P for p, q_ in zip(cvec[dst], cvec[src])]
+        for t, c in dpart[src].items(): dpart[dst][t] = (dpart[dst].get(t, 0) + c) % P
+    for o in reversed(ops):
+        if o[0] == 'add': addc(o[2], o[1])
+        elif o[0] == 'copy': addc(o[1], o[2])
+    scatter = [[(I21 - (HALF if i in trip[t] else 0)) % P for t in range(v)] for i in range(h)]
+    def readout(y, s, value, sign, seed=False):
+        # Logical readout macro: the literal audit combines its exact rational
+        # target coefficients and expands each numerator/42 into bounded shears.
+        cv = seed_c[s] if seed else cvec[s]; dp = seed_d[s] if seed else dpart[s]
+        if cv is not None:
+            for i, ci in enumerate(cv):
+                if ci:
+                    f = sign * ci * value % P; row = scatter[i]
+                    for t in range(v): y[t] = (y[t] + f * row[t]) % P
+        for t, c in dp.items(): y[t] = (y[t] + sign * c * value) % P
+    phase1 = sorted(Anc); rest = [i for i in range(len(ops)) if i not in Anc]
+    def replay(seed, omit_compensation=None):
+        rng = random.Random(seed)
+        x = [rng.randrange(P) for _ in range(v)]
+        z = [rng.randrange(P) for _ in live]
+        y0 = [rng.randrange(P) for _ in range(v)]; a = list(z); y = list(y0)
+        chronology = []
+        def inject(s, leaf):
+            dst = alias(s)
+            a[dst] = (a[dst] + x[leaf - 1]) % P
+            chronology.append(('src', dst, leaf - 1))
+        def run(i):
+            o = ops[i]
+            if o[0] == 'src': return
+            dst, src = (o[1], o[2]) if o[0] == 'add' else (o[2], o[1])
+            dst, src = alias(dst), alias(src)
+            require(dst != src, 'aliased gate ports')
+            a[dst] = (a[dst] + a[src]) % P
+            chronology.append(('work', dst, src))
+        for s in range(Rr):
+            if s not in dset: readout(y, s, a[alias(s)], -1)
+        for s, leaf in leaf_of.items():
+            if s not in dset: inject(s, leaf)
+        for i in phase1: run(i)
+        for s in centre_roles: readout(y, s, a[alias(s)], +1, True)
+        compensated = []
+        for s in deferred:
+            if s != omit_compensation:
+                readout(y, s, a[alias(s)], -1)
+                compensated.append(s)
+        if omit_compensation is None: check_compensation(merge, compensated)
+        for s in deferred:
+            if s in leaf_of: inject(s, leaf_of[s])
+        for i in rest: run(i)
+        for s, j in role_root.items():
+            if not kind[j]: readout(y, s, a[alias(s)], +1, True)
+        # Aliased births require reversal of the actual source/workspace
+        # chronology, including delayed source injections.
+        for kind_, dst, src in reversed(chronology):
+            value = x[src] if kind_ == 'src' else a[src]
+            a[dst] = (a[dst] - value) % P
+        return a == z, all((y[t] - y0[t] - x[t]) % P == 0 for t in range(v))
+    rep = [replay(seed) for seed in (1, 2)]
+    require(all(r == (True, True) for r in rep), 'aliased replay %s' % rep)
+    missing_compensation = replay(3, omit_compensation=reuse_pairs[0]['recipient'])
+    require(missing_compensation == (True, False), 'omitted compensation unexpectedly accepted')
+    compensated_births = sorted(merge)
+
+    # ------------------------------------------------------------ D. exact F2 frame facts
+    root_frame = {}
+    for s, j in role_root.items():
+        root_frame[s] = [1 << i for i in range(h) if i != centre_of[j]] if kind[j] else kernel([tmask[target[j]]], h)
+    FULLB = [1 << i for i in range(h)]
+    chain_dims = []
+    for s in range(Rr):
+        seq = [placed.get(s, [])] + role_frames[s]
+        if s in root_frame: seq.append(root_frame[s])
+        seq.append(FULLB)
+        require(all(contains(A, B) for A, B in zip(seq, seq[1:])), 'role chain nesting %d' % s)
+        require(all(nondeg(B) for B in seq[1:-1] if B), 'degenerate frame on role %d' % s)
+        chain_dims.append([len(reduce(B)) for B in seq])
+    for s, X in placed.items():
+        require(contains(X, F0(s)) and nondeg(X), 'deferral frame %d' % s)
+        require(all(not any(dot(xv, tmask[t]) for xv in X) for t in reach[s]), 'target frame %d' % s)
+    byT2 = defaultdict(list)
+    for s in deferred:
+        for t in reach[s]: byT2[t].append(s)
+    for t, ss in byT2.items():
+        ss.sort(key=lambda s: (len(placed[s]), s))
+        require(all(contains(placed[p], placed[q_]) for p, q_ in zip(ss, ss[1:])), 'target chain %d' % t)
+
+    # ------------------------------------------------------------ E. one-child histogram
+    z = Counter()
+    for s in range(Rr):
+        ds = chain_dims[s]
+        for a, b in zip(ds[:-1], ds[1:-1]):
+            if b > a: z[b - a] += 2 * v
+        lastd = ds[-2]
+        if s in role_root and kind[role_root[s]]: z[h - 1] += 2 * v           # copied centre transform
+        z[h - lastd] += 2 * v                                                  # final growth to F
+        z[m - h + ds[0]] += 2 * v                                              # exterior, gauged by sigma
+    levels = defaultdict(set)
+    for s in deferred:
+        for t in reach[s]: levels[t].add(len(placed[s]))
+    for t in range(v):
+        ds = sorted(levels[t] | {0, h - 1})
+        for a, b in zip(ds, ds[1:]): z[b - a] += 2 * v                       # target fronts
+    z[h - 1] += 2 * N                                                          # data-wire fronts
+    z[(h - 1) ** 2] += 2 * N                                                   # data macros
+    z[1] += N                                                                  # endpoint copies
+    z.pop(0, None)
+    virtual_R = R
+    for pair in reuse_pairs:
+        e, f = pair['e'], pair['s']
+        z[h - e] -= 2 * v
+        z[m - h + f] -= 2 * v
+        if f > e: z[f - e] += 2 * v
+    require(all(count >= 0 for count in z.values()), 'reuse histogram subtraction')
+    z = Counter({width: count for width, count in z.items() if count})
+    physical_R = len(live)
+    # Canonical inventory of actual physical source gauges. Recipients have
+    # disappeared; every reused donor remains a separate zero-gauge slot.
+    source_counts = Counter(sat_basis(placed.get(s, ())) for s in live)
+    physical_auxiliary_source_frames = [dict(basis=list(F), count=count)
+        for F, count in sorted(source_counts.items(), key=lambda item: (len(item[0]), item[0]))]
+    require(sum(source_counts.values()) == physical_R, 'physical source inventory')
+    require(all(not placed.get(a) for a in merge.values()), 'reused donor source gauge')
+    W = 2 * N + 2 * v * physical_R; L = 2 * v * h * (h - 1); s_ = W * m - N + L
+    require(sum(t * c for t, c in z.items()) == s_, 'complex rank mass')
+    require(all(0 < t < m for t in z), 'children below m')
+    out = dict(h=h, v=v, additions=c_add, roots=q, links=len(links), R=physical_R, virtual_R=virtual_R, reused_roles=len(reuse_pairs), m=m, N=N, W=W, L=L, total_rank=s_,
+               deficit=N - L, maxchild=max(z), phase_one_ops=len(Anc), phase_one_roles=len(touched),
+               physical_auxiliary_source_frames=physical_auxiliary_source_frames,
+               deferred_roles=len(deferred),
+               deferred_dims=dict(sorted(Counter(len(X) for X in placed.values()).items())),
+               lifted_additions=sum(1 for i,o in enumerate(ops) if o[0]=='add' and len(op_frames[i])>ranks[o[3]]),
+               physical_gate_frame_changes=sum(op_frames[i]!=frame_U[o[3]] for i,o in enumerate(ops) if o[0]!='src'),
+               replay=dict(seeds=[1, 2], scratch_restored=True, y_plus_x=True, field='Z/(2^61-1)',
+                           physically_aliased=True, inverse_order='true source/workspace chronology',
+                           omitted_compensation_rejected=True),
+               reuse_rejected_controls=reuse_rejected_controls,
+               child_multiplicities=dict(sorted(z.items())))
+    (HERE / 'complex-profile.json').write_text(json.dumps(out, indent=1) + '\n')
+    (HERE / 'reuse-pairs.json').write_text(json.dumps(reuse_pairs, indent=1) + '\n')
+    print('PASS complex: R=%d, deferred roles %d, maxchild %d, rank mass %d' % (physical_R, len(deferred), max(z), s_))
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/cyclic-deferred/gauge-phase-audit.json b/research/cyclic-deferred/gauge-phase-audit.json
new file mode 100644
index 0000000000..e25ce963c9
--- /dev/null
+++ b/research/cyclic-deferred/gauge-phase-audit.json
@@ -0,0 +1,10484 @@
+{
+  "R": 26597,
+  "all_grouped_complement_scalars_one": true,
+  "all_local_projectors_exact": true,
+  "all_source_frames_nondegenerate": true,
+  "checker_sha256": "a4308a0aeef9ee3c726b7fc4bd887bac2262b419a1c02f90a5cbc16f7be030fa",
+  "complement_formula": "(sigma tensor span J) orthogonal-sum (F24 tensor (span J)^perp), or transpose",
+  "complement_width": "576-|J|*(24-dim(sigma))",
+  "distinct_frames": 470,
+  "exact_physical_source_inventory_bound": true,
+  "group_sizes": {
+    "8": 4,
+    "24": 83
+  },
+  "grouped_exterior_histogram": {
+    "48": 143258,
+    "96": 58930,
+    "144": 95118,
+    "192": 18592,
+    "240": 1992,
+    "384": 191992,
+    "400": 6904,
+    "416": 2840,
+    "432": 38282,
+    "448": 896,
+    "456": 33532,
+    "464": 96,
+    "480": 33864,
+    "504": 1826,
+    "528": 12082,
+    "536": 1616,
+    "544": 1632,
+    "552": 88,
+    "560": 504
+  },
+  "h": 24,
+  "local_profile_sha256": "63e4794bc1deeee10c1294c7b7a0a951dd79b6e978472dc18d026ee607b9cc8c",
+  "nonzero_source_roles": 2598,
+  "normalform_scalar_exponents": {
+    "0": 24729,
+    "1": 1665,
+    "2": 203
+  },
+  "paid_normalform": "One full residual-width child with paid binary address adapters and quadratic fourth-root phase passes",
+  "partition_sha256": "9117d3f010354fa2f32d5c5fb324b280977e82e76060a70a867767a90989ccaa",
+  "phase_coefficient_points": 301,
+  "physical_auxiliary_source_frames": [
+    {
+      "basis": [],
+      "count": 23999
+    },
+    {
+      "basis": [
+        9,
+        5
+      ],
+      "count": 134
+    },
+    {
+      "basis": [
+        36,
+        20
+      ],
+      "count": 2
+    },
+    {
+      "basis": [
+        144,
+        80
+      ],
+      "count": 88
+    },
+    {
+      "basis": [
+        2304,
+        1280
+      ],
+      "count": 23
+    },
+    {
+      "basis": [
+        36864,
+        20480
+      ],
+      "count": 16
+    },
+    {
+      "basis": [
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+        327680
+      ],
+      "count": 98
+    },
+    {
+      "basis": [
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+        5242880
+      ],
+      "count": 502
+    },
+    {
+      "basis": [
+        8,
+        4,
+        2,
+        1
+      ],
+      "count": 4
+    },
+    {
+      "basis": [
+        33,
+        17,
+        9,
+        5
+      ],
+      "count": 72
+    },
+    {
+      "basis": [
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+        17
+      ],
+      "count": 6
+    },
+    {
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+      ],
+      "count": 8
+    },
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+      "count": 28
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+      "count": 42
+    },
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+  "scope": "Finite exact frame/projector/Gauss-phase audit and physical core bindings. Residual-to-child, copied-center, endpoint, precision and all-size contracts remain inherited",
+  "signed_stage_residuals_equal": true,
+  "source_inventory_sha256": "af8831a76a499d33ce374f5c361b1a1903334dd34d01de696fb5218fee34b514",
+  "status": "PASS physical source gauges, exact Gauss phases and signed grouped complements",
+  "wrong_source_sign_rejected": true
+}
diff --git a/research/cyclic-deferred/gauge-phase-witness.json b/research/cyclic-deferred/gauge-phase-witness.json
new file mode 100644
index 0000000000..e70534c78c
--- /dev/null
+++ b/research/cyclic-deferred/gauge-phase-witness.json
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diff --git a/research/cyclic-deferred/gauge_phase.py b/research/cyclic-deferred/gauge_phase.py
new file mode 100644
index 0000000000..6a2caa5a07
--- /dev/null
+++ b/research/cyclic-deferred/gauge_phase.py
@@ -0,0 +1,179 @@
+#!/usr/bin/env python3
+"""Exact phases for completed cores with physical deferred source gauges.
+
+Completed-core sharing and signed partition: an664 PR128, commit
+530588a019b4a74f09180680c9e3961bf649ec89, with OpenAI Codex assistance.
+The partition implements Xiande Zhang and Gennian Ge (2010).
+Generalized gauge/physical-reuse composition and this independent audit were
+prepared for eumemic with OpenAI Codex assistance. Apache-2.0.
+The one-child nondegenerate Gauss normal form remains the inherited PR104
+interface; finite phase verification does not establish its all-size compiler.
+"""
+from pathlib import Path
+from itertools import combinations
+from collections import Counter
+from hashlib import sha256
+import json,sys
+assert not sys.flags.optimize, 'Assertions must remain enabled'
+HERE=Path(__file__).resolve().parent
+PARTITION_PIN='9117d3f010354fa2f32d5c5fb324b280977e82e76060a70a867767a90989ccaa'
+UNITS=((1,0),(0,1),(-1,0),(0,-1))
+
+def canonical(vectors):
+    piv={}
+    for x in vectors:
+        for j in sorted(piv,reverse=True):
+            if x>>j&1:x^=piv[j]
+        if x:
+            j=x.bit_length()-1
+            for k in piv:
+                if piv[k]>>j&1:piv[k]^=x
+            piv[j]=x
+    return tuple(piv[j]for j in sorted(piv,reverse=True))
+
+def inventory_digest(inventory):
+    return sha256(json.dumps(inventory,sort_keys=True,separators=(',',':')).encode()).hexdigest()
+
+def validate_inventory(inventory,R,h):
+    seen=set();counts=Counter()
+    for row in inventory:
+        assert set(row)=={'basis','count'}, 'Source inventory fields'
+        B=row['basis'];n=row['count']
+        assert isinstance(B,list) and all(type(x)is int and 0<x<(1<<h)for x in B), 'Source inventory vectors'
+        assert type(n)is int and n>0, 'Source inventory multiplicity'
+        F=canonical(B)
+        assert tuple(B)==F and F not in seen, 'Canonical distinct source inventory'
+        seen.add(F);counts[F]=n
+        assert len(F)<h, 'Completed core must retain a positive active space'
+    assert inventory==[dict(basis=list(F),count=n)for F,n in sorted(counts.items(),key=lambda item:(len(item[0]),item[0]))], 'Source inventory ordering'
+    assert sum(counts.values())==R, 'Physical source inventory role count'
+    return counts
+
+def dot(x,y):return (x&y).bit_count()&1
+
+def multiply(a,b):return(a[0]*b[0]-a[1]*b[1],a[0]*b[1]+a[1]*b[0])
+
+def power(a,n):
+    out=(1,0)
+    while n:
+        if n&1:out=multiply(out,a)
+        a=multiply(a,a);n>>=1
+    return out
+
+def orthogonal_blocks(frame):
+    pool=list(frame);blocks=[]
+    while pool:
+        k=next((j for j,x in enumerate(pool)if dot(x,x)),None)
+        if k is not None:
+            a=pool.pop(k);blocks.append((a,))
+            pool=[x^(a if dot(x,a)else 0)for x in pool]
+        else:
+            a=pool.pop(0);k=next((j for j,b in enumerate(pool)if dot(a,b)),None)
+            assert k is not None, 'Degenerate physical source frame'
+            b=pool.pop(k);blocks.append((a,b))
+            pool=[x^(a if dot(x,b)else 0)^(b if dot(x,a)else 0)for x in pool]
+        assert len(canonical(pool))==len(pool)
+    assert canonical(x for block in blocks for x in block)==frame
+    for i,A in enumerate(blocks):
+        for B in blocks[i+1:]:assert all(not dot(x,y)for x in A for y in B)
+    return blocks
+
+def project(x,blocks):
+    out=0
+    for block in blocks:
+        if len(block)==1:out^=block[0]if dot(x,block[0])else 0
+        else:
+            a,b=block;out^=(a if dot(x,b)else 0)^(b if dot(x,a)else 0)
+    return out
+
+def gauss(blocks,sign):
+    out=(1,0)
+    for block in blocks:
+        vs=[0,block[0]]if len(block)==1 else[0,block[0],block[1],block[0]^block[1]]
+        vals=[UNITS[sign*x.bit_count()%4]for x in vs]
+        z=(sum(x for x,y in vals),sum(y for x,y in vals))
+        assert z in (((1,1),(1,-1))if len(block)==1 else((2,0),(-2,0)))
+        out=multiply(out,z)
+    return out
+
+def audit(local,reflection,partition):
+    h=local['h'];R=local['R'];assert h==24 and R==26597
+    inventory=local['physical_auxiliary_source_frames']
+    counts=validate_inventory(inventory,R,h)
+    assert reflection['physical_auxiliary_source_frames']==inventory, 'Reflection source inventory binding'
+    for field in ('completed_core_source_inventory_bound','completed_core_pre_exterior_frames_full','reflected_core_active_frames_complement_source','exact_birth_cut_invariants','physical_aliased_numeric_replay','exact_arbitrary_dirty_cancellation_by_dependency_cut'):
+        assert reflection[field]is True, 'Completed-core interface: '+field
+    assert reflection['R']==R and reflection['child_multiplicities']==local['child_multiplicities'], 'Reflection local profile binding'
+    trip=list(combinations(range(h),3));masks=[sum(1<<i for i in T)for T in trip]
+    assert sorted(t for G in partition['groups']for t in G)==list(range(len(trip))), 'Outer label coverage'
+    tests=[0]+[1<<i for i in range(h)]+[(1<<i)|(1<<j)for i,j in combinations(range(h),2)]
+    records=[];normalphases=Counter();wrong_sign_found=False
+    for frame,count in counts.items():
+        blocks=orthogonal_blocks(frame);rank=len(frame)
+        columns=[project(1<<i,blocks)for i in range(h)]
+        assert canonical(columns)==frame and all(project(x,blocks)==x for x in frame)
+        assert all(dot(columns[i],1<<j)==dot(columns[j],1<<i)for i in range(h)for j in range(h))
+        for x in tests:
+            u=project(x,blocks);e=x^u
+            assert project(e,blocks)==0 and not dot(u,e)
+            assert (u.bit_count()+e.bit_count()-x.bit_count())%4==0
+            assert (u.bit_count()-sum(project(x,[block]).bit_count()for block in blocks))%4==0
+            # C_F C_sigma^-1 = C_sigma-perp; the reflected stage has the
+            # same positive E phase, not its inverse.
+            assert (x.bit_count()-u.bit_count())%4==e.bit_count()%4
+            wrong_sign_found|=(-e.bit_count())%4!=e.bit_count()%4
+        G=gauss(blocks,-1);z=multiply(G,power((1,-1),rank));den=1<<rank
+        assert z[0]%den==z[1]%den==0
+        mu=(z[0]//den,z[1]//den);assert mu in UNITS
+        assert power(mu,8)==power(mu,24)==(1,0)
+        normalphases[UNITS.index(mu)]+=count
+        records.append(dict(basis=list(frame),count=count,rank=rank,
+            orthogonal_blocks=[list(B)for B in blocks],negative_gauss=list(G),normalform_scalar=list(mu)))
+    partial=iter(partition['partial_complement_basis_supports']);sizes=Counter();corrections=Counter()
+    for J in partition['groups']:
+        B=[masks[t]for t in J];sizes[len(B)]+=1
+        if len(B)==24:C=[]
+        else:
+            supports=next(partial)
+            assert len(supports)==16 and all(len(set(S))==len(S)and all(type(i)is int and 0<=i<h for i in S)for S in supports)
+            C=[sum(1<<i for i in S)for S in supports]
+        Q=B+C
+        assert len(Q)==h and all(dot(x,y)==(i==j)for i,x in enumerate(Q)for j,y in enumerate(Q)), 'Signed outer complement basis'
+        assert all(x.bit_count()%4==3 for x in B)
+        assert not C or Counter(x.bit_count()%4 for x in C)=={1:8,3:8}
+        for x in tests:assert (sum(c.bit_count()*dot(x,c)for c in Q)-x.bit_count())%4==0
+        for rec in records:
+            s=rec['rank'];g=len(B);width=g*s+h*len(C)
+            assert width==h*h-g*(h-s)
+            scalar=multiply(power(tuple(rec['normalform_scalar']),g),power((0,-1),h*sum(c.bit_count()%4==3 for c in C)))
+            assert scalar==(1,0), 'Grouped complement scalar phase'
+            if width:corrections[width]+=2*rec['count']
+    assert next(partial,None)is None and sizes=={24:83,8:4}
+    assert wrong_sign_found, 'Wrong source-sign control must detect a mismatch'
+    return dict(status='PASS physical source gauges, exact Gauss phases and signed grouped complements',
+        h=h,R=R,source_inventory_sha256=inventory_digest(inventory),physical_auxiliary_source_frames=inventory,
+        distinct_frames=len(counts),nonzero_source_roles=sum(n for F,n in counts.items()if F),
+        phase_coefficient_points=len(tests),group_sizes=dict(sorted(sizes.items())),
+        normalform_scalar_exponents=dict(sorted(normalphases.items())),
+        grouped_exterior_histogram=dict(sorted(corrections.items())),
+        all_local_projectors_exact=True,all_source_frames_nondegenerate=True,
+        all_grouped_complement_scalars_one=True,signed_stage_residuals_equal=True,
+        exact_physical_source_inventory_bound=True,wrong_source_sign_rejected=True,
+        complement_formula='(sigma tensor span J) orthogonal-sum (F24 tensor (span J)^perp), or transpose',
+        complement_width='576-|J|*(24-dim(sigma))',
+        paid_normalform='One full residual-width child with paid binary address adapters and quadratic fourth-root phase passes',
+        scope='Finite exact frame/projector/Gauss-phase audit and physical core bindings. Residual-to-child, copied-center, endpoint, precision and all-size contracts remain inherited'),records
+
+def main():
+    lp=HERE/'complex-profile.json';rp=HERE/'reflection-audit.json';pp=HERE/'inputs/shared-partition.json'
+    assert sha256(pp.read_bytes()).hexdigest()==PARTITION_PIN
+    local=json.loads(lp.read_text());reflection=json.loads(rp.read_text())
+    assert reflection['source_sha256']==sha256((HERE/'complex_deferred.py').read_bytes()).hexdigest()
+    assert reflection['audit_sha256']==sha256((HERE/'reflection_audit.py').read_bytes()).hexdigest()
+    result,records=audit(local,reflection,json.loads(pp.read_text()))
+    result.update(partition_sha256=PARTITION_PIN,local_profile_sha256=sha256(lp.read_bytes()).hexdigest(),
+        reflection_receipt_sha256=sha256(rp.read_bytes()).hexdigest(),checker_sha256=sha256(Path(__file__).read_bytes()).hexdigest())
+    (HERE/'gauge-phase-audit.json').write_text(json.dumps(result,indent=2,sort_keys=True)+'\n')
+    (HERE/'gauge-phase-witness.json').write_text(json.dumps(records,sort_keys=True,separators=(',',':'))+'\n')
+    print('PASS exact physical gauge sharing',result['R'],result['distinct_frames'],result['nonzero_source_roles'])
+if __name__=='__main__':main()
diff --git a/research/cyclic-deferred/inputs/complex-dag.json.gz b/research/cyclic-deferred/inputs/complex-dag.json.gz
new file mode 100644
index 0000000000..a5b0598e4c
Binary files /dev/null and b/research/cyclic-deferred/inputs/complex-dag.json.gz differ
diff --git a/research/cyclic-deferred/inputs/deferred_23.json.gz b/research/cyclic-deferred/inputs/deferred_23.json.gz
new file mode 100644
index 0000000000..4a4431bd42
Binary files /dev/null and b/research/cyclic-deferred/inputs/deferred_23.json.gz differ
diff --git a/research/cyclic-deferred/inputs/shared-partition.json b/research/cyclic-deferred/inputs/shared-partition.json
new file mode 100644
index 0000000000..51aea7dd44
--- /dev/null
+++ b/research/cyclic-deferred/inputs/shared-partition.json
@@ -0,0 +1,5718 @@
+{
+  "status": "PASS: exact coverage, parallel classes and binary GramI",
+  "h": 24,
+  "triple_indexing": "zero-based lexicographic combinations(range(24),3)",
+  "source": {
+    "authors": "Xiande Zhang and Gennian Ge",
+    "year": 2010,
+    "doi": "10.1002/jcd.20234",
+    "lemmas": [
+      "2.2",
+      "2.4"
+    ],
+    "url": "https://staff.ustc.edu.cn/~drzhangx/papers/mrpacking.pdf",
+    "pdf_sha256": "b6758567fd48844b708fd2164f75e0957d5880c4e037adb27826a4f545a27998"
+  },
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+}
diff --git a/research/cyclic-deferred/inputs/witness_23.json.gz b/research/cyclic-deferred/inputs/witness_23.json.gz
new file mode 100644
index 0000000000..3d86f83b87
Binary files /dev/null and b/research/cyclic-deferred/inputs/witness_23.json.gz differ
diff --git a/research/cyclic-deferred/phase_check.json b/research/cyclic-deferred/phase_check.json
new file mode 100644
index 0000000000..bae95cc643
--- /dev/null
+++ b/research/cyclic-deferred/phase_check.json
@@ -0,0 +1,953 @@
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+    {
+      "group": 53,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 54,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 55,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 56,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 57,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 58,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 59,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 60,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 61,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 62,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 63,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 64,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 65,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 66,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 67,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 68,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 69,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 70,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 71,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 72,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 73,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 74,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 75,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 76,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 77,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 78,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 79,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 80,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 81,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 82,
+      "size": 24,
+      "complement_basis": [],
+      "negative_C_axes": [],
+      "positive_C_axes": [],
+      "tensor_width": 0,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 83,
+      "size": 8,
+      "complement_basis": [
+        35,
+        37,
+        16,
+        8,
+        2240,
+        2368,
+        1024,
+        512,
+        143360,
+        151552,
+        65536,
+        32768,
+        9175040,
+        9699328,
+        4194304,
+        2097152
+      ],
+      "negative_C_axes": [
+        0,
+        1,
+        4,
+        5,
+        8,
+        9,
+        12,
+        13
+      ],
+      "positive_C_axes": [
+        2,
+        3,
+        6,
+        7,
+        10,
+        11,
+        14,
+        15
+      ],
+      "tensor_width": 384,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 84,
+      "size": 8,
+      "complement_basis": [
+        49,
+        37,
+        2,
+        8,
+        3136,
+        2368,
+        128,
+        512,
+        200704,
+        151552,
+        8192,
+        32768,
+        12845056,
+        9699328,
+        524288,
+        2097152
+      ],
+      "negative_C_axes": [
+        0,
+        1,
+        4,
+        5,
+        8,
+        9,
+        12,
+        13
+      ],
+      "positive_C_axes": [
+        2,
+        3,
+        6,
+        7,
+        10,
+        11,
+        14,
+        15
+      ],
+      "tensor_width": 384,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 85,
+      "size": 8,
+      "complement_basis": [
+        35,
+        41,
+        16,
+        4,
+        2240,
+        2624,
+        1024,
+        256,
+        143360,
+        167936,
+        65536,
+        16384,
+        9175040,
+        10747904,
+        4194304,
+        1048576
+      ],
+      "negative_C_axes": [
+        0,
+        1,
+        4,
+        5,
+        8,
+        9,
+        12,
+        13
+      ],
+      "positive_C_axes": [
+        2,
+        3,
+        6,
+        7,
+        10,
+        11,
+        14,
+        15
+      ],
+      "tensor_width": 384,
+      "tensor_scalar_phase_exponent_mod4": 0
+    },
+    {
+      "group": 86,
+      "size": 8,
+      "complement_basis": [
+        49,
+        41,
+        2,
+        4,
+        3136,
+        2624,
+        128,
+        256,
+        200704,
+        167936,
+        8192,
+        16384,
+        12845056,
+        10747904,
+        524288,
+        1048576
+      ],
+      "negative_C_axes": [
+        0,
+        1,
+        4,
+        5,
+        8,
+        9,
+        12,
+        13
+      ],
+      "positive_C_axes": [
+        2,
+        3,
+        6,
+        7,
+        10,
+        11,
+        14,
+        15
+      ],
+      "tensor_width": 384,
+      "tensor_scalar_phase_exponent_mod4": 0
+    }
+  ],
+  "scope": "Geometry and phase are standalone; numerical R comes from separately replayed PR117 producer"
+}
diff --git a/research/cyclic-deferred/phase_check.py b/research/cyclic-deferred/phase_check.py
new file mode 100644
index 0000000000..12bcf6911e
--- /dev/null
+++ b/research/cyclic-deferred/phase_check.py
@@ -0,0 +1,99 @@
+"""Standalone exact signed-MRP87 phase, complement and Gaussian checks; stdlib."""
+from pathlib import Path
+from itertools import combinations
+from collections import Counter
+import hashlib,json,sys
+
+path=Path(sys.argv[1]) if len(sys.argv)>1 else Path(__file__).parent/'inputs/shared-partition.json'
+raw=path.read_bytes();data=json.loads(raw)
+digest=hashlib.sha256(raw).hexdigest()
+h=24;triples=list(combinations(range(h),3));masks=[sum(1<<j for j in T) for T in triples]
+groups=data['groups'];assert Counter(map(len,groups))=={24:83,8:4}
+assert sorted(t for G in groups for t in G)==list(range(len(triples)))
+parity=lambda x:x.bit_count()&1
+def project(x,B):
+    out=0
+    for b in B:
+        if parity(x&b):out^=b
+    return out
+tests=[0]+[1<<i for i in range(h)]+[(1<<i)|(1<<j) for i,j in combinations(range(h),2)]
+partial=iter(data['partial_complement_basis_supports']);records=[];gram_entries=0
+for number,G in enumerate(groups):
+    B=[masks[t] for t in G]
+    if len(G)==24:C=[]
+    else:
+        supports=next(partial)
+        assert len(supports)==16 and all(len(s)==len(set(s)) and all(0<=j<h for j in s) for s in supports)
+        C=[sum(1<<j for j in s) for s in supports]
+        assert Counter(c.bit_count() for c in C)=={1:8,3:8}
+    Q=C+B;assert len(Q)==h
+    assert all(parity(x&y)==(i==j) for i,x in enumerate(Q) for j,y in enumerate(Q)), 'non-orthonormal completed group basis'
+    gram_entries+=len(Q)**2
+    for x in tests:
+        u=project(x,B);z=project(x,C)
+        assert u^z==x
+        assert (u.bit_count()-sum(3*parity(x&b) for b in B))%4==0
+        assert (z.bit_count()-sum(c.bit_count()*parity(x&c) for c in C))%4==0
+        assert (u.bit_count()+z.bit_count()-x.bit_count())%4==0
+    negative=[i for i,c in enumerate(C) if c.bit_count()%4==3]
+    assert (-24*len(negative))%4==0
+    records.append(dict(group=number,size=len(B),complement_basis=C,
+        negative_C_axes=negative,positive_C_axes=[i for i,c in enumerate(C) if c.bit_count()%4==1],
+        tensor_width=24*len(C),tensor_scalar_phase_exponent_mod4=0))
+assert next(partial,None) is None
+for b in masks:
+    for x in tests:
+        u=b if parity(x&b) else 0
+        assert (x.bit_count()-(x^u).bit_count()-u.bit_count())%4==0
+
+# Exact one-coordinate inverse normalization, numerators over denominator2.
+C1=[[(1,1),(1,-1)],[(1,-1),(1,1)]]
+assert [[(a,-b) for a,b in row] for row in C1]==[
+    [((1-2*i)*(1-2*j)*b,-(1-2*i)*(1-2*j)*a) for j,(a,b) in enumerate(row)]
+    for i,row in enumerate(C1)]
+
+# Actual Gaussian kernels with denominator2^n; no floating complex numbers.
+units=[(1,0),(0,1),(-1,0),(0,-1)]
+def kernel(n,phase):
+    N=1<<n;A=[]
+    for x in range(N):
+        row=[]
+        for y in range(N):
+            re=im=0
+            for z in range(N):
+                a,b=units[phase(z)%4];sgn=1-2*parity(z&(x^y));re+=sgn*a;im+=sgn*b
+            row.append((re,im))
+        A.append(row)
+    return A
+def multiply(A,B):
+    return [[(sum(a*c-b*d for (a,b),(c,d) in zip(row,col)),
+              sum(a*d+b*c for (a,b),(c,d) in zip(row,col))) for col in zip(*B)] for row in A]
+def scaled(A,n):return [[(n*a,n*b) for a,b in row] for row in A]
+n=4;N0=1<<n;u=7;v0=11
+qU=lambda x:3*parity(x&u)
+qV=lambda x:3*parity(x&v0)
+qG=lambda x:project(x,[u,v0]).bit_count()
+qD=lambda x:(x^project(x,[u])).bit_count()
+qE=lambda x:(x^project(x,[u,v0])).bit_count()
+qI=lambda x:x.bit_count()
+CU,CV,CG,CD,CE,CI=[kernel(n,q) for q in (qU,qV,qG,qD,qE,qI)]
+assert multiply(CU,CV)==scaled(CG,N0)
+assert multiply(CE,CG)==scaled(CI,N0)
+assert multiply(CI,kernel(n,lambda x:-qD(x)))==scaled(CU,N0)
+assert multiply(CI,CD)!=scaled(CU,N0)
+
+R=28705;v=len(triples);m=h*h;N=v*v;ell=552;W=2*N+2*len(groups)*R
+s=W*m-N+2*v*ell
+out=dict(status='PASS signed MRP87 actual phases, explicit paid complement adapters and stage-two signs',
+    partition_sha256=digest,groups=len(groups),group_size_histogram={24:83,8:4},
+    all_triples_once=True,all_full_basis_Gram_entries_checked=gram_entries,
+    phase_coefficient_points_per_group=len(tests),all_actual_Z4_phase_identities=True,
+    all_complement_bases_explicit=True,all_tensor_scalar_phases_one=True,
+    exact_inverse_direction_identity=True,exact_gaussian_composition=True,
+    exact_stage2_gauge=True,wrong_source_sign_negative_control=True,
+    R=R,W=W,rank_mass=s,deficit=N-2*v*ell,
+    exterior_histogram={384:2*R*4},zero_rank_bridges_omitted=2*R*83,
+    records=records,
+    scope='Geometry and phase are standalone; numerical R comes from separately replayed PR117 producer')
+Path(__file__).with_suffix('.json').write_text(json.dumps(out,indent=2)+'\n')
+print(json.dumps({k:v for k,v in out.items() if k!='records'},indent=2))
diff --git a/research/cyclic-deferred/producer.py b/research/cyclic-deferred/producer.py
new file mode 100644
index 0000000000..a93a265f01
--- /dev/null
+++ b/research/cyclic-deferred/producer.py
@@ -0,0 +1,17 @@
+"""Pinned PR117 scalar DAG adapter for the saturated deferred compiler.
+
+The searched graph is upstream PR117 work credited to eumemic with Anthropic
+Claude assistance. This adapter preserves that witness and producer unchanged;
+saturation integration is separate OpenAI Codex-assisted work. Apache-2.0.
+"""
+from hashlib import sha256
+from pathlib import Path
+from replayed_producer import build as replayed_build
+
+WITNESS_SHA256 = '3c034d0aae388ef567a454826f4f48b26fd8a94c71e8ffed4835271b349a783b'
+
+def build(h, prefix, central_disjoint, base=2):
+    assert (h, central_disjoint, base) == (24,24,2)
+    witness=Path(__file__).resolve().parent/'inputs/complex-dag.json.gz'
+    assert sha256(witness.read_bytes()).hexdigest() == WITNESS_SHA256, 'PR117 DAG pin mismatch'
+    return replayed_build(witness,prefix)
diff --git a/research/cyclic-deferred/reflection-audit.json b/research/cyclic-deferred/reflection-audit.json
new file mode 100644
index 0000000000..8b40ca11a6
--- /dev/null
+++ b/research/cyclic-deferred/reflection-audit.json
@@ -0,0 +1,10524 @@
+{
+  "R": 26597,
+  "all_frames_nondegenerate": true,
+  "audit_sha256": "9615b651ed42b7687ce4f5cc242cccdfa9abd638666bd2fc7ee1f56a488e76d7",
+  "bounded_chunk_coefficients": true,
+  "bounded_readout_chunks_sha256": "bb2ad510e5f5113eba2fd520f99bb05725087ae47e1566b7e63eafdefc69d809",
+  "child_multiplicities": {
+    "1": 408795376,
+    "10": 21118416,
+    "11": 4938560,
+    "12": 20826960,
+    "13": 17786912,
+    "14": 1922800,
+    "15": 2493568,
+    "16": 684112,
+    "17": 489808,
+    "18": 546480,
+    "19": 477664,
+    "2": 291621968,
+    "20": 550528,
+    "21": 1230592,
+    "22": 64768,
+    "23": 8363168,
+    "3": 52656384,
+    "4": 51041232,
+    "5": 21498928,
+    "529": 8193152,
+    "552": 97147952,
+    "554": 3493424,
+    "556": 1437040,
+    "558": 2319504,
+    "560": 453376,
+    "562": 48576,
+    "570": 821744,
+    "571": 817696,
+    "572": 825792,
+    "573": 44528,
+    "574": 255024,
+    "6": 17507600,
+    "7": 9229440,
+    "8": 12633808,
+    "9": 7869312
+  },
+  "completed_core_pre_exterior_frames_full": true,
+  "completed_core_source_inventory_bound": true,
+  "conservative_local_G": 14907958248,
+  "dirty_reuse_identity": "Every recipient is untouched before its compensated birth; donor has no future gate/root reads. Recipient old-response coefficient cancels its arbitrary current donor value. All physical shears are reversed in true chronology.",
+  "exact_arbitrary_dirty_cancellation_by_dependency_cut": true,
+  "exact_birth_cut_invariants": true,
+  "exact_chunk_reconstruction": true,
+  "exact_fresh_source_map": true,
+  "exact_integer_old_readout_transpose": true,
+  "expanded_old_readout_additions": 11862944,
+  "expanded_scalar_operations_per_stage": 12160614,
+  "forward_incidence_sha256": "516b499ebc23f69c8075a625181d67ac3b398479365e84e690b3abbdb8d89b12",
+  "h": 24,
+  "inverse_source_order": "True reverse chronological source/workspace word",
+  "largest_bounded_chunk_numerator_over_42": 42,
+  "largest_readout_numerator_over_42": 55,
+  "last_uses_recomputed": true,
+  "literal_frame_incidences_both_directions": true,
+  "literal_global_scalar_groups": 49230262048,
+  "old_readout_coefficients_over_42_sha256": "e0666f81e1d35e002d61b8f2b2b1ca0d1d79cc7fbcbc76063eb333294d875e5e",
+  "omitted_compensation_numeric_control": true,
+  "physical_aliased_numeric_replay": true,
+  "physical_auxiliary_source_frames": [
+    {
+      "basis": [],
+      "count": 23999
+    },
+    {
+      "basis": [
+        9,
+        5
+      ],
+      "count": 134
+    },
+    {
+      "basis": [
+        36,
+        20
+      ],
+      "count": 2
+    },
+    {
+      "basis": [
+        144,
+        80
+      ],
+      "count": 88
+    },
+    {
+      "basis": [
+        2304,
+        1280
+      ],
+      "count": 23
+    },
+    {
+      "basis": [
+        36864,
+        20480
+      ],
+      "count": 16
+    },
+    {
+      "basis": [
+        589824,
+        327680
+      ],
+      "count": 98
+    },
+    {
+      "basis": [
+        9437184,
+        5242880
+      ],
+      "count": 502
+    },
+    {
+      "basis": [
+        8,
+        4,
+        2,
+        1
+      ],
+      "count": 4
+    },
+    {
+      "basis": [
+        33,
+        17,
+        9,
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+        1024,
+        512,
+        256,
+        128,
+        64,
+        32,
+        16,
+        2,
+        1
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        8388620,
+        4194304,
+        2097152,
+        1048576,
+        524288,
+        262144,
+        131084,
+        65536,
+        32768,
+        16384,
+        8192,
+        4096,
+        2048,
+        1024,
+        512,
+        256,
+        128,
+        64,
+        32,
+        16,
+        2,
+        1
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        8388620,
+        4194304,
+        2097152,
+        1048588,
+        524288,
+        262144,
+        131072,
+        65536,
+        32768,
+        16384,
+        8192,
+        4096,
+        2048,
+        1024,
+        512,
+        256,
+        128,
+        64,
+        32,
+        16,
+        2,
+        1
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        8388656,
+        4194304,
+        2097152,
+        1048576,
+        524288,
+        262144,
+        131072,
+        65536,
+        32768,
+        16384,
+        8192,
+        4096,
+        2096,
+        1024,
+        512,
+        256,
+        128,
+        64,
+        8,
+        4,
+        2,
+        1
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        8388656,
+        4194304,
+        2097152,
+        1048576,
+        524336,
+        262144,
+        131072,
+        65536,
+        32768,
+        16384,
+        8192,
+        4096,
+        2048,
+        1024,
+        512,
+        256,
+        128,
+        64,
+        8,
+        4,
+        2,
+        1
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        8388656,
+        4194304,
+        2097152,
+        1048624,
+        524288,
+        262144,
+        131072,
+        65536,
+        32768,
+        16384,
+        8192,
+        4096,
+        2048,
+        1024,
+        512,
+        256,
+        128,
+        64,
+        8,
+        4,
+        2,
+        1
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        8388800,
+        4194304,
+        2097152,
+        1048576,
+        524288,
+        262144,
+        131072,
+        65536,
+        32768,
+        16384,
+        8384,
+        4096,
+        2048,
+        1024,
+        512,
+        256,
+        32,
+        16,
+        8,
+        4,
+        2,
+        1
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        8388800,
+        4194304,
+        2097152,
+        1048576,
+        524288,
+        262144,
+        131072,
+        65536,
+        32960,
+        16384,
+        8192,
+        4096,
+        2048,
+        1024,
+        512,
+        256,
+        32,
+        16,
+        8,
+        4,
+        2,
+        1
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        8388800,
+        4194304,
+        2097152,
+        1048576,
+        524288,
+        262144,
+        131264,
+        65536,
+        32768,
+        16384,
+        8192,
+        4096,
+        2048,
+        1024,
+        512,
+        256,
+        32,
+        16,
+        8,
+        4,
+        2,
+        1
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        8388800,
+        4194304,
+        2097152,
+        1048576,
+        524480,
+        262144,
+        131072,
+        65536,
+        32768,
+        16384,
+        8192,
+        4096,
+        2048,
+        1024,
+        512,
+        256,
+        32,
+        16,
+        8,
+        4,
+        2,
+        1
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        8388800,
+        4194304,
+        2097152,
+        1048768,
+        524288,
+        262144,
+        131072,
+        65536,
+        32768,
+        16384,
+        8192,
+        4096,
+        2048,
+        1024,
+        512,
+        256,
+        32,
+        16,
+        8,
+        4,
+        2,
+        1
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        8389376,
+        4194304,
+        2097152,
+        1048576,
+        524288,
+        262912,
+        131072,
+        65536,
+        32768,
+        16384,
+        8192,
+        4096,
+        2048,
+        1024,
+        128,
+        64,
+        32,
+        16,
+        8,
+        4,
+        2,
+        1
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        8389632,
+        4194304,
+        2097152,
+        1048576,
+        524288,
+        262144,
+        131072,
+        65536,
+        32768,
+        16384,
+        13312,
+        2048,
+        512,
+        256,
+        128,
+        64,
+        32,
+        16,
+        8,
+        4,
+        2,
+        1
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        8389632,
+        4194304,
+        2097152,
+        1048576,
+        524288,
+        262144,
+        131072,
+        65536,
+        50176,
+        8192,
+        4096,
+        2048,
+        512,
+        256,
+        128,
+        64,
+        32,
+        16,
+        8,
+        4,
+        2,
+        1
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        8389632,
+        4194304,
+        2097152,
+        1048576,
+        787456,
+        131072,
+        65536,
+        32768,
+        16384,
+        8192,
+        4096,
+        2048,
+        512,
+        256,
+        128,
+        64,
+        32,
+        16,
+        8,
+        4,
+        2,
+        1
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        8400896,
+        4194304,
+        2097152,
+        1048576,
+        524288,
+        262144,
+        131072,
+        65536,
+        32768,
+        28672,
+        2048,
+        1024,
+        512,
+        256,
+        128,
+        64,
+        32,
+        16,
+        8,
+        4,
+        2,
+        1
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        8404992,
+        4194304,
+        2097152,
+        1048576,
+        802816,
+        131072,
+        65536,
+        32768,
+        8192,
+        4096,
+        2048,
+        1024,
+        512,
+        256,
+        128,
+        64,
+        32,
+        16,
+        8,
+        4,
+        2,
+        1
+      ],
+      "count": 1
+    },
+    {
+      "basis": [
+        8585216,
+        4194304,
+        2097152,
+        1048576,
+        524288,
+        458752,
+        32768,
+        16384,
+        8192,
+        4096,
+        2048,
+        1024,
+        512,
+        256,
+        128,
+        64,
+        32,
+        16,
+        8,
+        4,
+        2,
+        1
+      ],
+      "count": 1
+    }
+  ],
+  "reflected_core_active_frames_complement_source": true,
+  "reflected_incidence_sha256": "71725b7c7c77687b0d6cbbda138e62470a0db48ac99516b4857eecf6910425b3",
+  "reflected_residual_rank_histogram_equal": true,
+  "reflected_scalar_map": "X becomes X-Y; Y unchanged; all auxiliary coordinates restored.",
+  "reflected_word_rule": "Reverse literal order including each bounded readout chunk, negate every shear, swap X/Y ports, complement every frame; auxiliary bank is separate.",
+  "rejected_controls": [
+    "nondead-donor",
+    "unavailable-birth",
+    "pair-alias",
+    "omitted-compensation",
+    "oversized-readout-chunk",
+    "split-readout-sum",
+    "reflected-chunk-order",
+    "inverse-sign",
+    "uncomplemented-frame",
+    "signed-root-readout"
+  ],
+  "reused_roles": 2108,
+  "scalar_readout_normalization": "Each exact numerator n/42 becomes signed42 chunks plus a signed remainder. Exact reconstruction and chunk magnitude<=1 checked; every same-frame shear is charged; reflection reverses chunk order.",
+  "scope": "Exact finite source coefficients, dirty cancellation algebra and literal reflected frame/charge ledger. Copied-centre transforms, endpoint correction and residual-to-child/all-size transfer retain their stated inherited contracts.",
+  "source_sha256": "6c0e4439ba523c8ad2d8043760cf2d0146a678e2b870b73d0d14bb543a644596",
+  "split_readout_macros": 32,
+  "two_stage_scalar_map": "(X,Y) becomes (-Y,X+Y) before inherited endpoint correction.",
+  "unsplit_old_readout_macros": 11862912,
+  "v": 2024,
+  "virtual_R": 28705,
+  "word_blocks": 277775
+}
diff --git a/research/cyclic-deferred/reflection_audit.py b/research/cyclic-deferred/reflection_audit.py
new file mode 100644
index 0000000000..540bd4db4e
--- /dev/null
+++ b/research/cyclic-deferred/reflection_audit.py
@@ -0,0 +1,411 @@
+#!/usr/bin/env python3
+"""Compensated reuse: literal inverse/reflection and exact finite scalar/frame audit.
+
+Birth-cut reuse follows jamesyc PR124; original contributors retained.
+Composition with saturated and physical frames prepared with OpenAI Codex.
+
+Captures an unchanged deferred producer before its output write, then checks
+the scalar computation on every source symbol, the exact signed old-readout
+transpose, the dependency cut and the complete complemented incidence ledger.
+This is finite verification; the inherited residual-to-child implementation,
+all-size transfer and analytic interfaces are not proved by this script.
+"""
+import ast
+from collections import Counter
+from functools import lru_cache
+from hashlib import sha256
+from itertools import combinations
+import argparse,json,pickle,struct,sys
+from pathlib import Path
+from reuse import check_pairs, check_compensation, mutation_controls
+sys.dont_write_bytecode=True
+assert not sys.flags.optimize,'Assertions must remain enabled'
+
+
+def capture(source):
+    source=Path(source).resolve();sys.path.insert(0,str(source.parent))
+    tree=ast.parse(source.read_text(),filename=str(source))
+    main=next(n for n in tree.body if isinstance(n,ast.FunctionDef) and n.name=='main')
+    at=next(i for i,n in enumerate(main.body) if isinstance(n,ast.Expr) and isinstance(n.value,ast.Call) and isinstance(n.value.func,ast.Attribute) and n.value.func.attr=='write_text')
+    main.body[at:]=[ast.Return(ast.Call(ast.Name('locals',ast.Load()),[],[]))]
+    ast.fix_missing_locations(tree)
+    ns={'__file__':str(source),'__name__':'reflection_capture'}
+    exec(compile(tree,str(source)+' [read-only capture]','exec'),ns)
+    data=ns['main']();data['source']=str(source);return data
+
+
+def basis(vectors):
+    piv={}
+    for x in vectors:
+        for p in sorted(piv,reverse=True):
+            if x>>p&1:x^=piv[p]
+        if x:
+            p=x.bit_length()-1
+            for q in piv:
+                if piv[q]>>p&1:piv[q]^=x
+            piv[p]=x
+    return tuple(piv[p] for p in sorted(piv,reverse=True))
+
+
+def dot(a,b):return (a&b).bit_count()&1
+
+
+@lru_cache(None)
+def contained(A,B):return len(basis(A+B))==len(B)
+
+
+@lru_cache(None)
+def complement(A,h):
+    A=basis(A);piv={x.bit_length()-1:x for x in A};result=[]
+    for j in range(h):
+        if j not in piv:
+            x=1<<j
+            for p,row in piv.items():
+                if row>>j&1:x|=1<<p
+            result.append(x)
+    return basis(result)
+
+
+@lru_cache(None)
+def nondeg(A):
+    return len(basis(sum(dot(x,y)<<j for j,y in enumerate(A)) for x in A))==len(A)
+
+
+def root_readout(d,s):
+    h,v,j=d['h'],d['v'],d['role_root'][s];p=(1<<61)-1
+    if d['kind'][j]:
+        assert d['seed_c'][s]==[int(i==d['centre_of'][j]) for i in range(h)],'Root centre readout'
+        assert d['seed_d'][s]=={},'Root centre extra target'
+    else:
+        assert d['seed_c'][s] is None,'Root extra centre'
+        expected=pow(2,-1,p)*(1 if j<v else -1)%p
+        assert d['seed_d'][s]=={d['target'][j]:expected},'Root signed readout'
+
+
+@lru_cache(None)
+def dual_frames(A,B,h):
+    return len(A)+len(B)==h and all(not dot(x,y) for x in A for y in B)
+
+
+def check_chunks(numerator,parts):
+    assert sum(parts)==numerator,'Split readout reconstruction'
+    assert all(0<abs(x)<=42 for x in parts),'Bounded readout chunk'
+
+def bounded_chunks(numerator):
+    q,r=divmod(abs(numerator),42);sign=1 if numerator>=0 else -1
+    parts=(sign*42,)*q+((sign*r,) if r else ())
+    check_chunks(numerator,parts)
+    return parts
+
+def reverse_row(row):
+    if row[0]=='old':return (row[0],row[1],row[3],row[2])
+    return row
+
+
+def check_reflection(forward,backward,h,v):
+    def bank(s):return s+v if s<v else s-v if s<2*v else s
+    assert len(forward)==len(backward),'Reflected event count'
+    for left,right in zip(reversed(forward),backward):
+        k,a,b,c,F,*tail=left;kk,aa,bb,cc,FF,*tt=right
+        assert kk==k and aa==bank(a) and cc==-c,'Reflected inverse shear'
+        assert bb==(bank(b) if k=='gate' else tuple(bank(t) for t in reversed(b))),'Reflected port incidence'
+        assert dual_frames(F,FF,h),'Reflected frame complement'
+        if k=='centre':
+            assert dual_frames(tail[0],tt[0],h) and tail[1:]==tt[1:],'Reflected centre copy'
+        elif k=='read':assert tt==[reverse_row(tail[0])],'Reflected bounded coefficient-row binding'
+        else:assert tail==tt,'Reflected coefficient-row binding'
+
+
+def audit(d):
+    h,v,R=d['h'],d['v'],d['R'];ops=d['ops'];q=d['q'];p=(1<<61)-1
+    trip=list(combinations(range(h),3));assert d['trip']==trip
+    for s in d['role_root']:root_readout(d,s)
+    full=(1<<v)-1;point=[sum(1<<j for j,T in enumerate(trip) if i in T) for i in range(h)]
+    # All fresh-source coefficients, exactly over the integers. No modular
+    # replay is used for this check, and no stored DAG support is trusted.
+    nodes={};values=[0]*R
+    for n in d['order']:
+        a,b=d['args'][n]
+        if not a:nodes[n]=1<<(n-1)
+        else:
+            assert not nodes[a]&nodes[b],('DAG overlap',n)
+            nodes[n]=nodes[a]|nodes[b]
+    for s,n in d['leaf_of'].items():values[s]=1<<(n-1)
+    for o in ops:
+        if o[0]=='src':continue
+        a,b=(o[1],o[2]) if o[0]=='add' else (o[2],o[1])
+        assert not values[a]&values[b],('literal signal overlap',o)
+        values[a]|=values[b]
+        assert values[a]==nodes[o[3]],('literal source coefficients',o)
+    negative=[0]*v;positive=[None]*v;centres={}
+    for s,j in d['role_root'].items():
+        assert values[s]==nodes[d['roots'][j]]
+        if d['kind'][j]:centres[d['centre_of'][j]]=values[s]
+        elif j<v:
+            t=d['target'][j];assert positive[t] is None;positive[t]=values[s]
+        else:
+            t=d['target'][j];assert not negative[t]&values[s];negative[t]|=values[s]
+    assert set(centres)==set(range(h))
+    for i,S in centres.items():assert S==full^point[i]
+    for t,T in enumerate(trip):
+        A,B,C=(point[i] for i in T)
+        assert positive[t]==full^(A|B|C)
+        assert negative[t]==((A&B&~C)|(A&C&~B)|(B&C&~A))
+    assert h-3==21
+    assert [((k-1)+(k==0)-(k==2)) for k in range(4)]==[0,0,0,2]
+    # Every centre decoder coefficient is 1/21-[i in T]/2. Thus the
+    # preceding source-support identities prove J L V = I over Q.
+
+    # Independently form J L on arbitrary scratch as integer centre rows
+    # and signed half-output rows. Compare every coefficient, not hashes.
+    cc=[None]*R;dd=[{} for _ in range(R)]
+    for s,j in d['role_root'].items():
+        if d['kind'][j]:cc[s]=[int(i==d['centre_of'][j]) for i in range(h)]
+        else:dd[s][d['target'][j]]=1 if j<v else -1
+        assert cc[s]==d['seed_c'][s],'Root centre readout'
+        assert {t:(x*pow(2,-1,p))%p for t,x in dd[s].items()}==d['seed_d'][s],'Root signed readout'
+    def addrow(a,b):
+        if cc[b] is not None:
+            cc[a]=list(cc[b]) if cc[a] is None else [x+y for x,y in zip(cc[a],cc[b])]
+        for t,x in dd[b].items():dd[a][t]=dd[a].get(t,0)+x
+    for o in reversed(ops):
+        if o[0]=='add':addrow(o[2],o[1])
+        elif o[0]=='copy':addrow(o[1],o[2])
+    maxcoef=0;readcount=0;unsplit_readcount=0;split_macros=0;actual_reach={};chunk_rows={};coefficient_digest=sha256();chunk_digest=sha256();maxchunk=0
+    for s in range(R):
+        assert cc[s] is None and d['cvec'][s] is None or cc[s] is not None and [x%p for x in cc[s]]==d['cvec'][s]
+        assert {t:(x*pow(2,-1,p))%p for t,x in dd[s].items()}==d['dpart'][s]
+        # Exact aggregate coefficients are logical macros. Compile each
+        # numerator/42 into signed unit chunks and one signed remainder. All
+        # chunks share the same ports/frame and their exact sum is checked.
+        if cc[s] is None:
+            numerators=[(t,21*x) for t,x in sorted(dd[s].items()) if x]
+            coefficient_digest.update(repr((s,numerators)).encode()+b'\n')
+        else:
+            total=2*sum(cc[s]);numerators=[]
+            for t,T in enumerate(trip):
+                num=total-21*sum(cc[s][i] for i in T)+21*dd[s].get(t,0)
+                if num:
+                    numerators.append((t,num))
+                    coefficient_digest.update(s.to_bytes(4,'little')+t.to_bytes(4,'little')+num.to_bytes(8,'little',signed=True))
+        reached=[];records=[]
+        for t,num in numerators:
+            maxcoef=max(maxcoef,abs(num));parts=bounded_chunks(num)
+            unsplit_readcount+=1;split_macros+=int(len(parts)>1)
+            readcount+=len(parts);maxchunk=max(maxchunk,max(map(abs,parts)))
+            for part in parts:
+                reached.append(t);records.append(struct.pack('<Ih',t,part))
+        forward_bytes=b''.join(records);backward_bytes=b''.join(reversed(records))
+        chunk_rows[s]=(sha256(forward_bytes).hexdigest(),sha256(backward_bytes).hexdigest())
+        chunk_digest.update(s.to_bytes(4,'little')+forward_bytes)
+        actual_reach[s]=tuple(reached)
+    assert maxchunk<=42,'Expanded scalar chunk exceeds one in absolute value'
+    # Early/remainder order is a legal commutation of independent shears.
+    # Deferred inputs are untouched by the early word; completed centre
+    # roles are untouched by its remainder. These facts make deferred
+    # readout placement equivalent to the canonical -JL, V, L, J,L^-1,-V.
+    early=set(d['phase1']);assert early==set(d['Anc'])
+    assert sorted(d['phase1'])==d['phase1'] and d['rest']==[i for i in range(len(ops)) if i not in early]
+    actual_last={}
+    for i,o in enumerate(ops):
+        if o[0] in ('add','copy'):
+            for role in o[1:3]:actual_last[role]=i
+    assert actual_last==d['last'],'Exact recomputed last-use map'
+    seen_late=set();early_touched=set();late_touched=set()
+    for i,o in enumerate(ops):
+        touched=set(o[1:3]) if o[0] in ('add','copy') else set()
+        if i in early:
+            assert not touched&seen_late;early_touched.update(touched)
+        else:seen_late.update(touched);late_touched.update(touched)
+    assert not set(d['deferred'])&early_touched
+    assert not set(d['centre_roles'])&late_touched
+    assert all(cc[s] is None for s in d['deferred'])
+    for s in d['deferred']:assert set(actual_reach[s])<=set(d['reach'][s])
+
+    # Frame-labelled literal chronological word. Readout blocks mean the
+    # explicit commuting additions to their individual named target ports;
+    # their scalar count and target incidences are expanded above.
+    ZERO=();FULL=basis(1<<i for i in range(h))
+    U={n:basis(B) for n,B in d['U'].items()}
+    sigma={s:basis(B) for s,B in d['placed'].items()}
+    rootframe={s:basis(B) for s,B in d['root_frame'].items()}
+    tm=[basis((sum(1<<i for i in T),)) for T in trip]
+    merge=check_pairs(d,d['reuse_pairs'])
+    reuse_rejected=mutation_controls(d,d['reuse_pairs'])
+    assert d['compensated_births']==sorted(merge),'Compensated birth inventory'
+    assert d['out']['reuse_rejected_controls']==reuse_rejected
+    assert all(result==(True,True) for result in d['rep']),'Physical aliased numeric replay'
+    assert d['missing_compensation']==(True,False),'Omitted compensation numerical control'
+    live=[s for s in range(R)if s not in merge]
+    assert len(set(merge.values()))==len(merge) and not set(merge)&set(merge.values())
+    for b,a in merge.items():
+        assert a not in sigma and b in sigma and b not in d['leaf_of']
+        assert a not in d['role_root'] and d['last'][a] in early
+        assert b not in early_touched
+        assert all(a not in ops[i][1:3] for i in d['rest'] if ops[i][0]!='src')
+        lastframe=basis(d['op_frames'][d['last'][a]]) if 'op_frames'in d else U[d['holds'][a][-1]]
+        assert contained(lastframe,sigma[b])
+    physical_id={s:i for i,s in enumerate(live)}
+    X=lambda t:t;Y=lambda t:v+t;A=lambda s:2*v+physical_id[merge.get(s,s)]
+    word=[]
+    forward_workspace=[]
+    def gate(a,b,c,F):
+        assert a!=b
+        word.append(('gate',a,b,c,F))
+        if c==1:forward_workspace.append((a,b,c,F))
+    def read(s,sgn,F,targets,row):word.append(('read',A(s),tuple(Y(t) for t in targets),sgn,F,row))
+    def run(i,sign=1,F=None):
+        o=ops[i]
+        if o[0]=='add':gate(A(o[1]),A(o[2]),sign,(basis(d['op_frames'][i]) if 'op_frames' in d else U[o[3]]) if F is None else F)
+        elif o[0]=='copy':gate(A(o[2]),A(o[1]),sign,(basis(d['op_frames'][i]) if 'op_frames' in d else U[o[3]]) if F is None else F)
+    deferred=set(d['deferred'])
+    for s in range(R):
+        if s not in deferred:read(s,-1,ZERO,actual_reach[s],('old',s,*chunk_rows[s]))
+    for s,n in d['leaf_of'].items():
+        if s not in deferred:gate(A(s),X(n-1),1,U[n])
+    for i in d['phase1']:run(i)
+    for s in d['centre_roles']:word.append(('centre',A(s),tuple(Y(t) for t in range(v)),1,rootframe[s],ZERO,d['centre_of'][d['role_root'][s]]))
+    compensated=[]
+    for s in d['deferred']:
+        read(s,-1,sigma[s],actual_reach[s],('old',s,*chunk_rows[s]))
+        compensated.append(s)
+    check_compensation(merge,compensated)
+    for s in d['deferred']:
+        if s in d['leaf_of']:n=d['leaf_of'][s];gate(A(s),X(n-1),1,U[n])
+    for i in d['rest']:run(i)
+    for s,j in d['role_root'].items():
+        if not d['kind'][j]:read(s,1,rootframe[s],(d['target'][j],),('root',21 if j<v else -21))
+    for a,b,c,F in reversed(forward_workspace):gate(a,b,-c,FULL)
+
+    def swap(s):return s+v if s<v else s-v if s<2*v else s
+    def reflect(event):
+        k,a,b,c,F,*rest=event
+        if k=='gate':return (k,swap(a),swap(b),-c,complement(F,h))
+        tail=(complement(rest[0],h),rest[1]) if k=='centre' else (reverse_row(rest[0]),)
+        return (k,swap(a),tuple(swap(t) for t in reversed(b)),-c,complement(F,h),*tail)
+    reverse=[reflect(e) for e in reversed(word)]
+    assert [reflect(e) for e in reversed(reverse)]==word
+    check_reflection(word,reverse,h,v)
+    # Reuse this exact captured word for targeted failure controls.
+    rejected=list(reuse_rejected)
+    for failure,num,parts in (('oversized-readout-chunk',55,(55,)),('split-readout-sum',55,(42,12))):
+        try:check_chunks(num,parts)
+        except AssertionError:rejected.append(failure)
+        else:raise AssertionError('Accepted bounded readout mutation: '+failure)
+    damaged=list(reverse)
+    i=next(i for i,e in enumerate(damaged) if e[0]=='read' and e[-1][0]=='old' and e[-1][2]!=e[-1][3])
+    event=list(damaged[i]);event[-1]=reverse_row(event[-1]);damaged[i]=tuple(event)
+    try:check_reflection(word,damaged,h,v)
+    except AssertionError:rejected.append('reflected-chunk-order')
+    else:raise AssertionError('Accepted reflected chunk-order mutation')
+    for failure in ('inverse-sign','uncomplemented-frame'):
+        damaged=list(reverse);first=list(damaged[0])
+        if failure=='inverse-sign':first[3]*=-1
+        else:first[4]=word[-1][4]
+        damaged[0]=tuple(first)
+        try:check_reflection(word,damaged,h,v)
+        except AssertionError:rejected.append(failure)
+        else:raise AssertionError('Accepted reflection mutation: '+failure)
+    s=next(s for s,j in d['role_root'].items() if not d['kind'][j]);original=d['seed_d'][s]
+    d['seed_d'][s]={t:(-c)%p for t,c in original.items()}
+    try:
+        try:root_readout(d,s)
+        except AssertionError:rejected.append('signed-root-readout')
+        else:raise AssertionError('Accepted signed root mutation')
+    finally:d['seed_d'][s]=original
+
+    def framescan(events,initial,expected):
+        current=list(initial);hist=Counter();copies=Counter();scalar=0;digest=sha256()
+        def promote(s,F):
+            old=current[s];assert contained(old,F),(s,old,F)
+            assert nondeg(F);assert nondeg(complement(F,h))
+            if len(F)>len(old):hist[len(F)-len(old)]+=1
+            current[s]=F
+        for event in events:
+            k,a,b,c,F,*rest=event
+            assert c in (-1,1)
+            promote(a,F)
+            if k=='gate':promote(b,F);scalar+=1
+            elif k=='read':
+                for t in b:promote(t,F)
+                scalar+=len(b)
+            else:
+                assert k=='centre';G=rest[0]
+                assert contained(G,F) or contained(F,G)
+                assert nondeg(G);copies[abs(len(F)-len(G))]+=1
+                for t in b:promote(t,G)
+                scalar+=len(b)+2 # temporary copy and erasure
+            digest.update(repr(event).encode()+b'\n')
+        # Each literal role cleanup reaches full space; any unused final
+        # endpoint is still represented explicitly in the expected chain.
+        for s,F in enumerate(expected):promote(s,F)
+        assert current==expected
+        return hist,copies,scalar,digest.hexdigest()
+    start=tm+[ZERO]*v+[sigma.get(s,ZERO) for s in live]
+    finish=[FULL]*v+[complement(T,h) for T in tm]+[FULL]*len(live)
+    # Independently bind the inventory to the actual literal physical ports.
+    # This reconstruction uses the frame scan's starting list, not the
+    # producer's virtual role inventory or its selected-pair accounting.
+    source_counts=Counter(basis(F) for F in start[2*v:])
+    physical_auxiliary_source_frames=[dict(basis=list(F),count=count)
+        for F,count in sorted(source_counts.items(),key=lambda item:(len(item[0]),item[0]))]
+    assert sum(source_counts.values())==len(live)
+    assert physical_auxiliary_source_frames==d['out']['physical_auxiliary_source_frames'], 'Literal physical source frame inventory'
+    assert all(start[A(a)]==ZERO for a in merge.values()), 'Reused donor starts at zero'
+    assert all(F==FULL for F in finish[2*v:]), 'Completed physical scratch finish'
+    H,C,scalar,digest=framescan(word,start,finish)
+    revstart=[None]*len(start);revfinish=[None]*len(start)
+    for s in range(len(start)):
+        revstart[swap(s)]=complement(finish[s],h);revfinish[swap(s)]=complement(start[s],h)
+    HR,CR,scalarR,reverse_digest=framescan(reverse,revstart,revfinish)
+    assert (H,C,scalar)==(HR,CR,scalarR)
+    z=Counter({r:2*v*n for r,n in (H+C).items()})
+    for s in live:z[h*h-h+len(sigma.get(s,ZERO))]+=2*v
+    z[(h-1)**2]+=2*v*v;z[1]+=v*v
+    z.pop(0,None)
+    assert dict(sorted(z.items()))=={int(r):n for r,n in d['out']['child_multiplicities'].items()},'Literal reflected child histogram'
+    assert sum(r*n for r,n in z.items())==d['out']['total_rank']
+    G=v*v+2*v*scalar
+    safe=8*(d['c_add']+2*R+(R+q)*v*(h+1)+h*h+h+1) if 'c_add' in d else 8*(d['out']['additions']+2*R+(R+q)*v*(h+1)+h*h+h+1)
+    assert safe>=scalar
+    return dict(h=h,v=v,R=len(live),virtual_R=R,reused_roles=len(merge),exact_birth_cut_invariants=True,last_uses_recomputed=True,physical_aliased_numeric_replay=True,omitted_compensation_numeric_control=True,inverse_source_order='True reverse chronological source/workspace word',exact_fresh_source_map=True,exact_integer_old_readout_transpose=True,
+                physical_auxiliary_source_frames=physical_auxiliary_source_frames,
+                completed_core_source_inventory_bound=True,completed_core_pre_exterior_frames_full=True,
+                reflected_core_active_frames_complement_source=all(revfinish[2*v+i]==complement(start[2*v+i],h) for i in range(len(live))),
+                exact_arbitrary_dirty_cancellation_by_dependency_cut=True,
+                dirty_reuse_identity="Every recipient is untouched before its compensated birth; donor has no future gate/root reads. Recipient old-response coefficient cancels its arbitrary current donor value. All physical shears are reversed in true chronology.",
+                reflected_word_rule='Reverse literal order including each bounded readout chunk, negate every shear, swap X/Y ports, complement every frame; auxiliary bank is separate.',
+                reflected_scalar_map='X becomes X-Y; Y unchanged; all auxiliary coordinates restored.',
+                two_stage_scalar_map='(X,Y) becomes (-Y,X+Y) before inherited endpoint correction.',
+                literal_frame_incidences_both_directions=True,all_frames_nondegenerate=True,
+                reflected_residual_rank_histogram_equal=True,child_multiplicities=dict(sorted(z.items())),
+                word_blocks=len(word),expanded_scalar_operations_per_stage=scalar,
+                expanded_old_readout_additions=readcount,largest_readout_numerator_over_42=maxcoef,
+                unsplit_old_readout_macros=unsplit_readcount,split_readout_macros=split_macros,
+                largest_bounded_chunk_numerator_over_42=maxchunk,
+                bounded_readout_chunks_sha256=chunk_digest.hexdigest(),
+                scalar_readout_normalization='Each exact numerator n/42 becomes signed42 chunks plus a signed remainder. Exact reconstruction and chunk magnitude<=1 checked; every same-frame shear is charged; reflection reverses chunk order.',
+                exact_chunk_reconstruction=True,bounded_chunk_coefficients=True,
+                old_readout_coefficients_over_42_sha256=coefficient_digest.hexdigest(),
+                literal_global_scalar_groups=G,conservative_local_G=safe,
+                forward_incidence_sha256=digest,reflected_incidence_sha256=reverse_digest,
+                rejected_controls=rejected,
+                scope='Exact finite source coefficients, dirty cancellation algebra and literal reflected frame/charge ledger. Copied-centre transforms, endpoint correction and residual-to-child/all-size transfer retain their stated inherited contracts.')
+
+
+def main():
+    ap=argparse.ArgumentParser();ap.add_argument('--source',type=Path);ap.add_argument('--state',type=Path)
+    ap.add_argument('--output',type=Path);ap.add_argument('--check',type=Path);a=ap.parse_args()
+    assert bool(a.source)!=bool(a.state)
+    d=capture(a.source) if a.source else pickle.loads(a.state.read_bytes())
+    result=audit(d);source=Path(d['source'])
+    result['source_sha256']=sha256(source.read_bytes()).hexdigest()
+    result['audit_sha256']=sha256(Path(__file__).read_bytes()).hexdigest()
+    result=json.loads(json.dumps(result));encoded=json.dumps(result,indent=2,sort_keys=True)+'\n'
+    if a.check:assert result==json.loads(a.check.read_text())
+    if a.output:a.output.write_text(encoded)
+    else:print(encoded,end='')
+    print('PASS exact scalar and literal complement reflection audit',file=sys.stderr)
+
+
+if __name__=='__main__':main()
diff --git a/research/cyclic-deferred/replayed_producer.py b/research/cyclic-deferred/replayed_producer.py
new file mode 100644
index 0000000000..673fbbba70
--- /dev/null
+++ b/research/cyclic-deferred/replayed_producer.py
@@ -0,0 +1,179 @@
+"""Replayed h=24 complex producer with all centers disjoint (rational divisor 21).
+
+Copyright 2026 eumemic, Apache-2.0. AI-assisted (Claude). The DAG was found by an
+automated search over producers of the retained paired-triple, pair-star and
+center-disjoint interfaces. The checks below are adapted from
+stopped_product/complex.py (copyright 2026 icekylinx, Apache-2.0, PR #104).
+
+The witness complex-dag.json.gz lists the ordinary
+additions after the v triple inputs and the retained roots: the disjoint sums
+D[t], the pair stars P(a,b,i) and the center-disjoint sums A_i. Every support,
+core, cover, type and rank is recomputed here from the operands; nothing is
+read from the witness except the operand pairs and the root indices.
+"""
+import gzip, json, struct, sys, array
+from fractions import Fraction
+from itertools import combinations
+from pathlib import Path
+
+
+def write_array(f, values):
+    """Little-endian array writer, as partial_swap.binary.write_array."""
+    if sys.byteorder != 'little' and values.itemsize > 1:
+        values = array.array(values.typecode, values)
+        values.byteswap()
+    values.tofile(f)
+
+
+def build(witness, prefix):
+    w = json.loads(gzip.decompress(Path(witness).read_bytes()))
+    h, v, d = w['h'], w['v'], w['central_disjoint']
+    assert (h, d, w['center_denominator']) == (24, 24, 21)
+    triples = list(combinations(range(h), 3))
+    assert v == len(triples)
+    args = [(0, 0)]; support = [0]; core = [0]; cover = [0]; types = [0]; ranks = [0]
+    for j, t in enumerate(triples):
+        mask = sum(1 << a for a in t)
+        args.append((0, 0)); support.append(1 << j); core.append(mask); cover.append(mask)
+        types.append(1); ranks.append(1)
+    flat = w['args']
+    assert len(flat) % 2 == 0
+    for k in range(0, len(flat), 2):
+        a, b, x = flat[k], flat[k+1], len(args)
+        assert 0 < a < x and 0 < b < x, 'Operands must precede their sum'
+        assert not support[a] & support[b], 'Ordinary additions must be cancellation-free'
+        args.append((a, b)); support.append(support[a] | support[b])
+        core.append(core[a] & core[b]); cover.append(cover[a] | cover[b])
+        if core[x].bit_count() >= 2:
+            types.append(1); ranks.append(support[x].bit_count())
+        else:
+            types.append(2); ranks.append(cover[x].bit_count())
+    n = len(args)
+    point_masks = [sum(1 << j for j, t in enumerate(triples) if a in t) for a in range(h)]
+    full_support = (1 << v) - 1
+    roots, kind = [], []
+    assert len(w['D']) == v
+    for t, node in zip(triples, w['D']):
+        assert 0 < node < n
+        assert support[node] == full_support & ~(point_masks[t[0]]|point_masks[t[1]]|point_masks[t[2]])
+        target_mask = sum(1 << a for a in t)
+        if types[node] == 2:
+            assert not cover[node] & target_mask
+        else:
+            assert types[node] == 1
+            assert not support[node] & (point_masks[t[0]]^point_masks[t[1]]^point_masks[t[2]])
+        roots.append(node); kind.append(0)
+    stars = [(a, b, i) for a, b in combinations(range(h), 2) for i in range(h) if i not in (a, b)]
+    assert len(w['P']) == len(stars)
+    pair_outputs = {}
+    for (a, b, i), node in zip(stars, w['P']):
+        assert 0 < node < n
+        assert support[node] == point_masks[a] & point_masks[b] & ~point_masks[i]
+        assert types[node] == 1
+        assert not support[node] & (point_masks[a]^point_masks[b]^point_masks[i])
+        pair_outputs[a, b, i] = node
+        roots.append(node); kind.append(0)
+    assert len(w['A']) == d
+    for i, node in enumerate(w['A']):
+        assert 0 < node < n
+        assert support[node] == full_support & ~point_masks[i]
+        assert types[node] == 2 and cover[node] == ((1 << h)-1) ^ (1 << i)
+        assert ranks[node] == h-1
+        roots.append(node); kind.append(1)
+    # Every A_i avoids i, so sum A_i = (h-3)T. The decoder is T=sum A_i/21
+    # and the scatter is T-(1/2)sum_{i in target} A_i.
+    decoder = Fraction(1, h-3)
+    assert decoder == Fraction(1, 21)
+    scatter_coefficients = [decoder, decoder-Fraction(1, 2)]
+    for coefficient in scatter_coefficients:
+        odd_denominator = coefficient.denominator
+        while odd_denominator % 2 == 0:
+            odd_denominator //= 2
+        assert 21 % odd_denominator == 0 and abs(coefficient) <= 1
+    for source in triples:
+        assert sum(i not in source for i in range(h))*decoder == 1
+    Dset = set(range(d))
+    for triple in triples:
+        k = len(Dset.intersection(triple))
+        assert 2*(3-k)-2*(d-k) == 2*(3-d)
+    patterns = {}
+    for target in triples:
+        chosen = len(Dset.intersection(target))
+        if chosen not in patterns:
+            patterns[chosen] = target
+    for chosen, target in patterns.items():
+        target_set = set(target)
+        for source in triples:
+            k = len(Dset.intersection(source))
+            total_coefficient = Fraction(2*(3-k)-2*(d-k), 2*(3-d))
+            scalar = Fraction(len((target_set-Dset).intersection(source)), 2)
+            scalar -= Fraction(len((target_set & Dset)-set(source)), 2)
+            scalar += Fraction(chosen-1, 2)*total_coefficient
+            assert scalar == Fraction(len(target_set.intersection(source))-1, 2)
+    for j, t in enumerate(triples):
+        masks = [point_masks[a] for a in t]
+        two = (masks[0]&masks[1]&~masks[2]) | (masks[0]&masks[2]&~masks[1]) | (masks[1]&masks[2]&~masks[0])
+        actual = 0
+        for a, b in combinations(t, 2):
+            excluded = next(i for i in t if i not in (a, b))
+            actual |= support[pair_outputs[a, b, excluded]]
+        assert actual == two
+        assert masks[0]&masks[1]&masks[2] == 1 << j
+    assert [((k-1)+(k == 0)-(k == 2)) for k in range(4)] == [0, 0, 0, 2]
+
+    def contained(x, y):
+        tx, ty = types[x], types[y]
+        if tx == 1 and ty == 1: return not(core[y]&~core[x] or cover[x]&~cover[y])
+        if tx in (1, 2) and ty == 2: return not cover[x]&~cover[y]
+        return False
+    active = [0]*n; stack = list(roots)
+    while stack:
+        x = stack.pop()
+        if not x or active[x]: continue
+        active[x] = 1
+        if args[x][0]: stack.extend(args[x])
+    degree = [0]*n
+    for x in range(1, n):
+        if active[x]:
+            if types[x] == 1:
+                assert core[x].bit_count() >= 2 and ranks[x] == support[x].bit_count()
+            else:
+                assert types[x] == 2 and ranks[x] == cover[x].bit_count()
+        if active[x] and args[x][0]:
+            for y in args[x]:
+                assert y < x and contained(y, x), 'Binary frame nesting failed'
+                degree[y] += 1
+    for x in roots: degree[x] += 1
+    H = [0]*(h+1); c = 0; loss = 0
+    for x in range(1, n):
+        if not active[x]: continue
+        r = ranks[x]
+        if args[x][0]:
+            c += 1; H[r] += degree[x]-1; H[h-r] += 1
+            for y in args[x]:
+                assert r >= ranks[y]
+                H[r-ranks[y]] += 1
+        else: H[1] += degree[x]
+    for x, k in zip(roots, kind):
+        r = ranks[x]
+        if k: H[r] += 1; H[h] += 1; loss += r
+        else: H[h-1-r] += 1; H[1] += 1
+    q = len(roots); R = c+q
+    result = dict(h=h, v=v, c=c, q=q, R=R, loss=loss, central_disjoint=d, histogram=H,
+                  rank_sum=sum(r*m for r, m in enumerate(H)))
+    assert loss == h*(h-1) and result['rank_sum'] == h*R+2*loss
+    result['center_denominator'] = 21
+    result['scalar_validation'] = dict(ordinary_supports_exact=True, centers_exact=True,
+        center_decoder='sum A_i / 21', scatter_coefficients=list(map(str, scatter_coefficients)),
+        rational_scalar_identity_exact=True, mixed_center_scatter_exact=True, binary_frames_nested=True,
+        binary_frames_nondegenerate=True)
+    with open(str(prefix)+'.bin', 'wb') as f:
+        f.write(struct.pack('<4I', h, v, n, q))
+        write_array(f, array.array('I', (a for pair in args for a in pair)))
+        write_array(f, array.array('Q', core)); write_array(f, array.array('Q', cover))
+        write_array(f, array.array('I', roots)); write_array(f, array.array('I', kind))
+        write_array(f, array.array('B', active))
+    with open(str(prefix)+'.labels', 'wb') as f:
+        write_array(f, array.array('I', ranks)); write_array(f, array.array('B', types))
+    Path(str(prefix)+'.json').write_text(json.dumps(result, indent=2)+'\n')
+    return result
diff --git a/research/cyclic-deferred/reuse-pairs.json b/research/cyclic-deferred/reuse-pairs.json
new file mode 100644
index 0000000000..5e32cba92a
--- /dev/null
+++ b/research/cyclic-deferred/reuse-pairs.json
@@ -0,0 +1,73016 @@
+[
+ {
+  "donor": 1719,
+  "recipient": 23014,
+  "donor_frame": [
+   8,
+   4,
+   2,
+   1
+  ],
+  "birth_frame": [
+   8,
+   4,
+   2,
+   1
+  ],
+  "e": 4,
+  "s": 4
+ },
+ {
+  "donor": 13472,
+  "recipient": 25594,
+  "donor_frame": [
+   8388608,
+   4194304,
+   2097152,
+   1048576
+  ],
+  "birth_frame": [
+   8388608,
+   4194304,
+   2097152,
+   1048576
+  ],
+  "e": 4,
+  "s": 4
+ },
+ {
+  "donor": 6812,
+  "recipient": 25569,
+  "donor_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   128,
+   64,
+   32,
+   16
+  ],
+  "birth_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   128,
+   64,
+   32,
+   16
+  ],
+  "e": 8,
+  "s": 8
+ },
+ {
+  "donor": 6061,
+  "recipient": 25866,
+  "donor_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   128,
+   64,
+   32,
+   16
+  ],
+  "birth_frame": [
+   524288,
+   262144,
+   131072,
+   65536,
+   128,
+   64,
+   32,
+   16
+  ],
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+  ],
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+ },
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+   128,
+   2,
+   1
+  ],
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+  "s": 19
+ },
+ {
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+   4
+  ],
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+ },
+ {
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diff --git a/research/cyclic-deferred/reuse.py b/research/cyclic-deferred/reuse.py
new file mode 100644
index 0000000000..2c4a6daa29
--- /dev/null
+++ b/research/cyclic-deferred/reuse.py
@@ -0,0 +1,167 @@
+"""Deterministic compensated birth-cut scratch reuse.
+
+Birth-cut reuse follows jamesyc PR124. Composition with saturated deferrals
+and physical hull frames prepared for eumemic with OpenAI Codex assistance.
+Apache-2.0; inherited authorship remains in the surrounding package.
+"""
+from collections import defaultdict
+from functools import lru_cache
+
+
+def basis(vectors):
+    piv = {}
+    for x in vectors:
+        for p in sorted(piv, reverse=True):
+            if x >> p & 1:
+                x ^= piv[p]
+        if x:
+            p = x.bit_length() - 1
+            for q in piv:
+                if piv[q] >> p & 1:
+                    piv[q] ^= x
+            piv[p] = x
+    return tuple(piv[p] for p in sorted(piv, reverse=True))
+
+
+def contained(A, B):
+    return len(basis(tuple(A) + tuple(B))) == len(B)
+
+
+def complement(A, h):
+    A = basis(A)
+    piv = {x.bit_length() - 1: x for x in A}
+    out = []
+    for j in range(h):
+        if j not in piv:
+            x = 1 << j
+            for p, row in piv.items():
+                if row >> j & 1:
+                    x |= 1 << p
+            out.append(x)
+    return basis(out)
+
+
+def nondeg(A):
+    return len(basis(sum(((x & y).bit_count() & 1) << j
+                         for j, y in enumerate(A)) for x in A)) == len(A)
+
+
+def check_pairs(data, pairs):
+    """Reject aliases, unfinished donors and unavailable recipient births."""
+    donors = [row['donor'] for row in pairs]
+    recipients = [row['recipient'] for row in pairs]
+    assert (len(set(donors)) == len(donors) and
+            len(set(recipients)) == len(recipients) and
+            not set(donors) & set(recipients)), 'Reuse pair alias'
+    early = set(data['Anc'])
+    first = {}
+    late_touched = set()
+    for i, op in enumerate(data['ops']):
+        if op[0] == 'src':
+            first.setdefault(op[1], i)
+            continue
+        for s in op[1:3]:
+            first.setdefault(s, i)
+        if i not in early:
+            late_touched.update(op[1:3])
+    for row in pairs:
+        a, b = row['donor'], row['recipient']
+        assert type(a) is int and type(b) is int, 'Reuse role type'
+        assert 0 <= a < data['R'] and 0 <= b < data['R'], 'Reuse role range'
+        assert (a not in data['placed'] and a not in data['role_root'] and
+                data['last'].get(a) in early and a not in late_touched), 'Reuse nondead donor'
+        assert (b in data['placed'] and b not in data['leaf_of'] and
+                b not in data['touched'] and first.get(b) not in early), 'Reuse unavailable birth'
+        birth = data['ops'][first[b]]
+        assert birth[0] == 'copy' and birth[2] == b, 'Reuse unavailable birth'
+        A = basis(data['op_frames'][data['last'][a]])
+        F = basis(data['placed'][b])
+        B = basis(data['op_frames'][first[b]])
+        assert nondeg(A) and nondeg(F) and nondeg(B), 'Reuse degenerate frame'
+        assert contained(A, F) and contained(F, B), 'Reuse frame containment'
+        assert basis(row['donor_frame']) == A and basis(row['birth_frame']) == F, 'Reuse recorded frames'
+        assert (row['e'], row['s']) == (len(A), len(F)), 'Reuse recorded dimensions'
+    return dict(zip(recipients, donors))
+
+
+def check_compensation(merge, compensated):
+    assert set(merge) <= set(compensated), 'Reuse omitted compensation'
+
+
+def select_reuse(data):
+    groups = defaultdict(list)
+    for a, i in data['last'].items():
+        if i in data['Anc'] and a not in data['role_root']:
+            A = basis(data['op_frames'][i])
+            assert nondeg(A)
+            groups[A].append(a)
+    frames = sorted(groups, key=lambda F: (-len(F), F))
+    pool = [groups[F][:] for F in frames]
+    births = defaultdict(list)
+    for b, F in data['placed'].items():
+        if b not in data['leaf_of']:
+            births[basis(F)].append(b)
+    allgroups = (1 << len(frames)) - 1
+
+    @lru_cache(None)
+    def annihilated(q):
+        return sum(1 << i for i, A in enumerate(frames)
+                   if all(not ((a & q).bit_count() & 1) for a in A))
+
+    eligible, capacity = {}, {}
+    for F in births:
+        mask = allgroups
+        for q in complement(F, data['h']):
+            mask &= annihilated(q)
+            if not mask:
+                break
+        eligible[F] = mask
+        capacity[F] = sum(len(pool[i]) for i in range(len(frames)) if mask >> i & 1)
+    available, pairs = allgroups, []
+    for F in sorted(births, key=lambda F: (capacity[F], len(F), F)):
+        for b in sorted(births[F]):
+            mask = eligible[F] & available
+            if not mask:
+                break
+            i = (mask & -mask).bit_length() - 1
+            a = pool[i].pop()
+            A = frames[i]
+            if not pool[i]:
+                available ^= 1 << i
+            pairs.append(dict(donor=a, recipient=b, donor_frame=A,
+                              birth_frame=F, e=len(A), s=len(F)))
+    check_pairs(data, pairs)
+    return pairs
+
+
+def mutation_controls(data, pairs):
+    """Mutate the actual selected finite pairing, not a synthetic inventory."""
+    import copy
+    rejected = []
+    occupied = {row[key] for row in pairs for key in ('donor', 'recipient')}
+    for failure in ('nondead-donor', 'unavailable-birth', 'pair-alias'):
+        changed = copy.deepcopy(pairs)
+        if failure == 'nondead-donor':
+            changed[0]['donor'] = next(s for s in data['role_root'] if s not in occupied)
+        elif failure == 'unavailable-birth':
+            changed[0]['recipient'] = next(s for s in data['leaf_of'] if s not in occupied)
+        else:
+            changed[0]['recipient'] = changed[0]['donor']
+        try:
+            check_pairs(data, changed)
+        except AssertionError as error:
+            expected = {'nondead-donor': 'nondead donor',
+                        'unavailable-birth': 'unavailable birth',
+                        'pair-alias': 'pair alias'}[failure]
+            assert expected in str(error), (failure, error)
+            rejected.append(failure)
+        else:
+            raise AssertionError('Accepted reuse mutation: ' + failure)
+    merge = check_pairs(data, pairs)
+    try:
+        check_compensation(merge, set(merge) - {pairs[0]['recipient']})
+    except AssertionError:
+        rejected.append('omitted-compensation')
+    else:
+        raise AssertionError('Accepted omitted birth compensation')
+    return rejected
diff --git a/research/cyclic-deferred/shared-bit-profile.json b/research/cyclic-deferred/shared-bit-profile.json
new file mode 100644
index 0000000000..be4def1883
--- /dev/null
+++ b/research/cyclic-deferred/shared-bit-profile.json
@@ -0,0 +1,247 @@
+{
+  "h": 23,
+  "m": 529,
+  "v": 1771,
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+  "R": 28866,
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+  "groups": 161,
+  "group_sizes": {
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+  },
+  "W": 15567734,
+  "total_rank": 8233987097,
+  "deficit": 1344189,
+  "maxchild": 518,
+  "child_multiplicities": {
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+    "3": 51022510,
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+  "exact_coverage": true,
+  "exact_pairwise_H_orthogonality": true,
+  "exact_projector_idempotence": true,
+  "exact_projector_self_adjointness": true,
+  "scope": "Exact completed-core rational projector identities and complete one-child profile. Requires unchanged PR128 minimal-V literal-word contract, common generic-basis/opposite-bank compiler and scalar/adapter work charge. Each recorded projector call pays its fixed PR104 atom adapters; the generic-basis existence argument does not specify a numerical primitive-operation constant."
+}
diff --git a/research/cyclic-deferred/shared-complex-profile.json b/research/cyclic-deferred/shared-complex-profile.json
new file mode 100644
index 0000000000..8c756789b6
--- /dev/null
+++ b/research/cyclic-deferred/shared-complex-profile.json
@@ -0,0 +1,10576 @@
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+ ],
+ "deferred_roles": 4706,
+ "deferred_dims": {
+  "2": 863,
+  "4": 357,
+  "6": 573,
+  "8": 144,
+  "10": 25,
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+  "20": 208,
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+  "22": 120
+ },
+ "lifted_additions": 38114,
+ "physical_gate_frame_changes": 19323,
+ "replay": {
+  "seeds": [
+   1,
+   2
+  ],
+  "scratch_restored": true,
+  "y_plus_x": true,
+  "field": "Z/(2^61-1)",
+  "physically_aliased": true,
+  "inverse_order": "true source/workspace chronology",
+  "omitted_compensation_rejected": true
+ },
+ "reuse_rejected_controls": [
+  "nondead-donor",
+  "unavailable-birth",
+  "pair-alias",
+  "omitted-compensation"
+ ],
+ "child_multiplicities": {
+  "1": 408795376,
+  "2": 291621968,
+  "3": 52656384,
+  "4": 51041232,
+  "5": 21498928,
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+  "21": 1230592,
+  "22": 64768,
+  "23": 8363168,
+  "48": 143258,
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+  "192": 18592,
+  "240": 1992,
+  "384": 191992,
+  "400": 6904,
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+  "528": 12082,
+  "529": 8193152,
+  "536": 1616,
+  "544": 1632,
+  "552": 88,
+  "560": 504
+ },
+ "shared_groups": 87,
+ "group_sizes": {
+  "8": 4,
+  "24": 83
+ },
+ "old_exterior_histogram": {
+  "552": 97147952,
+  "554": 3493424,
+  "556": 1437040,
+  "558": 2319504,
+  "560": 453376,
+  "562": 48576,
+  "570": 821744,
+  "571": 817696,
+  "572": 825792,
+  "573": 44528,
+  "574": 255024
+ },
+ "grouped_exterior_histogram": {
+  "48": 143258,
+  "96": 58930,
+  "144": 95118,
+  "192": 18592,
+  "240": 1992,
+  "384": 191992,
+  "400": 6904,
+  "416": 2840,
+  "432": 38282,
+  "448": 896,
+  "456": 33532,
+  "464": 96,
+  "480": 33864,
+  "504": 1826,
+  "528": 12082,
+  "536": 1616,
+  "544": 1632,
+  "552": 88,
+  "560": 504
+ },
+ "core_source_inventory_sha256": "af8831a76a499d33ce374f5c361b1a1903334dd34d01de696fb5218fee34b514",
+ "core_pre_exterior_inner_sink_rank": 24,
+ "core_active_rank_formula": "24-dim(physical source frame)",
+ "unshared_W": 115857808,
+ "unshared_rank": 66732235328
+}
diff --git a/research/cyclic-deferred/sharing.py b/research/cyclic-deferred/sharing.py
new file mode 100644
index 0000000000..2a13d59ffd
--- /dev/null
+++ b/research/cyclic-deferred/sharing.py
@@ -0,0 +1,67 @@
+#!/usr/bin/env python3
+"""Complete shared-core profile from the audited physical source-frame inventory.
+
+Completed-core sharing and signed outer partition: an664 PR128, commit
+530588a019b4a74f09180680c9e3961bf649ec89, with OpenAI Codex assistance.
+Partition: Xiande Zhang and Gennian Ge, JCD 18 (2010), 209-223.
+Generalized physical gauges and compensated reuse composition: eumemic with
+OpenAI Codex assistance. Reuse follows jamesyc PR124; original notices remain.
+Apache-2.0. This finite profile retains the stated inherited core interfaces.
+"""
+from pathlib import Path
+from collections import Counter
+from hashlib import sha256
+import json,sys
+from gauge_phase import inventory_digest,validate_inventory
+assert not sys.flags.optimize, 'Assertions must remain enabled'
+HERE=Path(__file__).resolve().parent
+PARTITION_PIN='9117d3f010354fa2f32d5c5fb324b280977e82e76060a70a867767a90989ccaa'
+
+def profile(local,partition,phase,gauge):
+    h,v,R,m,N=(local[k]for k in ('h','v','R','m','N'))
+    assert(h,v,R,m,N)==(24,2024,26597,576,4096576)
+    assert local['virtual_R']==28705 and local['reused_roles']==2108
+    inventory=local['physical_auxiliary_source_frames'];counts=validate_inventory(inventory,R,h)
+    groups=Counter(map(len,partition['groups']))
+    assert groups=={24:83,8:4}
+    assert sorted(t for G in partition['groups']for t in G)==list(range(v))
+    assert phase['partition_sha256']==gauge['partition_sha256']==PARTITION_PIN
+    for key in ('all_triples_once','all_actual_Z4_phase_identities','all_complement_bases_explicit','all_tensor_scalar_phases_one','exact_inverse_direction_identity','exact_gaussian_composition','exact_stage2_gauge','wrong_source_sign_negative_control'):
+        assert phase[key]is True, 'Signed outer partition phase contract: '+key
+    for key in ('all_local_projectors_exact','all_source_frames_nondegenerate','all_grouped_complement_scalars_one','signed_stage_residuals_equal','exact_physical_source_inventory_bound','wrong_source_sign_rejected'):
+        assert gauge[key]is True, 'Physical gauge phase contract: '+key
+    assert gauge['R']==R and gauge['physical_auxiliary_source_frames']==inventory
+    assert gauge['source_inventory_sha256']==inventory_digest(inventory)
+    assert {int(g):n for g,n in gauge['group_sizes'].items()}==dict(groups)
+    z=Counter({int(t):n for t,n in local['child_multiplicities'].items()})
+    old=Counter();new=Counter()
+    for F,n in counts.items():
+        old[m-h+len(F)]+=2*v*n
+        for g,k in groups.items():
+            width=m-g*(h-len(F))
+            assert 0<=width<m
+            if width:new[width]+=2*k*n
+    assert {r:n for r,n in z.items()if r>=m-h}==dict(old), 'Complete local auxiliary exterior inventory'
+    assert sum(old.values())==2*v*R
+    z.subtract(old);z.update(new);z=Counter({r:n for r,n in z.items()if n})
+    assert dict(new)=={int(r):n for r,n in gauge['grouped_exterior_histogram'].items()}, 'Exact gauge complement profile'
+    W=2*N+2*sum(groups.values())*R;mass=sum(r*n for r,n in z.items())
+    assert all(0<r<m and type(n)is int and n>0 for r,n in z.items())
+    assert mass==W*m-N+local['L'] and W*m-mass==local['deficit']
+    out=dict(local);out.update(W=W,total_rank=mass,maxchild=max(z),child_multiplicities=dict(sorted(z.items())),
+        shared_groups=sum(groups.values()),group_sizes=dict(sorted(groups.items())),
+        old_exterior_histogram=dict(sorted(old.items())),grouped_exterior_histogram=dict(sorted(new.items())),
+        core_source_inventory_sha256=inventory_digest(inventory),core_pre_exterior_inner_sink_rank=h,
+        core_active_rank_formula='24-dim(physical source frame)',unshared_W=local['W'],unshared_rank=local['total_rank'])
+    return out
+
+def main():
+    lp=HERE/'complex-profile.json';pp=HERE/'inputs/shared-partition.json';gp=HERE/'gauge-phase-audit.json';rp=HERE/'reflection-audit.json'
+    assert sha256(pp.read_bytes()).hexdigest()==PARTITION_PIN
+    gauge=json.loads(gp.read_text())
+    assert gauge['local_profile_sha256']==sha256(lp.read_bytes()).hexdigest()
+    assert gauge['reflection_receipt_sha256']==sha256(rp.read_bytes()).hexdigest()
+    result=profile(json.loads(lp.read_text()),json.loads(pp.read_text()),json.loads((HERE/'phase_check.json').read_text()),gauge)
+    (HERE/'shared-complex-profile.json').write_text(json.dumps(result,indent=1)+'\n')
+    print('PASS shared physical-gauge profile',result['R'],result['W'],result['total_rank'],result['maxchild'])
+if __name__=='__main__':main()
diff --git a/research/cyclic-deferred/test_controls.py b/research/cyclic-deferred/test_controls.py
new file mode 100644
index 0000000000..26ceb091bd
--- /dev/null
+++ b/research/cyclic-deferred/test_controls.py
@@ -0,0 +1,343 @@
+#!/usr/bin/env python3
+"""Negative controls for the cyclic/deferred frozen finite witness.
+
+These controls target source omissions, altered inputs, paid rank accounting,
+noncontracting children and invalid exact arithmetic. They do not substitute
+for the full producer and literal reflection checks. Prepared for eumemic with
+OpenAI Codex assistance; Apache-2.0, inherited notices retained.
+"""
+import sys
+sys.dont_write_bytecode = True
+if sys.flags.optimize:
+    raise ValueError('Negative controls require assertions')
+
+import argparse
+import copy
+import gzip
+import importlib.util
+import json
+from pathlib import Path
+import shutil
+import subprocess
+import tempfile
+import unittest
+
+from verify import (check_completed_core_audit, check_sources, compare_json,
+                    digest, safe_file, validate_profile, validate_shared_composition,
+                    validate_bit_composition)
+
+HERE = Path(__file__).resolve().parent
+REPOSITORY = HERE.parents[1]
+
+
+class FrozenControls(unittest.TestCase):
+    @classmethod
+    def setUpClass(cls):
+        cls.manifest = check_sources(HERE, REPOSITORY)
+        cls.profiles = {label: json.loads((HERE / (label + '-profile.json')).read_text())
+                        for label in ('bit', 'shared-bit', 'complex', 'shared-complex')}
+        sys.path.insert(0, str(REPOSITORY / 'scripts'))
+        spec = importlib.util.spec_from_file_location('checked_cyclic_certificate', HERE / 'certificate.py')
+        cls.certificate = importlib.util.module_from_spec(spec)
+        spec.loader.exec_module(cls.certificate)
+        cls.certificate.ROOT = REPOSITORY
+
+    def test_all_frozen_profiles_satisfy_physical_ledgers(self):
+        for label, profile in self.profiles.items():
+            validate_profile(profile, label)
+
+    def test_frozen_exact_certificate_recomputed(self):
+        expected = json.loads((HERE / 'certificate.json').read_text())
+        actual = self.certificate.js(self.certificate.exact())
+        self.assertEqual(actual, expected)
+
+    def test_shared_profile_reconstructed_from_paid_local_cores(self):
+        validate_shared_composition(HERE)
+        validate_bit_composition(HERE, json.loads((HERE / 'certificate.json').read_text()))
+
+    def test_wrong_shared_width_rejected(self):
+        changed = copy.deepcopy(self.profiles['shared-complex'])
+        changed['W'] += 2 * changed['R']
+        with self.assertRaisesRegex(ValueError, 'physical ledger'):
+            validate_profile(changed, 'shared-complex')
+
+    def test_missing_shared_exterior_rejected(self):
+        changed = copy.deepcopy(self.profiles['shared-complex'])
+        changed['child_multiplicities']['384'] -= 2 * changed['R']
+        with self.assertRaisesRegex(ValueError, 'rank mass'):
+            validate_profile(changed, 'shared-complex')
+
+    def test_nonorthogonal_partition_rejected(self):
+        from itertools import combinations
+        partition = json.loads((HERE / 'inputs/shared-partition.json').read_text())
+        masks = [sum(1 << j for j in triple) for triple in combinations(range(24), 3)]
+        groups = partition['groups']
+        i, j = next((i, j) for i in range(1, len(groups)) for j, t in enumerate(groups[i])
+                    if any((masks[t] & masks[u]).bit_count() & 1 for u in groups[0][1:]))
+        groups[0][0], groups[i][j] = groups[i][j], groups[0][0]
+        with tempfile.TemporaryDirectory(prefix='shared-core-nonorthogonal-') as directory:
+            path = Path(directory)
+            shutil.copyfile(HERE / 'phase_check.py', path / 'phase_check.py')
+            (path / 'partition.json').write_text(json.dumps(partition))
+            result = subprocess.run([sys.executable, '-B', str(path / 'phase_check.py'),
+                                     str(path / 'partition.json')], cwd=path,
+                                    capture_output=True, text=True)
+            self.assertNotEqual(result.returncode, 0)
+            self.assertIn('non-orthonormal completed group basis', result.stderr)
+
+    def test_understated_scalar_guard_and_unbound_core_source_rejected(self):
+        audit = json.loads((HERE / 'reflection-audit.json').read_text())
+        certificate = json.loads((HERE / 'certificate.json').read_text())
+        local = self.profiles['complex']
+        check_completed_core_audit(audit, local, certificate)
+        changed = copy.deepcopy(certificate)
+        changed['finite_bridge']['complex']['scalar_group_upper'] = audit['literal_global_scalar_groups'] - 1
+        with self.assertRaisesRegex(ValueError, 'scalar charge exceeds'):
+            check_completed_core_audit(audit, local, changed)
+        changed = copy.deepcopy(certificate)
+        account = changed['finite_bridge']['complex']
+        account['scalar_group_upper'] -= account['shared_basis_phase_group_upper']
+        account['shared_basis_phase_group_upper'] = 0
+        with self.assertRaisesRegex(ValueError, 'Missing paid shared basis'):
+            check_completed_core_audit(audit, local, changed)
+        changed = copy.deepcopy(audit)
+        changed['completed_core_source_inventory_bound'] = False
+        with self.assertRaisesRegex(ValueError, 'bound source'):
+            check_completed_core_audit(changed, local, certificate)
+
+    def test_resurrected_role_and_duplicate_source_frame_rejected(self):
+        from gauge_phase import validate_inventory
+        local = self.profiles['complex']
+        inventory = copy.deepcopy(local['physical_auxiliary_source_frames'])
+        inventory[0]['count'] += 1
+        with self.assertRaisesRegex(AssertionError, 'role count'):
+            validate_inventory(inventory, local['R'], local['h'])
+        inventory = copy.deepcopy(local['physical_auxiliary_source_frames'])
+        inventory.append(copy.deepcopy(inventory[0]))
+        with self.assertRaisesRegex(AssertionError, 'distinct source inventory'):
+            validate_inventory(inventory, local['R'], local['h'])
+
+    def test_ungauged_old_exterior_inventory_rejected(self):
+        from sharing import profile
+        local = copy.deepcopy(self.profiles['complex'])
+        local['child_multiplicities'] = {r: n for r, n in local['child_multiplicities'].items() if int(r) < 552}
+        local['child_multiplicities']['552'] = 2 * local['v'] * local['R']
+        partition = json.loads((HERE / 'inputs/shared-partition.json').read_text())
+        phase = json.loads((HERE / 'phase_check.json').read_text())
+        gauge = json.loads((HERE / 'gauge-phase-audit.json').read_text())
+        with self.assertRaisesRegex(AssertionError, 'auxiliary exterior inventory'):
+            profile(local, partition, phase, gauge)
+
+    def test_missing_grouped_gauge_complement_rejected(self):
+        from sharing import profile
+        partition = json.loads((HERE / 'inputs/shared-partition.json').read_text())
+        phase = json.loads((HERE / 'phase_check.json').read_text())
+        gauge = json.loads((HERE / 'gauge-phase-audit.json').read_text())
+        width = next(iter(gauge['grouped_exterior_histogram']))
+        gauge['grouped_exterior_histogram'][width] -= 2
+        with self.assertRaisesRegex(AssertionError, 'gauge complement profile'):
+            profile(self.profiles['complex'], partition, phase, gauge)
+
+    def test_one_stage_bit_loss_rejected(self):
+        changed = copy.deepcopy(self.profiles['shared-bit'])
+        changed['L'] //= 2
+        with self.assertRaisesRegex(ValueError, 'physical ledger'):
+            validate_profile(changed, 'shared-bit')
+
+    def test_omitted_paid_bit_adapter_calls_rejected(self):
+        changed = json.loads((HERE / 'certificate.json').read_text())
+        changed['finite_bridge']['bit_coarse']['paid_projector_adapter_calls'] -= 1
+        with self.assertRaisesRegex(ValueError, 'paid adapter call binding'):
+            validate_bit_composition(HERE, changed)
+
+    def test_omitted_reuse_role_charge_rejected(self):
+        changed = copy.deepcopy(self.profiles['complex'])
+        changed['reused_roles'] -= 1
+        with self.assertRaisesRegex(ValueError, 'matching role ledger'):
+            validate_profile(changed, 'complex')
+
+    def test_omitted_endpoint_charge_rejected(self):
+        for label, original in self.profiles.items():
+            changed = copy.deepcopy(original)
+            changed['child_multiplicities']['1'] -= changed['N']
+            with self.assertRaisesRegex(ValueError, 'rank mass'):
+                validate_profile(changed, label)
+
+    def test_nonshrinking_recursive_child_rejected(self):
+        for label, original in self.profiles.items():
+            changed = copy.deepcopy(original)
+            changed['child_multiplicities'][str(changed['m'])] = 1
+            with self.assertRaisesRegex(ValueError, 'recursive child'):
+                validate_profile(changed, label)
+
+    def test_understated_role_count_rejected(self):
+        changed = copy.deepcopy(self.profiles['complex'])
+        changed['R'] -= 1
+        with self.assertRaisesRegex(ValueError, 'physical ledger'):
+            validate_profile(changed, 'complex')
+
+    def test_overstated_matching_rejected(self):
+        changed = copy.deepcopy(self.profiles['complex'])
+        changed['links'] += 1
+        with self.assertRaisesRegex(ValueError, 'matching role ledger'):
+            validate_profile(changed, 'complex')
+
+    def test_missing_transitive_dependency_rejected(self):
+        changed = copy.deepcopy(self.manifest)
+        del changed['repository_files']['scripts/exclusion_circuit.py']
+        with self.assertRaisesRegex(ValueError, 'dependency closure'):
+            check_sources(HERE, REPOSITORY, changed)
+
+    def test_omitted_physical_input_rejected(self):
+        changed = copy.deepcopy(self.manifest)
+        del changed['package_files']['inputs/deferred_23.json.gz']
+        with self.assertRaisesRegex(ValueError, 'finite-input closure'):
+            check_sources(HERE, REPOSITORY, changed)
+
+    def test_changed_input_digest_rejected(self):
+        changed = copy.deepcopy(self.manifest)
+        changed['package_files']['inputs/witness_23.json.gz'] = '0' * 64
+        with self.assertRaisesRegex(ValueError, 'hash mismatch'):
+            check_sources(HERE, REPOSITORY, changed)
+
+    def test_omitted_pinned_complex_dag_rejected(self):
+        changed = copy.deepcopy(self.manifest)
+        del changed['package_files']['inputs/complex-dag.json.gz']
+        with self.assertRaisesRegex(ValueError, 'finite-input closure'):
+            check_sources(HERE, REPOSITORY, changed)
+
+    def test_omitted_pinned_partition_rejected(self):
+        changed = copy.deepcopy(self.manifest)
+        del changed['package_files']['inputs/shared-partition.json']
+        with self.assertRaisesRegex(ValueError, 'finite-input closure'):
+            check_sources(HERE, REPOSITORY, changed)
+
+    def test_omitted_nested_bit_dependency_rejected(self):
+        changed = copy.deepcopy(self.manifest)
+        del changed['package_files']['bit-sharing/profile161.json']
+        with self.assertRaisesRegex(ValueError, 'nested bit-package closure'):
+            check_sources(HERE, REPOSITORY, changed)
+
+    def test_changed_nested_bit_input_and_rehashed_manifest_rejected(self):
+        from bit_adapter import MINIMAL_V_PATH, package_inventory
+        with tempfile.TemporaryDirectory(prefix='dual-core-altered-bit-') as directory:
+            package = Path(directory)
+            shutil.copytree(HERE / 'bit-sharing', package / 'bit-sharing')
+            path = package / 'bit-sharing' / MINIMAL_V_PATH
+            path.write_bytes(path.read_bytes() + b'corrupted')
+            with self.assertRaisesRegex(ValueError, 'Nested bit source hash mismatch'):
+                package_inventory(package)
+            manifest_path = package / 'bit-sharing/MANIFEST.json'
+            manifest = json.loads(manifest_path.read_text())
+            manifest['sha256'][MINIMAL_V_PATH] = digest(path)
+            manifest_path.write_text(json.dumps(manifest))
+            with self.assertRaisesRegex(ValueError, 'Immutable bit-package manifest digest mismatch'):
+                package_inventory(package)
+
+    def test_changed_partition_rejected_even_when_rehashed(self):
+        with tempfile.TemporaryDirectory(prefix='shared-core-altered-partition-') as directory:
+            package = Path(directory)
+            for name in self.manifest['package_files']:
+                target = package / name
+                target.parent.mkdir(parents=True, exist_ok=True)
+                shutil.copyfile(HERE / name, target)
+            name = 'inputs/shared-partition.json'
+            path = package / name
+            partition = json.loads(path.read_text())
+            partition['groups'][0][0] = partition['groups'][0][1]
+            path.write_text(json.dumps(partition))
+            with self.assertRaisesRegex(ValueError, 'hash mismatch'):
+                check_sources(package, REPOSITORY, self.manifest)
+            changed = copy.deepcopy(self.manifest)
+            changed['package_files'][name] = digest(path)
+            with self.assertRaisesRegex(ValueError, 'partition provenance digest mismatch'):
+                check_sources(package, REPOSITORY, changed)
+
+    def test_changed_complex_dag_digest_rejected(self):
+        with tempfile.TemporaryDirectory(prefix='cyclic-deferred-altered-dag-') as directory:
+            package = Path(directory)
+            for name in self.manifest['package_files']:
+                target = package / name
+                target.parent.mkdir(parents=True, exist_ok=True)
+                shutil.copyfile(HERE / name, target)
+            name = 'inputs/complex-dag.json.gz'
+            path = package / name
+            graph = json.loads(gzip.decompress(path.read_bytes()))
+            graph['args'][0] = graph['args'][1]
+            path.write_bytes(gzip.compress(json.dumps(graph).encode(), mtime=0))
+            with self.assertRaisesRegex(ValueError, 'hash mismatch'):
+                check_sources(package, REPOSITORY, self.manifest)
+            # Rehashing a substituted DAG must not erase its immutable provenance.
+            changed = copy.deepcopy(self.manifest)
+            changed['package_files'][name] = digest(path)
+            with self.assertRaisesRegex(ValueError, 'scalar-DAG provenance digest mismatch'):
+                check_sources(package, REPOSITORY, changed)
+
+    def test_replayed_complex_dag_rejects_duplicated_operand(self):
+        from replayed_producer import build
+        with tempfile.TemporaryDirectory(prefix='cyclic-deferred-invalid-dag-') as directory:
+            graph = json.loads(gzip.decompress((HERE / 'inputs/complex-dag.json.gz').read_bytes()))
+            graph['args'][0] = graph['args'][1]
+            path = Path(directory) / 'invalid.json.gz'
+            path.write_bytes(gzip.compress(json.dumps(graph).encode(), mtime=0))
+            with self.assertRaisesRegex(AssertionError, 'cancellation-free'):
+                build(path, Path(directory) / 'invalid-producer')
+
+    def test_manifest_parent_escape_rejected(self):
+        with self.assertRaisesRegex(ValueError, 'Unsafe manifest path'):
+            safe_file(HERE, '../certificate.json')
+
+    def test_changed_regenerated_profile_rejected(self):
+        with tempfile.TemporaryDirectory(prefix='cyclic-deferred-negative-') as directory:
+            changed = copy.deepcopy(self.profiles['complex'])
+            changed['child_multiplicities']['1'] += 1
+            path = Path(directory) / 'changed.json'
+            path.write_text(json.dumps(changed))
+            with self.assertRaisesRegex(ValueError, 'Frozen generated record mismatch'):
+                compare_json(path, HERE / 'complex-profile.json')
+
+    def test_hash_bound_regenerated_formatting_change_rejected(self):
+        with tempfile.TemporaryDirectory(prefix='cyclic-deferred-formatting-') as directory:
+            for name in ('complex-profile.json', 'reflection-audit.json'):
+                expected = HERE / name
+                path = Path(directory) / name
+                path.write_text(json.dumps(json.loads(expected.read_text()), separators=(',', ':')))
+                with self.assertRaisesRegex(ValueError, 'Frozen generated byte mismatch'):
+                    compare_json(path, expected, exact_bytes=True)
+
+    def test_frozen_reflection_receipt_uses_writer_serialization(self):
+        path = HERE / 'reflection-audit.json'
+        canonical = json.dumps(json.loads(path.read_text()), indent=2, sort_keys=True) + '\n'
+        self.assertEqual(path.read_bytes(), canonical.encode())
+
+    def test_next_bit_and_complex_grid_points_rejected(self):
+        from fractions import Fraction as Q
+        module = self.certificate
+        bit = module.profile('shared-bit-profile.json')
+        self.assertTrue(module.contracts(bit, module.COARSE))
+        self.assertFalse(module.contracts(bit, module.COARSE + Q(1, 10**12)))
+        shared = module.profile('shared-complex-profile.json')
+        self.assertTrue(module.contracts(shared, module.COMPLEX))
+        self.assertFalse(module.contracts(shared, module.COMPLEX + Q(1, 10**12)))
+
+    def test_optimized_verifier_rejected_before_any_regeneration(self):
+        result = subprocess.run([sys.executable, '-O', str(HERE / 'verify.py')],
+                                capture_output=True, text=True)
+        self.assertNotEqual(result.returncode, 0)
+        self.assertIn('optimized Python is forbidden', result.stderr)
+
+
+def main():
+    global REPOSITORY
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--repository-root', type=Path, default=REPOSITORY)
+    args = parser.parse_args()
+    REPOSITORY = args.repository_root.resolve()
+    suite = unittest.defaultTestLoader.loadTestsFromTestCase(FrozenControls)
+    result = unittest.TextTestRunner(verbosity=2).run(suite)
+    if not result.wasSuccessful():
+        raise SystemExit(1)
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/cyclic-deferred/verify.py b/research/cyclic-deferred/verify.py
new file mode 100644
index 0000000000..38541e17da
--- /dev/null
+++ b/research/cyclic-deferred/verify.py
@@ -0,0 +1,385 @@
+#!/usr/bin/env python3
+"""Read-only verification of the frozen cyclic/deferred finite package.
+
+Regeneration occurs only in a temporary repository-shaped copy. Source hashes,
+complete profile ledgers, the exact certificate, and the independent reflection
+audit are required. General transfer and common-basis/address-adapter assumptions remain
+separate dependencies; the nested finite bit word is independently replayed. Prepared for eumemic
+with OpenAI Codex assistance; inherited authorship and licenses are retained.
+"""
+import sys
+sys.dont_write_bytecode = True
+if sys.flags.optimize:
+    raise ValueError('Verification requires assertions; optimized Python is forbidden')
+
+import argparse
+import ast
+from hashlib import sha256
+import json
+from math import comb
+import os
+from pathlib import Path
+import shutil
+import subprocess
+import tempfile
+from bit_adapter import package_inventory as bit_package_inventory
+
+HERE = Path(__file__).resolve().parent
+COMPLEX_DAG_PIN = '3c034d0aae388ef567a454826f4f48b26fd8a94c71e8ffed4835271b349a783b'
+PARTITION_PIN = '9117d3f010354fa2f32d5c5fb324b280977e82e76060a70a867767a90989ccaa'
+BIT_PINS = {
+    'witness_23.json.gz': 'b2486aa2bb222bacea6e52162a3920780e45dd8edc5d7cb03eabea336a1c6588',
+    'deferred_23.json.gz': '8c38e947ff9e021e308000dd82bb5e2194eb265d8b75194e22ce783d3e75317c',
+}
+REQUIRED = {
+    'verify.py', 'test_controls.py', 'producer.py', 'complex_deferred.py',
+    'bit_round7.py', 'certificate.py', 'bit-profile.json',
+    'bit_adapter.py', 'shared-bit-profile.json',
+    'complex-profile.json', 'certificate.json', 'replayed_producer.py',
+    'inputs/complex-dag.json.gz',
+    'sharing.py', 'phase_check.py', 'phase_check.json',
+    'shared-complex-profile.json', 'inputs/shared-partition.json',
+    'reuse.py', 'reuse-pairs.json', 'gauge_phase.py',
+    'gauge-phase-audit.json', 'gauge-phase-witness.json',
+    'README.md', 'SHARING.md', 'BITSHARING.md', 'PR128-NOTICE',
+    *('inputs/' + name for name in BIT_PINS),
+}
+DATA_DEPENDENCIES = {'certificates/copied-centers-network.json'}
+
+
+def require(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def safe_file(directory, name):
+    relative = Path(name)
+    require(not relative.is_absolute() and '..' not in relative.parts,
+            'Unsafe manifest path: ' + name)
+    path = (directory / relative).resolve()
+    require(path.is_relative_to(directory.resolve()) and path.is_file(),
+            'Missing or escaping manifest file: ' + name)
+    return path
+
+
+def digest(path):
+    return sha256(path.read_bytes()).hexdigest()
+
+
+def dependency_closure(package, repository, package_scripts):
+    """Include every local Python import, including package __init__ modules.
+
+    All outer package scripts are roots, covering explicit importlib loading.
+    The independently frozen nested bit package is checked through its exact
+    manifest/verifier boundary; unused inherited APIs are not entry points.
+    Only stdlib outer imports may be unresolved. The sole external arithmetic
+    data read is listed separately.
+    """
+    pending = [package / name for name in package_scripts]
+    seen, dependencies = set(), set(DATA_DEPENDENCIES)
+    while pending:
+        path = pending.pop().resolve()
+        if path in seen:
+            continue
+        seen.add(path)
+        imports = []
+        for node in ast.walk(ast.parse(path.read_text(), filename=str(path))):
+            if isinstance(node, ast.Import):
+                imports.extend((alias.name, 0) for alias in node.names)
+            elif isinstance(node, ast.ImportFrom) and node.module:
+                imports.append((node.module, node.level))
+        for name, level in imports:
+            parts = name.split('.')
+            if not level and parts[0] in sys.stdlib_module_names:
+                continue
+            found = None
+            bases = ((path.parent.parents[level-2] if level > 1 else path.parent),) if level else (package, repository / 'scripts')
+            for base in bases:
+                module = base.joinpath(*parts)
+                for candidate in (module.with_suffix('.py'), module / '__init__.py'):
+                    if candidate.is_file():
+                        found = candidate.resolve()
+                        break
+                if found is not None:
+                    pending.append(found)
+                    if found.is_relative_to((repository / 'scripts').resolve()):
+                        dependencies.add(str(found.relative_to(repository.resolve())))
+                        parent = found.parent
+                        while parent != (repository / 'scripts').resolve():
+                            initializer = parent / '__init__.py'
+                            if initializer.is_file():
+                                pending.append(initializer)
+                                dependencies.add(str(initializer.relative_to(repository)))
+                            parent = parent.parent
+                    break
+            require(found is not None, 'Unpinned non-stdlib Python import: ' + name)
+    return dependencies
+
+
+def create_manifest(package, repository, reflection_script, reflection_receipt):
+    scripts = sorted(path.name for path in package.glob('*.py'))
+    nested_bit = bit_package_inventory(package)
+    names = REQUIRED | set(scripts) | {reflection_script, reflection_receipt} | set(nested_bit)
+    dependencies = dependency_closure(package, repository, scripts)
+    return dict(
+        schema=1,
+        scope='Frozen finite producer, profiles, exact assembly and reflection audit; inherited all-size transfer remains conditional.',
+        package_files={name: digest(safe_file(package, name)) for name in sorted(names)},
+        repository_files={name: digest(safe_file(repository, name)) for name in sorted(dependencies)},
+        reflection=dict(script=reflection_script, receipt=reflection_receipt),
+        provenance=dict(
+            base='https://github.com/CrocSwap/integer-mult-bounds/pull/110',
+            cyclic_strips='https://github.com/CrocSwap/integer-mult-bounds/pull/111',
+            scalar_dag=dict(pull_request='https://github.com/CrocSwap/integer-mult-bounds/pull/117',
+                            commit='cbb05ce504d571546d9b7794c186a613c659c3bf',
+                            file='inputs/complex-dag.json.gz', sha256=COMPLEX_DAG_PIN),
+            completed_core_partition=dict(pull_request='https://github.com/CrocSwap/integer-mult-bounds/pull/128',
+                            commit='530588a019b4a74f09180680c9e3961bf649ec89',
+                            file='inputs/shared-partition.json', sha256=PARTITION_PIN),
+            compensated_birth_reuse='https://github.com/CrocSwap/integer-mult-bounds/pull/124',
+            bit161=dict(manifest='bit-sharing/MANIFEST.json',
+                        manifest_sha256=nested_bit['bit-sharing/MANIFEST.json'],
+                        verifier='bit-sharing/verify.py',
+                        scope='Independent frozen full geometry, physical word, partition and moment replay'),
+            round7_repository='https://github.com/Swapnil-jain/integer-mult-kappa',
+            round7_commit='741e7aa078392553815df7926ee17ac5e25a8c38',
+            round7_sha256=BIT_PINS),
+        attribution='Avi Eisenberg / ikeboy (PR62 and PR110, Anthropic Claude assistance); Rohan Arun (PR111, Anthropic Claude assistance); Swapnil Jain (round-seven bit word); icekylinx (retained stopped-product, copied-center and finite assembly interfaces); Zhihao Chen and RaD (retained assembly). PR117 scalar DAG is separate upstream work by eumemic with Anthropic Claude assistance, retained byte for byte with its original attribution. Completed-core sharing and signed orthogonal partition: an664 PR128 with OpenAI Codex assistance, using Xiande Zhang and Gennian Ge (2010). Compensated birth-cut reuse follows jamesyc PR124. Physical hull frames, generalized gauge sharing, 161-group bit sharing and verification for eumemic with OpenAI Codex assistance. Original source notices remain authoritative.')
+
+
+def check_sources(package, repository, manifest=None):
+    manifest = manifest or json.loads((package / 'SOURCE.json').read_text())
+    require(manifest['schema'] == 1, 'Unsupported source-manifest schema')
+    names = set(manifest['package_files'])
+    reflection = manifest['reflection']
+    require(REQUIRED | {reflection['script'], reflection['receipt']} <= names,
+            'Incomplete package source/finite-input closure')
+    bit_files = bit_package_inventory(package)
+    require({name for name in names if name.startswith('bit-sharing/')} == set(bit_files),
+            'Incomplete or stale nested bit-package closure')
+    require(all(manifest['package_files'][name] == expected for name, expected in bit_files.items()),
+            'Nested bit-package manifest binding mismatch')
+    for collection, base in ((manifest['package_files'], package),
+                             (manifest['repository_files'], repository)):
+        for name, expected in collection.items():
+            require(digest(safe_file(base, name)) == expected, 'Source hash mismatch: ' + name)
+    scripts = sorted(name for name in names if name.endswith('.py') and not name.startswith('bit-sharing/'))
+    expected = dependency_closure(package, repository, scripts)
+    require(set(manifest['repository_files']) == expected,
+            'Incomplete or stale transitive repository dependency closure')
+    for name, expected in BIT_PINS.items():
+        require(manifest['package_files']['inputs/' + name] == expected,
+                'Round-seven provenance digest mismatch: ' + name)
+    require(manifest['package_files']['inputs/complex-dag.json.gz'] == COMPLEX_DAG_PIN,
+            'PR117 scalar-DAG provenance digest mismatch')
+    require(manifest['package_files']['inputs/shared-partition.json'] == PARTITION_PIN,
+            'PR128 completed-core partition provenance digest mismatch')
+    return manifest
+
+
+def validate_profile(profile, label):
+    fields = ('h', 'v', 'R', 'm', 'N', 'W', 'L', 'total_rank', 'deficit', 'maxchild')
+    require(all(type(profile[key]) is int and profile[key] > 0 for key in fields),
+            label + ': nonpositive or noninteger ledger value')
+    h, v, R, m, N, W, L = (profile[key] for key in fields[:7])
+    require(v == comb(h, 3) and m == h*h and N == v*v, label + ': dimensions')
+    if label == 'shared-complex':
+        groups = {int(g): count for g, count in profile['group_sizes'].items()}
+        require(groups == {8: 4, 24: 83} and profile['shared_groups'] == sum(groups.values()),
+                label + ': completed-core group ledger')
+        require(profile['core_pre_exterior_inner_sink_rank'] == h,
+                label + ': completed-core gauges')
+        outer_width = profile['shared_groups']
+    elif label == 'shared-bit':
+        groups = {int(g): count for g, count in profile['group_sizes'].items()}
+        require(groups == {11: 161} and profile['groups'] == 161,
+                label + ': completed-core group ledger')
+        outer_width = profile['groups']
+    else:
+        outer_width = v
+    require(W == 2*N + 2*outer_width*R and L == 2*v*h*(h-1), label + ': physical ledger')
+    rows = {int(t): count for t, count in profile['child_multiplicities'].items()}
+    require(len(rows) == len(profile['child_multiplicities']), label + ': duplicate child widths')
+    require(all(0 < t < m and type(count) is int and count > 0 for t, count in rows.items()),
+            label + ': invalid recursive child')
+    require(sum(t*count for t, count in rows.items()) == profile['total_rank'] == W*m-N+L,
+            label + ': rank mass')
+    require(profile['deficit'] == N-L and profile['maxchild'] == max(rows),
+            label + ': deficit or maximum child')
+    require(rows.get((h-1)**2) == 2*N and rows.get(1, 0) >= N,
+            label + ': omitted data projector or endpoint charge')
+    if label in ('complex', 'shared-complex'):
+        require(profile['virtual_R'] == profile['additions']+profile['roots']-profile['links'] and
+                type(profile['reused_roles']) is int and profile['reused_roles'] >= 0 and
+                R + profile['reused_roles'] == profile['virtual_R'],
+                'complex: addition/roots/matching role ledger')
+        require(sum(profile['deferred_dims'].values()) == profile['deferred_roles'],
+                'complex: deferral inventory')
+        from gauge_phase import validate_inventory
+        validate_inventory(profile['physical_auxiliary_source_frames'], R, h)
+    if label == 'shared-bit':
+        require(profile['paid_projector_calls'] == sum(rows.values()),
+                label + ': paid projector adapter calls')
+        classes = profile['classes']
+        require(sum(part['rank'] for part in classes.values()) == profile['total_rank'] and
+                sum(part['calls'] for part in classes.values()) == profile['paid_projector_calls'],
+                label + ': class-wise paid rank/call ledger')
+    return rows
+
+
+def validate_shared_composition(package):
+    from sharing import profile as shared_profile
+    local = json.loads((package / 'complex-profile.json').read_text())
+    shared = json.loads((package / 'shared-complex-profile.json').read_text())
+    partition = json.loads((package / 'inputs/shared-partition.json').read_text())
+    phase = json.loads((package / 'phase_check.json').read_text())
+    gauge = json.loads((package / 'gauge-phase-audit.json').read_text())
+    require(gauge['local_profile_sha256'] == digest(package / 'complex-profile.json') and
+            gauge['reflection_receipt_sha256'] == digest(package / 'reflection-audit.json') and
+            gauge['checker_sha256'] == digest(package / 'gauge_phase.py'),
+            'Physical gauge audit does not bind the local source evidence')
+    reconstructed = shared_profile(local, partition, phase, gauge)
+    require(json.loads(json.dumps(reconstructed)) == shared,
+            'Shared profile does not match local completed cores and paid exterior')
+
+
+def validate_bit_composition(package, certificate=None):
+    bit = json.loads((package / 'shared-bit-profile.json').read_text())
+    nested = json.loads((package / 'bit-sharing/profile161.json').read_text())
+    nested_certificate = json.loads((package / 'bit-sharing/certificate.json').read_text())
+    require(bit == nested, 'Shared bit profile differs from its independently verified supplier')
+    validate_profile(bit, 'shared-bit')
+    if certificate is not None:
+        account = certificate['finite_bridge']['bit_coarse']
+        require(certificate['coarse_bit_saving'] == nested_certificate['certified_bit_saving'] and
+                account['halving_degree'] == nested_certificate['halving_degree'],
+                'Shared bit exact saving or halving degree mismatch')
+        require(account['completed_core_groups'] == bit['groups'] and
+                account['paid_projector_adapter_calls'] == bit['paid_projector_calls'],
+                'Shared bit paid adapter call binding mismatch')
+
+
+def check_completed_core_audit(audit, local, certificate):
+    require(audit['completed_core_source_inventory_bound'] is True and
+            audit['completed_core_pre_exterior_frames_full'] is True and
+            audit['reflected_core_active_frames_complement_source'] is True and
+            audit['exact_birth_cut_invariants'] is True and
+            audit['physical_aliased_numeric_replay'] is True,
+            'Completed cores require bound source gauges and reflected active frames')
+    require(audit['physical_auxiliary_source_frames'] == local['physical_auxiliary_source_frames'] and
+            audit['virtual_R'] == local['virtual_R'] and audit['reused_roles'] == local['reused_roles'],
+            'Physical source inventory or reused-role binding mismatch')
+    require((audit['h'], audit['v'], audit['R']) == (local['h'], local['v'], local['R']) and
+            audit['child_multiplicities'] == local['child_multiplicities'],
+            'Local reflection audit does not bind the completed-core profile')
+    actual = audit['literal_global_scalar_groups']
+    account = certificate['finite_bridge']['complex']
+    upper = account['scalar_group_upper']
+    core_upper = account['core_scalar_group_upper']
+    phase_upper = account['shared_basis_phase_group_upper']
+    shared = certificate['complex_profile']
+    expected_phase = 64 * shared['m']**2 * (2 * shared['shared_groups'] * shared['R'])
+    require(type(actual) is int and type(upper) is int and type(core_upper) is int and
+            0 < actual <= core_upper and upper == core_upper + phase_upper,
+            'Literal reflected scalar charge exceeds exact certificate guard')
+    require(type(phase_upper) is int and phase_upper == expected_phase,
+            'Missing paid shared basis and phase-wrapper reserve')
+
+
+def compare_json(actual, expected, *, exact_bytes=False):
+    require(json.loads(actual.read_text()) == json.loads(expected.read_text()),
+            'Frozen generated record mismatch: ' + expected.name)
+    if exact_bytes:
+        require(actual.read_bytes() == expected.read_bytes(),
+                'Frozen generated byte mismatch: ' + expected.name)
+
+
+def snapshot(package, repository, manifest):
+    return {
+        **{'package:' + n: digest(package / n) for n in manifest['package_files']},
+        **{'repository:' + n: digest(repository / n) for n in manifest['repository_files']},
+        'manifest': digest(package / 'SOURCE.json'),
+    }
+
+
+def verify(package, repository):
+    manifest = check_sources(package, repository)
+    before = snapshot(package, repository, manifest)
+    for label in ('bit', 'shared-bit', 'complex', 'shared-complex'):
+        validate_profile(json.loads((package / (label + '-profile.json')).read_text()), label)
+    validate_shared_composition(package)
+    validate_bit_composition(package, json.loads((package / 'certificate.json').read_text()))
+    environment = dict(os.environ, PYTHONDONTWRITEBYTECODE='1')
+    environment.pop('PYTHONPATH', None)
+    with tempfile.TemporaryDirectory(prefix='cyclic-deferred-verify-') as directory:
+        root = Path(directory)
+        target = root / 'research/cyclic-deferred'
+        for collection, source, destination in (
+                (manifest['package_files'], package, target),
+                (manifest['repository_files'], repository, root)):
+            for name in collection:
+                out = destination / name
+                out.parent.mkdir(parents=True, exist_ok=True)
+                shutil.copyfile(source / name, out)
+        shutil.copyfile(package / 'SOURCE.json', target / 'SOURCE.json')
+
+        def run(script, *arguments):
+            subprocess.run([sys.executable, str(target / script), *map(str, arguments)],
+                           cwd=root, env=environment, check=True)
+
+        run('bit_adapter.py')
+        compare_json(target / 'shared-bit-profile.json', package / 'shared-bit-profile.json')
+        print('PASS independent full bit replay and 161-group profile regenerated', flush=True)
+        run('bit_round7.py', target / 'inputs')
+        compare_json(target / 'bit-profile.json', package / 'bit-profile.json')
+        print('PASS pinned round-seven bit ledger regenerated', flush=True)
+        run('complex_deferred.py')
+        compare_json(target / 'complex-profile.json', package / 'complex-profile.json',
+                     exact_bytes=True)
+        compare_json(target / 'reuse-pairs.json', package / 'reuse-pairs.json')
+        print('PASS complete complex producer and deferred profile regenerated', flush=True)
+        reflection = manifest['reflection']
+        actual_reflection = target / reflection['receipt']
+        run(reflection['script'], '--source', target / 'complex_deferred.py',
+            '--output', actual_reflection)
+        compare_json(actual_reflection, package / reflection['receipt'], exact_bytes=True)
+        run('phase_check.py', target / 'inputs/shared-partition.json')
+        compare_json(target / 'phase_check.json', package / 'phase_check.json')
+        run('gauge_phase.py')
+        for name in ('gauge-phase-audit.json', 'gauge-phase-witness.json'):
+            compare_json(target / name, package / name)
+        run('sharing.py')
+        compare_json(target / 'shared-complex-profile.json', package / 'shared-complex-profile.json')
+        validate_shared_composition(target)
+        print('PASS signed partition phases and completed-core shared profile regenerated', flush=True)
+        run('certificate.py')
+        validate_bit_composition(target, json.loads((target / 'certificate.json').read_text()))
+        check_completed_core_audit(json.loads(actual_reflection.read_text()),
+                                  json.loads((package / 'complex-profile.json').read_text()),
+                                  json.loads((package / 'certificate.json').read_text()))
+        print('PASS independent literal reflection and scalar-charge audit', flush=True)
+        run('test_controls.py', '--repository-root', root)
+    require(snapshot(package, repository, manifest) == before,
+            'Verification changed the frozen package or a dependency')
+    print('PASS read-only finite package verification; inherited transfer remains conditional', flush=True)
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--repository-root', type=Path, default=HERE.parents[1])
+    parser.add_argument('--freeze-manifest', action='store_true',
+                        help='Explicit authoring operation: replace SOURCE.json from reviewed files')
+    parser.add_argument('--reflection-script', default='reflection_audit.py')
+    parser.add_argument('--reflection-receipt', default='reflection-audit.json')
+    args = parser.parse_args()
+    repository = args.repository_root.resolve()
+    if args.freeze_manifest:
+        manifest = create_manifest(HERE, repository, args.reflection_script, args.reflection_receipt)
+        (HERE / 'SOURCE.json').write_text(json.dumps(manifest, indent=2, sort_keys=True) + '\n')
+        print('Frozen source manifest; this operation does not certify the package')
+    else:
+        verify(HERE, repository)
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/deferred-stopped/PROOF.md b/research/deferred-stopped/PROOF.md
new file mode 100644
index 0000000000..58237eccf0
--- /dev/null
+++ b/research/deferred-stopped/PROOF.md
@@ -0,0 +1,114 @@
+# Deferred readouts on both networks of the stopped product-ring interchange
+
+## 1. Claim
+
+Under every hypothesis that PR #104 retains, the stopped product-ring interchange with
+
+- the bit network replaced by Swapnil Jain's round-seven deferred word (h = 23), and
+- PR #104's own complex network (h = 24) given lifted binary frames and deferred readouts,
+
+gives the conditional bound T(n) = O(n (log n)^(1-κ)) with
+
+κ = 429239/5000000000 = 8.58478 × 10^-5,
+
+which is +10.1% over PR #104 (7.79476 × 10^-5). The next κ grid point at denominator 10^10 is rejected.
+This is a finite conditional certificate only.
+
+| | PR #104 | here |
+|---|---|---|
+| bit network | positive-label producer, R = 38,776 | Swapnil round seven, R = 28,866, 11,565 deferred slots |
+| coarse bit saving | 4019/50000000 = 8.038e-5 | 620523/5000000000 = 1.241046e-4 |
+| stopped bit saving | 8.034e-5 | 1.240190e-4 |
+| complex network | aligned rational centres, R = 44,918 | same DAG and roles, lifted frames, 8,893 deferred roles |
+| complex saving | 1949/25000000 = 7.796e-5 | 214657/2500000000 = 8.58628e-5 |
+| κ | 7.79476e-5 | 8.58478e-5 |
+
+## 2. What changes
+
+Nothing in PR #104's interface changes: opposite-bank factorization, stopped atom recursion, ordinary
+wrapper, rational centres, common odd grid, product rows and the balanced assembly are used as they are.
+Two finite inputs change.
+
+**Bit network.** Swapnil Jain's round-seven witness (commit 741e7aa078392553815df7926ee17ac5e25a8c38, files pinned by SHA-256 in
+`bit_round7.py`): PR #62's producer, lifted frames, late copies, deferred readouts and V leaves. Swapnil's own
+checkers verify the word and every frame exactly, and PR #97 replays its forward and reflected ledgers. We do not
+change it. Under PR #104's rule every residual of rank r is one child of width r, so the slot chains, the gauged
+exteriors (width m - (h - dim σ_u)), the copied-centre transforms, the target and data-wire chains, the data
+residuals and the endpoint copies give the child list in `bit-profile.json` (rank mass 57,403,754,177, the
+value PR #97 reports).
+
+**Complex network.** PR #104's DAG (`scripts/stopped_product/complex.py`, unchanged) and its role count
+R = c + q - |M| = 45,764 + 8,120 - 8,966 = 44,918. New:
+
+1. *Matching.* A maximum carrier matching under the inherited matcher's adjacency (order by rank and node, the
+   inherited label-inclusion rules), size 8,966.
+2. *Roles.* An explicit compile: one role per free leaf use, the pivot continues each addition, copies serve the
+   other free uses, a linked operand stays on the non-pivot and continues to its linked use.
+3. *Lifted frames.* For an addition n, U_n is the common kernel over F2 of the functionals of every root reachable
+   along slot successors (t_S for an output read into y_S, e_i for the retained centre A_i, whose frame is
+   span{e_j : j ≠ i}). Every label lies in its kernel (outputs have even intersection with their target; A_i
+   avoids i). The lift is used where U_n is nondegenerate and kept monotone along successors; elsewhere the
+   inherited label stays. 20,234 additions are lifted.
+4. *Deferred readouts* (Swapnil's design B on the complex word). Phase one is the closure of the 24 centre
+   completions under per-role precedence. A role untouched by phase one, whose garbage does not reach a centre
+   (a centre scatters to every target), reads its garbage out at
+   σ_u ⊆ F0(u) ∩ ⋂ t_S^⊥ over its reached targets, nondegenerate. An insertion rule (largest first) shrinks each
+   new frame only as far as nesting on its targets requires. Readouts run in increasing (dim σ_u, role).
+
+The stage-one word is: readouts of the non-deferred roles at frame 0, early V gates, phase one, centre copies
+read into their targets, deferred readouts in order, deferred V gates, the rest of L, output reads, L^-1, V^-1.
+The readout coefficients come from the transpose sweep of L applied to the root reads (output reads ±1/2, centre
+scatter 1/21 - [i ∈ S]/2); their only odd denominator is PR #104's 21.
+
+## 3. What is checked (`complex_deferred.py`)
+
+- **A.** Matching size 8,966 and R = 44,918.
+- **B.** With zero scratch the compiled word computes every one of the 8,120 root values.
+- **C.** The deferred word, replayed over Z/(2^61 - 1) with arbitrary scratch and data (two seeds), restores every
+  role and adds exactly x_S to every target.
+- **D.** Exactly over F2: every role chain (start σ_u or 0, start frame, every gate, linked continuations, root
+  frame, F) is nested; every lifted and deferral frame is nondegenerate; every σ_u lies in its start frame and in
+  t_S^⊥ for each reached target; on every target the deferral frames are nested in readout order.
+- **E.** The one-child histogram from the explicit role chains has total rank W m - N + L = 109,450,358,336,
+  every child is below m = 576 (the largest is 574).
+
+`certificate.py` recomputes both moments exactly with PR #104's routines, the stopped bit saving
+(1 - 1/1000)·coarse + (1/1000)·384599/10^10, the finite bridge and the 47-constraint assembly.
+
+## 4. Endpoints, stage two and the bridge
+
+- *Endpoints.* A deferred auxiliary role starts at the gauge of σ_u and ends at the complementary gauge, so its
+  endpoint action is the identity shear on arbitrary inputs: the bit identity D_{I-P} D_P = D_I and its phase
+  analogue, the translated source C_{D_0} with sink C_I C_{D_0} of `notes/copied-centers-complex.tex` and
+  `notes/endpoint-gauge-complex.tex` (PR21/PR24). Its exterior residual has rank m - h + dim σ_u.
+- *Stage two.* As in Swapnil's round seven and PR #104's two-stage counts, stage two is the complement time
+  reversal of stage one with the same edge multiset. PR #97 replays the reflected ledger for the bit word; for the
+  complex deferred word this is stated here, not machine-checked.
+- *Bridge.* The larger children (bit 528 of 529, complex 574 of 576) raise the halving degrees to 367 and 200,
+  so the product-row stock becomes p^33000 (PR #104 used p^4000; PR29 p^47000). As in PR #104, the stock enters
+  only the row-product slack and the eventual cutoffs, not κ. The semantic induction 2B(m - r) ≥ s + E holds.
+
+## 5. Scope
+
+All of PR #104's inherited hypotheses remain: the opposite-bank factorization for every residual class in the
+finite lists (including the new gauged exteriors, deferral starts and target levels on both networks, and the
+binary one-child normal form on the complex side), atom streaming and the ordinary wrapper, the common exact grid,
+routing, recovery, the fixed-tape transfer and the analytic interfaces. The stage-two statement above is the one
+new written assumption for the complex network. No global optimality is claimed.
+
+## 6. Credits
+
+- **icekylinx**: PR #104's stopped product-ring interchange, opposite-bank factorization, rational centres and the
+  h = 24 complex producer used unchanged; PR #10/#18/#24/#32/#36 lineage.
+- **Swapnil Jain**: deferred readouts (design B), V leaves, lifted frames, late copies and the round-seven bit
+  witness used unchanged (https://github.com/Swapnil-jain/integer-mult-kappa, commit
+  741e7aa078392553815df7926ee17ac5e25a8c38).
+- **Zhihao Chen (jacklightChen)**: PR #97's integration of the deferred bit word and its reflected ledgers;
+  PR21/PR29 translated gauges and two-stage accounting.
+- **Aurel Prosz (Paureel)**: the two-stage topology and paid endpoint correction.
+- All earlier authors credited in PR #104's NOTICE and SOURCES.json, including Avi Eisenberg/ikeboy (PR #53, #62,
+  whose producer is the round-seven bit DAG), RaD/hipotures, Rohan Arun, eumemic, Dominik Scholz, Chafik
+  Boukhalfa, James Chang and Douglas Colkitt.
+
+The complex deferral construction, its checks and this certificate were prepared by Avi Eisenberg (ikeboy) with
+Anthropic Claude assistance.
diff --git a/research/deferred-stopped/README.md b/research/deferred-stopped/README.md
new file mode 100644
index 0000000000..a46910f4ed
--- /dev/null
+++ b/research/deferred-stopped/README.md
@@ -0,0 +1,37 @@
+# Deferred readouts on both stopped networks
+
+Conditional κ = 429239/5000000000 = **8.58478 × 10^-5** (+10.1% over PR #104's 7.79476 × 10^-5).
+
+PR #104's stopped product-ring interchange compiles every projector residual of rank r as one child of width r.
+Under that rule deferred readouts pay on both networks:
+
+- **bit**: Swapnil Jain's round-seven deferred word (h = 23), unchanged, gives a stopped saving of 1.2402 × 10^-4;
+- **complex**: PR #104's h = 24 network, with lifted binary frames and deferred readouts on 8,893 of its 44,918
+  roles, gives 214657/2500000000 = 8.58628 × 10^-5, which is now the binding side.
+
+See [PROOF.md](PROOF.md) for the construction, the checks and the inherited assumptions.
+
+## Reproduce
+
+From the repository root (standard-library Python 3; about two minutes):
+
+```sh
+python3 research/deferred-stopped/complex_deferred.py
+python3 research/deferred-stopped/bit_round7.py DIR
+python3 research/deferred-stopped/certificate.py
+```
+
+`DIR` holds Swapnil's two round-seven files, `witness_23.json.gz` and `deferred_23.json.gz`, for example
+`certificates/round7/` of https://github.com/Swapnil-jain/integer-mult-kappa at commit
+`741e7aa078392553815df7926ee17ac5e25a8c38`, or PR #97's copy in `research/deferred-signed/swapnil-round7/`.
+`bit_round7.py` checks both files against their SHA-256 pins. The three scripts write `complex-profile.json`,
+`bit-profile.json` and `certificate.json`.
+
+## Files
+
+| file | content |
+|---|---|
+| `complex_deferred.py` | complex DAG (PR #104 producer), matching, roles, lifted frames, deferral, replay and exact F2 checks |
+| `bit_round7.py` | one-child bit profile of the round-seven deferred word |
+| `certificate.py` | exact moments, stopped bit saving, finite bridge, 47-constraint assembly, κ |
+| `*-profile.json`, `certificate.json` | generated outputs |
diff --git a/research/deferred-stopped/bit-profile.json b/research/deferred-stopped/bit-profile.json
new file mode 100644
index 0000000000..36105e819a
--- /dev/null
+++ b/research/deferred-stopped/bit-profile.json
@@ -0,0 +1,69 @@
+{
+ "h": 23,
+ "v": 1771,
+ "R": 28866,
+ "m": 529,
+ "N": 3136441,
+ "W": 108516254,
+ "L": 1792252,
+ "total_rank": 57403754177,
+ "deficit": 1344189,
+ "maxchild": 528,
+ "deferred_slots": 11565,
+ "source": {
+  "repository": "https://github.com/Swapnil-jain/integer-mult-kappa",
+  "commit": "741e7aa078392553815df7926ee17ac5e25a8c38",
+  "sha256": {
+   "witness_23.json.gz": "b2486aa2bb222bacea6e52162a3920780e45dd8edc5d7cb03eabea336a1c6588",
+   "deferred_23.json.gz": "8c38e947ff9e021e308000dd82bb5e2194eb265d8b75194e22ce783d3e75317c"
+  }
+ },
+ "child_multiplicities": {
+  "1": 239511811,
+  "2": 238288050,
+  "3": 53377940,
+  "4": 46970462,
+  "5": 23632224,
+  "6": 7409864,
+  "7": 4105178,
+  "8": 5755750,
+  "9": 2082696,
+  "10": 878416,
+  "11": 1388464,
+  "12": 1080310,
+  "13": 945714,
+  "14": 226688,
+  "15": 2451064,
+  "16": 1508892,
+  "17": 4771074,
+  "18": 2149994,
+  "19": 7367360,
+  "20": 8182020,
+  "21": 1466388,
+  "22": 88550,
+  "484": 6272882,
+  "506": 61280142,
+  "507": 545468,
+  "508": 173558,
+  "509": 293986,
+  "510": 580888,
+  "511": 14168,
+  "512": 322322,
+  "513": 17710,
+  "514": 1129898,
+  "515": 127512,
+  "516": 410872,
+  "517": 99176,
+  "518": 1186570,
+  "519": 340032,
+  "520": 322322,
+  "521": 106260,
+  "522": 1583274,
+  "523": 626934,
+  "524": 2854852,
+  "525": 13711082,
+  "526": 4629394,
+  "527": 5653032,
+  "528": 6233920
+ }
+}
diff --git a/research/deferred-stopped/bit_round7.py b/research/deferred-stopped/bit_round7.py
new file mode 100644
index 0000000000..56f972277b
--- /dev/null
+++ b/research/deferred-stopped/bit_round7.py
@@ -0,0 +1,129 @@
+#!/usr/bin/env python3
+"""One-child bit profile of Swapnil Jain's round-seven deferred word (h=23), under PR104's opposite-bank rule.
+
+Reads the two frozen round-seven data files (not vendored; pinned by SHA-256 below) from a directory given on
+the command line, e.g. a checkout of https://github.com/Swapnil-jain/integer-mult-kappa at
+741e7aa078392553815df7926ee17ac5e25a8c38 (certificates/round7/), or PR97's imported copy
+(research/deferred-signed/swapnil-round7/certificates/round7/). The frame dimensions are the ones recorded in
+the data file; Swapnil's check_frames.py re-derives each of them exactly and PR97 replays the word.
+
+Every projector residual of rank r is one child of width r (PR104, notes/stopped-product-factorization.tex):
+  each slot: its chain steps from the deferral frame sigma_u to F, and one exterior child m - (h - dim sigma_u);
+  each retained centre: one copied transform of width h-1;
+  each target y_T: its chain 0 -> deferred readout levels -> t_T^perp;  each data wire X_S: its chain of V frames;
+  each data pair: one data residual of width (h-1)^2 and one endpoint copy.
+Usage: python3 research/deferred-stopped/bit_round7.py DIR      (writes bit-profile.json next to this file)
+Prepared by Avi Eisenberg (ikeboy) with Anthropic Claude assistance. Apache-2.0.
+"""
+import gzip
+import hashlib
+import json
+import sys
+from collections import Counter, defaultdict
+from itertools import combinations
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+PINS = {'witness_23.json.gz': 'b2486aa2bb222bacea6e52162a3920780e45dd8edc5d7cb03eabea336a1c6588',
+        'deferred_23.json.gz': '8c38e947ff9e021e308000dd82bb5e2194eb265d8b75194e22ce783d3e75317c'}
+
+
+def require(ok, msg):
+    if not ok:
+        raise SystemExit('FAIL: ' + msg)
+
+
+def main(directory):
+    data = {}
+    for name, digest in PINS.items():
+        raw = (Path(directory) / name).read_bytes()
+        require(hashlib.sha256(raw).hexdigest() == digest, 'pinned hash of ' + name)
+        data[name] = json.loads(gzip.decompress(raw))
+    W, D = data['witness_23.json.gz'], data['deferred_23.json.gz']
+    h = D['h']; R = D['R']; trip = list(combinations(range(h), 3)); v = len(trip); m = h * h; N = v * v
+    tid = {T: i for i, T in enumerate(trip)}
+    args = {int(n): (tuple(a) if a else None) for n, a in W['args'].items()}
+    late = {int(n): (list(e), list(l)) for n, (e, l) in W['late'].items()}
+    ops = [tuple(o[:2]) + (tuple(o[2]),) if o[0] == 'fan' else tuple(o) for o in D['ops']]
+    hold, start = D['hold'], D['start']
+    out = {s: (c, tuple(T)) for s, c, T in D['out']}; ret = {s: c for s, c in D['ret']}
+    sigma = dict(zip(D['readout_order'], D['sigma'])); sel = set(D['readout_order'])
+    f = [0] * R
+    for s, B in sigma.items(): f[s] = len(B)
+    vstart = dict(zip(D['vleaf_slots'], D['vleaf_start']))
+    node_dims = {n: d for n, d in D['node_dims']}
+    late_dims = {(n, tuple(lt)): d for n, lt, d in D['late_dims']}
+    pivslot = {o[3]: o[1] for o in ops if o[0] == 'add'}
+
+    def start_key(s):
+        n0 = hold[s][0]
+        if args[n0] is None: return ('v', s)
+        st = start[s]; k = st[2] if len(st) > 2 else None; dec = late.get(n0)
+        if dec is None or k in dec[0]: return ('n', n0)
+        lt = dec[1]; return ('c', n0, tuple(lt[lt.index(k):]))
+
+    def chain(s):
+        ch = [('sigma', s) if s in sel else ('0',), start_key(s)]
+        for idx, n in enumerate(hold[s]):
+            if idx > 0: ch.append(('n', n))
+            if args[n] is not None and pivslot.get(n) == s and n in late:
+                lt = late[n][1]
+                for i in range(len(lt) - 1): ch.append(('c', n, tuple(lt[i:])))
+        if s in out: ch.append(('out',))
+        if s in ret: ch.append(('ret',))
+        ch.append(('F',)); return ch
+
+    def dim(key):
+        k = key[0]
+        if k == '0': return 0
+        if k == 'F': return h
+        if k in ('out', 'ret'): return h - 1
+        if k == 'n': return node_dims[key[1]]
+        if k == 'c': return late_dims[key[1], key[2]]
+        if k == 'sigma': return f[key[1]]
+        if k == 'v': return len(vstart[key[1]])
+        raise KeyError(key)
+
+    # garbage supports (transpose sweep) for the deferred levels on each target
+    cov = [set() for _ in range(R)]
+    for s, (c, T) in out.items(): cov[s].add(tid[T])
+    for s, c in ret.items(): cov[s].update(t for t, T in enumerate(trip) if c in T)
+    for o in reversed(ops):
+        if o[0] == 'add': cov[o[2]] |= cov[o[1]]
+        elif o[0] == 'fan':
+            for g in o[2]: cov[o[1]] |= cov[g]
+    levels = defaultdict(set)
+    for s in sel:
+        for t in cov[s]: levels[t].add(f[s])
+
+    z = Counter()
+    for s in range(R):
+        ds = [dim(k) for k in chain(s)]
+        require(all(b >= a for a, b in zip(ds, ds[1:])) and ds[-1] == h, 'monotone chain %d' % s)
+        for a, b in zip(ds, ds[1:]):
+            if b > a: z[b - a] += 2 * v
+        z[m - (h - f[s])] += 2 * v
+    z[h - 1] += 2 * v * h
+    for t in range(v):
+        ls = sorted(levels[t] | {0, h - 1})
+        for a, b in zip(ls, ls[1:]): z[b - a] += 2 * v
+    for n, us in D['xs_order']:
+        ds = [1] + [len(vstart[s]) for s in us] + [h]
+        ds = [d for i, d in enumerate(ds) if i == 0 or d != ds[i - 1]]
+        require(all(b > a for a, b in zip(ds, ds[1:])), 'data-wire chain')
+        for a, b in zip(ds, ds[1:]): z[b - a] += 2 * v
+    z[(h - 1) ** 2] += 2 * N
+    z[1] += N
+    Wb = 2 * N + 2 * v * R; L = 2 * v * h * (h - 1); s_ = Wb * m - N + L
+    require(sum(t * c for t, c in z.items()) == s_, 'bit rank mass')
+    require(all(0 < t < m for t in z), 'children below m')
+    prof = dict(h=h, v=v, R=R, m=m, N=N, W=Wb, L=L, total_rank=s_, deficit=N - L, maxchild=max(z),
+                deferred_slots=len(sel), source=dict(repository='https://github.com/Swapnil-jain/integer-mult-kappa',
+                commit='741e7aa078392553815df7926ee17ac5e25a8c38', sha256=PINS),
+                child_multiplicities=dict(sorted(z.items())))
+    (HERE / 'bit-profile.json').write_text(json.dumps(prof, indent=1) + '\n')
+    print('PASS bit: R=%d, deferred slots %d, maxchild %d, rank mass %d' % (R, len(sel), max(z), s_))
+
+
+if __name__ == '__main__':
+    main(sys.argv[1] if len(sys.argv) > 1 else '.')
diff --git a/research/deferred-stopped/certificate.json b/research/deferred-stopped/certificate.json
new file mode 100644
index 0000000000..00b4ec1e4a
--- /dev/null
+++ b/research/deferred-stopped/certificate.json
@@ -0,0 +1,175 @@
+{
+  "assembly": {
+    "absorption_gap": "4333030679469840929768124093977/62510732826606157597419467323285000000000",
+    "margins": {
+      "Gaussian_arithmetic": "1287976684152650437608480516106535343/10001717252256985215587114771725600000000",
+      "bulk_exposure": "214656535343/2500000000000000",
+      "compact_phase_layer": "1073282644516519913206535343/12502146565321231519483893464657",
+      "coordinate_movement": "214656535343/2500000000000000",
+      "dimension": "12499999875000000000000000000000/12502146565321231519483893464657",
+      "original_prefix": "107340766598217312552173018193464657/1250214656532123151948389346465700000000",
+      "scalar_work": "1717339754838650265883252241706535343/10001717252256985215587114771725600000000"
+    },
+    "minimum_margin": "1073282644516519913206535343/12502146565321231519483893464657",
+    "parameters": {
+      "C0": 2994465792675104952757349965746759652008778122322142627817848832,
+      "C1": 1,
+      "a_bit": "214656535343/2500000000000000",
+      "a_complex": "214657/2500000000",
+      "actual_bit_saving": "1240189553/10000000000000",
+      "alpha_squared_power": "1609986482874019698550000000/12502146565321231519483893464657",
+      "beta": "1/1000000",
+      "c": "1073282665982173018193464657/12500000000000000000000000000000",
+      "delta": "1/800000000",
+      "epsilon": "12499999875000000000000000000000/12502146565321231519483893464657",
+      "eta": "1/100000000",
+      "kappa": "429239/5000000000",
+      "lambda_": "249978534346680356535343/250000000000000000000000",
+      "lambda_prime": "124989267173447506535343/125000000000000000000000",
+      "q": "10732826552493464657/125000000000000000000000",
+      "sigma": "2499785343/2500000000",
+      "tau": "2499785343464657/2500000000000000"
+    },
+    "recurrence": {
+      "internal": "2499785343464657/2500000000000000",
+      "leaf": "2499785343214657/2500000000000000",
+      "reservations": "12498926717334017826981806535343/12500000000000000000000000000000"
+    },
+    "strict_constraints": {
+      "Gaussian_arithmetic_above_kappa": "10733781545608580555445025115159759483/250042931306424630389677869293140000000000",
+      "K_dominates_log": "107328265524934635837173447506535343/1250214656532123151948389346465700000000",
+      "K_geometry": "107340766598217312552173018193464657/1250214656532123151948389346465700000000",
+      "alpha_below_one": "12500536578838357499785343464657/12502146565321231519483893464657",
+      "alpha_below_one_fourth": "12495706619389735440689693464657/50008586261284926077935573858628",
+      "alpha_positive": "1609986482874019698550000000/12502146565321231519483893464657",
+      "artificial_boundary": "70014167468624421159203961426046506535343/10001717252256985215587114771725600000000",
+      "beta_below_one": "999999/1000000",
+      "beta_positive": "1/1000000",
+      "bit_positive": "214656535343/2500000000000000",
+      "bulk_exposure_above_kappa": "37035343/2500000000000000",
+      "c_below_one": "12498926717334017826981806535343/12500000000000000000000000000000",
+      "c_positive": "1073282665982173018193464657/12500000000000000000000000000000",
+      "cell_above_band": "12499463171161642500214656535343/25004293130642463038967786929314",
+      "compact_leaf": "12714656535343/125000000000000000000000",
+      "compact_phase_layer_above_kappa": "4333030679469840929768124093977/62510732826606157597419467323285000000000",
+      "compact_reservations": "10732826552493464657/12500000000000000000000000000000",
+      "complex_above_bit": "464657/2500000000000000",
+      "complex_below_one_over32": "77910343/2500000000",
+      "coordinate_movement_above_kappa": "37035343/2500000000000000",
+      "delta_below_one_eighth": "99999999/800000000",
+      "delta_positive": "1/800000000",
+      "dimension_above_kappa": "62494632966110448079903808253053124093977/62510732826606157597419467323285000000000",
+      "epsilon_below_one": "2146690321231519483893464657/12502146565321231519483893464657",
+      "epsilon_positive": "12499999875000000000000000000000/12502146565321231519483893464657",
+      "gamma_sublinear": "536703838357499785343464657/12502146565321231519483893464657",
+      "guard_width": "2146690321231519483893464657/12502146565321231519483893464657",
+      "lambda_above_internal": "214656535343/250000000000000000000000",
+      "lambda_above_sigma": "46680356535343/250000000000000000000000",
+      "lambda_above_tau": "214656535343/250000000000000000000000",
+      "lambda_prime_above_lambda": "214656535343/250000000000000000000000",
+      "lambda_prime_below_one": "10732826552493464657/125000000000000000000000",
+      "leaf_saving_above_bit": "1/10000000000",
+      "literal_scalar_guard": "402943837929348086347490890940",
+      "original_prefix_above_kappa": "629440358945531416903962797326827/62510732826606157597419467323285000000000",
+      "phase_boundary": "1287976684152650437608480516106535343/10001717252256985215587114771725600000000",
+      "phase_local": "1717339754838650265883252241706535343/10001717252256985215587114771725600000000",
+      "prime_interval_packing": "2146690321231519483893464657/12502146565321231519483893464657",
+      "q_below_internal": "214656535343/125000000000000000000000",
+      "q_below_leaf": "12714656535343/125000000000000000000000",
+      "q_below_reservations": "10732826552493464657/12500000000000000000000000000000",
+      "q_positive": "10732826552493464657/125000000000000000000000",
+      "record_suffix": "2146690321231519483893464657/12502146565321231519483893464657",
+      "row_product_gap": "21189/25",
+      "scalar_work_above_kappa": "21467858312758576262314318255159759483/250042931306424630389677869293140000000000",
+      "short_record_fallback": "31247316020033937757008189620029384607958127649/31255366413303078798709733661642500000000000000",
+      "small_field_exposure": "1073407676714999570686929314/12502146565321231519483893464657"
+    }
+  },
+  "base": "PR104 head 854ba7dca98651eab2050402384e1a7784534f0a",
+  "bit": {
+    "coarse_saving": "620523/5000000000",
+    "counts": {
+      "R": 28866,
+      "W": 108516254,
+      "deferred_slots": 11565,
+      "h": 23,
+      "m": 529,
+      "maxchild": 528,
+      "total_rank": 57403754177,
+      "v": 1771
+    },
+    "moment_gap": "210811394225115117568018038513239592855337768061996351784199300828041381607148420475711921673436778332750996651783152394205578153858435663243959795913385215868392831844509655226993164072062830500832781615792327068936897853818968534771730481866850519439187068953062899333771640996075361901014500735179763112430083120460814689994116646689670445286414265265018468311435502209195246240538069975559360544007937885404793780372701951022292660282948582656517071290400479729799954952580125532795274568021525151839594883237275569314958070368193662075730567130277424969924134425921633872848924585639423998410253779062359718371205767101510073996880454144801457072151080008705648366603197414406872289670642046369219405455128737300640897667220951127018547660455087533581513233999285275099394371781290865408274484542745086391849285198194905751058992419590161634431610100995441112613134240446609058758039909343152776920263419910572835208525277494067408376961456317/225678505065634612156474913616400633538202019254665477026878220214422133931127460164253533068001699469116755578965090566731480236912032281717920888140691998261537790258547549397061712443744511202265657747547567084090291301783866427704803156891369853685579999500594559805069402727752752227038426964007028952158078563045054314475625278140097385695241851533235641996467089273692960018822280793234080014352234618229252999217015923728814468917104222554059646947960375336061268431587018084270516494408656203498548153054913267707164839939006583338918817922618882667407408888179421052689449193499742148706302504238261936554897181289081072466758282367100305194479622149878636307463869297507337915811001244472279386614413535081936912313821531543735588465392499062010570720287729070905437080967709221039366272598541269000920375996285867953823816142239661027100714288630069735295241672325865922423925213948613346461034734632365263739073012128769510020000000000000000000000",
+    "stopped_saving": "1240189553/10000000000000"
+  },
+  "complex": {
+    "counts": {
+      "R": 44918,
+      "W": 190021216,
+      "deferred_roles": 8893,
+      "h": 24,
+      "m": 576,
+      "maxchild": 574,
+      "total_rank": 109450358336,
+      "v": 2024
+    },
+    "moment_gap": "88301284141942852349786272358911595120000420578552187561142071843083140835081165954598269953603159972525663693558868972689146185155411834676228714955329059432394778298100545891381302421394794303237626549527713862261276656942246868839949845155337175702657628564964600909613181267405991527055947538283019556031587158098318798755637269980823599499009552235241463891126134912516374961621697669023792194774305349574435245146518419799694178664643594108583316149754557776251922460810128999706784941841740847389653952435807583800062182165911883618980285421081541915510402227182490035361011191674206574792045268082637368945733008419490087505310344811508745393581449139761913559952494272322223808007836813175902382766819729906991789797044140283347524500198153199375890634896941931622814963367507249808681003857076806191317833919071452335898024170995119906284068985721462680222566297431686072161795896181548150927924607598240580782210064261914540490641568966426778147251/38567646380471990066846323528257836937304900418571665458488647335738525964839401421162181723482413035129710616602006898867035617030508100341220985070552885919489004210392740293654574535114858149929488168983623251817921624399761542587906296152985111009541140777876971526639421362325581838554355340456733298659299558103166845735904917705525545917977763439849396606704910357032922787157529838149607648539023383581492909176298859827611384377284000467495755998769776989475028380041994440168558308474514943668961421778304739693567885686710949313112429918061388147558166502860588101321963799112097800740240024875708063144007409056818296375362429458681617944441266401553094477220896677057264283075058375992973747610948147134524276528806960610780471475495100666041243705609728626257304886601414783085155723501606613314074979295066720153503485896357460909958593539684309751023373339104134081577618563270169710365314887102059427502899064010200818867353355720320000000000000000000000",
+    "saving": "214657/2500000000"
+  },
+  "finite_bridge": {
+    "bit_coarse": {
+      "W": 108516254,
+      "halving_degree": 367,
+      "m": 529,
+      "maxchild": 528,
+      "wire_bits": 27
+    },
+    "complex": {
+      "W": 190021216,
+      "halving_degree": 200,
+      "m": 576,
+      "maxchild": 574,
+      "s": 109450358336,
+      "scalar_group_upper": 1656684480,
+      "scalar_terms": [
+        {
+          "c": 45764,
+          "copied_center_extra_groups": 48,
+          "h": 24,
+          "invocations": 2024,
+          "local_group_upper": 408200,
+          "v": 2024
+        },
+        {
+          "c": 45764,
+          "copied_center_extra_groups": 48,
+          "h": 24,
+          "invocations": 2024,
+          "local_group_upper": 408200,
+          "v": 2024
+        }
+      ],
+      "wire_bits": 28
+    },
+    "ordinary_leaf_row_degree": 252,
+    "rows": {
+      "coefficient": 15761,
+      "contract": "W_complex^D_complex * W_coarse^D_coarse * W_old^D_old; one preceding prefix and one padding; sequential reuse",
+      "degree": 33000,
+      "degree_gap": "21189/25",
+      "suffix_slope": 132000
+    },
+    "semantic": {
+      "B": 403063477262941121553211537024,
+      "C0": 2994465792675104952757349965746759652008778122322142627817848832,
+      "C1": 1,
+      "E": 403063477262941121443761178688,
+      "exact_grid": "One common dyadic grid times the fixed odd divisor to K=G*(D_complex+1); no child rounding",
+      "fixed_odd_divisor": 21,
+      "induction_gap": 1209190431788823364659634611072,
+      "literal_charge": 119639333593035096270287748,
+      "strict_literal_gap": 402943837929348086347490890940
+    }
+  },
+  "kappa": "429239/5000000000",
+  "status": "Conditional witness: deferred readouts on both networks of the stopped product-ring interchange"
+}
diff --git a/research/deferred-stopped/certificate.py b/research/deferred-stopped/certificate.py
new file mode 100644
index 0000000000..d7685100c1
--- /dev/null
+++ b/research/deferred-stopped/certificate.py
@@ -0,0 +1,126 @@
+#!/usr/bin/env python3
+"""Exact certificate: deferred readouts on both networks of PR104's stopped product-ring interchange.
+
+Inputs (regenerated by the two scripts in this directory):
+  bit-profile.json      Swapnil Jain's round-seven deferred bit word, one child per residual (bit_round7.py);
+  complex-profile.json  PR104's h=24 complex network with lifted binary frames and deferred readouts
+                        (complex_deferred.py).
+Computes, with PR104's inherited exact routines: the largest coarse bit saving and complex saving on the
+10^-10 grid with a contracting moment, the stopped ordinary bit saving (PR104 stopping at theta = 1/1000),
+the finite semantic/product-row bridge (row stock raised as the larger children require; it enters only the
+row slack and cutoffs) and the 47-constraint balanced assembly at the largest kappa on the 10^-10 grid.
+Usage: python3 research/deferred-stopped/certificate.py      (writes certificate.json)
+Prepared by Avi Eisenberg (ikeboy) with Anthropic Claude assistance. Apache-2.0.
+"""
+import json
+import sys
+from fractions import Fraction as Q
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+sys.path.insert(0, str(ROOT / 'scripts'))
+from partial_swap_network import moment                                    # noqa: E402
+from copied_centers_network import finite_bridge as copied_bridge          # noqa: E402
+from structured_bulk_assembly import assembly, js                          # noqa: E402
+
+GRID = 10**10
+ATOM = Q(1, 1000)                       # PR104 stopping exponent theta
+OLD = Q(384599, 10**10)                 # PR104 retained ordinary leaf saving
+PHASE_STOP = Q(1, 10**6)                # PR104 beta
+
+
+def require(ok, msg):
+    if not ok:
+        raise SystemExit('FAIL: ' + msg)
+
+
+def profile(name):
+    p = json.loads((HERE / name).read_text())
+    p['child_multiplicities'] = {int(t): c for t, c in p['child_multiplicities'].items()}
+    z = p['child_multiplicities']
+    require(sum(t * c for t, c in z.items()) == p['total_rank'] == p['W'] * p['m'] - p['N'] + p['L'], name + ' rank mass')
+    require(all(0 < t < p['m'] and c > 0 for t, c in z.items()), name + ' children')
+    require(max(z) == p['maxchild'], name + ' maxchild')
+    return p
+
+
+def contracts(p, a):
+    try:
+        moment(p['m'], p['W'], p['child_multiplicities'], a, True)
+        return True
+    except SystemExit:
+        return False
+    except Exception:
+        return False
+
+
+def largest(p, lo, hi):
+    a, b = int(lo * GRID), int(hi * GRID)
+    require(contracts(p, Q(a, GRID)) and not contracts(p, Q(b, GRID)), 'bracket')
+    while b - a > 1:
+        mid = (a + b) // 2
+        a, b = (mid, b) if contracts(p, Q(mid, GRID)) else (a, mid)
+    return Q(a, GRID)
+
+
+def main():
+    bit, phase = profile('bit-profile.json'), profile('complex-profile.json')
+    coarse = largest(bit, Q(7, 10**5), Q(2, 10**4))
+    ac = largest(phase, Q(7, 10**5), Q(2, 10**4))
+    bm = moment(bit['m'], bit['W'], bit['child_multiplicities'], coarse, True)
+    cm = moment(phase['m'], phase['W'], phase['child_multiplicities'], ac, True)
+    require(not contracts(bit, coarse + Q(1, GRID)) and not contracts(phase, ac + Q(1, GRID)), 'next grid points rejected')
+    require(0 < OLD <= coarse < ATOM < 1, 'stopped-adapter exponents')
+    ab = (1 - ATOM) * coarse + ATOM * OLD
+    require(ATOM > ab, 'adapter toll subordinate to the stopped saving')
+    abit = min(ab, (1 - PHASE_STOP) * ac - Q(1, GRID))
+
+    row = dict(h=phase['h'], v=phase['v'], c=phase['additions'], central_disjoint=24)
+    previous = json.loads((ROOT / 'certificates' / 'copied-centers-network.json').read_text())
+    bridge = copied_bridge(bit, phase, [row, row])
+    old_bit = previous['finite_bridge']['bit']
+    old_degree = old_bit['halving_degree'] * old_bit['wire_bits']
+    require(old_degree == 252, 'inherited ordinary leaf row stock')
+    coarse_bridge = bridge.pop('bit'); cpx = bridge['complex']
+    coefficient = (coarse_bridge['halving_degree'] * coarse_bridge['wire_bits'] + old_degree +
+                   cpx['halving_degree'] * cpx['wire_bits'])
+    degree = 1000 * ((51 * coefficient) // 25000 + 1)
+    require(Q(degree) > Q(51 * coefficient, 25), 'product stock strictly sufficient')
+    bridge.update(bit_coarse=coarse_bridge, ordinary_leaf_row_degree=old_degree)
+    bridge['rows'] = dict(coefficient=coefficient, degree=degree, suffix_slope=4 * degree,
+                          degree_gap=Q(degree) - Q(51 * coefficient, 25),
+                          contract='W_complex^D_complex * W_coarse^D_coarse * W_old^D_old; one preceding prefix and one padding; sequential reuse')
+    bridge['semantic'].update(fixed_odd_divisor=21,
+                              exact_grid='One common dyadic grid times the fixed odd divisor to K=G*(D_complex+1); no child rounding')
+
+    def accepted(k):
+        try:
+            assembly(abit, ac, bridge, Q(k, GRID), beta=PHASE_STOP)
+            return True
+        except AssertionError:
+            return False
+    lo, hi = int(Q(7, 10**5) * GRID), int(ac * GRID) + 1
+    require(accepted(lo) and not accepted(hi), 'kappa bracket')
+    while hi - lo > 1:
+        mid = (lo + hi) // 2
+        lo, hi = (mid, hi) if accepted(mid) else (lo, mid)
+    kappa = Q(lo, GRID)
+    assembled = assembly(abit, ac, bridge, kappa, beta=PHASE_STOP)
+    require(not accepted(lo + 1), 'next kappa grid point rejected')
+    require(len(assembled['strict_constraints']) == 47 and len(assembled['margins']) == 7, 'complete assembly')
+    assembled['parameters']['actual_bit_saving'] = ab
+    result = dict(status='Conditional witness: deferred readouts on both networks of the stopped product-ring interchange',
+                  kappa=kappa, bit=dict(coarse_saving=coarse, stopped_saving=ab, moment_gap=bm['strict_gap'],
+                                        counts={k: bit[k] for k in ('h', 'v', 'R', 'm', 'W', 'total_rank', 'maxchild', 'deferred_slots')}),
+                  complex=dict(saving=ac, moment_gap=cm['strict_gap'],
+                               counts={k: phase[k] for k in ('h', 'v', 'R', 'm', 'W', 'total_rank', 'maxchild', 'deferred_roles')}),
+                  finite_bridge=bridge, assembly=assembled,
+                  base='PR104 head 854ba7dca98651eab2050402384e1a7784534f0a')
+    (HERE / 'certificate.json').write_text(json.dumps(js(result), indent=2, sort_keys=True) + '\n')
+    print('PASS kappa = %s = %.10e' % (kappa, float(kappa)))
+    print('coarse bit saving %s, stopped %s, complex saving %s' % (coarse, ab, ac))
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/deferred-stopped/complex-profile.json b/research/deferred-stopped/complex-profile.json
new file mode 100644
index 0000000000..400e10fbbe
--- /dev/null
+++ b/research/deferred-stopped/complex-profile.json
@@ -0,0 +1,99 @@
+{
+ "h": 24,
+ "v": 2024,
+ "additions": 45764,
+ "roots": 8120,
+ "links": 8966,
+ "R": 44918,
+ "m": 576,
+ "N": 4096576,
+ "W": 190021216,
+ "L": 2234496,
+ "total_rank": 109450358336,
+ "deficit": 1862080,
+ "maxchild": 574,
+ "phase_one_ops": 17811,
+ "phase_one_roles": 15082,
+ "deferred_roles": 8893,
+ "deferred_dims": {
+  "1": 6,
+  "2": 206,
+  "4": 259,
+  "5": 1,
+  "6": 680,
+  "7": 1,
+  "8": 542,
+  "9": 8,
+  "10": 681,
+  "11": 40,
+  "12": 725,
+  "13": 133,
+  "14": 717,
+  "15": 143,
+  "16": 614,
+  "17": 233,
+  "18": 1007,
+  "19": 526,
+  "20": 612,
+  "21": 237,
+  "22": 1522
+ },
+ "lifted_additions": 20234,
+ "replay": {
+  "seeds": [
+   1,
+   2
+  ],
+  "scratch_restored": true,
+  "y_plus_x": true,
+  "field": "Z/(2^61-1)"
+ },
+ "child_multiplicities": {
+  "1": 242993344,
+  "2": 116792896,
+  "3": 37808320,
+  "4": 59355824,
+  "5": 46325312,
+  "6": 32234224,
+  "7": 19661136,
+  "8": 30449056,
+  "9": 9715200,
+  "10": 21725616,
+  "11": 6861360,
+  "12": 20555744,
+  "13": 5145008,
+  "14": 21919920,
+  "15": 4048000,
+  "16": 18390064,
+  "17": 8055520,
+  "18": 17013744,
+  "19": 8739632,
+  "20": 9460176,
+  "21": 1716352,
+  "22": 8937984,
+  "23": 8290304,
+  "529": 8193152,
+  "552": 145829200,
+  "553": 24288,
+  "554": 833888,
+  "556": 1048432,
+  "557": 4048,
+  "558": 2752640,
+  "559": 4048,
+  "560": 2194016,
+  "561": 32384,
+  "562": 2756688,
+  "563": 161920,
+  "564": 2934800,
+  "565": 538384,
+  "566": 2902416,
+  "567": 578864,
+  "568": 2485472,
+  "569": 943184,
+  "570": 4076336,
+  "571": 2129248,
+  "572": 2477376,
+  "573": 959376,
+  "574": 6161056
+ }
+}
diff --git a/research/deferred-stopped/complex_deferred.py b/research/deferred-stopped/complex_deferred.py
new file mode 100644
index 0000000000..9eb572013c
--- /dev/null
+++ b/research/deferred-stopped/complex_deferred.py
@@ -0,0 +1,458 @@
+#!/usr/bin/env python3
+"""Deferred readouts on PR104's h=24 complex network, with lifted binary frames.
+
+Builds PR104's aligned rational-center DAG (scripts/stopped_product/complex.py, unchanged), a maximum carrier
+matching under the inherited matcher's adjacency rules, an explicit role compile, lifted binary frames,
+phase one, deferred readout frames and the stage-one word. Checks:
+  A. the matching has the inherited size (8966) and the role count R = c + q - links = 44918;
+  B. the compiled word computes every root value (zero scratch, random data, mod p);
+  C. replay of the deferred word with arbitrary scratch and data (two seeds, mod p): every role restored and
+     every target y_S receives exactly x_S;
+  D. exact F2 frame facts: lifted frames nondegenerate and nested along every role chain (gates, links, copies,
+     root frames), every deferral frame inside its start frame and inside t_S^perp for every reached target,
+     nondegenerate, and nested on every target in readout order;
+  E. the one-child histogram from the explicit role chains, with total rank W m - N + L.
+Writes complex-profile.json next to this file.  Usage: python3 research/deferred-stopped/complex_deferred.py
+Prepared by Avi Eisenberg (ikeboy) with Anthropic Claude assistance. Apache-2.0.
+"""
+import array
+import json
+import random
+import struct
+import sys
+import tempfile
+from collections import Counter, defaultdict
+from itertools import combinations
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+sys.path.insert(0, str(ROOT / 'scripts'))
+from stopped_product.complex import build  # noqa: E402  (PR104 producer, unchanged)
+
+P = (1 << 61) - 1
+
+
+def require(ok, msg):
+    if not ok:
+        raise SystemExit('FAIL: ' + msg)
+
+
+def load(prefix):
+    def rd(fh, typ, cnt):
+        a = array.array(typ); a.fromfile(fh, cnt); return a
+    with open(str(prefix) + '.bin', 'rb') as f:
+        h, v, n, q = struct.unpack('<4I', f.read(16))
+        flat = rd(f, 'I', 2 * n); core = rd(f, 'Q', n); cover = rd(f, 'Q', n)
+        roots = rd(f, 'I', q); kind = rd(f, 'I', q); active = rd(f, 'B', n)
+    with open(str(prefix) + '.labels', 'rb') as f:
+        ranks = rd(f, 'I', n); types = rd(f, 'B', n)
+    args = [(flat[2 * i], flat[2 * i + 1]) for i in range(n)]
+    return h, v, n, q, args, core, cover, roots, kind, active, ranks, types
+
+
+# ---------------------------------------------------------------- F2 linear algebra on bit vectors
+def dot(a, b): return bin(a & b).count('1') & 1
+
+
+def reduce(basis):
+    rows = []; piv = []
+    for vec in basis:
+        for p, r in zip(piv, rows):
+            if vec >> p & 1: vec ^= r
+        if vec:
+            p = vec.bit_length() - 1
+            for i in range(len(rows)):
+                if rows[i] >> p & 1: rows[i] ^= vec
+            rows.append(vec); piv.append(p)
+    return rows
+
+
+def kernel(funcs, h):
+    K = reduce(funcs); piv = [r.bit_length() - 1 for r in K]
+    out = []
+    for c in (i for i in range(h) if i not in piv):
+        vec = 1 << c
+        for r, p in zip(K, piv):
+            if r >> c & 1: vec |= 1 << p
+        out.append(vec)
+    return out
+
+
+def restrict(basis, funcs):
+    B = reduce(basis)
+    for f in funcs:
+        if not B: break
+        vals = [dot(b, f) for b in B]
+        if not any(vals): continue
+        k = vals.index(1); pv = B[k]
+        B = [b ^ pv if val else b for b, val in zip(B, vals)]; B.pop(k)
+    return reduce(B)
+
+
+def contains(A, B):
+    RB = reduce(B)
+    return len(reduce(RB + list(A))) == len(RB)
+
+
+def cap(A, B):
+    A = reduce(A); B = reduce(B)
+    if not A or not B: return []
+    ker = []; red = []
+    for i, vec in enumerate(A + B):
+        tag = 1 << i
+        for rv, rt in red:
+            if vec >> (rv.bit_length() - 1) & 1: vec ^= rv; tag ^= rt
+        if vec: red.append((vec, tag))
+        else:
+            comb_ = 0
+            for k in range(len(A)):
+                if tag >> k & 1: comb_ ^= A[k]
+            ker.append(comb_)
+    return reduce(ker)
+
+
+def nondeg(B):
+    B = reduce(B); k = len(B)
+    if not k: return True
+    M = [sum(dot(B[i], B[j]) << j for j in range(k)) for i in range(k)]
+    return len(reduce(M)) == k
+
+
+def hopcroft_karp(left, adj):
+    """maximum bipartite matching; adj[x] = sorted list of right vertices."""
+    INF = 1 << 60
+    mate_l = {x: None for x in left}; mate_r = {}
+    while True:
+        dist = {}; queue = []
+        for x in left:
+            if mate_l[x] is None: dist[x] = 0; queue.append(x)
+            else: dist[x] = INF
+        found = INF; head = 0
+        while head < len(queue):
+            x = queue[head]; head += 1
+            if dist[x] >= found: continue
+            for r in adj[x]:
+                y = mate_r.get(r)
+                if y is None: found = min(found, dist[x] + 1)
+                elif dist[y] == INF: dist[y] = dist[x] + 1; queue.append(y)
+        if found == INF: break
+        gained = 0
+        for x0 in left:
+            if mate_l[x0] is not None: continue
+            stack = [(x0, iter(adj[x0]))]; path = []
+            while stack:
+                x, it = stack[-1]
+                advanced = False
+                for r in it:
+                    y = mate_r.get(r)
+                    if (y is None and dist[x] + 1 == found) or (y is not None and dist[y] == dist[x] + 1):
+                        path.append((x, r))
+                        if y is None:
+                            for xx, rr in path: mate_l[xx] = rr; mate_r[rr] = xx
+                            gained += 1; stack = []; break
+                        stack.append((y, iter(adj[y]))); advanced = True; break
+                else:
+                    dist[x] = INF; stack.pop()
+                    if path: path.pop()
+                    continue
+                if not advanced: break
+        if not gained: break
+    return mate_l
+
+
+def main():
+    with tempfile.TemporaryDirectory(prefix='deferred-stopped-') as work:
+        prefix = Path(work) / 'complex'
+        build(24, prefix, central_disjoint=24)
+        h, v, n, q, args, core, cover, roots, kind, active, ranks, types = load(prefix)
+    trip = list(combinations(range(h), 3)); require(len(trip) == v, 'triple count')
+    m = h * h; N = v * v; FULL = (1 << h) - 1
+    tmask = [sum(1 << p for p in T) for T in trip]
+    tid = {T: i for i, T in enumerate(trip)}
+
+    # ------------------------------------------------------------ uses and matching (inherited adjacency)
+    degree = [0] * n; uses = defaultdict(list); c_add = 0
+    for x in range(1, n):
+        if active[x] and args[x][0]:
+            c_add += 1
+            for pos, y in enumerate(args[x]): degree[y] += 1; uses[y].append(('gate', x, pos))
+    for j, x in enumerate(roots): degree[x] += 1; uses[x].append(('root', j))
+    def unode(u): return roots[u[1]] if u[0] == 'root' else u[1]
+    def before(x, u):
+        y = unode(u)
+        return (ranks[x], x) < (ranks[y], (n + u[1]) if u[0] == 'root' else y)
+    def incl(x, y):
+        tx, ty = types[x], types[y]
+        if tx == 1 and ty == 1: return not (core[y] & ~core[x]) and not (cover[x] & ~cover[y])
+        if tx in (1, 2) and ty == 2: return not (cover[x] & ~cover[y])
+        if tx == 1 and ty == 3: return bool(core[x] & core[y])
+        if tx == 2 and ty == 3: return not (cover[x] & ~core[y])
+        if tx == 3 and ty == 2: return cover[y] == FULL
+        if tx == 3 and ty == 3: return core[x] == core[y]
+        return False
+    left = []; adj = {}
+    for x in range(1, n):
+        if not (active[x] and args[x][0]): continue
+        rs = sorted({(y, k) for y in args[x] for k, u in enumerate(uses[y]) if before(x, u) and incl(x, unode(u))})
+        if rs: left.append(x); adj[x] = rs
+    mate = hopcroft_karp(left, adj)
+    links = {x: r for x, r in mate.items() if r is not None}
+    R = c_add + q - len(links)
+    require(len(links) == 8966 and R == 44918, 'matching size %d, roles %d' % (len(links), R))
+    linked_use = {(y, k): x for x, (y, k) in links.items()}
+
+    # ------------------------------------------------------------ root targets, read coefficients, root functionals
+    target = [None] * q
+    for j in range(v): target[j] = j
+    idx = v
+    for a, b in combinations(range(h), 2):
+        others = sorted((i for i in range(h) if i not in (a, b)), key=lambda i: ((i ^ 1) in (a, b), i))
+        for i in others: target[idx] = tid[tuple(sorted((a, b, i)))]; idx += 1
+    centre_of = {}
+    for j in range(q):
+        if kind[j]:
+            miss = FULL & ~cover[roots[j]]; require(bin(miss).count('1') == 1, 'centre cover')
+            centre_of[j] = miss.bit_length() - 1
+    require(idx + len(centre_of) == q, 'root order')
+    rootfun = [(1 << centre_of[j]) if kind[j] else tmask[target[j]] for j in range(q)]
+
+    # ------------------------------------------------------------ lifted binary frames, monotone
+    succ = defaultdict(set); droot = defaultdict(set)
+    for x in range(1, n):
+        if not active[x]: continue
+        for k, u in enumerate(uses[x]):
+            if (x, k) in linked_use: continue
+            (succ[x].add(u[1]) if u[0] == 'gate' else droot[x].add(u[1]))
+    for x, (y, k) in links.items():
+        u = uses[y][k]; (succ[x].add(u[1]) if u[0] == 'gate' else droot[x].add(u[1]))
+    order_desc = sorted((x for x in range(1, n) if active[x]), key=lambda x: (ranks[x], x), reverse=True)
+    K = {}
+    for x in order_desc:
+        rows = [rootfun[j] for j in droot[x]]
+        for t in succ[x]: rows += K[t]
+        K[x] = reduce(rows)
+    def envelope(x):
+        if not args[x][0]: return [tmask[x - 1]]
+        if types[x] == 2: return [1 << i for i in range(h) if cover[x] >> i & 1]
+        require(types[x] == 1, 'only ordinary and common-pair labels occur')
+        ks = [i for i in range(h) if (cover[x] & ~core[x]) >> i & 1]
+        require(len(ks) == ranks[x], 'common-pair support')
+        return [core[x] | (1 << k) for k in ks]
+    U = {}
+    for x in range(1, n):
+        if not active[x]: continue
+        if not args[x][0]: U[x] = envelope(x); continue
+        B = kernel(K[x], h)
+        require(contains(envelope(x), B), 'label outside lifted frame')
+        U[x] = B if nondeg(B) else envelope(x)
+    changed = True
+    while changed:
+        changed = False
+        for x in order_desc:
+            if args[x][0] and any(not contains(U[x], U[t]) for t in succ[x]):
+                env = envelope(x)
+                if len(reduce(U[x])) != len(reduce(env)) or not contains(U[x], env): U[x] = env; changed = True
+    dimU = {x: len(reduce(B)) for x, B in U.items()}
+
+    # ------------------------------------------------------------ explicit role compile
+    order = sorted((x for x in range(1, n) if active[x]), key=lambda x: (ranks[x], x))
+    holds = []; first_node = []; ops = []; edge_key = {}; role_root = {}
+    def new_role(node): holds.append([node]); first_node.append(node); return len(holds) - 1
+    def serve(s, y, k):
+        u = uses[y][k]
+        if u[0] == 'gate': edge_key[(y, k)] = s
+        else: role_root[s] = u[1]
+    for x in order:
+        free = [k for k in range(len(uses[x])) if (x, k) not in linked_use]
+        if not args[x][0]:
+            for k in free:
+                s = new_role(x); ops.append(('src', s, x)); serve(s, x, k)
+            continue
+        a, b = args[x]
+        ka = next(k for k, u in enumerate(uses[a]) if u == ('gate', x, 0))
+        kb = next(k for k, u in enumerate(uses[b]) if u == ('gate', x, 1))
+        sa, sb = edge_key.pop((a, ka)), edge_key.pop((b, kb))
+        if x in links and links[x][0] == a: sa, sb = sb, sa
+        piv, oth = sa, sb
+        ops.append(('add', piv, oth, x)); holds[piv].append(x); holds[oth].append(x)
+        if x in links: serve(oth, *links[x])
+        require(free, 'every use linked at node %d' % x)
+        serve(piv, x, free[0])
+        for k in free[1:]:
+            f = new_role(x); ops.append(('copy', piv, f, x)); serve(f, x, k)
+    require(not edge_key and len(holds) == R, 'compile')
+    Rr = R
+
+    # ------------------------------------------------------------ B. the word computes every root value
+    rng = random.Random(20261008)
+    xs = [rng.randrange(P) for _ in range(v)]
+    val = [0] * n
+    for i in range(v): val[i + 1] = xs[i]
+    for x in order:
+        if args[x][0]: val[x] = (val[args[x][0]] + val[args[x][1]]) % P
+    leaf_of = {s: first_node[s] for s in range(Rr) if not args[first_node[s]][0]}
+    a_ = [0] * Rr
+    for s, leaf in leaf_of.items(): a_[s] = xs[leaf - 1]
+    for o in ops:
+        if o[0] == 'add': a_[o[1]] = (a_[o[1]] + a_[o[2]]) % P
+        elif o[0] == 'copy': a_[o[2]] = (a_[o[2]] + a_[o[1]]) % P
+    require(all(a_[s] == val[roots[j]] for s, j in role_root.items()), 'root values')
+
+    # ------------------------------------------------------------ phase one and garbage reach
+    def touch(o): return [o[1], o[2]] if o[0] in ('add', 'copy') else []
+    prev = {}; pred = defaultdict(list); last = {}
+    for i, o in enumerate(ops):
+        for s in touch(o):
+            if s in prev: pred[i].append(prev[s])
+            prev[s] = i; last[s] = i
+    centre_roles = [s for s, j in role_root.items() if kind[j]]
+    Anc = set(); st = [last[s] for s in centre_roles]
+    while st:
+        i = st.pop()
+        if i not in Anc: Anc.add(i); st.extend(pred[i])
+    touched = set(centre_roles)
+    for i in Anc: touched.update(touch(ops[i]))
+    require(all(last[s] in Anc for s in centre_roles), 'centres complete in phase one')
+    reach = [set() for _ in range(Rr)]; reach_all = [False] * Rr
+    for s, j in role_root.items():
+        if kind[j]: reach_all[s] = True
+        else: reach[s].add(target[j])
+    for o in reversed(ops):
+        if o[0] == 'add': reach[o[2]] |= reach[o[1]]; reach_all[o[2]] = reach_all[o[2]] or reach_all[o[1]]
+        elif o[0] == 'copy': reach[o[1]] |= reach[o[2]]; reach_all[o[1]] = reach_all[o[1]] or reach_all[o[2]]
+
+    # ------------------------------------------------------------ deferral frames, insertion nesting
+    def F0(s): return U[first_node[s]]
+    cand = {}
+    for s in range(Rr):
+        if s in touched or reach_all[s]: continue
+        S_ = restrict(F0(s), [tmask[t] for t in reach[s]])
+        if S_ and nondeg(S_): cand[s] = S_
+    byT = defaultdict(list); placed = {}
+    for s in sorted(cand, key=lambda s: (-len(cand[s]), s)):
+        X = cand[s]; changed = True
+        while changed and X:
+            changed = False
+            for t in reach[s]:
+                for w in byT[t]:
+                    B = placed[w]
+                    if (len(B) >= len(X) and not contains(X, B)) or (len(B) < len(X) and not contains(B, X)):
+                        X = cap(X, B); changed = True
+                    if not X: break
+                if not X: break
+        if X and nondeg(X):
+            placed[s] = X
+            for t in reach[s]: byT[t].append(s)
+    deferred = sorted(placed, key=lambda s: (len(placed[s]), s)); dset = set(deferred)
+
+    # ------------------------------------------------------------ C. replay with arbitrary scratch and data
+    inv = lambda a: pow(a % P, P - 2, P); HALF = inv(2); I21 = inv(21)
+    cvec = [None] * Rr; dpart = [dict() for _ in range(Rr)]
+    for s, j in role_root.items():
+        if kind[j]: c = [0] * h; c[centre_of[j]] = 1; cvec[s] = c
+        else: dpart[s][target[j]] = (P - HALF) if j >= v else HALF
+    seed_c = {s: (None if cvec[s] is None else list(cvec[s])) for s in role_root}
+    seed_d = {s: dict(dpart[s]) for s in role_root}
+    def addc(dst, src):
+        if cvec[src] is not None:
+            cvec[dst] = list(cvec[src]) if cvec[dst] is None else [(p + q_) % P for p, q_ in zip(cvec[dst], cvec[src])]
+        for t, c in dpart[src].items(): dpart[dst][t] = (dpart[dst].get(t, 0) + c) % P
+    for o in reversed(ops):
+        if o[0] == 'add': addc(o[2], o[1])
+        elif o[0] == 'copy': addc(o[1], o[2])
+    scatter = [[(I21 - (HALF if i in trip[t] else 0)) % P for t in range(v)] for i in range(h)]
+    def readout(y, s, value, sign, seed=False):
+        cv = seed_c[s] if seed else cvec[s]; dp = seed_d[s] if seed else dpart[s]
+        if cv is not None:
+            for i, ci in enumerate(cv):
+                if ci:
+                    f = sign * ci * value % P; row = scatter[i]
+                    for t in range(v): y[t] = (y[t] + f * row[t]) % P
+        for t, c in dp.items(): y[t] = (y[t] + sign * c * value) % P
+    phase1 = sorted(Anc); rest = [i for i in range(len(ops)) if i not in Anc]
+    def replay(seed):
+        rng = random.Random(seed)
+        x = [rng.randrange(P) for _ in range(v)]; z = [rng.randrange(P) for _ in range(Rr)]
+        y0 = [rng.randrange(P) for _ in range(v)]; a = list(z); y = list(y0)
+        for s in range(Rr):
+            if s not in dset: readout(y, s, a[s], -1)
+        for s, leaf in leaf_of.items():
+            if s not in dset: a[s] = (a[s] + x[leaf - 1]) % P
+        def run(i, sign=1):
+            o = ops[i]
+            if o[0] == 'add': a[o[1]] = (a[o[1]] + sign * a[o[2]]) % P
+            elif o[0] == 'copy': a[o[2]] = (a[o[2]] + sign * a[o[1]]) % P
+        for i in phase1: run(i)
+        for s in centre_roles: readout(y, s, a[s], +1, True)
+        for s in deferred: readout(y, s, a[s], -1)
+        for s in deferred:
+            if s in leaf_of: a[s] = (a[s] + x[leaf_of[s] - 1]) % P
+        for i in rest: run(i)
+        for s, j in role_root.items():
+            if not kind[j]: readout(y, s, a[s], +1, True)
+        for i in reversed(range(len(ops))): run(i, -1)
+        for s, leaf in leaf_of.items(): a[s] = (a[s] - x[leaf - 1]) % P
+        return a == z, all((y[t] - y0[t] - x[t]) % P == 0 for t in range(v))
+    rep = [replay(seed) for seed in (1, 2)]
+    require(all(r == (True, True) for r in rep), 'replay %s' % rep)
+
+    # ------------------------------------------------------------ D. exact F2 frame facts
+    root_frame = {}
+    for s, j in role_root.items():
+        root_frame[s] = [1 << i for i in range(h) if i != centre_of[j]] if kind[j] else kernel([tmask[target[j]]], h)
+    FULLB = [1 << i for i in range(h)]
+    chain_dims = []
+    for s in range(Rr):
+        seq = [placed.get(s, []), F0(s)] + [U[x] for x in holds[s][1:]]
+        if s in root_frame: seq.append(root_frame[s])
+        seq.append(FULLB)
+        require(all(contains(A, B) for A, B in zip(seq, seq[1:])), 'role chain nesting %d' % s)
+        require(all(nondeg(B) for B in seq[1:-1] if B), 'degenerate frame on role %d' % s)
+        chain_dims.append([len(reduce(B)) for B in seq])
+    for s, X in placed.items():
+        require(contains(X, F0(s)) and nondeg(X), 'deferral frame %d' % s)
+        require(all(not any(dot(xv, tmask[t]) for xv in X) for t in reach[s]), 'target frame %d' % s)
+    byT2 = defaultdict(list)
+    for s in deferred:
+        for t in reach[s]: byT2[t].append(s)
+    for t, ss in byT2.items():
+        ss.sort(key=lambda s: (len(placed[s]), s))
+        require(all(contains(placed[p], placed[q_]) for p, q_ in zip(ss, ss[1:])), 'target chain %d' % t)
+
+    # ------------------------------------------------------------ E. one-child histogram
+    z = Counter()
+    for s in range(Rr):
+        ds = chain_dims[s]
+        for a, b in zip(ds[:-1], ds[1:-1]):
+            if b > a: z[b - a] += 2 * v
+        lastd = ds[-2]
+        if s in role_root and kind[role_root[s]]: z[h - 1] += 2 * v           # copied centre transform
+        z[h - lastd] += 2 * v                                                  # final growth to F
+        z[m - h + ds[0]] += 2 * v                                              # exterior, gauged by sigma
+    levels = defaultdict(set)
+    for s in deferred:
+        for t in reach[s]: levels[t].add(len(placed[s]))
+    for t in range(v):
+        ds = sorted(levels[t] | {0, h - 1})
+        for a, b in zip(ds, ds[1:]): z[b - a] += 2 * v                       # target fronts
+    z[h - 1] += 2 * N                                                          # data-wire fronts
+    z[(h - 1) ** 2] += 2 * N                                                   # data macros
+    z[1] += N                                                                  # endpoint copies
+    z.pop(0, None)
+    W = 2 * N + 2 * v * R; L = 2 * v * h * (h - 1); s_ = W * m - N + L
+    require(sum(t * c for t, c in z.items()) == s_, 'complex rank mass')
+    require(all(0 < t < m for t in z), 'children below m')
+    out = dict(h=h, v=v, additions=c_add, roots=q, links=len(links), R=R, m=m, N=N, W=W, L=L, total_rank=s_,
+               deficit=N - L, maxchild=max(z), phase_one_ops=len(Anc), phase_one_roles=len(touched),
+               deferred_roles=len(deferred),
+               deferred_dims=dict(sorted(Counter(len(X) for X in placed.values()).items())),
+               lifted_additions=sum(1 for x in U if args[x][0] and dimU[x] > ranks[x]),
+               replay=dict(seeds=[1, 2], scratch_restored=True, y_plus_x=True, field='Z/(2^61-1)'),
+               child_multiplicities=dict(sorted(z.items())))
+    (HERE / 'complex-profile.json').write_text(json.dumps(out, indent=1) + '\n')
+    print('PASS complex: R=%d, deferred roles %d, maxchild %d, rank mass %d' % (R, len(deferred), max(z), s_))
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/joint-dual/SOURCE.json b/research/joint-dual/SOURCE.json
index 80c0f29dc4..a119bae377 100644
--- a/research/joint-dual/SOURCE.json
+++ b/research/joint-dual/SOURCE.json
@@ -3,9 +3,9 @@
   "base_pr57_commit": "cd350f76c9bc01489ec83568bded532cb69be938",
   "base_pr58_commit": "bc2f7ed4c20dc18898305ab17165c0c995cbb804",
   "files": {
-    "Makefile": "97620e0205736d9c0fce736e8ac6ab67f8ed0e490c381f6cac152cc6a003dca7",
-    "NOTICE": "1c53970fb4face631a1b331ec66d33111eccd280a129f9b64c2da6df96861c3f",
-    "README.md": "fd92de31aece76b1f5ff897e6e0dbc38dd3bf67c140579bbcb03a1a11a792668",
+    "Makefile": "0bd8071c6a4899b9e6975b53690069b4ffc70ac3514c019485b59f02ed994a6b",
+    "NOTICE": "46b84a2254d554c8adbe67ddaf33640e1b99e31168a1d90305d470e4fdab8e23",
+    "README.md": "083799d50375b5a16bd5af0eb1b0e4f630a6d05b414fe7010bc266206d9eda11",
     "certificates/joint-dual-compiler.json": "516c4d85158553dabd0a98216a064b2a7397957a9d5dd0df1d6e5aec53a4e805",
     "certificates/joint-dual-profiles-23.json": "159fa8fee175fc1d24f63e72d3c2f1dc9f4fcd78b938c5e7d2f735fd0fd37209",
     "certificates/joint-dual-profiles-25.json": "8d190ce42d097e828c5756918c52f2963df576cf4c6600b0fc7f2b7fdb6ad223",
diff --git a/scripts/stopped_product/README.md b/scripts/stopped_product/README.md
new file mode 100644
index 0000000000..6c18be051f
--- /dev/null
+++ b/scripts/stopped_product/README.md
@@ -0,0 +1,36 @@
+# Selected rational-center producer
+
+Run from the repository root:
+
+```sh
+python3 scripts/stopped_product_producer.py --output /tmp/stopped-product-producers.json
+```
+
+Only the new aligned complex producer at h=24 is rebuilt. It retains all
+24 disjoint-point sums and decodes the total by exact division by21. The
+pair-star input order is aligned with global pairs. Its generator adapts
+`structured_bulk.complex`, preserving exact support, output orthogonality,
+binary frame nesting and nondegeneracy checks. Rational decoder and scatter
+coefficients are checked with `Fraction`; their only odd denominator is21.
+The unchanged deterministic matcher compares every count and histogram
+entry against `stopped-product-complex-input.json`.
+
+The h23 bit record is compared with the preceding validated copied-center
+input. No old bit DAG, matching, or corner is regenerated. Both copied-center
+histogram transformations reuse the established exact rank-mass checker.
+
+All DAGs, labels and the compiled matcher are temporary by default;
+`--work-dir` retains them. Python assertions must be enabled. Standard-library
+Python and a C++17 compiler suffice. No binary archive is an input.
+
+The odd-grid extension is a written recursive invariant, not a claim proved
+by the finite scalar tests. With dyadic root inputs, K=G(D+1) supplies enough
+odd-divisor reserve along the active chain. A completed child preserves its
+incoming odd valuation; it does not generally make rational child inputs
+dyadic. Only the completed outer dyadic operator restores the original
+dyadic grid. No child return is rounded.
+
+Copyright 2026 icekylinx, Apache-2.0. New selected alignment, rational centers
+and integration were prepared with OpenAI GPT-6 Astra/Codex assistance.
+Underlying paired producers and deterministic carrier matching retain the
+previous contributors' attribution in `NOTICE` and `SOURCES.json`.
diff --git a/scripts/stopped_product/__init__.py b/scripts/stopped_product/__init__.py
new file mode 100644
index 0000000000..f7a6e90661
--- /dev/null
+++ b/scripts/stopped_product/__init__.py
@@ -0,0 +1 @@
+"""Selected stopped-product inputs and rational producer; icekylinx, 2026."""
diff --git a/scripts/stopped_product/complex.py b/scripts/stopped_product/complex.py
new file mode 100644
index 0000000000..ef1af25507
--- /dev/null
+++ b/scripts/stopped_product/complex.py
@@ -0,0 +1,228 @@
+"""Selected aligned rational-center producer, h=24 with all centers disjoint.
+
+Copyright 2026 icekylinx, Apache-2.0. AI-assisted adaptation of the research
+round-five handoff, using the credited retained PairedTriple circuit; see NOTICE.
+Adapted from structured_bulk.complex: align pair-star inputs and permit the
+fixed rational divisor21; retain all exact support/frame/scalar checks.
+
+Builds a concrete scalar addition DAG, its nondegenerate binary frames, and
+the unmixed carrier histogram. No historical audit suite is invoked.
+"""
+import json, struct, array
+from itertools import combinations
+from pathlib import Path
+from partial_swap.binary import write_array
+from paired_triple_circuit import PairedTriple
+
+def build(h, prefix, central_disjoint, base=2):
+    assert (h, central_disjoint, base) == (24,24,2)
+    d=central_disjoint
+    assert 0<d<=h
+    divisor=abs(3-d)
+    assert divisor == 21
+    triples=list(combinations(range(h),3)); v=len(triples)
+    args=[(0,0)]; support=[0]; core=[0]; cover=[0]; types=[0]; ranks=[0]
+    lookup={}; inputs={}; center_coeff={}
+    # Each central tree adds h-1 ordinary pair totals, so every intermediate
+    # coefficient is at most h-1. Wider lanes rule out hidden integer carries.
+    coefficient_bits=h.bit_length()
+    def packed(mask):
+        result=0
+        while mask:
+            bit=mask&-mask; mask-=bit
+            result |= 1 << (coefficient_bits*(bit.bit_length()-1))
+        return result
+    for j,t in enumerate(triples):
+        mask=sum(1<<a for a in t); x=len(args); inputs[t]=x
+        args.append((0,0));support.append(1<<j);core.append(mask);cover.append(mask)
+        types.append(1);ranks.append(1);lookup[1<<j]=x
+    def add(a,b,center=None):
+        if not a:return b
+        if not b:return a
+        if center is None:
+            assert not support[a]&support[b], "Ordinary additions must be cancellation-free"
+        s=support[a]|support[b]
+        if center is None and s in lookup:return lookup[s]
+        x=len(args);args.append((a,b));support.append(s)
+        core.append(core[a]&core[b]);cover.append(cover[a]|cover[b])
+        if center is not None:
+            types.append(3);ranks.append(h-1);core[x]=1<<center
+            ca=center_coeff[a] if a in center_coeff else packed(support[a])
+            cb=center_coeff[b] if b in center_coeff else packed(support[b])
+            center_coeff[x]=ca+cb
+        elif core[x].bit_count()>=2:
+            types.append(1);ranks.append(s.bit_count());lookup[s]=x
+        else:
+            types.append(2);ranks.append(cover[x].bit_count());lookup[s]=x
+        return x
+    def total(xs,center=None):
+        xs=[x for x in xs if x]
+        while len(xs)>1:
+            xs=[add(xs[i],xs[i+1],center) if i+1<len(xs) else xs[i] for i in range(0,len(xs),2)]
+        return xs[0] if xs else 0
+    paired=PairedTriple(h,base)
+    allresults=paired.triple(list(range(h)),paired.variables)
+    selected=list(paired.outputs.values())+[allresults[(i,)] for i in range(d)]
+    active=set();stack=list(selected)
+    while stack:
+        x=stack.pop()
+        if not x or x in active:continue
+        active.add(x)
+        if paired.args[x]:stack.extend(paired.args[x])
+    mapping={0:0}|{j+1:inputs[t] for j,t in enumerate(triples)}
+    for x in sorted(active):
+        if paired.args[x]:
+            a,b=paired.args[x];mapping[x]=add(mapping[a],mapping[b])
+    roots=[mapping[paired.outputs[t]] for t in triples];kind=[0]*v
+    center_disjoint_roots=[mapping[allresults[(i,)]] for i in range(d)]
+    del paired, mapping, active, allresults
+    pairtotals={}; pair_outputs={}
+    point_masks=[sum(1<<j for j,t in enumerate(triples) if a in t) for a in range(h)]
+    full_support=(1<<v)-1
+    for t,node in zip(triples,roots):
+        assert support[node] == full_support & ~(point_masks[t[0]]|point_masks[t[1]]|point_masks[t[2]])
+        target_mask=sum(1<<a for a in t)
+        if types[node]==2:
+            assert not cover[node]&target_mask
+        else:
+            assert types[node]==1
+            assert not support[node]&(point_masks[t[0]]^point_masks[t[1]]^point_masks[t[2]])
+    for i,node in enumerate(center_disjoint_roots):
+        assert support[node] == full_support & ~point_masks[i]
+        assert types[node]==2 and cover[node]==((1<<h)-1)^(1<<i)
+        assert ranks[node]==h-1
+    for a,b in combinations(range(h),2):
+        others=[i for i in range(h) if i not in (a,b)]
+        others.sort(key=lambda i:((i^1) in (a,b),i))
+        pre=[0]
+        for i in others:pre.append(add(pre[-1],inputs[tuple(sorted((a,b,i)))]))
+        suf=[0]*(len(others)+1)
+        for j in range(len(others)-1,-1,-1):suf[j]=add(inputs[tuple(sorted((a,b,others[j])))],suf[j+1])
+        for j,i in enumerate(others):
+            node=add(pre[j],suf[j+1])
+            roots.append(node);kind.append(0)
+            pair_outputs[a,b,i]=node
+            assert support[node] == point_masks[a]&point_masks[b]&~point_masks[i]
+            assert types[node]==1
+            assert not support[node]&(point_masks[a]^point_masks[b]^point_masks[i])
+        pairtotals[a,b]=pre[-1]
+        assert support[pre[-1]] == point_masks[a]&point_masks[b]
+    for a in range(d,h):
+        root=total([pairtotals[tuple(sorted((a,b)))] for b in range(h) if b!=a],center=a)
+        assert center_coeff[root] == 2*packed(point_masks[a]), "Center coefficients must be exactly doubled"
+        roots.append(root);kind.append(1)
+    roots.extend(center_disjoint_roots);kind.extend([1]*d)
+    # Mixed retained centers recover the total by
+    # (sum B_i - 2 sum D_i)/(2(3-d)); check each source coefficient.
+    Dset=set(range(d))
+    for triple in triples:
+        k=len(Dset.intersection(triple))
+        assert 2*(3-k)-2*(d-k)==2*(3-d)
+    # Check the rational substituted center scatter for every realizable pair of
+    # target/source memberships. Its coefficient is (intersection-1)/2.
+    from fractions import Fraction
+    # Every A_i avoids i, so sum A_i = (h-3)T. The selected decoder is
+    # T=sum A_i/21 and scatter T-(1/2)sum_{i in target} A_i.
+    decoder=Fraction(1,h-3)
+    assert decoder==Fraction(1,21)
+    scatter_coefficients=[decoder,decoder-Fraction(1,2)]
+    for coefficient in scatter_coefficients:
+        odd_denominator=coefficient.denominator
+        while odd_denominator%2==0:
+            odd_denominator//=2
+        assert 21%odd_denominator==0 and abs(coefficient)<=1
+    for source in triples:
+        assert sum(i not in source for i in range(h))*decoder==1
+    patterns={}
+    for target in triples:
+        chosen=len(Dset.intersection(target))
+        if chosen not in patterns:
+            patterns[chosen]=target
+    for chosen,target in patterns.items():
+        target_set=set(target)
+        for source in triples:
+            k=len(Dset.intersection(source))
+            total_coefficient=Fraction(2*(3-k)-2*(d-k),2*(3-d))
+            scalar=Fraction(len((target_set-Dset).intersection(source)),2)
+            scalar-=Fraction(len((target_set&Dset)-set(source)),2)
+            scalar+=Fraction(chosen-1,2)*total_coefficient
+            assert scalar==Fraction(len(target_set.intersection(source))-1,2)
+    # Check the scalar correction for each target, using the verified exact
+    # roots. Twice the coefficient is (intersection-1)+[intersection=0]
+    # -[intersection=2], which is two precisely for the diagonal.
+    for j,t in enumerate(triples):
+        masks=[point_masks[a] for a in t]
+        two=(masks[0]&masks[1]&~masks[2]) | (masks[0]&masks[2]&~masks[1]) | (masks[1]&masks[2]&~masks[0])
+        actual=0
+        for a,b in combinations(t,2):
+            excluded=next(i for i in t if i not in (a,b))
+            actual |= support[pair_outputs[a,b,excluded]]
+        assert actual == two
+        assert masks[0]&masks[1]&masks[2] == 1<<j
+    assert [((k-1)+(k==0)-(k==2)) for k in range(4)] == [0,0,0,2]
+    def contained(x,y):
+        tx,ty=types[x],types[y]
+        if tx==1 and ty==1: return not(core[y]&~core[x] or cover[x]&~cover[y])
+        if tx in (1,2) and ty==2: return not cover[x]&~cover[y]
+        if tx==1 and ty==3: return bool(core[x]&core[y])
+        if tx==2 and ty==3: return not cover[x]&~core[y]
+        if tx==3 and ty==2: return cover[y]==(1<<h)-1
+        if tx==3 and ty==3: return core[x]==core[y]
+        return False
+    active=[0]*len(args);stack=list(roots)
+    while stack:
+        x=stack.pop()
+        if not x or active[x]:continue
+        active[x]=1
+        if args[x][0]:stack.extend(args[x])
+    degree=[0]*len(args)
+    for x in range(1,len(args)):
+        if active[x]:
+            if types[x]==1:
+                # Shared-pair triple indicators have self-dot one and
+                # pairwise dot zero over F2; a source is a single such line.
+                assert core[x].bit_count()>=2 and ranks[x]==support[x].bit_count()
+            elif types[x]==2:
+                assert ranks[x]==cover[x].bit_count()
+            else:
+                # E_a is the orthogonal complement of the odd vector 1+e_a.
+                assert types[x]==3 and core[x].bit_count()==1 and ranks[x]==h-1
+        if active[x] and args[x][0]:
+            for y in args[x]:
+                assert y<x and contained(y,x), "Binary frame nesting failed"
+                degree[y]+=1
+    for x in roots:degree[x]+=1
+    H=[0]*(h+1);c=0;loss=0
+    for x in range(1,len(args)):
+        if not active[x]:continue
+        r=ranks[x]
+        if args[x][0]:
+            c+=1;H[r]+=degree[x]-1;H[h-r]+=1
+            for y in args[x]:
+                assert r>=ranks[y]
+                H[r-ranks[y]]+=1
+        else:H[1]+=degree[x]
+    for x,k in zip(roots,kind):
+        r=ranks[x]
+        if k:H[r]+=1;H[h]+=1;loss+=r
+        else:H[h-1-r]+=1;H[1]+=1
+    q=len(roots);R=c+q
+    result=dict(h=h,v=v,c=c,q=q,R=R,loss=loss,central_disjoint=d,histogram=H,rank_sum=sum(r*n for r,n in enumerate(H)))
+    assert loss == h*(h-1) and result['rank_sum'] == h*R+2*loss
+    prefix=Path(prefix)
+    result['center_denominator']=21
+    result['scalar_validation']=dict(ordinary_supports_exact=True, centers_exact=True,
+        center_decoder='sum A_i / 21', scatter_coefficients=list(map(str,scatter_coefficients)),
+        rational_scalar_identity_exact=True, mixed_center_scatter_exact=True, binary_frames_nested=True,
+        binary_frames_nondegenerate=True)
+    with open(str(prefix)+'.bin','wb') as f:
+        f.write(struct.pack('<4I',h,v,len(args),q))
+        write_array(f,array.array('I',(a for pair in args for a in pair)))
+        write_array(f,array.array('Q',core));write_array(f,array.array('Q',cover))
+        write_array(f,array.array('I',roots));write_array(f,array.array('I',kind))
+        write_array(f,array.array('B',active))
+    with open(str(prefix)+'.labels','wb') as f:
+        write_array(f,array.array('I',ranks));write_array(f,array.array('B',types))
+    Path(str(prefix)+'.json').write_text(json.dumps(result,indent=2)+'\n')
+
+    return result
diff --git a/scripts/stopped_product_network.py b/scripts/stopped_product_network.py
new file mode 100644
index 0000000000..5f349a2676
--- /dev/null
+++ b/scripts/stopped_product_network.py
@@ -0,0 +1,163 @@
+#!/usr/bin/env python3
+"""Exact stopped product-ring moments, rational-center guard and assembly.
+
+Copyright 2026 icekylinx. Apache-2.0.
+Developed with substantial OpenAI GPT-6 Astra and Codex assistance.
+Retains the copied-center two-stage construction, Paureel/PR29 topology,
+PR23 semantic assembly and RaD analytic interfaces; see SOURCES.json.
+"""
+import argparse
+from collections import Counter
+from fractions import Fraction as Q
+from hashlib import sha256
+from math import comb
+from pathlib import Path
+import json
+import sys
+
+from certify import require
+from copied_centers.physical import copied_histogram
+from copied_centers_network import finite_bridge as copied_bridge, grid_upper
+from partial_swap_network import moment
+from structured_bulk_assembly import assembly, js
+
+ROOT = Path(__file__).resolve().parents[1]
+COARSE = Q(4019, 50000000)
+ATOM = Q(1, 1000)
+OLD = Q(384599, 10**10)
+AB = (1-ATOM)*COARSE+ATOM*OLD
+AC = Q(1949, 25000000)
+PHASE_STOP = Q(1, 10**6)
+ASSEMBLY_BIT = min(AB, (1-PHASE_STOP)*AC-Q(1, 10**10))
+KAPPA = Q(194869, 2500000000)
+
+
+def profile(row):
+    h, v, R, ell = (row[k] for k in ('h', 'v', 'R', 'loss'))
+    require(v == comb(h, 3), 'Triple count')
+    H = copied_histogram(row)['histogram']
+    m, N, B = h*h, v*v, v*R
+    W, L = 2*N+2*B, 2*v*ell
+    z = Counter({m-h:2*B, (h-1)**2:2*N, h-1:4*N, 1:N})
+    for r, n in enumerate(H):
+        if r and n:
+            z[r] += 2*v*n
+    z = dict(sorted((t,n) for t,n in z.items() if n))
+    s = W*m-N+L
+    require(all(0 < t < m and n > 0 for t,n in z.items()), 'Shrinking children')
+    require(sum(t*n for t,n in z.items()) == s, 'Complete rank mass')
+    return dict(dimensions=[h,h], m=m, N=N, B1=B, B2=B, W=W, L=L,
+                total_rank=s, deficit=N-L, maxchild=max(z),
+                copied_histogram=H, child_multiplicities=z)
+
+
+def coarse_moment(p):
+    def log_upper(x):
+        power = 0
+        while x >= 2:
+            x /= 2
+            power += 1
+        def small(y):
+            z = (y-1)/(y+1)
+            return 2*sum((z**(2*j+1)/(2*j+1) for j in range(24)), Q(0)) + 2*z**49/(49*(1-z*z))
+        return grid_upper(power*small(Q(2))+small(x))
+    total, logs = Q(0), {}
+    for t, n in p['child_multiplicities'].items():
+        logs[t] = log_upper(Q(p['m'],t))
+        u = COARSE*logs[t]
+        require(0 < u < 3, 'Coarse exponential enclosure')
+        total += n*t*grid_upper(1+u+u*u/(2*(1-u/3)))
+    upper = total/(p['W']*p['m'])
+    mass = Q(p['total_rank'],p['W']*p['m'])
+    require(mass < 1 and upper < 1, 'Both stopped-recurrence moments')
+    require(0 < OLD <= COARSE < ATOM < 1, 'Stopped-adapter exponents')
+    require(ATOM > AB, 'Adapter toll subordinate to the stopped saving')
+    return dict(saving=COARSE, exponent=1-COARSE, moment_upper=upper,
+                strict_gap=1-upper, rank_moment=mass,
+                logarithm_upper_bounds=logs, rounding_denominator=1 << 100,
+                stopped_adapter=dict(atom_exponent=ATOM, base_ordinary_saving=OLD,
+                    effective_saving=AB, coarse_toll_exponent=1-ATOM,
+                    coarse_toll_subordinate_gap=ATOM-COARSE))
+
+
+def finite_bridge(bit, phase, row, previous):
+    bridge = copied_bridge(bit, phase, [row,row])
+    old_bit = previous['finite_bridge']['bit']
+    old_degree = old_bit['halving_degree']*old_bit['wire_bits']
+    require(old_degree == 252, 'Inherited ordinary leaf row stock')
+    coarse = bridge.pop('bit')
+    c = bridge['complex']
+    coefficient = (coarse['halving_degree']*coarse['wire_bits']+old_degree+
+                   c['halving_degree']*c['wire_bits'])
+    degree = 4000
+    require(Q(degree) > Q(51*coefficient,25), 'Product stock strictly sufficient')
+    bridge.update(bit_coarse=coarse, ordinary_leaf_row_degree=old_degree)
+    bridge['rows'] = dict(coefficient=coefficient, degree=degree, suffix_slope=4*degree,
+        degree_gap=Q(degree)-Q(51*coefficient,25),
+        contract='W_complex^D_complex * W_coarse^D_coarse * W_old^D_old; one preceding prefix and one padding; sequential reuse')
+    require(row['h'] == row['central_disjoint'] == 24, 'Selected all-disjoint centers')
+    divisor = row['h']-3
+    require(divisor == 21, 'Selected fixed odd divisor')
+    bridge['semantic'].update(fixed_odd_divisor=divisor,
+        exact_grid='One common dyadic grid times the fixed odd divisor to K=G*(D_complex+1); no child rounding')
+    return bridge
+
+
+def certificate():
+    require(not sys.flags.optimize, 'Run without -O')
+    def read(name):
+        return json.loads((ROOT/'certificates'/name).read_text())
+    bit_row = read('stopped-product-bit-axis.json')
+    complex_row = read('stopped-product-complex-input.json')
+    expected = read('stopped-product-input.json')
+    previous = read('copied-centers-network.json')
+    require(Q(previous['bit']['saving']) == OLD, 'Certified ordinary leaf saving')
+    require(bit_row['h'] == 23, 'Selected bit dimension')
+    bit, phase = profile(bit_row), profile(complex_row)
+    for label,p in (('bit',bit),('complex',phase)):
+        for key,value in expected[label].items():
+            require(js(p[key]) == value, label+' saved '+key)
+    bm = coarse_moment(bit)
+    cm = moment(phase['m'],phase['W'],phase['child_multiplicities'],AC,True)
+    require(bm['strict_gap'] == Q(expected['bit_moment_gap']), 'Saved coarse moment')
+    require(cm['strict_gap'] == Q(expected['complex_moment_gap']), 'Saved complex moment')
+    bridge = finite_bridge(bit,phase,complex_row,previous)
+    assembled = assembly(ASSEMBLY_BIT,AC,bridge,KAPPA,beta=PHASE_STOP)
+    require(ASSEMBLY_BIT <= AB, 'Assembly saving supported by actual bit interface')
+    assembled['parameters']['actual_bit_saving'] = AB
+    require(js(bridge) == expected['finite_bridge'], 'Saved semantic and product bridge')
+    require(js(assembled) == expected['assembly'], 'Saved exact assembly')
+    require(len(assembled['strict_constraints']) == 47 and len(assembled['margins']) == 7,
+            'Complete assembly constraints')
+    sources = sorted(p for p in (ROOT/'certificates').glob('stopped-product-*.json')
+                     if p.name != 'stopped-product-network.json')
+    sources += sorted((ROOT/'scripts').glob('stopped_product*.py'))
+    sources += sorted(p for p in (ROOT/'scripts/stopped_product').rglob('*')
+                      if p.is_file() and '__pycache__' not in p.parts)
+    sources += sorted((ROOT/'notes').glob('stopped-product-*.tex'))
+    sources += [ROOT/path for path in (
+        'scripts/copied_centers/physical.py','scripts/copied_centers_network.py',
+        'scripts/partial_swap_network.py','scripts/structured_bulk_assembly.py',
+        'certificates/copied-centers-network.json','certificates/copied-centers-bit-axes.json')]
+    return dict(status='Conditional stopped product-ring multiplication witness',kappa=KAPPA,
+        bit=dict(counts=bit,**bm),complex=dict(counts=phase,**cm),
+        finite_bridge=bridge,assembly=assembled,
+        base_commit='56b66d58297deca1d7dd130247d720e960f77a37',
+        proof_predecessor_commit='11817ccacb564bb7f98789c20dc11d3fece207e3',
+        source_sha256={str(p.relative_to(ROOT)):sha256(p.read_bytes()).hexdigest() for p in sources},
+        scope='Exact finite moments, stopped savings and assembly; the new complex producer is checked separately. '
+              'Opposite-bank factorization, atom streaming, ordinary conversion, common exact grid and retained analytic/tape interfaces are written proof dependencies.')
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument('--output',type=Path,default=ROOT/'certificates/stopped-product-network.json')
+    args = parser.parse_args()
+    result = certificate()
+    args.output.write_text(json.dumps(js(result),indent=2,sort_keys=True)+'\n')
+    print('PASS kappa=194869/2500000000 = 7.79476e-5')
+    print('Coarse and complex moments; stopped bit saving; exact rational-center guard; p^4000 stock; 47 strict constraints')
+
+
+if __name__ == '__main__':
+    main()
diff --git a/scripts/stopped_product_producer.py b/scripts/stopped_product_producer.py
new file mode 100644
index 0000000000..aefc91d9a7
--- /dev/null
+++ b/scripts/stopped_product_producer.py
@@ -0,0 +1,92 @@
+#!/usr/bin/env python3
+"""Regenerate only the new h24 aligned all-disjoint rational-center producer.
+
+Copyright 2026 icekylinx, Apache-2.0. Prepared with OpenAI GPT-6 Astra and
+Codex assistance. Prior paired circuit and matcher attribution is retained
+in NOTICE. No old bit producer is regenerated.
+"""
+import argparse
+import gc
+import json
+import os
+from pathlib import Path
+import shlex
+import subprocess
+import sys
+import tempfile
+
+from copied_centers.corners import require
+from copied_centers.physical import copied_histogram
+from endpoint_gauge_producer import compare
+from stopped_product.complex import build
+
+ROOT = Path(__file__).resolve().parents[1]
+
+
+def read(path):
+    return json.loads(path.read_text())
+
+
+def regenerate(work, bit, expected, previous_bits, compiler):
+    require(sys.flags.optimize == 0, 'Run without -O: mathematical assertions must remain enabled')
+    require(bit['h'] == 23 and bit['label_source'] == 'positive', 'Wrong selected bit axis')
+    old = next(row for row in previous_bits if row['h'] == 23)
+    for key in ('h','v','c','q','baseline_R','matching','R','loss','histogram','label_source'):
+        require(bit[key] == old[key], 'Reused bit producer mismatch: '+key)
+    require(bit['R'] == bit['c']+bit['q']-bit['matching'], 'Bit role identity failed')
+    bit_copied = copied_histogram(bit)
+    require((expected['h'],expected['central_disjoint'],expected['center_denominator']) ==
+            (24,24,21), 'Wrong rational-center parameters')
+    print('Regenerating only complex h=24: aligned pair stars, all24 disjoint centers, divisor21',
+          file=sys.stderr, flush=True)
+    prefix = work/'complex_rational_d24_24'
+    construction = build(24, prefix, central_disjoint=24)
+    gc.collect()
+    program = work/'match_complex_general'
+    source = ROOT/'scripts'/'endpoint_gauge'/'match_complex_general.cpp'
+    subprocess.run([*shlex.split(compiler), '-O3', '-std=c++17', str(source),
+                    '-o', str(program)], check=True)
+    matched = json.loads(subprocess.check_output(
+        [str(program), str(prefix)+'.bin', str(prefix)+'.labels'], text=True))
+    compare(matched, expected, 'complex h=24 rational centers')
+    require(construction['R'] == matched['baseline_R'], 'Wrong raw complex roles')
+    phase_copied = copied_histogram(matched)
+    return dict(bit=dict(reused_validated_producer=True, certificate_equal=True,
+                         copied=bit_copied),
+                complex=dict(**matched, central_disjoint=24, center_denominator=21,
+                             construction=construction, copied=phase_copied,
+                             certificate_equal=True),
+                old_bit_regeneration_skipped=True,
+                scope='Exact new scalar supports, rational scatter, nested frames and matching histogram. '
+                      'The common odd-denominator grid and recursive tape transfer remain written proof interfaces.')
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    for name, filename in (
+            ('bit-axis','stopped-product-bit-axis.json'),
+            ('complex-input','stopped-product-complex-input.json'),
+            ('previous-bits','copied-centers-bit-axes.json')):
+        parser.add_argument('--'+name, type=Path, default=ROOT/'certificates'/filename)
+    parser.add_argument('--cxx', default=os.environ.get('CXX','c++'))
+    parser.add_argument('--work-dir', type=Path, help='Keep generated intermediate files')
+    parser.add_argument('--output', type=Path, help='Write full incremental verification JSON')
+    args = parser.parse_args()
+    inputs = [read(path) for path in (args.bit_axis,args.complex_input,args.previous_bits)]
+    def run(work):
+        return regenerate(work,*inputs,args.cxx)
+    if args.work_dir:
+        args.work_dir.mkdir(parents=True,exist_ok=True)
+        result = run(args.work_dir.resolve())
+    else:
+        with tempfile.TemporaryDirectory(prefix='stopped-product-') as directory:
+            result = run(Path(directory))
+    encoded = json.dumps(result,indent=2)+'\n'
+    if args.output:
+        args.output.write_text(encoded)
+    else:
+        print(encoded,end='')
+
+
+if __name__ == '__main__':
+    main()