diff --git a/.github/workflows/reversed-rational-centers.yml b/.github/workflows/reversed-rational-centers.yml
new file mode 100644
index 0000000000..95cdb10a6f
--- /dev/null
+++ b/.github/workflows/reversed-rational-centers.yml
@@ -0,0 +1,15 @@
+name: Verify reversed rational centers
+on: [push, pull_request]
+jobs:
+  focused:
+    runs-on: ubuntu-latest
+    strategy:
+      matrix:
+        python: ['3.11', '3.13', '3.14']
+    steps:
+      - uses: actions/checkout@v4
+      - uses: actions/setup-python@v5
+        with:
+          python-version: ${{ matrix.python }}
+      - run: python research/reversed-rational-centers/verify.py
+      - run: python research/reversed-rational-centers/test_controls.py
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": {
+      "original_prefix": "974622765755022147550220501004449/12501948995521765066540100444900000000",
+      "coordinate_movement": "1948995551/25000000000000",
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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,
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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-strips-both-axes/README.md b/research/cyclic-strips-both-axes/README.md
new file mode 100644
index 0000000000..156df6d3e2
--- /dev/null
+++ b/research/cyclic-strips-both-axes/README.md
@@ -0,0 +1,55 @@
+# Cyclic interval strips on both axes of the stopped product-ring bound
+
+Under all of PR104's inherited and newly proposed interfaces (as used by PR107–PR109),
+this finite replacement gives conditional
+**κ = 1715630240171/(2·10^16) = 8.578151200855×10⁻⁵** —
+**7.278% above PR109** (7.9961717358×10⁻⁵), 7.384% above PR108 and 9.999% above PR107.
+
+## Idea
+
+Both axes of PR104 compute their leave-one-out ("strip") sums with the ordinary
+prefix/suffix identity `s_j = A_{j-1} + B_{j+1}`. We replace them with **cyclic interval
+strips** (the interval construction of PR62 on the h23/h25 frame-compiler lineage):
+every cyclic interval `I(a,m) = I(a,m-1) + v_(a+m-1)` is built and `s_j` is the
+complementary interval. Each interval has a later consumer with a containing cover, so the
+**unchanged** legal carrier matchers join many more additions. Nothing else changes.
+
+| | PR107/108 | This |
+|---|---:|---:|
+| **bit h23** (retained-total producer, positive labels) | | |
+| additions / matched / roles | 34,764 / 1,324 / 38,776 | 57,162 / 30,197 / **32,301** |
+| certified stopped coarse saving | 4019/5·10^7 = 8.038e-5 | **898920417/10^13 = 8.98920e-5** |
+| **complex h24** (all-disjoint rational centers) | PR108 | |
+| matched / roles | 15,422 / 43,270 | 37,802 / **38,506** |
+| strict complex saving | 7989584037/10^14 | **8579641984/10^14** |
+| **κ** | 7.988289e-5 (PR108) | **8.578151e-5** |
+
+- Bit axis: only `PairedExclusionCircuit.block`'s strip layout changes (cyclic, carry last),
+  loaded privately (`bit_paired.py`); positive labels are recomputed for the new graph by the
+  unchanged labeler, and the positive matching equals the envelope matching.
+- Complex axis: PairedTriple strips use cyclic intervals (`producer.py: cyclic_vector`),
+  pair stars keep PR108's dual-suffix identity, and the coordinate order returns to PR104's `((i^1) in (a,b), i)`.
+- The complex leaf still binds: bit parameter = min(AB, (1−10⁻⁶)·b − 10⁻¹⁰) exactly as in PR107.
+
+## Checks
+
+```sh
+python3 research/cyclic-strips-both-axes/verify.py
+(cd research/cyclic-strips-both-axes && python3 test_controls.py)
+make verify
+```
+
+`verify.py` regenerates **both** rows from source (bit: graph → scalar/frame verify → export →
+unchanged `match_exported_dag` → recomputed positive labels → unchanged `match_positive_dag`;
+complex: every exact support, center, rational-scatter and frame-nesting assertion, unchanged
+`match_complex_general`), certifies the stopped coarse moment at the new COARSE and rejects
+COARSE+10⁻¹³, certifies the complex moment and rejects b+10⁻¹⁴, and checks all 47 strict
+constraints and seven margins, rejecting κ+10⁻¹⁷. Controls reject the PR104 bit row, the
+PR108 complex row, unpaid cleanup, omitted loss and an unpaid role. Full verification passed on `fb550b9b1a3c60cc419701e530494ed754d08cef` in 2409.831 seconds, completed 2026-10-09T00:45:01.380047+00:00: `make -j1 verify` (78 isolated test modules and 20 historical patch checks), both producer regenerations, exact assembly and six adversarial controls. All 39 research-head GitHub checks passed; no source drift. See `research/cyclic-strips-both-axes/validation.json` for evidence hashes.
+
+Scope: finite conditional witness; PR104's opposite-bank factorization, stopped streaming and
+odd-denominator grid interfaces remain written proof dependencies. No optimality/runtime claim.
+
+Credit: icekylinx (PR104), Avi Eisenberg (PR62 interval strips, PR53), the PR55 dual-suffix
+authors, PR107–PR109, and all predecessors in PR104's NOTICE/SOURCES. Prepared by Rohan Arun
+with Anthropic Claude assistance, Apache-2.0.
diff --git a/research/cyclic-strips-both-axes/SOURCE.json b/research/cyclic-strips-both-axes/SOURCE.json
new file mode 100644
index 0000000000..426af9e3e9
--- /dev/null
+++ b/research/cyclic-strips-both-axes/SOURCE.json
@@ -0,0 +1,341 @@
+{
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+  }
+}
diff --git a/research/cyclic-strips-both-axes/bit-row.json b/research/cyclic-strips-both-axes/bit-row.json
new file mode 100644
index 0000000000..6dd3f69694
--- /dev/null
+++ b/research/cyclic-strips-both-axes/bit-row.json
@@ -0,0 +1,38 @@
+{
+  "h": 23,
+  "v": 1771,
+  "c": 57162,
+  "q": 5336,
+  "baseline_R": 62498,
+  "R": 32301,
+  "loss": 506,
+  "histogram": [
+    10530,
+    38685,
+    78087,
+    10442,
+    16357,
+    2634,
+    3081,
+    816,
+    1405,
+    415,
+    577,
+    219,
+    612,
+    173,
+    796,
+    1154,
+    1057,
+    1545,
+    883,
+    3423,
+    9215,
+    2110,
+    23,
+    23
+  ],
+  "matching": 30197,
+  "label_source": "positive",
+  "description": "Retained-total aligned triple producer with cyclic interval strips (carry last), base-two recursion, original maximum carrier matching, positive labels."
+}
\ No newline at end of file
diff --git a/research/cyclic-strips-both-axes/bit_paired.py b/research/cyclic-strips-both-axes/bit_paired.py
new file mode 100644
index 0000000000..6e88b8c910
--- /dev/null
+++ b/research/cyclic-strips-both-axes/bit_paired.py
@@ -0,0 +1,89 @@
+# Cyclic interval strips (PR62 interval idea) in the h23 retained-total bit producer.
+# Copy of scripts/partial_swap/paired.py with ONLY the strip leave-one-out replaced; loaded privately.
+# Rohan Arun with Anthropic Claude assistance, Apache-2.0; original notices retained.
+# Copyright 2026 icekylinx. Licensed under Apache-2.0.
+# Adapted with AI assistance from the archived partial-swap research producer.
+# Underlying circuit modules: jacklightChen/integer-mult-bounds, PR7
+# commit 6725c6a17b17871a35353fd29157f4ed851bc114; original credits retained.
+#!/usr/bin/env python3
+"""Cancellation-free pair exclusion with recursively aggregated vertex weights.
+
+At each level group vertices in pairs. Internal edges become vertex weights
+in the smaller graph. Keeping those weights in the recursion avoids a separate
+leave-two-out circuit for each level. All coefficients are checked exactly.
+"""
+from exclusion_circuit import ExclusionCircuit
+from itertools import combinations
+import os
+LAYOUT='cyclic';ORDER='id';CARRY='last'
+class PairedExclusionCircuit(ExclusionCircuit):
+ base_threshold = 2
+ def loo(self,values):
+  """Total and leave-one-out sums; layout selected by BITX_LAYOUT (bitx research)."""
+  add=self.add;k=len(values);v=[0]+list(values)  # 1-based
+  if LAYOUT=='prefix':
+   st,one,_=self.vector(values,False);return st,one
+  if k==0:return 0,[]
+  if LAYOUT=='dual':
+   # s_j=(A_{j-1}+v_{j+1})+B_{j+2}, s_k=A_{k-1}
+   A=[0]*(k+1)
+   for t in range(1,k):A[t]=add(A[t-1],v[t])
+   B=[0]*(k+3)
+   for t in range(k,1,-1):B[t]=add(v[t],B[t+1])
+   one=[add(add(A[t-1],v[t+1]),B[t+2]) for t in range(1,k)]+[A[k-1]]
+   return add(A[k-1],v[k]),one
+  if LAYOUT=='skip':
+   # s_j=A_{j-2}+(v_{j-1}+B_{j+1}), s_1=B_2
+   A=[0]*(k+1)
+   for t in range(1,k-1):A[t]=add(A[t-1],v[t])
+   B=[0]*(k+3)
+   for t in range(k,1,-1):B[t]=add(v[t],B[t+1])
+   one=[B[2]]+[add(A[t-2],add(v[t-1],B[t+1])) for t in range(2,k+1)]
+   return add(v[1],B[2]),one
+  if LAYOUT=='cyclic':
+   # I(a,m)=I(a,m-1)+v_{a+m-1} (cyclic, 0-based a); s_j=I(j+1,k-1)
+   vals=list(values);I={}
+   for a in range(k):
+    cur=0
+    for m in range(1,k):
+     cur=add(cur,vals[(a+m-1)%k]);I[a,m]=cur
+   one=[I[(j+1)%k,k-1] if k>1 else 0 for j in range(k)]
+   return add(one[0],vals[0]),one
+  raise ValueError(LAYOUT)
+ def pair(self,points):return self.block(points,{(a,b):self.variables[a,b] for a,b in combinations(points,2)},{a:0 for a in points})
+ def grouping(self,points):return [points[i:i+2] for i in range(0,len(points),2)]
+ def block(self,points,edges,weights):
+  if len(points)<=self.base_threshold:
+   total=lambda omit:self.total([x for p,x in edges.items() if not set(p)&set(omit)]+[x for p,x in weights.items() if p not in omit])
+   return total(()),{a:total((a,)) for a in points},{(a,b):total((a,b)) for a,b in combinations(points,2)}
+  groups=self.grouping(points);ng=len(groups);e=lambda a,b:edges[tuple(sorted((a,b)))]
+  coarse={(i,j):self.total([e(a,b) for a in groups[i] for b in groups[j]]) for i,j in combinations(range(ng),2)}
+  wt={i:self.total([weights[a] for a in g]+[e(a,b) for a,b in combinations(g,2)]) for i,g in enumerate(groups)}
+  total,outside,far=self.block(list(range(ng)),coarse,wt)
+  strips={};sums={}
+  for i,g in enumerate(groups):
+   other=[j for j in range(ng) if j!=i]
+   if ORDER=='rev':other.sort(key=lambda j:-j)
+   elif ORDER=='partner':other.sort(key=lambda j:((j^1)==i,-j))
+   elif ORDER=='partnerfirst':other.sort(key=lambda j:((j^1)!=i,j))
+   for a in g:
+    carry=self.total([weights[u] for u in g if u!=a])
+    vals=[self.total([e(u,v) for u in g if u!=a for v in groups[j]]) for j in other]
+    if CARRY=='last':
+     st,one=self.loo(vals+[carry]);one=one[:-1]
+    else:
+     st,one=self.loo([carry]+vals);one=one[1:]
+    strips[a]={j:z for j,z in zip(other,one)};sums[a]=st
+  out={};single={a:self.add(outside[i],sums[a]) for i,g in enumerate(groups) for a in g}
+  for i,g in enumerate(groups):
+   for a,b in combinations(g,2):out[a,b]=outside[i]
+  for i,j in combinations(range(ng),2):
+   for a in groups[i]:
+    left=self.add(far[i,j],strips[a][j])
+    for b in groups[j]:
+     cross=self.total([e(u,v) for u in groups[i] if u!=a for v in groups[j] if v!=b])
+     out[a,b]=self.add(left,self.add(strips[b][i],cross))
+  return total,single,out
+if __name__=='__main__':
+ for n in (15,25,45,49):
+  c=PairedExclusionCircuit(n);print(n,c.verify(),flush=True)
diff --git a/research/cyclic-strips-both-axes/certificate.json b/research/cyclic-strips-both-axes/certificate.json
new file mode 100644
index 0000000000..131c425997
--- /dev/null
+++ b/research/cyclic-strips-both-axes/certificate.json
@@ -0,0 +1,324 @@
+{
+  "kappa": "1715630240171/20000000000000000",
+  "coarse_bit_saving": "898920417/10000000000000",
+  "effective_bit_saving": "898406095583/10000000000000000",
+  "complex_saving": "67028453/781250000000",
+  "bit_parameter": "67028307846547/781250000000000000",
+  "bit_profile": {
+    "dimensions": [
+      23,
+      23
+    ],
+    "m": 529,
+    "N": 3136441,
+    "B1": 57205071,
+    "B2": 57205071,
+    "W": 120683024,
+    "L": 1792252,
+    "total_rank": 63839975507,
+    "deficit": 1344189,
+    "maxchild": 506,
+    "copied_histogram": [
+      10530,
+      38708,
+      78087,
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+      16357,
+      2634,
+      3081,
+      816,
+      1405,
+      415,
+      577,
+      219,
+      612,
+      173,
+      796,
+      1154,
+      1057,
+      1545,
+      883,
+      3423,
+      9215,
+      2110,
+      23,
+      0
+    ],
+    "child_multiplicities": {
+      "1": 140240177,
+      "2": 276584154,
+      "3": 36985564,
+      "4": 57936494,
+      "5": 9329628,
+      "6": 10912902,
+      "7": 2890272,
+      "8": 4976510,
+      "9": 1469930,
+      "10": 2043734,
+      "11": 775698,
+      "12": 2167704,
+      "13": 612766,
+      "14": 2819432,
+      "15": 4087468,
+      "16": 3743894,
+      "17": 5472390,
+      "18": 3127586,
+      "19": 12124266,
+      "20": 32639530,
+      "21": 7473620,
+      "22": 12627230,
+      "484": 6272882,
+      "506": 114410142
+    }
+  },
+  "coarse_moment": {
+    "saving": "898920417/10000000000000",
+    "exponent": "9999101079583/10000000000000",
+    "moment_upper": "11424122985883093306581821078762724597/11424122985883213408381955141789548544",
+    "strict_gap": "120101800134063026823947/11424122985883213408381955141789548544",
+    "rank_moment": "1567279/1567312",
+    "logarithm_upper_bounds": {
+      "1": "1987355562417364065669439618237/316912650057057350374175801344",
+      "2": "3535376905093068344529747466859/633825300114114700748351602688",
+      "3": "6556765722521218186395115000937/1267650600228229401496703205376",
+      "4": "6192085370702817115441231394489/1267650600228229401496703205376",
+      "5": "1477304328485575102713485747439/316912650057057350374175801344",
+      "6": "1419524320759474653194212865427/316912650057057350374175801344",
+      "7": "2741344040614642504793860160533/633825300114114700748351602688",
+      "8": "5313416931219497541822967855259/1267650600228229401496703205376",
+      "9": "161378412355405628441014735279/39614081257132168796771975168",
+      "10": "2515274437229490418617839725263/633825300114114700748351602688",
+      "11": "4909728867822167960512763283247/1267650600228229401496703205376",
+      "12": "2399714421777289519579293961239/633825300114114700748351602688",
+      "13": "4697962657128327885574713503543/1267650600228229401496703205376",
+      "14": "1151004910436491358992364195459/316912650057057350374175801344",
+      "15": "2258280393397031167285649759463/633825300114114700748351602688",
+      "16": "2217374245868088984102352158015/633825300114114700748351602688",
+      "17": "4357897653501964928993364503311/1267650600228229401496703205376",
+      "18": "4285440755889660536494207989699/1267650600228229401496703205376",
+      "19": "1054225602598430741931079593279/316912650057057350374175801344",
+      "20": "4151880434975661263617415911297/1267650600228229401496703205376",
+      "21": "4090031554081046933305076849055/1267650600228229401496703205376",
+      "22": "4031060428338848386894499744017/1267650600228229401496703205376",
+      "484": "28174651752060127777810254067/316912650057057350374175801344",
+      "506": "28174651752060127777810254067/633825300114114700748351602688"
+    },
+    "rounding_denominator": 1267650600228229401496703205376,
+    "stopped_adapter": {
+      "atom_exponent": "1/1000",
+      "base_ordinary_saving": "384599/10000000000",
+      "effective_saving": "898406095583/10000000000000000",
+      "coarse_toll_exponent": "999/1000",
+      "coarse_toll_subordinate_gap": "9101079583/10000000000000"
+    }
+  },
+  "complex_profile": {
+    "dimensions": [
+      24,
+      24
+    ],
+    "m": 576,
+    "N": 4096576,
+    "B1": 77936144,
+    "B2": 77936144,
+    "W": 164065440,
+    "L": 2234496,
+    "total_rank": 94499831360,
+    "deficit": 1862080,
+    "maxchild": 552,
+    "copied_histogram": [
+      9770,
+      59192,
+      68812,
+      13879,
+      15704,
+      8270,
+      6289,
+      4193,
+      3764,
+      2066,
+      2359,
+      1616,
+      2101,
+      1552,
+      3139,
+      1832,
+      2461,
+      3464,
+      3785,
+      4220,
+      2644,
+      84,
+      288,
+      24,
+      0
+    ],
+    "child_multiplicities": {
+      "1": 243705792,
+      "2": 278550976,
+      "3": 56182192,
+      "4": 63569792,
+      "5": 33476960,
+      "6": 25457872,
+      "7": 16973264,
+      "8": 15236672,
+      "9": 8363168,
+      "10": 9549232,
+      "11": 6541568,
+      "12": 8504848,
+      "13": 6282496,
+      "14": 12706672,
+      "15": 7415936,
+      "16": 9962128,
+      "17": 14022272,
+      "18": 15321680,
+      "19": 17082560,
+      "20": 10702912,
+      "21": 340032,
+      "22": 1165824,
+      "23": 16483456,
+      "529": 8193152,
+      "552": 155872288
+    }
+  },
+  "bridge": {
+    "complex": {
+      "m": 576,
+      "W": 164065440,
+      "s": 94499831360,
+      "maxchild": 552,
+      "halving_degree": 17,
+      "wire_bits": 28,
+      "scalar_terms": [
+        {
+          "h": 24,
+          "v": 2024,
+          "c": 68188,
+          "invocations": 2024,
+          "local_group_upper": 497896,
+          "copied_center_extra_groups": 48
+        },
+        {
+          "h": 24,
+          "v": 2024,
+          "c": 68188,
+          "invocations": 2024,
+          "local_group_upper": 497896,
+          "copied_center_extra_groups": 48
+        }
+      ],
+      "scalar_group_upper": 2019773888
+    },
+    "semantic": {
+      "E": 666564786024792104413897128000,
+      "literal_charge": 108734400428349651811564676,
+      "strict_literal_gap": 666456051624363754762085563324,
+      "B": 666564786024792104508396959360,
+      "C0": 8189496372663279519858398035052300773010373215228808037348147200,
+      "C1": 1,
+      "induction_gap": 31328544943165228911894657089920,
+      "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": 1160,
+      "degree": 4000,
+      "suffix_slope": 16000,
+      "degree_gap": "8168/5",
+      "contract": "W_complex^D_complex * W_coarse^D_coarse * W_old^D_old; one preceding prefix and one padding; sequential reuse"
+    },
+    "bit_coarse": {
+      "m": 529,
+      "W": 120683024,
+      "maxchild": 506,
+      "halving_degree": 16,
+      "wire_bits": 27
+    },
+    "ordinary_leaf_row_degree": 252
+  },
+  "assembly": {
+    "parameters": {
+      "a_bit": "67028307846547/781250000000000000",
+      "a_complex": "67028453/781250000000",
+      "tau": "781182971692153453/781250000000000000",
+      "sigma": "781182971547/781250000000",
+      "eta": "1/100000000",
+      "beta": "1/1000000",
+      "q": "3351415325299042153453/39062500000000000000000000",
+      "c": "335141535881319540644342153453/3906250000000000000000000000000000",
+      "epsilon": "38675742187500000000000000000000/38682379040281299245108808338153",
+      "lambda_": "78118297169282373607846547/78125000000000000000000000",
+      "lambda_prime": "39059148584674700957846547/39062500000000000000000000",
+      "alpha_squared_power": "4977542871731250722950000000/38682379040281299245108808338153",
+      "delta": "1/800000000",
+      "C0": 8189496372663279519858398035052300773010373215228808037348147200,
+      "C1": 1,
+      "kappa": "1715630240171/20000000000000000"
+    },
+    "strict_constraints": {
+      "bit_positive": "67028307846547/781250000000000000",
+      "complex_above_bit": "145153453/781250000000000000",
+      "complex_below_one_over32": "24347034047/781250000000",
+      "beta_positive": "1/1000000",
+      "beta_below_one": "999999/1000000",
+      "leaf_saving_above_bit": "1/10000000000",
+      "q_positive": "3351415325299042153453/39062500000000000000000000",
+      "q_below_internal": "67028307846547/39062500000000000000000000",
+      "q_below_leaf": "3973278307846547/39062500000000000000000000",
+      "c_positive": "335141535881319540644342153453/3906250000000000000000000000000000",
+      "c_below_one": "3905914858464118680459355657846547/3906250000000000000000000000000000",
+      "q_below_reservations": "3351415325299042153453/3906250000000000000000000000000000",
+      "lambda_above_tau": "67028307846547/78125000000000000000000000",
+      "lambda_above_sigma": "14582373607846547/78125000000000000000000000",
+      "lambda_above_internal": "67028307846547/78125000000000000000000000",
+      "lambda_prime_above_lambda": "67028307846547/78125000000000000000000000",
+      "compact_leaf": "3973278307846547/39062500000000000000000000",
+      "compact_reservations": "3351415325299042153453/3906250000000000000000000000000000",
+      "lambda_prime_below_one": "3351415325299042153453/39062500000000000000000000",
+      "epsilon_positive": "38675742187500000000000000000000/38682379040281299245108808338153",
+      "epsilon_below_one": "6636852781299245108808338153/38682379040281299245108808338153",
+      "guard_width": "6636852781299245108808338153/38682379040281299245108808338153",
+      "K_geometry": "331861978595365915008411293508338153/3868237904028129924510880833815300000000",
+      "K_dominates_log": "19519017619679917404260148340685991/227543406119301760265345931400900000000",
+      "record_suffix": "6636852781299245108808338153/38682379040281299245108808338153",
+      "phase_local": "5309443542660355805747425413591661847/30945903232225039396087046670522400000000",
+      "phase_boundary": "3981995615005960297060754891191661847/30945903232225039396087046670522400000000",
+      "gamma_sublinear": "97606465268705552109314009/2275434061193017602653459314009",
+      "cell_above_band": "38674082877590432005614141661847/77364758080562598490217616676306",
+      "prime_interval_packing": "6636852781299245108808338153/38682379040281299245108808338153",
+      "alpha_positive": "4977542871731250722950000000/38682379040281299245108808338153",
+      "alpha_below_one": "38677401497409567994385858338153/38682379040281299245108808338153",
+      "alpha_below_one_fourth": "38662468868794374242217008338153/154729516161125196980435233352612",
+      "delta_positive": "1/800000000",
+      "delta_below_one_eighth": "99999999/800000000",
+      "short_record_fallback": "30212830769573826207778426965514256994530790592309/30220608625219765035241256514182031250000000000000",
+      "small_field_exposure": "195212930537411104218628018/2275434061193017602653459314009",
+      "artificial_boundary": "216627959517045590523923760789592791661847/30945903232225039396087046670522400000000",
+      "literal_scalar_guard": "666456051624363754762085563324",
+      "row_product_gap": "8168/5",
+      "original_prefix_above_kappa": "7736475809721094396312855454591827455837/773647580805625984902176166763060000000000000000",
+      "coordinate_movement_above_kappa": "184025437877/12500000000000000000",
+      "compact_phase_layer_above_kappa": "1664834547291093786961227455837/773647580805625984902176166763060000000000000000",
+      "bulk_exposure_above_kappa": "184025437877/12500000000000000000",
+      "Gaussian_arithmetic_above_kappa": "33185231131885545519232926433578507402455837/773647580805625984902176166763060000000000000000",
+      "scalar_work_above_kappa": "66371429323245433236399689493578507402455837/773647580805625984902176166763060000000000000000",
+      "dimension_above_kappa": "773448479090756736538092714054153786961227455837/773647580805625984902176166763060000000000000000"
+    },
+    "margins": {
+      "original_prefix": "331861978595365915008411293508338153/3868237904028129924510880833815300000000",
+      "coordinate_movement": "67028307846547/781250000000000000",
+      "compact_phase_layer": "3318232962163256337091661847/38682379040281299245108808338153",
+      "bulk_exposure": "67028307846547/781250000000000000",
+      "Gaussian_arithmetic": "3981995615005960297060754891191661847/30945903232225039396087046670522400000000",
+      "scalar_work": "5309443542660355805747425413591661847/30945903232225039396087046670522400000000",
+      "dimension": "38675742187500000000000000000000/38682379040281299245108808338153"
+    },
+    "minimum_margin": "3318232962163256337091661847/38682379040281299245108808338153",
+    "absorption_gap": "1664834547291093786961227455837/773647580805625984902176166763060000000000000000",
+    "recurrence": {
+      "internal": "781182971692153453/781250000000000000",
+      "leaf": "781182971614028453/781250000000000000",
+      "reservations": "3905914858464118680459355657846547/3906250000000000000000000000000000"
+    }
+  }
+}
diff --git a/research/cyclic-strips-both-axes/producer.json b/research/cyclic-strips-both-axes/producer.json
new file mode 100644
index 0000000000..7aeee6a5df
--- /dev/null
+++ b/research/cyclic-strips-both-axes/producer.json
@@ -0,0 +1,41 @@
+{
+  "h": 24,
+  "v": 2024,
+  "c": 68188,
+  "q": 8120,
+  "baseline_R": 76308,
+  "matched": 37802,
+  "R": 38506,
+  "orientation_changes": 5924,
+  "rank_sum": 925248,
+  "loss": 552,
+  "central_disjoint": 24,
+  "center_denominator": 21,
+  "histogram": [
+    9770,
+    59168,
+    68812,
+    13879,
+    15704,
+    8270,
+    6289,
+    4193,
+    3764,
+    2066,
+    2359,
+    1616,
+    2101,
+    1552,
+    3139,
+    1832,
+    2461,
+    3464,
+    3785,
+    4220,
+    2644,
+    84,
+    288,
+    24,
+    24
+  ]
+}
\ No newline at end of file
diff --git a/research/cyclic-strips-both-axes/producer.py b/research/cyclic-strips-both-axes/producer.py
new file mode 100644
index 0000000000..80ff092dd7
--- /dev/null
+++ b/research/cyclic-strips-both-axes/producer.py
@@ -0,0 +1,270 @@
+# Cyclic-strip recursion plus dual-suffix pair stars, by Rohan Arun with Anthropic Claude assistance.
+# Parent: PR107 (Rohan Arun with OpenAI Codex) on icekylinx PR104 at 854ba7dca98651eab2050402384e1a7784534f0a.
+# Pair-star leave-one-out identity: PR55 dual-suffix strips (credited there).
+# Original copyright, Apache-2.0 notice and scalar/frame assertions retained.
+"""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
+from exclusion_circuit import ExclusionCircuit
+
+
+def cyclic_vector(self,values,two=True):
+    """Leave-one-out by cyclic interval strips (PR62 idea): I(a,m)=I(a,m-1)+v_(a+m-1).
+
+    Used for every strip of the PairedTriple recursion; leave-two-out vectors are unchanged.
+    Zero entries are omitted (their leave-one-out value is the total), exactly as before."""
+    if two:return ExclusionCircuit.vector(self,values,two)
+    vals=list(values);n=len(vals)
+    if n==0:return 0,[],{}
+    nz=[i for i,x in enumerate(vals) if x]
+    if not nz:return 0,[0]*n,{}
+    if len(nz)==1:
+        x=vals[nz[0]];return x,[0 if i==nz[0] else x for i in range(n)],{}
+    v=[vals[i] for i in nz];k=len(v)
+    if k<3:
+        tot=self.add(v[0],v[1]);one=[v[1],v[0]]
+    else:
+        full=(1<<k)-1;iv=lambda a,m:sum(1<<((a+t)%k) for t in range(m))
+        node={1<<i:x for i,x in enumerate(v)}
+        for m in range(2,k):
+            for a in range(k):node[iv(a,m)]=self.add(node[iv(a,m-1)],node[1<<((a+m-1)%k)])
+        node[full]=self.add(node[iv(0,k-1)],node[1<<(k-1)])
+        tot=node[full];one=[node[full^(1<<j)] for j in range(k)]
+    out=[tot]*n
+    for i,y in zip(nz,one):out[i]=y
+    return tot,out,{}
+
+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
+    PairedTriple.vector=cyclic_vector
+    try:paired=PairedTriple(h,base)
+    finally:del PairedTriple.vector
+    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))
+        # Dual-suffix leave-one-out (the PR55 strip identity, here on pair stars):
+        # s_j = (A_{j-1} + v_{j+1}) + B_{j+2}, s_k = A_{k-1}, with prefix sums A
+        # and suffix sums B. Each A_{j-1} feeds A_j and A_{j-1}+v_{j+1}; nested
+        # covers let the legal carrier matcher join far more additions.
+        vals=[inputs[tuple(sorted((a,b,i)))] for i in others];k=len(vals)
+        assert k>=3
+        A=[0]*(k+1)
+        for t in range(1,k):A[t]=add(A[t-1],vals[t-1])
+        B=[0]*(k+2)
+        for t in range(k,1,-1):B[t]=add(vals[t-1],B[t+1])
+        dual=[add(add(A[t-1],vals[t]),B[t+2]) for t in range(1,k)]+[A[k-1]]
+        pre=[A[k-1],add(A[k-1],vals[k-1])]
+        for j,i in enumerate(others):
+            node=dual[j]
+            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/research/cyclic-strips-both-axes/producer_bit.py b/research/cyclic-strips-both-axes/producer_bit.py
new file mode 100644
index 0000000000..7742952190
--- /dev/null
+++ b/research/cyclic-strips-both-axes/producer_bit.py
@@ -0,0 +1,43 @@
+"""Regenerate the h23 retained-total bit row with cyclic strips (PR104/#36 selected bit pipeline).
+
+Same pipeline as scripts/partial_swap_producer.py: graph -> scalar/frame verify -> export ->
+unchanged match_exported_dag -> positive labels recomputed for this graph -> unchanged
+match_positive_dag. Only PairedExclusionCircuit.block's leave-one-out layout is replaced, privately.
+Rohan Arun with Anthropic Claude assistance, Apache-2.0.
+"""
+import gc, importlib.util, json, shlex, os, subprocess, sys
+from pathlib import Path
+HERE=Path(__file__).resolve().parent;ROOT=HERE.parents[1]
+sys.path.insert(0,str(ROOT/'scripts'))
+import partial_swap.paired as base_paired
+from partial_swap.graph import graph, export
+from partial_swap.positive import run as positive_labels
+
+def build_row(work,cxx=None):
+    assert not sys.flags.optimize,'Assertions must remain enabled'
+    spec=importlib.util.spec_from_file_location('bit_paired',HERE/'bit_paired.py');layout=importlib.util.module_from_spec(spec);spec.loader.exec_module(layout)
+    assert (layout.LAYOUT,layout.ORDER,layout.CARRY)==('cyclic','id','last')
+    cls=base_paired.PairedExclusionCircuit;old_block=cls.block
+    cls.block=layout.PairedExclusionCircuit.block;cls.loo=layout.PairedExclusionCircuit.loo
+    try:
+        work=Path(work);work.mkdir(parents=True,exist_ok=True)
+        progs={}
+        for name in ('match_exported_dag','match_positive_dag'):
+            progs[name]=work/name
+            subprocess.run([*shlex.split(cxx or os.environ.get('CXX','c++')),'-O3','-std=c++17',str(ROOT/'scripts/partial_swap'/(name+'.cpp')),'-o',str(progs[name])],check=True)
+        h=23;circuit=graph(h);scalar=circuit.verify();frames=circuit.verify_frames()
+        dag=work/f'triple_base2_{h}.bin';export(circuit,dag);del circuit;gc.collect()
+        original=json.loads(subprocess.check_output([str(progs['match_exported_dag']),str(dag),str(dag)+'.links'],text=True))
+        positive_labels(str(dag))
+        matched=json.loads(subprocess.check_output([str(progs['match_positive_dag']),str(dag),str(dag)+'.positive'],text=True))
+    finally:
+        cls.block=old_block;del cls.loo
+    assert matched['loss']==h*(h-1) and matched['rank_sum']==h*matched['R']+2*h*(h-1)
+    assert original['matched']==matched['matched'],'Positive labels must not enlarge the envelope matching'
+    assert matched['R']==matched['c']+matched['q']-matched['matched']
+    return dict(h=h,v=matched['v'],c=matched['c'],q=matched['q'],baseline_R=matched['baseline_R'],matching=matched['matched'],
+                R=matched['R'],loss=matched['loss'],histogram=matched['histogram'],label_source='positive',
+                description='Retained-total aligned triple producer with cyclic interval strips (carry last), base-two recursion, original maximum carrier matching, positive labels.')
+if __name__=='__main__':
+    import tempfile
+    with tempfile.TemporaryDirectory() as d: print(json.dumps(build_row(d)))
diff --git a/research/cyclic-strips-both-axes/test_controls.py b/research/cyclic-strips-both-axes/test_controls.py
new file mode 100644
index 0000000000..3404f5cd62
--- /dev/null
+++ b/research/cyclic-strips-both-axes/test_controls.py
@@ -0,0 +1,24 @@
+"""Adversarial controls: old rows, unpaid cleanup and omitted loss cannot certify the new bound."""
+import copy, unittest
+import verify as v
+
+class Controls(unittest.TestCase):
+    def setUp(self):
+        self.bit=v.read(v.HERE/'bit-row.json');self.row=v.read(v.HERE/'producer.json')
+    def test_selected_rows_pass_without_certificate(self):
+        v.exact(self.bit,self.row,compare=False)
+    def test_old_bit_row_rejected(self):
+        with self.assertRaises(Exception):v.exact(v.read(v.ROOT/'certificates/stopped-product-bit-axis.json'),self.row,compare=False)
+    def test_pr108_complex_row_rejected(self):
+        with self.assertRaises(Exception):v.exact(self.bit,v.read(v.ROOT/'research/dual-suffix-centers/producer.json'),compare=False)
+    def test_unpaid_complex_cleanup_rejected(self):
+        r=copy.deepcopy(self.row);r['histogram'][24]-=1
+        with self.assertRaises(Exception):v.exact(self.bit,r,compare=False)
+    def test_omitted_bit_loss_rejected(self):
+        b=copy.deepcopy(self.bit);b['loss']=0
+        with self.assertRaises(Exception):v.exact(b,self.row,compare=False)
+    def test_unpaid_bit_role_rejected(self):
+        b=copy.deepcopy(self.bit);b['R']-=1
+        with self.assertRaises(Exception):v.exact(b,self.row,compare=False)
+
+if __name__=='__main__':unittest.main()
diff --git a/research/cyclic-strips-both-axes/validation.json b/research/cyclic-strips-both-axes/validation.json
new file mode 100644
index 0000000000..f3e44d9ab7
--- /dev/null
+++ b/research/cyclic-strips-both-axes/validation.json
@@ -0,0 +1,37 @@
+{
+  "state": "passed",
+  "tested_commit": "fb550b9b1a3c60cc419701e530494ed754d08cef",
+  "started_at": "2026-10-09T00:04:51.548632+00:00",
+  "pid": 118,
+  "commands": [
+    [
+      "make",
+      "-j1",
+      "verify"
+    ],
+    [
+      "python",
+      "research/cyclic-strips-both-axes/verify.py"
+    ],
+    [
+      "python",
+      "research/cyclic-strips-both-axes/test_controls.py"
+    ]
+  ],
+  "platform": "Linux Daytona python:3.11-slim with g++ and libboost-dev",
+  "compiler": "c++ -include algorithm",
+  "returncode": 0,
+  "elapsed_seconds": 2409.831456184387,
+  "completed_at": "2026-10-09T00:45:01.380047+00:00",
+  "log_sha256": "6e550320967103dccded235deda3d8fabfe081b6b72b266b498039a896910b34",
+  "git_status": "",
+  "drift": [],
+  "evidence_archive_sha256": "17f1f947ae16fc6ddb7cc3823c3c17937b6dd1ae88af2c7b0448c58aa5ad6fc8",
+  "coverage": {
+    "isolated_test_modules": 78,
+    "historical_patch_checks": 20,
+    "own_adversarial_controls": 6,
+    "research_head_github_checks_passed": 39
+  },
+  "post_validation_changes": "Only receipt, package README status and its own source fingerprint; executable and mathematical source unchanged."
+}
diff --git a/research/cyclic-strips-both-axes/verify.py b/research/cyclic-strips-both-axes/verify.py
new file mode 100644
index 0000000000..c250b44d99
--- /dev/null
+++ b/research/cyclic-strips-both-axes/verify.py
@@ -0,0 +1,83 @@
+"""Cyclic interval strips on both axes of PR104's stopped product-ring construction.
+
+Bit axis (h23 retained-total producer) and complex axis (h24 all-disjoint rational
+centers, with PR108 dual-suffix pair stars) both replace prefix/suffix leave-one-out
+sums by cyclic interval strips. Everything else - matchers, positive labels, exact
+support/frame/rational-scatter assertions, profiles, bridge and assembly - is PR104/PR107.
+Copyright 2026 Rohan Arun, Apache-2.0. Prepared with Anthropic Claude assistance.
+"""
+import argparse, hashlib, importlib.util, json, os, shlex, subprocess, sys, tempfile
+from pathlib import Path
+from fractions import Fraction as Q
+HERE=Path(__file__).resolve().parent;ROOT=HERE.parents[1]
+sys.path.insert(0,str(ROOT/'scripts'))
+import stopped_product_network as parent
+from structured_bulk_assembly import assembly, js
+
+COARSE=Q(898920417,10**13)          # certified stopped coarse saving of the new bit row
+B=Q(8579641984,10**14)              # strict complex saving of the new complex row
+KAPPA=Q(8578151200855,10**17)
+PRIOR={'PR107':Q(7798412662809,10**17),'PR108':Q(7988289348441,10**17),'PR109':Q(39980858679,500000000000000)}
+
+def read(p): return json.loads(Path(p).read_text())
+def pins():
+    assert not sys.flags.optimize,'Assertions must remain enabled'
+    for name,digest in read(HERE/'SOURCE.json')['files'].items():
+        assert hashlib.sha256((ROOT/name).read_bytes()).hexdigest()==digest,name
+def rejects(f):
+    try: f()
+    except Exception: return True
+    return False
+
+def exact(bit_row,row,write=False,compare=True):
+    old=(parent.COARSE,parent.AB)
+    try:
+        parent.COARSE=COARSE;parent.AB=(1-parent.ATOM)*COARSE+parent.ATOM*parent.OLD
+        bit=parent.profile(bit_row);coarse=parent.coarse_moment(bit)
+        parent.COARSE=COARSE+Q(1,10**13);parent.AB=(1-parent.ATOM)*parent.COARSE+parent.ATOM*parent.OLD
+        assert rejects(lambda:parent.coarse_moment(bit)),'Next coarse grid accepted'
+        parent.COARSE=COARSE;parent.AB=(1-parent.ATOM)*COARSE+parent.ATOM*parent.OLD
+        phase=parent.profile(row)
+        cm=parent.moment(phase['m'],phase['W'],phase['child_multiplicities'],B,True)
+        assert rejects(lambda:parent.moment(phase['m'],phase['W'],phase['child_multiplicities'],B+Q(1,10**14),True)),'Next complex grid accepted'
+        bridge=parent.finite_bridge(bit,phase,row,read(ROOT/'certificates/copied-centers-network.json'))
+        a=min(parent.AB,(1-parent.PHASE_STOP)*B-Q(1,10**10))
+        assembled=assembly(a,B,bridge,KAPPA,beta=parent.PHASE_STOP)
+        assert len(assembled['strict_constraints'])==47 and len(assembled['margins'])==7
+        assert rejects(lambda:assembly(a,B,bridge,KAPPA+Q(1,10**17),beta=parent.PHASE_STOP)),'Next kappa grid accepted'
+        for name,k in PRIOR.items(): assert KAPPA>k,name
+        computed=dict(kappa=KAPPA,coarse_bit_saving=COARSE,effective_bit_saving=parent.AB,complex_saving=B,bit_parameter=a,
+                      bit_profile=bit,coarse_moment=coarse,complex_profile=phase,bridge=bridge,assembly=assembled)
+        if write:(HERE/'certificate.json').write_text(json.dumps(js(computed),indent=2)+'\n')
+        if compare:
+            expected=read(HERE/'certificate.json')
+            for key,value in computed.items(): assert js(value)==expected[key],key
+        return dict(kappa=str(KAPPA),complex_saving=str(B),coarse_bit_saving=str(COARSE),constraints=47,margins=7,next_grid_controls=3,
+                    gains={k:float((KAPPA/v-1)*100) for k,v in PRIOR.items()})
+    finally: parent.COARSE,parent.AB=old
+
+def regenerate(work):
+    spec=importlib.util.spec_from_file_location('csb_bit',HERE/'producer_bit.py');bitmod=importlib.util.module_from_spec(spec);spec.loader.exec_module(bitmod)
+    bit=bitmod.build_row(work/'bit')
+    stored=read(HERE/'bit-row.json')
+    for k in ('h','v','c','q','baseline_R','matching','R','loss','histogram','label_source'): assert bit[k]==stored[k],('bit',k)
+    program=work/'matcher'
+    subprocess.run([*shlex.split(os.environ.get('CXX','c++')),'-O3','-std=c++17',str(ROOT/'scripts/endpoint_gauge/match_complex_general.cpp'),'-o',str(program)],check=True)
+    spec=importlib.util.spec_from_file_location('csb_complex',HERE/'producer.py');cx=importlib.util.module_from_spec(spec);spec.loader.exec_module(cx)
+    con=cx.build(24,work/'complex',24)
+    row=json.loads(subprocess.check_output([str(program),str(work/'complex')+'.bin',str(work/'complex')+'.labels'],text=True))
+    stored=read(HERE/'producer.json')
+    for k in ('h','v','c','q','R','matched','loss','histogram'): assert row[k]==stored[k],('complex',k)
+    assert con['R']==row['baseline_R'] and row['R']==row['c']+row['q']-row['matched']
+    return bit,row
+
+def main():
+    p=argparse.ArgumentParser();p.add_argument('--arithmetic-only',action='store_true');p.add_argument('--write',action='store_true');a=p.parse_args()
+    if not a.write:pins()
+    if a.write or not a.arithmetic_only:
+        with tempfile.TemporaryDirectory(prefix='cyclic-strips-') as tmp:
+            bit,row=regenerate(Path(tmp))
+    receipt=exact(read(HERE/'bit-row.json'),read(HERE/'producer.json'),write=a.write)
+    receipt['bit_and_complex_producers_regenerated']=not a.arithmetic_only or a.write
+    print(json.dumps(receipt,indent=2));print('PASS conditional cyclic strips on both axes')
+if __name__=='__main__':main()
diff --git a/research/dual-suffix-centers/README.md b/research/dual-suffix-centers/README.md
new file mode 100644
index 0000000000..34e6c401d2
--- /dev/null
+++ b/research/dual-suffix-centers/README.md
@@ -0,0 +1,61 @@
+# Dual-suffix pair stars in the rational-center producer
+
+Under **all of PR104's inherited and newly proposed interfaces** (as used by PR107),
+this finite replacement gives conditional
+**κ = 7988289348441/10^17 = 7.988289348441×10⁻⁵**, about **2.4348% above PR107**
+(7798412662809/10^17) and 2.483% above PR104.
+
+## What changes
+
+PR104/PR107 bind on the **complex** (h=24, all-disjoint rational-center) producer:
+its strict saving 7799647191/10^14 sits below the stopped bit saving
+803380799/10^13. Each pair star `(a,b)` of that producer computes all 22
+leave-one-out sums of `v_i = x_{a,b,i}` with the ordinary layout
+`s_j = A_{j-1} + B_{j+1}` (prefix A, suffix B).
+
+We replace only that block by the dual-suffix identity from the h23/h25 bit
+strips of PR55 (credited there):
+
+    s_j = (A_{j-1} + v_{j+1}) + B_{j+2}   (j < k),     s_k = A_{k-1}.
+
+Each prefix `A_{j-1}` now feeds `A_j` and `A_{j-1}+v_{j+1}` whose covers nest,
+so the unchanged legal carrier matcher (`scripts/endpoint_gauge/match_complex_general.cpp`)
+joins far more additions. The coordinate order, center trees, rational divisor 21,
+scatter, copied-center corrections, stopped bit axis and assembly are unchanged.
+
+| h24 complex producer | PR107 | This |
+|---|---:|---:|
+| additions c | 45,548 | 50,572 |
+| matched carriers | 8,750 | 15,422 |
+| roles R = c + q − matched | 44,918 | **43,270** |
+| strict complex saving | 7799647191/10^14 | **7989584037/10^14** |
+| κ | 7798412662809/10^17 | **7988289348441/10^17** |
+
+The bit parameter is min(stopped bit saving, (1−10⁻⁶)·b − 10⁻¹⁰) exactly as in
+PR107, so the new binding is still the complex leaf, with 0.57% headroom left
+below the bit axis.
+
+## Checks
+
+```sh
+python3 research/dual-suffix-centers/verify.py
+python3 research/dual-suffix-centers/test_controls.py
+make verify
+```
+
+The verifier reproduces PR104's certificate, regenerates the PR104, PR107 and
+new scalar/frame producers (every exact support, center, rational scatter and
+binary frame nesting assertion is retained), recompiles the inherited matcher,
+checks every histogram entry, then all 47 strict constraints, seven margins and
+both next-grid exclusions (complex 7989584038/10^14 and κ 7988289348442/10^17).
+Controls reject unpaid cleanup, omitted retained loss, and both the PR104 and
+PR107 producer histograms at the new bound. Full repository verification is pending.
+
+Scope: a finite conditional witness. The opposite-bank factorization, stopped
+atom streaming and odd-denominator grid interfaces of PR104 remain written proof
+dependencies, as in PR107. No global optimality or practical speedup is claimed.
+
+Credit icekylinx (PR104 stopped product-ring / rational centers), the PR55
+dual-suffix strip authors, Avi Eisenberg (PR53 skip-prefix strips), PR107, and
+all predecessors credited in PR104's NOTICE and SOURCES. Prepared by Rohan Arun
+with Anthropic Claude assistance, Apache-2.0.
diff --git a/research/dual-suffix-centers/SOURCE.json b/research/dual-suffix-centers/SOURCE.json
new file mode 100644
index 0000000000..3dde98d01b
--- /dev/null
+++ b/research/dual-suffix-centers/SOURCE.json
@@ -0,0 +1,337 @@
+{
+  "parent_commit": "b06b82a",
+  "files": {
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diff --git a/research/dual-suffix-centers/certificate.json b/research/dual-suffix-centers/certificate.json
new file mode 100644
index 0000000000..b596369c07
--- /dev/null
+++ b/research/dual-suffix-centers/certificate.json
@@ -0,0 +1,215 @@
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+      "lambda_above_tau": "7989566047415963/10000000000000000000000000000",
+      "lambda_above_sigma": "1806947969747415963/10000000000000000000000000000",
+      "lambda_above_internal": "7989566047415963/10000000000000000000000000000",
+      "lambda_prime_above_lambda": "7989566047415963/10000000000000000000000000000",
+      "compact_leaf": "507989566047415963/5000000000000000000000000000",
+      "compact_reservations": "399478294381232102584037/500000000000000000000000000000000000",
+      "lambda_prime_below_one": "399478294381232102584037/5000000000000000000000000000",
+      "epsilon_positive": "499999995000000000000000000000000000/500079895659275724714898039502584037",
+      "epsilon_below_one": "79900659275724714898039502584037/500079895659275724714898039502584037",
+      "guard_width": "79900659275724714898039502584037/500079895659275724714898039502584037",
+      "K_geometry": "3995282983760150863441874639635802584037/50007989565927572471489803950258403700000000",
+      "K_dominates_log": "234987231988960036844827977574582200939/2941646445054563086558223761779906100000000",
+      "record_suffix": "79900659275724714898039502584037/500079895659275724714898039502584037",
+      "phase_local": "63920027340684112642706887169190097415963/400063916527420579771918431602067229600000000",
+      "phase_boundary": "47938895245852193034359725101960497415963/400063916527420579771918431602067229600000000",
+      "gamma_sublinear": "1175083242267052912378467799061/29416464450545630865582237617799061",
+      "cell_above_band": "499980018584881460100489566047415963/1000159791318551449429796079005168074",
+      "prime_interval_packing": "79900659275724714898039502584037/500079895659275724714898039502584037",
+      "alpha_positive": "59924244157184815387605550000000/500079895659275724714898039502584037",
+      "alpha_below_one": "500019971415118539899510433952584037/500079895659275724714898039502584037",
+      "alpha_below_one_fourth": "499840198682646985453347617302584037/2000319582637102898859592158010336148",
+      "delta_positive": "1/800000000",
+      "delta_below_one_eighth": "99999999/800000000",
+      "short_record_fallback": "49996004078644645333255568412560243706676014994897217369/50007989565927572471489803950258403700000000000000000000",
+      "small_field_exposure": "2350166484534105824756935598122/29416464450545630865582237617799061",
+      "artificial_boundary": "2800527316851379574435680075263707026897415963/400063916527420579771918431602067229600000000",
+      "literal_scalar_guard": "451373808041401410361486475452",
+      "row_product_gap": "40024/25",
+      "original_prefix_above_kappa": "500079895671920321133699171957267374422563683/50007989565927572471489803950258403700000000000000000",
+      "coordinate_movement_above_kappa": "1276698974963/100000000000000000000",
+      "compact_phase_layer_above_kappa": "12644596418801132454683337422563683/50007989565927572471489803950258403700000000000000000",
+      "bulk_exposure_above_kappa": "1276698974963/100000000000000000000",
+      "Gaussian_arithmetic_above_kappa": "1997579001867045186174224697281216860332797563683/50007989565927572471489803950258403700000000000000000",
+      "scalar_work_above_kappa": "3995220513721035137217619955684916860332797563683/50007989565927572471489803950258403700000000000000000",
+      "dimension_above_kappa": "49996004717096135521056879259059536154683337422563683/50007989565927572471489803950258403700000000000000000"
+    },
+    "margins": {
+      "original_prefix": "3995282983760150863441874639635802584037/50007989565927572471489803950258403700000000",
+      "coordinate_movement": "7989566047415963/100000000000000000000",
+      "compact_phase_layer": "39947829038644915877171597415963/500079895659275724714898039502584037",
+      "bulk_exposure": "7989566047415963/100000000000000000000",
+      "Gaussian_arithmetic": "47938895245852193034359725101960497415963/400063916527420579771918431602067229600000000",
+      "scalar_work": "63920027340684112642706887169190097415963/400063916527420579771918431602067229600000000",
+      "dimension": "499999995000000000000000000000000000/500079895659275724714898039502584037"
+    },
+    "minimum_margin": "39947829038644915877171597415963/500079895659275724714898039502584037",
+    "absorption_gap": "12644596418801132454683337422563683/50007989565927572471489803950258403700000000000000000",
+    "recurrence": {
+      "internal": "99992010433952584037/100000000000000000000",
+      "leaf": "99992010423952584037/100000000000000000000",
+      "reservations": "499960052170162398495360364197415963/500000000000000000000000000000000000"
+    }
+  }
+}
diff --git a/research/dual-suffix-centers/producer.json b/research/dual-suffix-centers/producer.json
new file mode 100644
index 0000000000..936145d5cc
--- /dev/null
+++ b/research/dual-suffix-centers/producer.json
@@ -0,0 +1,41 @@
+{
+  "h": 24,
+  "v": 2024,
+  "c": 50572,
+  "q": 8120,
+  "baseline_R": 58692,
+  "matched": 15422,
+  "R": 43270,
+  "orientation_changes": 2790,
+  "rank_sum": 1039584,
+  "loss": 552,
+  "central_disjoint": 24,
+  "center_denominator": 21,
+  "histogram": [
+    8930,
+    58270,
+    30626,
+    14032,
+    11787,
+    8398,
+    8383,
+    4533,
+    6547,
+    2447,
+    4974,
+    2003,
+    5186,
+    1936,
+    5277,
+    1982,
+    4733,
+    3522,
+    3477,
+    4785,
+    3388,
+    252,
+    288,
+    24,
+    24
+  ]
+}
diff --git a/research/dual-suffix-centers/producer.py b/research/dual-suffix-centers/producer.py
new file mode 100644
index 0000000000..15d4b3b56b
--- /dev/null
+++ b/research/dual-suffix-centers/producer.py
@@ -0,0 +1,240 @@
+# Dual-suffix pair-star adaptation by Rohan Arun with Anthropic Claude assistance.
+# Parent: PR107 (Rohan Arun with OpenAI Codex) on icekylinx PR104 at 854ba7dca98651eab2050402384e1a7784534f0a.
+# Pair-star leave-one-out identity: PR55 dual-suffix strips (credited there).
+# Original copyright, Apache-2.0 notice and scalar/frame assertions retained.
+"""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))
+        # Dual-suffix leave-one-out (the PR55 strip identity, here on pair stars):
+        # s_j = (A_{j-1} + v_{j+1}) + B_{j+2}, s_k = A_{k-1}, with prefix sums A
+        # and suffix sums B. Each A_{j-1} feeds A_j and A_{j-1}+v_{j+1}; nested
+        # covers let the legal carrier matcher join far more additions.
+        vals=[inputs[tuple(sorted((a,b,i)))] for i in others];k=len(vals)
+        assert k>=3
+        A=[0]*(k+1)
+        for t in range(1,k):A[t]=add(A[t-1],vals[t-1])
+        B=[0]*(k+2)
+        for t in range(k,1,-1):B[t]=add(vals[t-1],B[t+1])
+        dual=[add(add(A[t-1],vals[t]),B[t+2]) for t in range(1,k)]+[A[k-1]]
+        pre=[A[k-1],add(A[k-1],vals[k-1])]
+        for j,i in enumerate(others):
+            node=dual[j]
+            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/research/dual-suffix-centers/test_controls.py b/research/dual-suffix-centers/test_controls.py
new file mode 100644
index 0000000000..9a7089534f
--- /dev/null
+++ b/research/dual-suffix-centers/test_controls.py
@@ -0,0 +1,33 @@
+"""Adversarial cost and strict-endpoint controls for the selected finite witness."""
+import copy, unittest
+from fractions import Fraction as Q
+import verify as v
+
+class Controls(unittest.TestCase):
+    def setUp(self): self.row=v.read(v.HERE/'producer.json')
+    def test_unpaid_cleanup_rejected(self):
+        self.row['histogram'][24]-=1
+        with self.assertRaises(ValueError):v.exact(self.row)
+    def test_omitted_retained_loss_rejected(self):
+        self.row['loss']=0
+        with self.assertRaises(ValueError):v.exact(self.row)
+    def test_parent_histogram_cannot_claim_new_bound(self):
+        row=v.read(v.ROOT/'certificates/stopped-product-complex-input.json')
+        with self.assertRaises((AssertionError,ValueError)):v.exact(row)
+    def test_next_complex_grid_rejected(self):
+        p=v.parent.profile(self.row)
+        with self.assertRaises(ValueError):
+            v.parent.moment(p['m'],p['W'],p['child_multiplicities'],Q(7989584038,10**14),True)
+    def test_next_kappa_grid_rejected(self):
+        c=v.read(v.HERE/'certificate.json');b=Q(c['complex_saving']);a=Q(c['bit_parameter'])
+        bit=v.parent.profile(v.read(v.ROOT/'certificates/stopped-product-bit-axis.json'))
+        bridge=v.parent.finite_bridge(bit,v.parent.profile(self.row),self.row,
+            v.read(v.ROOT/'certificates/copied-centers-network.json'))
+        with self.assertRaises(AssertionError):
+            v.assembly(a,b,bridge,Q(7988289348442,10**17),beta=v.parent.PHASE_STOP)
+
+    def test_pr107_producer_cannot_claim_new_bound(self):
+        row=v.read(v.ROOT/'research/reversed-rational-centers/producer.json')
+        with self.assertRaises((AssertionError,ValueError)):v.exact(row)
+
+if __name__=='__main__':unittest.main()
diff --git a/research/dual-suffix-centers/verify.py b/research/dual-suffix-centers/verify.py
new file mode 100644
index 0000000000..d9c0df28e9
--- /dev/null
+++ b/research/dual-suffix-centers/verify.py
@@ -0,0 +1,89 @@
+"""Dual-suffix pair stars in the h24 rational-center producer of PR107/PR104.
+
+Copyright 2026 Rohan Arun, Apache-2.0. Prepared with Anthropic Claude assistance;
+adapted from PR107's verifier (OpenAI Codex assistance).
+Does not formally verify the inherited analytic or all-size tape interfaces.
+"""
+import argparse, hashlib, importlib.util, json, os, shlex, subprocess, sys, tempfile
+from pathlib import Path
+from fractions import Fraction as Q
+HERE=Path(__file__).resolve().parent
+ROOT=HERE.parents[1]
+sys.path.insert(0,str(ROOT/'scripts'))
+import stopped_product_network as parent
+from stopped_product.complex import build as parent_build
+from structured_bulk_assembly import assembly, js
+
+def read(p): return json.loads(p.read_text())
+def pins():
+    assert not sys.flags.optimize, 'Assertions must remain enabled'
+    for name,digest in read(HERE/'SOURCE.json')['files'].items():
+        assert hashlib.sha256((ROOT/name).read_bytes()).hexdigest()==digest,name
+
+def exact(row):
+    original=parent.certificate()
+    assert original['kappa']==Q(194869,2500000000)
+    bit=parent.profile(read(ROOT/'certificates/stopped-product-bit-axis.json'))
+    phase=parent.profile(row)
+    bridge=parent.finite_bridge(bit,phase,row,read(ROOT/'certificates/copied-centers-network.json'))
+    b=Q(7989584037,10**14)
+    cm=parent.moment(phase['m'],phase['W'],phase['child_multiplicities'],b,True)
+    a=min(parent.AB,(1-parent.PHASE_STOP)*b-Q(1,10**10))
+    k=Q(7988289348441,10**17)
+    assembled=assembly(a,b,bridge,k,beta=parent.PHASE_STOP)
+    assert len(assembled['strict_constraints'])==47 and len(assembled['margins'])==7
+    assert k>original['kappa'] and k>Q(7798412662809,10**17) and a<=parent.AB
+    # Reject the next grid point for each selected strict endpoint.
+    try: parent.moment(phase['m'],phase['W'],phase['child_multiplicities'],b+Q(1,10**14),True)
+    except ValueError as e: assert str(e)=='Characteristic moment does not contract'
+    else: raise AssertionError('Complex next grid was accepted')
+    try: assembly(a,b,bridge,k+Q(1,10**17),beta=parent.PHASE_STOP)
+    except AssertionError: pass
+    else: raise AssertionError('Kappa next grid was accepted')
+    computed=dict(kappa=k,complex_saving=b,bit_parameter=a,actual_bit_saving=parent.AB,
+                  profile=phase,bridge=bridge,assembly=assembled)
+    if WRITE:(HERE/'certificate.json').write_text(json.dumps(js(computed),indent=2)+'\n')
+    expected=read(HERE/'certificate.json')
+    for key,value in computed.items():
+        assert js(value)==expected[key],key
+    return dict(kappa=str(k),complex_moment_gap=str(cm['strict_gap']),
+                constraints=47,margins=7,next_grid_controls=2)
+
+def regenerate(work):
+    program=work/'matcher'
+    subprocess.run([*shlex.split(os.environ.get('CXX','c++')),'-O3','-std=c++17',
+        str(ROOT/'scripts/endpoint_gauge/match_complex_general.cpp'),'-o',str(program)],check=True)
+    spec=importlib.util.spec_from_file_location('selected_rational_producer',HERE/'producer.py')
+    pspec=importlib.util.spec_from_file_location('pr107_producer',ROOT/'research/reversed-rational-centers/producer.py')
+    prior=importlib.util.module_from_spec(pspec);pspec.loader.exec_module(prior);prior_build=prior.build
+    module=importlib.util.module_from_spec(spec);spec.loader.exec_module(module)
+    results={}
+    for name,builder,expected in (
+        ('parent',parent_build,read(ROOT/'certificates/stopped-product-complex-input.json')),
+        ('pr107',prior_build,read(ROOT/'research/reversed-rational-centers/producer.json')),
+        ('selected',module.build,read(HERE/'producer.json'))):
+        prefix=work/name
+        construction=builder(24,prefix,24)
+        row=json.loads(subprocess.check_output([str(program),str(prefix)+'.bin',str(prefix)+'.labels'],text=True))
+        for key in ('h','v','c','q','R','loss','histogram'):
+            assert row[key]==expected[key],(name,key)
+        assert construction['R']==row['baseline_R']
+        assert row['R']==row['c']+row['q']-row['matched']
+        results[name]=row
+    assert results['selected']['R']<results['pr107']['R'] and results['selected']['matched']>results['pr107']['matched']
+    return results
+
+WRITE=False
+def main():
+    global WRITE
+    parser=argparse.ArgumentParser();parser.add_argument('--arithmetic-only',action='store_true');parser.add_argument('--write',action='store_true');args=parser.parse_args()
+    WRITE=args.write
+    if not WRITE:pins()
+    receipt=exact(read(HERE/'producer.json'))
+    if not args.arithmetic_only:
+        work=ROOT/'build/dual-suffix-centers';work.mkdir(parents=True,exist_ok=True)
+        with tempfile.TemporaryDirectory(prefix='replay-',dir=work) as tmp:
+            rows=regenerate(Path(tmp))
+        receipt['parent_and_selected_producers_regenerated']=True
+    print(json.dumps(receipt,indent=2));print('PASS conditional dual-suffix rational-center refinement')
+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/research/reversed-rational-centers/README.md b/research/reversed-rational-centers/README.md
new file mode 100644
index 0000000000..38a5103bea
--- /dev/null
+++ b/research/reversed-rational-centers/README.md
@@ -0,0 +1,52 @@
+# Reversed pair-star orders on PR104
+
+Under **all of PR104's inherited and newly proposed interfaces**, this finite
+refinement gives conditional **κ = 7798412662809/10^17 = 7.798412662809×10⁻⁵**,
+about **0.04686049% above PR104** at commit
+`854ba7dca98651eab2050402384e1a7784534f0a`.
+
+For each pair `(a,b)`, sort the other coordinates by
+`((i ^ 1) in (a,b), -i)` instead of `((i ^ 1) in (a,b), i)`.
+This reverses the order within each of the two aligned groups. It preserves
+the scalar identities, rank-label rules and legal carrier matching algorithm,
+while changing the distribution of paid child widths.
+
+The h24 producer still has 44,918 roles and 8,966 matches. Its new strict
+complex saving is `7799647191/10^14`. The h23 stopped bit construction,
+ordinary leaf, fixed divisor 21, copied-center corrections and three nested
+row reserves are unchanged. The complete cost profile is regenerated; no
+endpoint gates or cleanup calls are removed.
+
+```sh
+python3 research/reversed-rational-centers/verify.py
+python3 research/reversed-rational-centers/test_controls.py
+make verify
+```
+
+The focused verifier reproduces PR104's exact certificate, rebuilds both
+the parent and selected scalar/frame producers, recompiles the inherited
+matcher, checks every histogram entry, then checks all 47 strict constraints,
+seven margins and two next-grid exclusions. Five adversarial controls reject
+unpaid cleanup, omitted retained loss, reuse of the old profile, and excessive
+complex/final exponents. Full repository verification is pending.
+
+PR104 changed `Makefile`, `README.md` and `NOTICE` without refreshing
+the inherited joint source fingerprints. This branch refreshes only those
+three pins. The affected independent h23/h25 physical replay passed;
+its derived certificate and receipt differ only in source/checksum metadata.
+No mathematical inputs or checks were changed. Full verification remains pending.
+
+This is a small finite refinement, not a proof of global optimality or a
+practical runtime claim. In particular, **the new opposite-bank factorization,
+stopped atom streaming and odd-denominator grid interfaces in PR104 remain
+written proof dependencies, not independently established theorems here**.
+See [the argument and scope](proof.md).
+
+Credit icekylinx for PR104 and its stopped product-ring/rational-center
+construction; Zhihao Chen for the semantic/bulk assembly; RaD/hipotures for
+its analytic interfaces; Aurel Prosz/Paureel and the credited two-stage
+contributors; eumemic and the retained paired-circuit contributors;
+Douglas Colkitt, OpenAI, David Harvey, Joris van der Hoeven, and all
+predecessors credited in PR104's `NOTICE` and `SOURCES.json`.
+Original notices and licenses are retained. This order refinement was
+prepared by Rohan Arun with OpenAI Codex assistance, under Apache-2.0.
diff --git a/research/reversed-rational-centers/SOURCE.json b/research/reversed-rational-centers/SOURCE.json
new file mode 100644
index 0000000000..a5cc42f611
--- /dev/null
+++ b/research/reversed-rational-centers/SOURCE.json
@@ -0,0 +1,335 @@
+{
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diff --git a/research/reversed-rational-centers/certificate.json b/research/reversed-rational-centers/certificate.json
new file mode 100644
index 0000000000..e31330c72e
--- /dev/null
+++ b/research/reversed-rational-centers/certificate.json
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+      "beta_positive": "1/1000000",
+      "beta_below_one": "999999/1000000",
+      "leaf_saving_above_bit": "1/10000000000",
+      "q_positive": "389981461768011058647191/5000000000000000000000000000",
+      "q_below_internal": "7799629391352809/5000000000000000000000000000",
+      "q_below_leaf": "507799629391352809/5000000000000000000000000000",
+      "c_positive": "38998146566782567632730158647191/500000000000000000000000000000000000",
+      "c_below_one": "499961001853433217432367269841352809/500000000000000000000000000000000000",
+      "q_below_reservations": "389981461768011058647191/500000000000000000000000000000000000",
+      "lambda_above_tau": "7799629391352809/10000000000000000000000000000",
+      "lambda_above_sigma": "1787764348491352809/10000000000000000000000000000",
+      "lambda_above_internal": "7799629391352809/10000000000000000000000000000",
+      "lambda_prime_above_lambda": "7799629391352809/10000000000000000000000000000",
+      "compact_leaf": "507799629391352809/5000000000000000000000000000",
+      "compact_reservations": "389981461768011058647191/500000000000000000000000000000000000",
+      "lambda_prime_below_one": "389981461768011058647191/5000000000000000000000000000",
+      "epsilon_positive": "499999995000000000000000000000000000/500077996292743583673497449258647191",
+      "epsilon_below_one": "78001292743583673497449258647191/500077996292743583673497449258647191",
+      "guard_width": "78001292743583673497449258647191/500077996292743583673497449258647191",
+      "K_geometry": "3900314656678257153254477632730158647191/50007799629274358367349744925864719100000000",
+      "K_dominates_log": "3899814617680110196490448231988941352809/50007799629274358367349744925864719100000000",
+      "record_suffix": "78001292743583673497449258647191/500077996292743583673497449258647191",
+      "phase_local": "62400534116870646054375733420303541352809/400062397034194866938797959406917752800000000",
+      "phase_boundary": "46799275334165034294079246502550741352809/400062397034194866938797959406917752800000000",
+      "gamma_sublinear": "19501573478382014700370608647191/500077996292743583673497449258647191",
+      "cell_above_band": "499980493426521617985299629391352809/1000155992585487167346994898517294382",
+      "prime_interval_packing": "78001292743583673497449258647191/500077996292743583673497449258647191",
+      "alpha_positive": "58499719265201658797078650000000/500077996292743583673497449258647191",
+      "alpha_below_one": "500019496573478382014700370608647191/500077996292743583673497449258647191",
+      "alpha_below_one_fourth": "499843997415682777038309134658647191/2000311985170974334693989797034588764",
+      "delta_positive": "1/800000000",
+      "delta_below_one_eighth": "99999999/800000000",
+      "short_record_fallback": "49996099076962146296086959474975986461246601315262190481/50007799629274358367349744925864719100000000000000000000",
+      "small_field_exposure": "39003146956764029400741217294382/500077996292743583673497449258647191",
+      "artificial_boundary": "2800514781032263644829400388068762325941352809/400062397034194866938797959406917752800000000",
+      "literal_scalar_guard": "400658342468135428522454357180",
+      "row_product_gap": "40024/25",
+      "original_prefix_above_kappa": "500077996773432614989091414962443455621980481/50007799629274358367349744925864719100000000000000000",
+      "coordinate_movement_above_kappa": "1216728543809/100000000000000000000",
+      "compact_phase_layer_above_kappa": "480689031315593965703796264621980481/50007799629274358367349744925864719100000000000000000",
+      "bulk_exposure_above_kappa": "1216728543809/100000000000000000000",
+      "Gaussian_arithmetic_above_kappa": "1950094838089145566120417271503646465365746980481/50007799629274358367349744925864719100000000000000000",
+      "scalar_work_above_kappa": "3900252185927347036157478136222746465365746980481/50007799629274358367349744925864719100000000000000000",
+      "dimension_above_kappa": "49996099685421318516279360511458684803796264621980481/50007799629274358367349744925864719100000000000000000"
+    },
+    "margins": {
+      "original_prefix": "3900314656678257153254477632730158647191/50007799629274358367349744925864719100000000",
+      "coordinate_movement": "7799629391352809/100000000000000000000",
+      "compact_phase_layer": "38998145786819644096708041352809/500077996292743583673497449258647191",
+      "bulk_exposure": "7799629391352809/100000000000000000000",
+      "Gaussian_arithmetic": "46799275334165034294079246502550741352809/400062397034194866938797959406917752800000000",
+      "scalar_work": "62400534116870646054375733420303541352809/400062397034194866938797959406917752800000000",
+      "dimension": "499999995000000000000000000000000000/500077996292743583673497449258647191"
+    },
+    "minimum_margin": "38998145786819644096708041352809/500077996292743583673497449258647191",
+    "absorption_gap": "480689031315593965703796264621980481/50007799629274358367349744925864719100000000000000000",
+    "recurrence": {
+      "internal": "99992200370608647191/100000000000000000000",
+      "leaf": "99992200360608647191/100000000000000000000",
+      "reservations": "499961001853433217432367269841352809/500000000000000000000000000000000000"
+    }
+  },
+  "next_grid_rejected": true,
+  "relative_improvement_over104": "3652662809/7794760000000",
+  "scope": "Candidate only; finite producer and arithmetic checks passed. No new proof of inherited all-size interfaces."
+}
diff --git a/research/reversed-rational-centers/producer.json b/research/reversed-rational-centers/producer.json
new file mode 100644
index 0000000000..d68bff1297
--- /dev/null
+++ b/research/reversed-rational-centers/producer.json
@@ -0,0 +1,41 @@
+{
+  "h": 24,
+  "v": 2024,
+  "c": 45548,
+  "q": 8120,
+  "baseline_R": 53668,
+  "matched": 8750,
+  "R": 44918,
+  "orientation_changes": 3302,
+  "rank_sum": 1079136,
+  "loss": 552,
+  "histogram": [
+    8930,
+    49042,
+    30364,
+    16020,
+    11883,
+    8272,
+    6399,
+    4356,
+    6682,
+    2263,
+    5109,
+    1808,
+    5317,
+    1733,
+    5391,
+    1757,
+    4908,
+    3290,
+    5849,
+    4315,
+    4488,
+    264,
+    564,
+    24,
+    24
+  ],
+  "central_disjoint": 24,
+  "center_denominator": 21
+}
diff --git a/research/reversed-rational-centers/producer.py b/research/reversed-rational-centers/producer.py
new file mode 100644
index 0000000000..4055781c19
--- /dev/null
+++ b/research/reversed-rational-centers/producer.py
@@ -0,0 +1,231 @@
+# Order-only adaptation by Rohan Arun with OpenAI Codex assistance.
+# Parent: icekylinx PR104 at 854ba7dca98651eab2050402384e1a7784534f0a.
+# Original copyright, Apache-2.0 notice and scalar/frame assertions retained.
+"""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/research/reversed-rational-centers/proof.md b/research/reversed-rational-centers/proof.md
new file mode 100644
index 0000000000..10acfd25c9
--- /dev/null
+++ b/research/reversed-rational-centers/proof.md
@@ -0,0 +1,53 @@
+# Why the order-only refinement composes
+
+This argument is conditional on the complete interfaces in PR104. It proves
+the compatibility of the local change, not those interfaces themselves.
+
+For a fixed pair `(a,b)`, let the leaves be the independent scalar variables
+indexed by triples `{a,b,i}` with `i` different from `a,b`. For any permutation
+of these leaves, the prefix strictly before a leaf and suffix strictly after
+it have disjoint supports. Adding them produces exactly the sum omitting
+that leaf. Thus reordering changes neither a pair-total output nor any
+leave-one-out output. The h24 disjoint-center circuit is untouched.
+
+Every intermediate prefix or suffix still has the common pair `{a,b}`.
+Its triple indicator vectors are pairwise orthogonal over F2, each with
+self-pairing one. Their span is therefore nondegenerate with dimension
+equal to the number of included leaves. An addition joins disjoint leaf
+sets, so its source frames nest in its target frame. Existing support-based
+reuse is valid because identical support specifies the same scalar form
+and the same frame in this class. The retained generator verifies these
+facts for every active node, and the exact scalar scatter identity remains
+unchanged. In particular the center decoder is still division by 21.
+
+The selected permutation sorts by pair-partner membership first and by
+descending coordinate second. The inherited matcher works on the resulting
+DAG. Every selected carrier must share the appropriate input, satisfy frame
+inclusion, respect the inherited rank/temporal ordering, and use a distinct
+matching endpoint. Its residual histogram, including output and retained
+center calls, is recomputed from that concrete matching. No optimal-weight
+matching claim is made or needed.
+
+The same copied-center transformation replaces exactly 24 width-24 cleanup
+calls by width-one calls, retaining all width-23 transforms. For `h=24`,
+`v=binomial(24,3)`, `R=44918`, let `B=vR`, `N=v²`, `m=h²`,
+`W=2N+2B`, and `L=2v h(h−1)`. The complete child list has `2B` children
+of width `m−h`, `2N` of width `(h−1)²`, `4N` of width `h−1`,
+`2v H'[r]` of each producer width `r`, and `N` additional width-one calls.
+It has total rank `Wm−N+L`. All these classes remain paid.
+
+The source-pinned verifier checks this rank identity and an exact rational
+upper bound on the complex moment at `b=7799647191/10^14`. The enclosure
+uses the inherited atanh logarithm series and rational exponential bound.
+The next point on this specified grid fails that enclosure; this is an
+arithmetic boundary for this certificate, not a global lower bound.
+
+All geometry parameters, selected dimension, scalar node counts, row-stock
+rules and denominator remain those of PR104. Its finite bridge is recomputed
+from the complete selected profile. With actual bit saving
+`803380799/10^13`, phase stop `β=10^-6`, and assembly bit parameter
+`a=min(actual_bit,(1−β)b−10^-10)`, the inherited assembly supplies 47
+strict inequalities and seven margins. The minimum margin strictly exceeds
+`κ=7798412662809/10^17`. The following point on that final grid fails the
+selected assembly margin. The source-pinned exact certificate records all
+values. This concludes only the conditional finite refinement.
diff --git a/research/reversed-rational-centers/test_controls.py b/research/reversed-rational-centers/test_controls.py
new file mode 100644
index 0000000000..347275d571
--- /dev/null
+++ b/research/reversed-rational-centers/test_controls.py
@@ -0,0 +1,29 @@
+"""Adversarial cost and strict-endpoint controls for the selected finite witness."""
+import copy, unittest
+from fractions import Fraction as Q
+import verify as v
+
+class Controls(unittest.TestCase):
+    def setUp(self): self.row=v.read(v.HERE/'producer.json')
+    def test_unpaid_cleanup_rejected(self):
+        self.row['histogram'][24]-=1
+        with self.assertRaises(ValueError):v.exact(self.row)
+    def test_omitted_retained_loss_rejected(self):
+        self.row['loss']=0
+        with self.assertRaises(ValueError):v.exact(self.row)
+    def test_parent_histogram_cannot_claim_new_bound(self):
+        row=v.read(v.ROOT/'certificates/stopped-product-complex-input.json')
+        with self.assertRaises((AssertionError,ValueError)):v.exact(row)
+    def test_next_complex_grid_rejected(self):
+        p=v.parent.profile(self.row)
+        with self.assertRaises(ValueError):
+            v.parent.moment(p['m'],p['W'],p['child_multiplicities'],Q(7799647192,10**14),True)
+    def test_next_kappa_grid_rejected(self):
+        c=v.read(v.HERE/'certificate.json');b=Q(c['complex_saving']);a=Q(c['bit_parameter'])
+        bit=v.parent.profile(v.read(v.ROOT/'certificates/stopped-product-bit-axis.json'))
+        bridge=v.parent.finite_bridge(bit,v.parent.profile(self.row),self.row,
+            v.read(v.ROOT/'certificates/copied-centers-network.json'))
+        with self.assertRaises(AssertionError):
+            v.assembly(a,b,bridge,Q(7798412662810,10**17),beta=v.parent.PHASE_STOP)
+
+if __name__=='__main__':unittest.main()
diff --git a/research/reversed-rational-centers/verify.py b/research/reversed-rational-centers/verify.py
new file mode 100644
index 0000000000..96affed54d
--- /dev/null
+++ b/research/reversed-rational-centers/verify.py
@@ -0,0 +1,83 @@
+"""Exact finite refinement of icekylinx's conditional PR104 construction.
+
+Copyright 2026 Rohan Arun, Apache-2.0. Prepared with OpenAI Codex assistance.
+Does not formally verify the inherited analytic or all-size tape interfaces.
+"""
+import argparse, hashlib, importlib.util, json, os, shlex, subprocess, sys, tempfile
+from pathlib import Path
+from fractions import Fraction as Q
+HERE=Path(__file__).resolve().parent
+ROOT=HERE.parents[1]
+sys.path.insert(0,str(ROOT/'scripts'))
+import stopped_product_network as parent
+from stopped_product.complex import build as parent_build
+from structured_bulk_assembly import assembly, js
+
+def read(p): return json.loads(p.read_text())
+def pins():
+    assert not sys.flags.optimize, 'Assertions must remain enabled'
+    for name,digest in read(HERE/'SOURCE.json')['files'].items():
+        assert hashlib.sha256((ROOT/name).read_bytes()).hexdigest()==digest,name
+
+def exact(row):
+    original=parent.certificate()
+    assert original['kappa']==Q(194869,2500000000)
+    bit=parent.profile(read(ROOT/'certificates/stopped-product-bit-axis.json'))
+    phase=parent.profile(row)
+    bridge=parent.finite_bridge(bit,phase,row,read(ROOT/'certificates/copied-centers-network.json'))
+    b=Q(7799647191,10**14)
+    cm=parent.moment(phase['m'],phase['W'],phase['child_multiplicities'],b,True)
+    a=min(parent.AB,(1-parent.PHASE_STOP)*b-Q(1,10**10))
+    k=Q(7798412662809,10**17)
+    assembled=assembly(a,b,bridge,k,beta=parent.PHASE_STOP)
+    assert len(assembled['strict_constraints'])==47 and len(assembled['margins'])==7
+    assert k>original['kappa'] and a<=parent.AB
+    # Reject the next grid point for each selected strict endpoint.
+    try: parent.moment(phase['m'],phase['W'],phase['child_multiplicities'],b+Q(1,10**14),True)
+    except ValueError as e: assert str(e)=='Characteristic moment does not contract'
+    else: raise AssertionError('Complex next grid was accepted')
+    try: assembly(a,b,bridge,k+Q(1,10**17),beta=parent.PHASE_STOP)
+    except AssertionError: pass
+    else: raise AssertionError('Kappa next grid was accepted')
+    expected=read(HERE/'certificate.json')
+    for key,value in dict(kappa=k,complex_saving=b,bit_parameter=a,actual_bit_saving=parent.AB,
+                          profile=phase,bridge=bridge,assembly=assembled).items():
+        assert js(value)==expected[key],key
+    return dict(kappa=str(k),complex_moment_gap=str(cm['strict_gap']),
+                constraints=47,margins=7,next_grid_controls=2)
+
+def regenerate(work):
+    program=work/'matcher'
+    subprocess.run([*shlex.split(os.environ.get('CXX','c++')),'-O3','-std=c++17',
+        str(ROOT/'scripts/endpoint_gauge/match_complex_general.cpp'),'-o',str(program)],check=True)
+    spec=importlib.util.spec_from_file_location('selected_rational_producer',HERE/'producer.py')
+    module=importlib.util.module_from_spec(spec);spec.loader.exec_module(module)
+    results={}
+    for name,builder,expected in (
+        ('parent',parent_build,read(ROOT/'certificates/stopped-product-complex-input.json')),
+        ('selected',module.build,read(HERE/'producer.json'))):
+        prefix=work/name
+        construction=builder(24,prefix,24)
+        row=json.loads(subprocess.check_output([str(program),str(prefix)+'.bin',str(prefix)+'.labels'],text=True))
+        for key in ('h','v','c','q','R','loss','histogram'):
+            assert row[key]==expected[key],(name,key)
+        assert construction['R']==row['baseline_R']
+        assert row['R']==row['c']+row['q']-row['matched']
+        results[name]=row
+    # Pin the sole mathematical source edit relative to PR104's generator.
+    selected=(HERE/'producer.py').read_text().splitlines(keepends=True)[3:]
+    old=(ROOT/'scripts/stopped_product/complex.py').read_text()
+    assert ''.join(selected)==old.replace('others.sort(key=lambda i:((i^1) in (a,b),i))',
+                                          'others.sort(key=lambda i:((i^1) in (a,b),-i))')
+    return results
+
+def main():
+    parser=argparse.ArgumentParser();parser.add_argument('--arithmetic-only',action='store_true');args=parser.parse_args()
+    pins();receipt=exact(read(HERE/'producer.json'))
+    if not args.arithmetic_only:
+        work=ROOT/'build/reversed-rational-centers';work.mkdir(parents=True,exist_ok=True)
+        with tempfile.TemporaryDirectory(prefix='replay-',dir=work) as tmp:
+            rows=regenerate(Path(tmp))
+        receipt['parent_and_selected_producers_regenerated']=True
+    print(json.dumps(receipt,indent=2));print('PASS conditional reversed rational-center refinement')
+if __name__=='__main__': main()
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()