diff --git a/research/jp8-transposed-coefficient-work/CERTIFICATE.json b/research/jp8-transposed-coefficient-work/CERTIFICATE.json
new file mode 100644
index 0000000000..3490305d02
--- /dev/null
+++ b/research/jp8-transposed-coefficient-work/CERTIFICATE.json
@@ -0,0 +1,1098 @@
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+    {"start":27613,"stop":27615,"common_backward_target":355,"forward_routes":[["shift",-7]],"columns":4,"forward_primitive_sha256":"de73a723836190a786a662afb98077b7bc93579db6d3e60e6f944332279fec62","backward_primitives":[["clear",724],["sub",724,542],["halve",724],["halve",724],["halve",724],["halve",724],["halve",724],["halve",724],["halve",724],["sub",724,541],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["sub",355,724],["clear",724]],"backward_primitive_sha256":"87c8e4af6f06ba3942b25d81ff1c5cdeb1f5d5794b1a50836414351eaa73be8f","real_primitives_before":94,"real_primitives_after":70,"isolated_original_prefix":"8454145","isolated_companion_prefix":"8454145"},
+    {"start":27620,"stop":27626,"common_backward_target":355,"forward_routes":[["shift",-3],["shift",2],["shift",-2],["shift",0],["shift",-2]],"columns":8,"forward_primitive_sha256":"05a25faa6a55fb9ecb06aa0de1d34f468864e45ed9b161c4e4cfb4e2f1ded7a1","backward_primitives":[["clear",724],["add",724,561],["halve",724],["halve",724],["sub",724,560],["sub",724,559],["halve",724],["halve",724],["sub",724,555],["double",724],["double",724],["add",724,554],["halve",724],["halve",724],["halve",724],["add",724,552],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["sub",355,724],["clear",724]],"backward_primitive_sha256":"29386c867fa1a3b3d94d5eeba756af3b7729678fe0b3eb962466a58157c05b50","real_primitives_before":246,"real_primitives_after":74,"isolated_original_prefix":"999425","isolated_companion_prefix":"999425"},
+    {"start":27627,"stop":27629,"common_backward_target":355,"forward_routes":[["shift",11]],"columns":4,"forward_primitive_sha256":"a554ce90b3c4268b9bd13cd195ce420e0622d48777638b00bd8f49edcc1d8acc","backward_primitives":[["clear",724],["sub",724,604],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["sub",724,573],["double",724],["double",724],["double",724],["double",724],["sub",355,724],["clear",724]],"backward_primitive_sha256":"256daa86e440bff9cf644faab3565ecd98a87b345be4cb4be753fa5aa527013f","real_primitives_before":54,"real_primitives_after":40,"isolated_original_prefix":"32785","isolated_companion_prefix":"32785"},
+    {"start":27629,"stop":27631,"common_backward_target":355,"forward_routes":[["read",-12,12]],"columns":4,"forward_primitive_sha256":"2538f74be0848c51536048087c794efe47d4fe2ecd42081e6d1b137a6c9b3131","backward_primitives":[["clear",724],["sub",724,608],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["sub",355,724],["halve",724],["halve",724],["halve",724],["halve",724],["halve",724],["halve",724],["halve",724],["halve",724],["halve",724],["halve",724],["halve",724],["halve",724],["sub",724,606],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["sub",355,724],["double",724],["double",724],["double",724],["double",724],["double",724],["sub",355,724],["double",724],["sub",355,724],["double",724],["sub",355,724],["double",724],["double",724],["sub",355,724],["double",724],["sub",355,724],["double",724],["sub",355,724],["double",724],["sub",355,724],["clear",724]],"backward_primitive_sha256":"01a7eff9663b717915c752fd449e3e0e7edd128b304c45749ee15a62fe437a30","real_primitives_before":114,"real_primitives_after":110,"isolated_original_prefix":"1015937","isolated_companion_prefix":"1015937"},
+    {"start":27636,"stop":27638,"common_backward_target":355,"forward_routes":[["read",-3,12]],"columns":4,"forward_primitive_sha256":"ec306b54e6f606c4dc899babf65a4805d73431505bf4a4785c5a97190a8b20d3","backward_primitives":[["clear",724],["add",724,618],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["sub",355,724],["halve",724],["halve",724],["halve",724],["sub",724,617],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["sub",355,724],["clear",724]],"backward_primitive_sha256":"5e13a3808042a2d9f83b86d1eadfc67df992c1cb0676f17d132158637e05203e","real_primitives_before":84,"real_primitives_after":66,"isolated_original_prefix":"2105345","isolated_companion_prefix":"2105345"},
+    {"start":27644,"stop":27648,"common_backward_target":355,"forward_routes":[["shift",8],["read",0,-1],["read",0,-1]],"columns":6,"forward_primitive_sha256":"94d58349a22c8ec1948ff2bd75aaa4451ed8f16b5234e394f18ca63c27c8f77f","backward_primitives":[["clear",724],["sub",724,632],["halve",724],["sub",355,724],["add",724,630],["halve",724],["sub",355,724],["sub",724,628],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["sub",724,627],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["sub",355,724],["clear",724]],"backward_primitive_sha256":"dfb687572038245f63ca4b072b86ecc4bf60cf108101eda11c8df28d94d6ff0b","real_primitives_before":204,"real_primitives_after":66,"isolated_original_prefix":"29425673/4","isolated_companion_prefix":"29425673/4"},
+    {"start":27649,"stop":27651,"common_backward_target":355,"forward_routes":[["shift",5]],"columns":4,"forward_primitive_sha256":"cea877ca9316b388f9a666d0e3803a7754928b3e7fa57de83c0f44a969abf2e2","backward_primitives":[["clear",724],["sub",724,636],["double",724],["double",724],["double",724],["double",724],["double",724],["sub",724,635],["halve",724],["halve",724],["sub",355,724],["clear",724]],"backward_primitive_sha256":"31dc191e1c5baa456222cdb7378d764de17a25169d30ca7c57c760e2372b61f2","real_primitives_before":26,"real_primitives_after":24,"isolated_original_prefix":"37/4","isolated_companion_prefix":"33"},
+    {"start":27654,"stop":27662,"common_backward_target":355,"forward_routes":[["shift",-1],["shift",0],["shift",-9],["shift",-1],["shift",0],["shift",0],["shift",7]],"columns":10,"forward_primitive_sha256":"95f05d1227aa3924fdd3ca96054d8b691590180c51fd4144fdd2c30aed0d1c5e","backward_primitives":[["clear",724],["add",724,668],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["add",724,666],["add",724,664],["add",724,662],["halve",724],["sub",724,660],["halve",724],["halve",724],["halve",724],["halve",724],["halve",724],["halve",724],["halve",724],["halve",724],["halve",724],["sub",724,659],["sub",724,656],["halve",724],["sub",724,654],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["sub",355,724],["clear",724]],"backward_primitive_sha256":"06c79acec6eb3ef162a82a560f4b8166bd2e21d071fddcfec30a5fa2f316b4b6","real_primitives_before":238,"real_primitives_after":92,"isolated_original_prefix":"270657","isolated_companion_prefix":"270657"},
+    {"start":27662,"stop":27665,"common_backward_target":355,"forward_routes":[["shift",0],["shift",0]],"columns":5,"forward_primitive_sha256":"67c6c4ec065d0e4ee47ff793ef08d1e7fe11ba712a5655ef9d2deedf0fac0c8d","backward_primitives":[["clear",724],["add",724,674],["add",724,672],["add",724,670],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["double",724],["sub",355,724],["double",724],["sub",355,724],["double",724],["sub",355,724],["double",724],["sub",355,724],["double",724],["sub",355,724],["double",724],["sub",355,724],["double",724],["sub",355,724],["double",724],["sub",355,724],["clear",724]],"backward_primitive_sha256":"369a7724bac5f4cff9cc672bf168bb69ba68874bf7876dfa696612c48b4c7e2b","real_primitives_before":180,"real_primitives_after":64,"isolated_original_prefix":"3133441","isolated_companion_prefix":"3133441"}
+  ]
+}
diff --git a/research/jp8-transposed-coefficient-work/CLAIM.md b/research/jp8-transposed-coefficient-work/CLAIM.md
new file mode 100644
index 0000000000..d19e154027
--- /dev/null
+++ b/research/jp8-transposed-coefficient-work/CLAIM.md
@@ -0,0 +1,63 @@
+# Local sprint claim: transposed private work
+
+- Track: complex supplier; fresh OS-random seed **25** (`secrets.randbelow(100)`).
+- Exact base: `c290b7b919790300e14004a75d8d24b55ca3c5b5`.
+- Current accepted result: #741/#589, conditional kappa
+  `1661202643272239/1000000000000000000`; read its certificate, construction
+  and maintainer review. This experiment deliberately uses historical #626.
+- Exact object: the inverse-transpose companion of the fixed-source JP8
+  cleanup blocks constructed in #811, including varying magnitudes and
+  positive numerator-changing source reads.
+- Method: transpose each literal private-work circuit in reverse order,
+  reverse the sign of updates into the original source, and check all dirty
+  native/private inputs, complete prefix envelopes and all costs. Compare
+  with #751's equal-coefficient reverse-Horner result. No larger search.
+- Deliverable: standalone pinned companion construction, finite exact
+  certificate, independent block check, negative controls and cost note.
+- Expected finish: 06:45 UTC on 2026-10-13; handoff before 06:54 UTC.
+
+Three candidates: (0) CK numerator-changing forward reuse; (1) JP8
+inverse-transpose continuation; (2) transport coefficient reuse into #741.
+Seed `25 % 3 = 1` selects the second. Candidate 0 repeats the recent forward
+mechanism; candidate 2 requires a wider interface audit. This claim concerns
+the backward dual, not another forward coefficient search.
+
+Checked open metadata through #811, #751 at
+`7a1a3a319da359ec8b6e83230a7fcf7733fac8e5`, #739, #793 and the review ledger.
+#751 handles 999 equal-magnitude fan-in runs; #739/#811 report no backward
+same-source join. This extends the backward circuit class to the transposes
+of their varying-magnitude continuations. #793 changes compensation order.
+There is no pinned sprint issue; the issue list only contains #650 about
+Discussions. This file is the local coordination fallback. #811 will be
+pinned at its completed historical-assembly update, with full attribution.
+
+Contributor: Ryan Haines (@Ryan-Haines), with substantial OpenAI Codex
+assistance. Apache-2.0. Result and handoff will be recorded once after checks.
+
+## Result and handoff
+
+Completed: [PR #830](https://github.com/CrocSwap/integer-mult-bounds/pull/830).
+All 1,091 backward companions pass; complete backward count is 224,470,
+saving 32,810 real primitives and 13,122 beyond #751's equal-coefficient
+result. Paired counts are 224,422/224,470. Exact complete prefixes remain
+64/101 bits, while 377 isolated prefixes grow (worst ratio 40/9), explicitly
+recorded as a limit on blanket transposition arguments. No new kappa.
+
+Literal replay covers 725 arbitrary columns in both orientations; the
+independent integer checker covers 6,776 block input columns and the full
+backward prefix recurrence. Windows/Linux, 10+4+25 mutation controls, five
+optimized/relocated/recompressed packaging tests, Ruff, Git-byte source pins
+and current selected-record integrity pass. README.md gives the commands.
+
+Final overlap refresh: 638 open PRs through #825 at 06:33 UTC; no new
+transposed-coefficient/fan-in claim was present. #816 is a CK target forest;
+#821 audits a global prefix estimate, a different object. #811 and #751
+remain at the pinned heads. No pinned coordination issue exists. Main
+advanced to `7ee02b7c96d85f22b38d28bd7ed49bb8c1aeb261`; all six consumed
+input files are unchanged, and the selected #741 frontier remains separate.
+
+Initial research commit: `0a5e2034c0d537c05a2cc3bb45f5d35ae0e3a668`.
+This is the single final claim update. The local workspace also retains
+`jp8-transposed-s25.patch` and `jp8-transposed-HANDOFF.txt`, with the final
+commit, patch hash and reproduction command, as a submission-independent
+handoff. No issue was opened merely to reserve this scope.
diff --git a/research/jp8-transposed-coefficient-work/EXPECTED.json b/research/jp8-transposed-coefficient-work/EXPECTED.json
new file mode 100644
index 0000000000..bc901c9195
--- /dev/null
+++ b/research/jp8-transposed-coefficient-work/EXPECTED.json
@@ -0,0 +1,90 @@
+{
+  "schema": "jp8-transposed-private-work-replay/1",
+  "base_commit": "c290b7b919790300e14004a75d8d24b55ca3c5b5",
+  "negative_controls": [
+    "initial_clear",
+    "final_clear",
+    "missed_inverse_sign",
+    "omitted_target_write",
+    "forward_instead_of_reverse_order",
+    "inverted_scale",
+    "endpoint_neutral_excess_growth",
+    "frame_crossing",
+    "wrong_macro_dual",
+    "forward_source_write"
+  ],
+  "orientations": {
+    "forward": {
+      "columns": 725,
+      "real_primitives": 224422,
+      "prefix_l1": "4536094588619580433782972610869617/281474976710656",
+      "prefix_bits": 64,
+      "non_scalar_sha256": "1fcf711555d365968ff8b64172090960cf98559865bd858c521c2adc7b12f792",
+      "paid_profile": {
+        "1": 2512,
+        "2": 400,
+        "3": 448,
+        "4": 160,
+        "5": 70,
+        "6": 8,
+        "8": 8
+      },
+      "original_real_primitives": 257232,
+      "saved_real_primitives": 32810
+    },
+    "backward": {
+      "columns": 725,
+      "real_primitives": 224470,
+      "prefix_l1": "16753412757380464429924108070201913461/8388608",
+      "prefix_bits": 101,
+      "non_scalar_sha256": "299442a1d68206eb7f49f0c9789cc8fa410a0615cff9c1900c54b064e901aa21",
+      "paid_profile": {
+        "1": 2512,
+        "2": 400,
+        "3": 448,
+        "4": 160,
+        "5": 70,
+        "6": 8,
+        "8": 8
+      },
+      "original_real_primitives": 257280,
+      "saved_real_primitives": 32810
+    }
+  },
+  "certificate_sha256": "8b4a9f4502acc772ebb18e1fd84fba9cdaf479b9b2759a5cd921515bc1f3a171",
+  "blocks": 1091,
+  "block_input_columns": 6776,
+  "unintegrated_budget": {
+    "adapter_allowances": 437040,
+    "compiler_charge": 25934336,
+    "original_local_bound": 11334342462720,
+    "paired_candidate_local_bound": 11334342429910,
+    "backward_only_local_bound": 11334342462672,
+    "new_kappa_claimed": false
+  },
+  "comparators": {
+    "equal_magnitude": {
+      "blocks": 999,
+      "backward_saved": 19688,
+      "backward_primitives": 237592
+    },
+    "power_only": {
+      "blocks": 1015,
+      "backward_saved": 30026,
+      "backward_primitives": 227254
+    }
+  },
+  "prefix_scope": {
+    "isolated_growth_blocks": 377,
+    "worst_start": 24875,
+    "worst_ratio": "40/9",
+    "old_isolated_prefix": "9/2",
+    "new_isolated_prefix": "20"
+  },
+  "prefix_nonmonotonicity_witness": {
+    "sources": 4,
+    "coefficient": "-1/4",
+    "old_prefix": "2",
+    "transposed_prefix": "4"
+  }
+}
diff --git a/research/jp8-transposed-coefficient-work/INDEPENDENT.json b/research/jp8-transposed-coefficient-work/INDEPENDENT.json
new file mode 100644
index 0000000000..bd303d2d7d
--- /dev/null
+++ b/research/jp8-transposed-coefficient-work/INDEPENDENT.json
@@ -0,0 +1,18 @@
+{
+  "method": "independent integer block matrices and dyadic full-prefix recurrence",
+  "certificate_sha256": "8b4a9f4502acc772ebb18e1fd84fba9cdaf479b9b2759a5cd921515bc1f3a171",
+  "blocks": 1091,
+  "block_input_columns": 6776,
+  "real_saved": 32810,
+  "isolated_prefix_growth_blocks": 377,
+  "complete_backward": {
+    "real_primitives": 224470,
+    "prefix_l1": "16753412757380464429924108070201913461/8388608"
+  },
+  "negative_controls": [
+    "initial_clear",
+    "final_clear",
+    "inverse_sign",
+    "neutral_growth"
+  ]
+}
diff --git a/research/jp8-transposed-coefficient-work/NATIVE.json b/research/jp8-transposed-coefficient-work/NATIVE.json
new file mode 100644
index 0000000000..12d61cb9c7
--- /dev/null
+++ b/research/jp8-transposed-coefficient-work/NATIVE.json
@@ -0,0 +1,71 @@
+{
+  "status": "PASS_INDEPENDENT_LOCAL_SCALAR_FRAME_AND_WRAPPER_REPLAY",
+  "v": 112,
+  "helpers": 492,
+  "local": {
+    "forward": {
+      "all_exact_columns": 716,
+      "prefix_l1_upper": "4536094588619580433782972610869617/281474976710656",
+      "prefix_bits_upper": 64,
+      "charged_payload_bits": 192,
+      "private_work_erased": true
+    },
+    "backward": {
+      "all_exact_columns": 716,
+      "prefix_l1_upper": "16753412757380464429924108070201913461/8388608",
+      "prefix_bits_upper": 101,
+      "charged_payload_bits": 192,
+      "private_work_erased": true
+    }
+  },
+  "paid_profiles": {
+    "forward": {
+      "1": 2512,
+      "4": 160,
+      "8": 8,
+      "3": 448,
+      "2": 400,
+      "5": 70,
+      "6": 8
+    },
+    "backward": {
+      "1": 2512,
+      "4": 160,
+      "8": 8,
+      "3": 448,
+      "2": 400,
+      "5": 70,
+      "6": 8
+    }
+  },
+  "entering_helpers": 204,
+  "cold_hyperplane_ended_helpers": 22,
+  "negative_controls_rejected": [
+    "wrong_scale",
+    "missing_center",
+    "wrong_common_frame",
+    "wrong_child_rank",
+    "missing_pre_wrapper",
+    "missing_post_wrapper",
+    "wrapper_with_wrong_frame",
+    "wrapper_in_wrong_phase",
+    "wrapper_on_a_data_register",
+    "wrapper_on_a_target_register",
+    "wrapper_on_a_centre_register",
+    "hpe_pre_wrapper_resolved_to_full_recipe",
+    "hpe_post_wrapper_without_terminal_frame",
+    "hpe_post_wrapper_wrong_terminal_frame",
+    "hpe_tail_gate_at_full_frame",
+    "hpe_tail_climb_dropped",
+    "hpe_helper_exit_adapter_to_full",
+    "hpe_tail_gate_moved_into_full_cleanup",
+    "cold_hpe_pre_wrapper_replaced_by_identity_entry_adapter",
+    "cold_hpe_pre_wrapper_dropped",
+    "cold_hpe_post_wrapper_replaced_by_exit_adapter",
+    "cold_hpe_pre_wrapper_resolved_to_cold_full_convention",
+    "cold_hpe_wrapper_with_nonzero_entrance",
+    "cold_hpe_post_wrapper_wrong_terminal_frame",
+    "cold_hpe_premix_gate_before_pre_wrapper"
+  ],
+  "scope": "Literal forward and inverse-transpose replay on every arbitrary data/helper column, physical frame incidence, ordered paid wrappers, cleared private work and prefix coefficients. Global hosting and outer exponent are separate."
+}
diff --git a/research/jp8-transposed-coefficient-work/PROOF.md b/research/jp8-transposed-coefficient-work/PROOF.md
new file mode 100644
index 0000000000..1f218ff5be
--- /dev/null
+++ b/research/jp8-transposed-coefficient-work/PROOF.md
@@ -0,0 +1,121 @@
+# Transposing a cleared work stream
+
+## Endpoint identity
+
+Take a fixed-source block `y_j += q_j x`, with `x` distinct from every
+target. Targets may repeat. Its live-register matrix is `F = I + B`, where
+
+```text
+B = sum_j q_j e_(y_j) e_x^T.
+```
+
+Since no `y_j` is `x`, `B^2=0`; hence `F^(-T)=I-B^T`.
+The backward block is therefore `x -= sum_j q_j y_j`. Its source registers
+are unchanged, so the target updates commute within this block. No order
+change across block boundaries is used.
+
+The forward private-work program begins and ends with `clear(w)`. The
+pinned constructor proves its live map is `F` and its final work is zero
+for arbitrary incoming work. On the enlarged live-plus-work vector its
+matrix is consequently `P = diag(F,0)`.
+
+If the original primitive execution is `E_1,...,E_m`, then
+`P = E_m ... E_1`. Transposing gives `P^T = E_1^T ... E_m^T`, implemented
+by executing the transposed primitives in **reverse order**. Clear is a
+self-transpose coordinate projection. Private doubling and halving are
+self-transpose diagonal maps. A unit transvection transposes by swapping
+source and destination, retaining its sign.
+
+Let `D` change only the live coordinate `x` to `-x`. Then
+`D P^T D = diag(I-B^T,0)`, since `D B^T D = -B^T`.
+The original program never writes `x`, so its transpose never reads `x`
+as the source of an off-diagonal update. The conjugation is implemented
+without additional negation gates: simply reverse the sign of every
+transposed update **into `x`**. Other transposed update signs stay unchanged.
+
+Thus the concrete compiler is:
+
+1. Reverse the complete primitive list, including both clears.
+2. Keep private clears/doublings/halvings as written.
+3. Swap source and target of each signed unit update.
+4. Reverse that update's sign exactly when its new target is `x`.
+
+This realizes the inverse-transpose live block while killing arbitrary
+incoming work and returning it to zero. Each old primitive contributes one
+new primitive, so the count is identical, including all source reads and
+both clears. This algebraic argument is local to the stated register model;
+it does not establish the inherited all-size multiplication theorem.
+
+## Interfaces and complete costs
+
+All affected original events are consecutive scalar additions in the
+historical #626 JP8 `full_frame_cleanup`, at frame 1. The candidate generator
+checks each backward macro against the exact signed scalar dual of its
+forward counterpart. No source is written inside a forward block; no source
+is written inside the backward fan-in except its common target. No child,
+adapter, wrapper, frame change or private-center event lies inside a block.
+All endpoints used by the block have native terminal frame 1.
+
+The shared private stream is cleared before the next external event. Source
+and target registers agree at block boundaries for arbitrary dirty inputs.
+Non-scalar event streams, helper counts and recursive profiles stay exactly
+as pinned. The new schedule needs no extra register or restoration pass.
+
+The construction maps all 1,091 forward blocks, not a selected successful
+subset. Their total saving is 32,810 real-component primitives in each
+orientation. The backward equal-magnitude and power-only comparators save
+19,688 and 30,026 respectively. Including every unchanged scalar primitive
+gives 224,422 forward and 224,470 backward. The original adapter/compiler
+allowances are retained in the complete unintegrated budget in EXPECTED.json.
+No stronger kappa is inferred from this finite constant saving.
+
+## Prefix growth is a separate obligation
+
+Transposition preserves the endpoint and count; it does not in general
+preserve the positive prefix envelope. For a simple counterexample, let
+four independent source registers each have envelope 1 and the target have
+envelope 1. The original backward block performs four additions with
+coefficient `-1/4`. Separate preparation has envelope at most 1 and the
+target ends at envelope 2, so the maximum is 2. The transposed shared circuit
+first accumulates four sources in `w`, reaching envelope 4, then halves twice,
+subtracts into the target and clears `w`. Its endpoint is the same but its
+maximum is 4. The verifier executes this witness explicitly.
+
+The concrete corpus contains 377 such isolated growth cases. At event span
+`[24875,24886)`, common target 336 receives eleven contributions with
+magnitudes `1/4,1/4,1/2,1/8,1/4,1/8,1/4,1/2,1/2,1/4,1/2`.
+The original isolated maximum is `9/2`; the companion's maximum is `20`.
+The certificate records the literal signed circuit and both exact values.
+This is the largest isolated ratio in the corpus, `40/9`.
+
+For the complete program, propagate positive envelopes independently of
+signed matrix cancellations: additions and subtractions both add source
+envelopes; scaling uses the absolute multiplier; clear sets zero. The
+Fraction-based literal replay and the independent integer/dyadic recurrence
+agree on the exact backward value
+
+```text
+16753412757380464429924108070201913461 / 8388608.
+```
+
+It is precisely the original 101-bit bound, below `2^192`. The paired
+forward construction retains its original 64-bit bound. The finite full
+replay, rather than an invalid blanket prefix-invariance rule, discharges
+this obligation for these pinned programs.
+
+## Independent finite checks
+
+The second checker consumes literal backward circuits and original macros,
+without importing the transpose compiler or the forward constructor. Each
+block is executed on independent integer columns at scale `2^d`, where `d`
+is at least its total number of halving instructions and every reference
+coefficient denominator exponent. All intermediate columns are therefore
+integral at that scale; each division is checked for exactness. This checks
+6,776 block input columns, including dirty private work.
+
+Its complete-program prefix checker represents positive dyadics as reduced
+integer/exponent pairs and expands the original macros separately. It checks
+literal costs and the exact full backward prefix, with no Fraction arithmetic.
+Mutation controls include missing clears, inverse-sign errors and circuits
+with the same endpoint but excessive intermediate growth. These are scoped
+finite checks, not a formalization of the general multiplication theorem.
diff --git a/research/jp8-transposed-coefficient-work/README.md b/research/jp8-transposed-coefficient-work/README.md
new file mode 100644
index 0000000000..505c027893
--- /dev/null
+++ b/research/jp8-transposed-coefficient-work/README.md
@@ -0,0 +1,158 @@
+# Inverse-transpose JP8 coefficient work
+
+**Finite historical #626 circuit construction: 32,810 fewer backward
+real-component primitives; no new kappa.**
+
+Transposing the cleared private-work circuits from
+[#811](https://github.com/CrocSwap/integer-mult-bounds/pull/811) supplies their
+backward companions, including varying magnitudes and numerator-changing
+source reads. The complete backward count falls **257,280 -> 224,470**.
+This is **13,122 beyond #751's equal-coefficient reverse-Horner saving**.
+Every read, accumulation, shift and both boundary clears is charged.
+
+| Historical #626 scalar program | Forward | Backward |
+|---|---:|---:|
+| Original | 257,232 | 257,280 |
+| #811 forward construction alone | 224,422 | 257,280 |
+| #811 plus this companion | **224,422** | **224,470** |
+
+The backward comparator counts are independently reconstructed: 237,592
+for 999 equal-magnitude blocks, 227,254 for 1,015 power-only blocks, and
+224,470 for this 1,091-block class. These are arithmetic operation counts,
+not timing claims. No new work register, helper, frame change, restoration
+operation or recursive child is omitted from the count.
+
+The current accepted construction at this base is **#741/#589**, with
+conditional kappa **0.001661202643272239**. This deliberately scoped result
+uses the historical #626 JP8 word. It neither changes the selected frontier
+nor asserts transport to #741.
+
+## Reproduce
+
+Python 3.11+ and the standard library suffice, offline, from the repository
+root:
+
+```sh
+python3 -I -B research/jp8-transposed-coefficient-work/verify.py
+```
+
+Expected final line:
+
+```text
+PASS_JP8_TRANSPOSED_WORK backward_saved=32810 blocks=1091 columns=725 controls=10
+```
+
+The command verifies source/dependency pins, reconstructs the literal
+companion circuits, checks all block boundaries on arbitrary inputs, and
+executes every scalar primitive in both complete candidate programs. It
+checks both shear endpoints on **716 native plus nine arbitrary private
+columns**, all 492 dirty helpers, private-work erasure, costs and unchanged
+paid profiles. Ten controls catch missing clears, an incorrect inverse
+sign, wrong order/scaling, omitted writes, frame crossing and excessive
+endpoint-neutral growth. Ordinary verification is read-only.
+
+The independent integer implementation imports neither the constructor nor
+the main verifier:
+
+```sh
+python3 -I -B research/jp8-transposed-coefficient-work/independent.py
+```
+
+Expected: `PASS_INDEPENDENT_TRANSPOSED_WORK blocks=1091 columns=6776
+saved=32810 controls=4`. It executes every committed backward block matrix
+on an exact integer grid, checking **6,776 arbitrary block input columns**,
+then replays the complete backward positive prefix recurrence with a separate
+integer/dyadic representation. It also checks literal counts and four
+endpoint/clearing/prefix mutations.
+
+The unchanged original macro frame/wrapper contract has its own scratch
+replay:
+
+```sh
+python3 -I -B research/jp8-transposed-coefficient-work/native_replay.py --output NEW_DIRECTORY
+python3 -I -B research/jp8-transposed-coefficient-work/test_package.py
+```
+
+The first prints `PASS_UNCHANGED_JP8_NATIVE_CONTRACT columns=716 controls=25`.
+It checks the pinned original macro word, not an unimplemented production
+primitive-contract change. The five packaging tests check optimized Python,
+relocated minimal sources, equivalent recompression, read-only execution,
+source/vendor corruption, a wrong companion sign and an undercharged receipt.
+
+`verify.py --write NEW_DIRECTORY` explicitly generates CERTIFICATE.json and
+EXPECTED.json. `independent.py --write NEW_FILE` generates its receipt.
+Existing outputs are not overwritten.
+
+## Prefix caveat and full cost
+
+Transposition does **not** preserve every isolated prefix bound. Of the
+1,091 blocks, **377 have larger isolated positive prefixes**. Block
+`[24875,24886)` grows from `9/2` to `20`, a factor **40/9**. A four-source
+quarter-coefficient example grows from 2 to 4. These are concrete finite
+counterexamples to a blanket nonincrease claim, not failures of the final
+program. [PROOF.md](PROOF.md) explains the distinction.
+
+Both complete candidate programs retain the exact inherited prefixes:
+**64 bits forward and 101 bits backward**, within the existing 192-bit
+allowance. The independent dyadic replay confirms the complete backward
+value. This finite check is necessary; the endpoint-transposition identity
+alone would not establish the prefix contract.
+
+Keeping all **437,040 adapter allowances**, the **25,934,336 compiler charge**,
+all original macros and the complete paid recursive profile, the optimistic
+paired local invoice changes **11,334,342,462,720 -> 11,334,342,429,910**.
+The backward-only substitution changes that invoice by just **48**, since
+the original forward count then binds. These are explicit unintegrated
+budget calculations. They do not constitute revised global admission or
+an exponent improvement. The paid rank profile and all source geometry are
+unchanged.
+
+## Scope, sources and credit
+
+Exact base: `c290b7b919790300e14004a75d8d24b55ca3c5b5`.
+OS-random assignment seed **25** chose the complex track and, by `25 % 3`,
+this mechanism among three candidates in [CLAIM.md](CLAIM.md). This is a
+deterministic transformation of all 1,091 blocks from a pinned construction,
+not a new coefficient search or a claim of general circuit optimality.
+Its stopping rule is complete block and full-program replay.
+
+The pending #811 dependency is pinned at
+`b5c244c916dd0b30957332edad7ae7ae22b0eb9d`. Its constructor is retained
+verbatim in [vendor/construct_seed31.py](vendor/construct_seed31.py), with
+its SHA-256 in [SOURCES.json](SOURCES.json). No GitHub fetch or sibling PR
+checkout is required. The full forward route search was completed in #811;
+this package checks the new backward companion and its composition with
+that forward construction.
+
+Direct predecessors:
+
+- Ryan Haines's #811 positive one-read forward circuits; Jeffrey Cox
+  (@namredips)'s #739 power-of-two reuse; @ellipsi2's #722 equal-magnitude
+  forward reuse.
+- Sagar Thakrar (@thakrarsagar)'s
+  [#751](https://github.com/CrocSwap/integer-mult-bounds/pull/751), pinned at
+  `7a1a3a319da359ec8b6e83230a7fcf7733fac8e5`, supplies the equal-coefficient
+  backward reverse-Horner predecessor. The new contribution covers varying
+  magnitudes, one-read transitions and their exact complete prefix audit.
+- Jeremy Kothe's #622/#623 providers and Chafik Boukhalfa's #626 owner
+  allocation, retained emitter/auditor and PR514/PR433 lineage, icekylinx,
+  DreamingOfClouds, Rohan Arun, eumemic and other credited predecessors.
+
+#793's compensation permutation, #807's restored CK helper bases and
+#816's CK target forests change different mechanisms. Their savings are not
+combined here. The current #741 construction/review was read before choosing
+the historical object. No pending code besides the explicitly pinned #811
+constructor is imported.
+
+The coefficient-work clearing interface changes from one macro to one
+block. Production-emitter integration, revised typed contracts, global
+invoices/admission and inherited all-size/compiler, common-label/mutable-
+basis, whole-call hosting, precision/tape/recovery and analytic assumptions
+remain separate. Finite exact replay is not a formal proof of the full
+integer-multiplication theorem.
+
+Contributor: **Ryan Haines (@Ryan-Haines)**, with substantial **OpenAI Codex**
+assistance in research, implementation, verification and writing. No
+independent human review is claimed. New and vendored material retain the
+repository's **Apache-2.0** license. See [VALIDATION.md](VALIDATION.md) for
+completed checks and explicit limits.
diff --git a/research/jp8-transposed-coefficient-work/SOURCES.json b/research/jp8-transposed-coefficient-work/SOURCES.json
new file mode 100644
index 0000000000..a9d6a0b73a
--- /dev/null
+++ b/research/jp8-transposed-coefficient-work/SOURCES.json
@@ -0,0 +1,16 @@
+{
+  "base_commit": "c290b7b919790300e14004a75d8d24b55ca3c5b5",
+  "forward_pr811": "b5c244c916dd0b30957332edad7ae7ae22b0eb9d",
+  "equal_fanin_pr751": "7a1a3a319da359ec8b6e83230a7fcf7733fac8e5",
+  "historical_pr626": "660de893d27bb7abd1d5b01b1640207528a5c851",
+  "current_selected_pr741": "e516b1463606840a15f80aac1995d64ce0092e89",
+  "vendor_sha256": "43b0fd819a8f8c02189aaa13c39b1c55f960c8b3c90cf4b8672499ead3cea3f0",
+  "payload_sha256": {
+    "research/community-k492-audit/pr626/replay/mixed-global/base/providers/jp8/CANONICAL-WORD.json.gz": "fb4a8d14ed2a8048698bea6ae16f86163a1aa6c80f2eb622222eee5cdbce9b3b",
+    "research/community-k492-audit/pr626/replay/mixed-global/base/providers/jp8/LOCAL-PROGRAMS.json.gz": "efe45ce7fb025a6f53cc9eb54e951ac2211a5ed927e689a049e4f68ca680654b",
+    "research/complex-k492-owner-repair/data/SPEC.json": "2a2577ac526535826fed9d5f7f83f6e2e0ec058a146c64f7286720b995fa2323",
+    "research/complex-k492-owner-repair/code/assemble.py": "1b1a9024ac910dc9b9d9e30150684e1fa9cb228a4c1c275b75ab04b57fdef1dd",
+    "research/community-k492-audit/pr626/replay/ASSEMBLY.json": "6127732e2fcdd470dab349c1c51d03cd9f0787ad0309b1b4a7c811a908028f4e",
+    "research/complex-k492-owner-repair/code/hpe_audit_local.py": "6bd9a1e279c9c9e4f3767d9f82e537b559292906feb2ddab02f380fb6523a810"
+  }
+}
diff --git a/research/jp8-transposed-coefficient-work/VALIDATION.md b/research/jp8-transposed-coefficient-work/VALIDATION.md
new file mode 100644
index 0000000000..d02bdf1751
--- /dev/null
+++ b/research/jp8-transposed-coefficient-work/VALIDATION.md
@@ -0,0 +1,48 @@
+# Validation record
+
+Base `c290b7b919790300e14004a75d8d24b55ca3c5b5`; completed 2026-10-13.
+Windows Python 3.14.0 and Ubuntu 24.04 / WSL Python 3.12.3 both pass the
+literal replay, independent integer replay and five packaging tests.
+
+```sh
+python3 -I -B research/jp8-transposed-coefficient-work/verify.py
+python3 -I -B research/jp8-transposed-coefficient-work/independent.py
+python3 -I -B research/jp8-transposed-coefficient-work/test_package.py
+python3 -I -B research/jp8-transposed-coefficient-work/native_replay.py --output NEW_DIRECTORY
+```
+
+Completed coverage:
+
+- 1,091 transposed blocks and 6,776 arbitrary local input columns, including
+  dirty private work; exact full paired primitive replay on 725 columns in
+  each orientation, including all dirty helpers and private inputs.
+- Exact counts 224,422 / 224,470; complete positive prefix envelopes identical
+  to the pinned 64/101-bit original values; all paid children and non-scalar
+  event hashes retained.
+- Separate integer-grid matrices and integer/exponent full backward prefix
+  recurrence; both original and companion isolated prefixes checked.
+- All 377 isolated-prefix growth cases recorded, including worst ratio 40/9;
+  explicit four-quarter-source counterexample with original/new prefixes 2/4.
+- Ten construction controls and four independent controls reject incorrect
+  clearing, signs, order, scaling, endpoints and excessive intermediate growth.
+- The unchanged native auditor checks the original macro frame/wrapper
+  contract in a fresh scratch directory and rejects its 25 controls.
+- Five packaging tests execute with `-O -I -B` from a different working
+  directory. A minimal copied tree needs only this package and six pinned
+  source files. Equivalent gzip recompression passes without writes; source,
+  vendor, companion and receipt mutations are detected. The independent
+  checker remains runnable after deliberate corruption of the unused vendor.
+- All six runtime input bytes match Git blobs at the base. The vendored
+  constructor is byte-identical to the named file in PR811 commit
+  `b5c244c916dd0b30957332edad7ae7ae22b0eb9d`.
+- Ruff 0.17.0 passes for all new/vendored Python sources. Selected-record
+  integrity passes for the current #741 fraction
+  `1661202643272239/1000000000000000000`. Staged whitespace and focused diff
+  checks pass.
+
+No full repository suite, full selected cold reconstruction, global invoice
+or admission regeneration, Lean, or remote CI pass is claimed. The native
+check concerns the unchanged original macro word; the new private-work
+lowering is verified separately and has not been admitted into production.
+The finite cost changes remain historical #626 results, with no claimed
+transport into #741 and no new kappa.
diff --git a/research/jp8-transposed-coefficient-work/independent.py b/research/jp8-transposed-coefficient-work/independent.py
new file mode 100644
index 0000000000..4dbe2a0dbb
--- /dev/null
+++ b/research/jp8-transposed-coefficient-work/independent.py
@@ -0,0 +1,261 @@
+"""Integer-only block matrices and a separate complete dyadic prefix replay.
+
+No imports from verify.py or the vendored constructor. Ryan Haines,
+with substantial OpenAI Codex assistance. Apache-2.0.
+"""
+
+import argparse
+import copy
+import gzip
+import hashlib
+import json
+from math import gcd
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+WORK = 724
+
+
+def check(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def digest(value):
+    return hashlib.sha256(json.dumps(value, sort_keys=True,
+                                    separators=(",", ":")).encode()).hexdigest()
+
+
+def normalized(n, e):
+    if n == 0:
+        return 0, 0
+    zeros = min(e, (n & -n).bit_length() - 1)
+    return n >> zeros, e - zeros
+
+
+def add_norm(a, b):
+    e = max(a[1], b[1])
+    return normalized((a[0] << (e - a[1])) + (b[0] << (e - b[1])), e)
+
+
+def greater(a, b):
+    return a[0] << b[1] > b[0] << a[1]
+
+
+def fraction_text(pair):
+    n, e = normalized(*pair)
+    return str(n) if not e else f"{n}/{1 << e}"
+
+
+def reg(pair):
+    check(pair[0] in ("L", "C"), "invalid register tag")
+    return pair[1] if pair[0] == "L" else 716 + pair[1]
+
+
+def macro(op):
+    tag, target = op[0], reg(op[1])
+    if tag == "clear":
+        return [["clear", target]]
+    if tag == "copy":
+        return [["copy", target, reg(op[2])]]
+    if tag == "scale":
+        a, b = abs(op[2]), abs(op[3])
+        g = gcd(a, b)
+        a, b = a // g, b // g
+        check(a and a & (a - 1) == 0 and b & (b - 1) == 0, "non-dyadic scale")
+        k = a.bit_length() - b.bit_length()
+        return ([[("double" if k > 0 else "halve"), target]] * abs(k)
+                + ([["neg", target]] if (op[2] < 0) != (op[3] < 0) else []))
+    check(tag == "add", "unknown macro")
+    source = reg(op[2])
+    a, b = abs(op[3]), abs(op[4])
+    g = gcd(a, b)
+    a, b = a // g, b // g
+    check(a and b & (b - 1) == 0 and source != target, "invalid add macro")
+    result = [["clear", WORK], ["add", WORK, source]]
+    for bit in format(a, "b")[1:]:
+        result.append(["double", WORK])
+        if bit == "1":
+            result.append(["add", WORK, source])
+    result += [["halve", WORK]] * (b.bit_length() - 1)
+    negative = (op[3] < 0) != (op[4] < 0)
+    return result + [["sub" if negative else "add", target, WORK], ["clear", WORK]]
+
+
+def block_matrix(events, block, ops):
+    word = events[block["start"]:block["stop"]]
+    target = block["common_backward_target"]
+    check(all(e[0] == "scalar" and e[2] == 1 and e[1][0] == "add"
+              and e[1][1] == ["L", target] and e[1][2][0] == "L"
+              and e[1][2][1] != target for e in word), "invalid fan-in block")
+    regs = sorted({target, WORK} | {e[1][2][1] for e in word})
+    pos = {r: i for i, r in enumerate(regs)}
+    d = max(sum(o[0] == "halve" for o in ops),
+            max(e[1][4].bit_length() - 1 for e in word))
+    unit = 1 << d
+    rows = {r: [unit * (i == pos[r]) for i in range(len(regs))] for r in regs}
+    norms = {r: unit for r in regs}
+    expected = copy.deepcopy(rows)
+    old_target_norm, old_prefix = unit, unit
+    for event in word:
+        op = event[1]
+        check(unit % op[4] == 0, "insufficient grid for reference coefficient")
+        expected[target][pos[op[2][1]]] += op[3] * (unit // op[4])
+        a, b = abs(op[3]), abs(op[4])
+        g = gcd(a, b)
+        a, b = a // g, b // g
+        old_target_norm += a * (unit // b)
+        old_prefix = max(old_prefix, a * unit, old_target_norm)
+    expected[WORK] = [0] * len(regs)
+    check(fraction_text((old_prefix, d)) == block["isolated_original_prefix"],
+          "independent original isolated prefix mismatch")
+    prefix = unit
+    for instruction in ops:
+        tag, out = instruction[:2]
+        check(out in rows, "unknown block register")
+        if tag == "clear":
+            rows[out], norms[out] = [0] * len(regs), 0
+        elif tag in ("add", "sub"):
+            src = instruction[2]
+            check(out != src, "aliased block update")
+            sign = 1 if tag == "add" else -1
+            rows[out] = [a + sign * b for a, b in zip(rows[out], rows[src])]
+            norms[out] += norms[src]
+        else:
+            check(tag in ("double", "halve"), "unknown block primitive")
+            if tag == "double":
+                rows[out] = [2 * a for a in rows[out]]
+                norms[out] *= 2
+            else:
+                check(all(a % 2 == 0 for a in rows[out]) and norms[out] % 2 == 0,
+                      "integer-grid precision exhausted")
+                rows[out] = [a // 2 for a in rows[out]]
+                norms[out] //= 2
+        prefix = max(prefix, norms[out])
+    check(rows == expected, "independent matrix/clearing mismatch")
+    check(fraction_text((prefix, d)) == block["isolated_companion_prefix"],
+          "independent isolated prefix mismatch")
+    return len(regs)
+
+
+def prefix_replay(events, blocks):
+    norm = {r: (1, 0) for r in range(725)}
+    prefix, index, primitives = (1, 0), 0, 0
+    starts = {b["start"]: b for b in blocks}
+    check(len(starts) == len(blocks), "duplicate block starts")
+    prior = 0
+    for block in blocks:
+        check(prior <= block["start"] < block["stop"] <= len(events), "block overlap/bounds")
+        prior = block["stop"]
+    while index < len(events):
+        event = events[index]
+        if index in starts:
+            block = starts[index]
+            ops, index = block["backward_primitives"], block["stop"]
+        else:
+            ops = macro(event[1]) if event[0] == "scalar" else []
+            index += 1
+        for instruction in ops:
+            tag, target = instruction[:2]
+            if tag == "clear":
+                norm[target] = (0, 0)
+            elif tag == "copy":
+                norm[target] = norm[instruction[2]]
+            elif tag in ("add", "sub"):
+                norm[target] = add_norm(norm[target], norm[instruction[2]])
+            elif tag == "halve":
+                n, e = norm[target]
+                norm[target] = normalized(n, e + 1)
+            elif tag == "double":
+                n, e = norm[target]
+                norm[target] = normalized(2 * n, e)
+            else:
+                check(tag == "neg", "unknown norm primitive")
+            if greater(norm[target], prefix):
+                prefix = norm[target]
+        primitives += len(ops)
+    check(all(norm[r] == (0, 0) for r in range(716, 725)), "private norm not cleared")
+    return {"real_primitives": 2 * primitives, "prefix_l1": fraction_text(prefix)}
+
+
+def reject(call, name):
+    try:
+        call()
+    except (ValueError, KeyError):
+        return name
+    raise ValueError("independent negative control accepted: " + name)
+
+
+def build():
+    pins = json.loads((HERE / "SOURCES.json").read_text())
+    data = {}
+    for path, expected in pins["payload_sha256"].items():
+        raw = (ROOT / path).read_bytes()
+        if path.endswith(".gz"):
+            raw = gzip.decompress(raw)
+        check(hashlib.sha256(raw).hexdigest() == expected, "independent source pin")
+        if path.endswith((".json", ".json.gz")):
+            data[path] = json.loads(raw)
+    programs = next(v for k, v in data.items() if k.endswith("LOCAL-PROGRAMS.json.gz"))
+    spec = next(v for k, v in data.items() if k.endswith("/data/SPEC.json"))["providers"]["jp8"]
+    certificate = json.loads((HERE / "CERTIFICATE.json").read_text())
+    blocks = certificate["blocks"]
+    events = programs["backward"]
+    columns, saving, grew = 0, 0, 0
+    for block in blocks:
+        ops = block["backward_primitives"]
+        check(digest(ops) == block["backward_primitive_sha256"], "backward primitive hash")
+        columns += block_matrix(events, block, ops)
+        old = 2 * sum(len(macro(e[1])) for e in events[block["start"]:block["stop"]])
+        new = 2 * len(ops)
+        check((old, new) == (block["real_primitives_before"], block["real_primitives_after"]),
+              "independent block invoice mismatch")
+        saving += old - new
+        def parsed(s):
+            parts = [int(p) for p in s.split("/")]
+            return parts[0], parts[1].bit_length() - 1 if len(parts) == 2 else 0
+        grew += greater(parsed(block["isolated_companion_prefix"]),
+                        parsed(block["isolated_original_prefix"]))
+    full = prefix_replay(events, blocks)
+    original = spec["complete_contract"]["local"]["backward"]
+    check(full["real_primitives"] == original["real_component_primitives"] - saving,
+          "complete independent invoice mismatch")
+    check(full["prefix_l1"] == original["prefix_l1"], "complete independent prefix changed")
+    fixture = next(b for b in blocks if any(r[0] == "read" for r in b["forward_routes"]))
+    ops = fixture["backward_primitives"]
+    mutations = [("initial_clear", ops[1:]), ("final_clear", ops[:-1])]
+    bad = copy.deepcopy(ops)
+    j = next(j for j, op in enumerate(ops) if op[0] in ("add", "sub")
+             and op[1] == fixture["common_backward_target"])
+    bad[j][0] = "add" if bad[j][0] == "sub" else "sub"
+    mutations.append(("inverse_sign", bad))
+    mutations.append(("neutral_growth", ops[:-1] + [["double", WORK]] * 20
+                      + [["halve", WORK]] * 20 + ops[-1:]))
+    controls = [reject(lambda bad=bad: block_matrix(events, fixture, bad), name)
+                for name, bad in mutations]
+    return {"method": "independent integer block matrices and dyadic full-prefix recurrence",
+            "certificate_sha256": digest(certificate), "blocks": len(blocks),
+            "block_input_columns": columns, "real_saved": saving,
+            "isolated_prefix_growth_blocks": grew, "complete_backward": full,
+            "negative_controls": controls}
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument("--write", type=Path)
+    args = parser.parse_args()
+    result = build()
+    if args.write:
+        check(not args.write.exists(), "refusing to replace receipt")
+        args.write.write_text(json.dumps(result, indent=2) + "\n")
+    else:
+        check(json.loads((HERE / "INDEPENDENT.json").read_text()) == result,
+              "independent receipt mismatch")
+    print(f"PASS_INDEPENDENT_TRANSPOSED_WORK blocks={result['blocks']} "
+          f"columns={result['block_input_columns']} saved={result['real_saved']} "
+          f"controls={len(result['negative_controls'])}")
+
+
+if __name__ == "__main__":
+    main()
diff --git a/research/jp8-transposed-coefficient-work/native_replay.py b/research/jp8-transposed-coefficient-work/native_replay.py
new file mode 100644
index 0000000000..219bd40dbb
--- /dev/null
+++ b/research/jp8-transposed-coefficient-work/native_replay.py
@@ -0,0 +1,37 @@
+"""Check the unchanged historical macro frame/wrapper contract in scratch."""
+
+import argparse
+import json
+import shutil
+import subprocess
+import sys
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+sys.path.insert(0, str(HERE))
+
+from verify import PROVIDER, ROOT, load_inputs, require
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument("--output", type=Path, required=True)
+    args = parser.parse_args()
+    load_inputs()
+    directory = args.output.resolve()
+    require(not directory.exists(), "native replay requires a new directory")
+    directory.mkdir(parents=True)
+    for name in ("CANONICAL-WORD.json.gz", "LOCAL-PROGRAMS.json.gz"):
+        shutil.copyfile(ROOT / PROVIDER / name, directory / name)
+    auditor = ROOT / "research/complex-k492-owner-repair/code/hpe_audit_local.py"
+    subprocess.run([sys.executable, "-I", "-B", str(auditor), str(directory)], check=True)
+    result = json.loads((directory / "INDEPENDENT-LOCAL-REPLAY.json").read_text())
+    del result["canonical_word_sha256"]
+    del result["local_programs_sha256"]
+    require(result == json.loads((HERE / "NATIVE.json").read_text()), "native semantic receipt")
+    print("PASS_UNCHANGED_JP8_NATIVE_CONTRACT columns=716 "
+          f"controls={len(result['negative_controls_rejected'])}")
+
+
+if __name__ == "__main__":
+    main()
diff --git a/research/jp8-transposed-coefficient-work/test_package.py b/research/jp8-transposed-coefficient-work/test_package.py
new file mode 100644
index 0000000000..4d47d02507
--- /dev/null
+++ b/research/jp8-transposed-coefficient-work/test_package.py
@@ -0,0 +1,87 @@
+"""Focused portability and adversarial artifact tests, with assertions off."""
+
+import gzip
+import hashlib
+import json
+import shutil
+import subprocess
+import sys
+import tempfile
+import unittest
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+PACKAGE = HERE.relative_to(ROOT)
+
+
+class PackageTests(unittest.TestCase):
+    def setUp(self):
+        scratch = tempfile.TemporaryDirectory(prefix="jp8-transpose-qa-")
+        self.addCleanup(scratch.cleanup)
+        self.root = Path(scratch.name).resolve()
+        self.package = self.root / PACKAGE
+        shutil.copytree(HERE, self.package, ignore=shutil.ignore_patterns("__pycache__"))
+        for name in json.loads((HERE / "SOURCES.json").read_text())["payload_sha256"]:
+            target = self.root / name
+            target.parent.mkdir(parents=True, exist_ok=True)
+            shutil.copyfile(ROOT / name, target)
+
+    def run_script(self, name, success=True):
+        result = subprocess.run([sys.executable, "-O", "-I", "-B", str(self.package / name)],
+                                check=False, cwd=self.root.parent, text=True, capture_output=True)
+        if success:
+            self.assertEqual(result.returncode, 0, result.stdout + result.stderr)
+        else:
+            self.assertNotEqual(result.returncode, 0, "corruption accepted")
+        return result.stdout + result.stderr
+
+    def snapshot(self):
+        return {str(p.relative_to(self.root)): hashlib.sha256(p.read_bytes()).hexdigest()
+                for p in self.root.rglob("*") if p.is_file()}
+
+    def test_relocated_recompressed_read_only(self):
+        for path in self.root.rglob("*.gz"):
+            path.write_bytes(gzip.compress(gzip.decompress(path.read_bytes()),
+                                           compresslevel=1, mtime=123456789))
+        before = self.snapshot()
+        self.run_script("verify.py")
+        self.run_script("independent.py")
+        self.assertEqual(before, self.snapshot())
+
+    def test_pinned_source_corruption(self):
+        path = next(self.root.rglob("CANONICAL-WORD.json.gz"))
+        obj = json.loads(gzip.decompress(path.read_bytes()))
+        obj["R"] -= 1
+        path.write_bytes(gzip.compress(json.dumps(obj).encode(), mtime=0))
+        self.assertIn("source payload changed", self.run_script("verify.py", False))
+        self.assertIn("independent source pin", self.run_script("independent.py", False))
+
+    def test_vendored_dependency_corruption(self):
+        path = self.package / "vendor/construct_seed31.py"
+        path.write_bytes(path.read_bytes() + b"\n# deliberate pin mutation\n")
+        self.assertIn("vendored forward constructor changed", self.run_script("verify.py", False))
+        # The independent circuit replay does not execute or depend on this code.
+        self.run_script("independent.py")
+
+    def test_companion_sign_corruption(self):
+        path = self.package / "CERTIFICATE.json"
+        obj = json.loads(path.read_text())
+        block = obj["blocks"][0]
+        op = next(op for op in block["backward_primitives"]
+                  if op[0] in ("add", "sub") and op[1] == block["common_backward_target"])
+        op[0] = "sub" if op[0] == "add" else "add"
+        path.write_text(json.dumps(obj))
+        self.assertIn("CERTIFICATE.json mismatch", self.run_script("verify.py", False))
+        self.assertIn("backward primitive hash", self.run_script("independent.py", False))
+
+    def test_undercharged_receipt(self):
+        path = self.package / "EXPECTED.json"
+        obj = json.loads(path.read_text())
+        obj["orientations"]["backward"]["real_primitives"] -= 2
+        path.write_text(json.dumps(obj))
+        self.assertIn("EXPECTED.json mismatch", self.run_script("verify.py", False))
+
+
+if __name__ == "__main__":
+    unittest.main()
diff --git a/research/jp8-transposed-coefficient-work/vendor/construct_seed31.py b/research/jp8-transposed-coefficient-work/vendor/construct_seed31.py
new file mode 100644
index 0000000000..0b996246d4
--- /dev/null
+++ b/research/jp8-transposed-coefficient-work/vendor/construct_seed31.py
@@ -0,0 +1,180 @@
+"""Finite positive one-read routes for fixed-order JP8 cleanup.
+
+Ryan Haines, with substantial OpenAI Codex assistance. Apache-2.0.
+Original binary-Horner contract: attributed #626/#622 emitter.
+Power-of-two comparator: Jeffrey Cox (@namredips), PR #739; no code imported.
+"""
+
+from fractions import Fraction as Q
+from functools import lru_cache
+
+WORK = 724
+LIMIT = 16
+
+
+def require(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def register(pair):
+    tag, index = pair
+    require(isinstance(index, int), "noninteger register")
+    require((tag == "L" and 0 <= index < 716)
+            or (tag == "C" and 0 <= index < 8), "unknown register")
+    return index if tag == "L" else 716 + index
+
+
+def exponent(q):
+    a, b = q.numerator, q.denominator
+    if a > 0 and a & (a - 1) == 0 and b & (b - 1) == 0:
+        return a.bit_length() - b.bit_length()
+    return None
+
+
+def coefficient(op):
+    q = Q(op[3], op[4])
+    require(q and exponent(Q(1, q.denominator)) is not None,
+            "nonzero dyadic coefficient required")
+    return q
+
+
+def preparation_cost(q):
+    a, b = abs(q.numerator), q.denominator
+    return a.bit_length() + a.bit_count() + b.bit_length() - 1
+
+
+def shift(target, power):
+    return [["double" if power > 0 else "halve", target]
+            for _ in range(abs(power))]
+
+
+def prepare(source, magnitude):
+    require(magnitude > 0, "positive magnitude required")
+    a, b = magnitude.numerator, magnitude.denominator
+    require(b & (b - 1) == 0, "nondyadic preparation")
+    result = [["clear", WORK], ["add", WORK, source]]
+    for bit in bin(a)[3:]:
+        result.append(["double", WORK])
+        if bit == "1":
+            result.append(["add", WORK, source])
+    return result + shift(WORK, 1 - b.bit_length())
+
+
+def lower_macro(op):
+    tag, target = op[0], register(op[1])
+    if tag == "clear":
+        return [["clear", target]]
+    if tag == "copy":
+        return [["copy", target, register(op[2])]]
+    if tag == "scale":
+        q = Q(op[2], op[3])
+        k = exponent(abs(q))
+        require(k is not None, "scale is not a signed power of two")
+        return shift(target, k) + ([["neg", target]] if q < 0 else [])
+    require(tag == "add", "unknown macro")
+    source, q = register(op[2]), coefficient(op)
+    require(target != source, "source/target alias")
+    return (prepare(source, abs(q))
+            + [["add" if q > 0 else "sub", target, WORK], ["clear", WORK]])
+
+
+@lru_cache(None)
+def one_read_routes(a, b):
+    """Enumerate all feasible (u,v) in the claimed square, using unique v.
+
+    For each u, (a*2**u+1)*2**v=b determines v. Positive coefficients
+    make the maximum below an exact coefficient-work L1 envelope.
+    """
+    cap = max(a, Q(b.numerator))
+    routes = []
+    for u in range(-LIMIT, LIMIT + 1):
+        middle = a * Q(2)**u + 1
+        v = exponent(b / middle)
+        if (v is not None and -LIMIT <= v <= LIMIT
+                and max(a, middle, b) <= cap):
+            routes.append((abs(u) + 1 + abs(v), u, v))
+    return tuple(sorted(routes))
+
+
+def select_route(a, b, additive):
+    """Strict improvement over reset; ties keep the earlier/simpler choice."""
+    cost, route = 1 + preparation_cost(b), None
+    k = exponent(b / a)
+    if k is not None and abs(k) < cost:
+        cost, route = abs(k), ["shift", k]
+    if additive:
+        candidates = one_read_routes(a, b)
+        if candidates and candidates[0][0] < cost:
+            cost, u, v = candidates[0]
+            route = ["read", u, v]
+    return cost, route
+
+
+def eligible(event, phase):
+    if (phase != "full_frame_cleanup" or event[0] != "scalar"
+            or event[2] != 1 or event[1][0] != "add"):
+        return None
+    op = event[1]
+    q = coefficient(op)
+    if (op[1][0] != "L" or op[2][0] != "L" or op[1] == op[2]
+            or abs(q) == 1):
+        return None
+    return register(op[2]), abs(q)
+
+
+def plan(events, additive):
+    phase, previous = None, None
+    edges = {}
+    for index, event in enumerate(events):
+        if event[0] == "phase":
+            phase = event[1]
+        current = eligible(event, phase)
+        if previous and current and previous[0] == current[0]:
+            _, route = select_route(previous[1], current[1], additive)
+            if route is not None:
+                edges[index] = route
+        previous = current
+    blocks = []
+    for index, route in edges.items():
+        if blocks and blocks[-1]["stop"] == index:
+            blocks[-1]["stop"] = index + 1
+            blocks[-1]["routes"].append(route)
+        else:
+            blocks.append({"start": index - 1, "stop": index + 1,
+                           "routes": [route]})
+    return blocks
+
+
+def lower_block(events, block):
+    start, stop, routes = block["start"], block["stop"], block["routes"]
+    require(0 <= start < stop <= len(events) and stop - start >= 2,
+            "invalid block span")
+    require(len(routes) == stop - start - 1, "route count mismatch")
+    phase = next((e[1] for e in reversed(events[:start + 1])
+                  if e[0] == "phase"), None)
+    data = [eligible(e, phase) for e in events[start:stop]]
+    require(all(data), "block crossed phase/frame or ineligible operation")
+    source = data[0][0]
+    require(all(item[0] == source for item in data), "block changes source")
+    result = prepare(source, data[0][1])
+    for j, event in enumerate(events[start:stop]):
+        if j:
+            route = routes[j - 1]
+            a, b = data[j - 1][1], data[j][1]
+            if route[0] == "shift":
+                require(len(route) == 2 and exponent(b / a) == route[1],
+                        "invalid rescaling route")
+                result.extend(shift(WORK, route[1]))
+            else:
+                require(route[0] == "read" and len(route) == 3,
+                        "unknown route")
+                u, v = route[1:]
+                require((abs(u) + 1 + abs(v), u, v) in one_read_routes(a, b),
+                        "invalid one-read route or prefix excursion")
+                result.extend(shift(WORK, u))
+                result.append(["add", WORK, source])
+                result.extend(shift(WORK, v))
+        q = coefficient(event[1])
+        result.append(["add" if q > 0 else "sub", register(event[1][1]), WORK])
+    return result + [["clear", WORK]]
diff --git a/research/jp8-transposed-coefficient-work/verify.py b/research/jp8-transposed-coefficient-work/verify.py
new file mode 100644
index 0000000000..2da47e690b
--- /dev/null
+++ b/research/jp8-transposed-coefficient-work/verify.py
@@ -0,0 +1,363 @@
+"""Exact inverse-transpose companion to PR811's positive coefficient work.
+
+Ryan Haines, with substantial OpenAI Codex assistance. Apache-2.0.
+The pinned forward constructor is retained verbatim in vendor/.
+"""
+
+import argparse
+import copy
+import gzip
+import hashlib
+import json
+import sys
+from collections import Counter
+from fractions import Fraction as Q
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+sys.path.insert(0, str(HERE / "vendor"))
+
+import construct_seed31 as forward
+
+WORK = forward.WORK
+BASE = "c290b7b919790300e14004a75d8d24b55ca3c5b5"
+PROVIDER = "research/community-k492-audit/pr626/replay/mixed-global/base/providers/jp8/"
+SPEC = "research/complex-k492-owner-repair/data/SPEC.json"
+
+
+def require(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def digest(obj):
+    return hashlib.sha256(json.dumps(obj, sort_keys=True,
+                                    separators=(",", ":")).encode()).hexdigest()
+
+
+def load_inputs():
+    pins = json.loads((HERE / "SOURCES.json").read_text())
+    require(pins["base_commit"] == BASE, "base pin mismatch")
+    data = {}
+    for path, expected in pins["payload_sha256"].items():
+        raw = (ROOT / path).read_bytes()
+        if path.endswith(".gz"):
+            raw = gzip.decompress(raw)
+        require(hashlib.sha256(raw).hexdigest() == expected, "source payload changed: " + path)
+        if path.endswith((".json", ".json.gz")):
+            data[path] = json.loads(raw)
+    require(hashlib.sha256((HERE / "vendor/construct_seed31.py").read_bytes()).hexdigest()
+            == pins["vendor_sha256"], "vendored forward constructor changed")
+    spec = data[SPEC]["providers"]["jp8"]
+    native = data[PROVIDER + "CANONICAL-WORD.json.gz"]
+    programs = data[PROVIDER + "LOCAL-PROGRAMS.json.gz"]
+    for name in ("CANONICAL-WORD.json.gz", "LOCAL-PROGRAMS.json.gz"):
+        require(digest(data[PROVIDER + name]) == spec["artifact_semantic_sha256"][name],
+                "historical SPEC binding failed")
+    require((native["v"], native["R"], native["h"]) == (112, 492, 9), "dimensions changed")
+    for orientation, events in programs.items():
+        require(digest(events) == spec["program_hashes"][orientation], "program binding failed")
+    return programs, native, spec
+
+
+def dual_macro(op):
+    require(op[0] == "add", "dual block contains a non-addition")
+    return ["add", op[2], op[1], -op[3], op[4]]
+
+
+def transpose(ops, source):
+    """Transpose the cleared linear circuit; negate writes to original source."""
+    result = []
+    for op in reversed(ops):
+        tag, target = op[:2]
+        if tag in ("clear", "double", "halve"):
+            require(target == WORK, "non-private clear/scale in forward block")
+            result.append(list(op))
+        else:
+            require(tag in ("add", "sub"), "unsupported transposition primitive")
+            require(target != source and target != op[2], "forward block writes source/aliases")
+            new_target, new_source = op[2], target
+            if new_target == source:
+                tag = "sub" if tag == "add" else "add"
+            result.append([tag, new_target, new_source])
+    return result
+
+
+class ExactMachine:
+    """Literal primitive interpreter on arbitrary scalar input columns."""
+
+    def __init__(self, registers=range(725)):
+        self.rows = {r: {r: Q(1)} for r in registers}
+        self.norm = {r: Q(1) for r in registers}
+        self.prefix = Q(1)
+
+    def run(self, ops):
+        for op in ops:
+            tag, target = op[:2]
+            if tag == "clear":
+                self.rows[target], self.norm[target] = {}, Q(0)
+            elif tag == "copy":
+                self.rows[target] = dict(self.rows[op[2]])
+                self.norm[target] = self.norm[op[2]]
+            elif tag in ("add", "sub"):
+                source = op[2]
+                require(source != target, "aliased primitive")
+                sign = 1 if tag == "add" else -1
+                for col, value in self.rows[source].items():
+                    new = self.rows[target].get(col, Q(0)) + sign * value
+                    if new:
+                        self.rows[target][col] = new
+                    else:
+                        self.rows[target].pop(col, None)
+                self.norm[target] += self.norm[source]
+            else:
+                require(tag in ("double", "halve", "neg"), "unknown primitive")
+                factor = {"double": Q(2), "halve": Q(1, 2), "neg": Q(-1)}[tag]
+                self.rows[target] = {c: factor * x for c, x in self.rows[target].items()}
+                self.norm[target] *= abs(factor)
+            self.prefix = max(self.prefix, self.norm[target])
+
+
+def compare_boundary(f_events, b_events, block):
+    start, stop = block["start"], block["stop"]
+    for f, b in zip(f_events[start:stop], b_events[start:stop]):
+        require(f[0] == b[0] == "scalar" and f[2] == b[2] == 1,
+                "block crossed full-frame boundary")
+        require(b[1] == dual_macro(f[1]), "backward block is not the scalar dual")
+    source = f_events[start][1][2][1]
+    primitive = forward.lower_block(f_events, block)
+    companion = transpose(primitive, source)
+    regs = {source, WORK} | {e[1][1][1] for e in f_events[start:stop]}
+    reference, actual = ExactMachine(regs), ExactMachine(regs)
+    for event in b_events[start:stop]:
+        reference.run(forward.lower_macro(event[1]))
+    actual.run(companion)
+    require(actual.rows == reference.rows, "dirty-input companion boundary mismatch")
+    require(not actual.rows[WORK], "companion private work survives")
+    require(all(actual.norm[r] == reference.norm[r] for r in regs if r != WORK),
+            "companion changed live positive envelope")
+    require(len(primitive) == len(companion), "transposition changed cost")
+    old = sum(len(forward.lower_macro(e[1])) for e in b_events[start:stop])
+    return companion, {
+        "start": start, "stop": stop, "common_backward_target": source,
+        "forward_routes": block["routes"], "columns": len(regs),
+        "forward_primitive_sha256": digest(primitive),
+        "backward_primitives": companion,
+        "backward_primitive_sha256": digest(companion),
+        "real_primitives_before": 2 * old, "real_primitives_after": 2 * len(companion),
+        "isolated_original_prefix": str(reference.prefix),
+        "isolated_companion_prefix": str(actual.prefix),
+    }
+
+
+def full_replay(events, native, replacements, orientation):
+    machine, index, count = ExactMachine(), 0, 0
+    paid, non_scalar = Counter(), []
+    while index < len(events):
+        event = events[index]
+        if index in replacements:
+            stop, ops = replacements[index]
+            require(not machine.rows[WORK], "work dirty at replacement boundary")
+            for e in events[index:stop]:
+                require(e[0] == "scalar" and e[2] == 1
+                        and all(native["final"][r[1]] == 1 for r in e[1][1:3]),
+                        "replacement touches a partial frame")
+            index = stop
+        else:
+            ops = forward.lower_macro(event[1]) if event[0] == "scalar" else []
+            if event[0] != "scalar":
+                non_scalar.append(event)
+                if event[0] == "climb":
+                    paid[event[5]] += 1
+                elif event[0] == "center":
+                    paid[event[4]] += 1
+                if event[0] in ("climb", "center", "adapter", "residual_adapter"):
+                    require(not machine.rows[WORK], "work crosses an interface")
+            index += 1
+        machine.run(ops)
+        count += len(ops)
+    for r in range(716):
+        expected = {r: Q(1)}
+        if orientation == "forward" and 112 <= r < 224:
+            expected[r - 112] = Q(1)
+        elif orientation == "backward" and r < 112:
+            expected[r + 112] = Q(-1)
+        require(machine.rows[r] == expected, "complete shear endpoint mismatch")
+    require(all(not machine.rows[r] for r in range(716, 725)), "private input leaked")
+    require(dict(paid) == {int(k): v for k, v in native["local_profile"].items()},
+            "paid recursive profile changed")
+    require(machine.prefix < 2**192, "complete prefix exceeds inherited allowance")
+    return {"columns": 725, "real_primitives": 2 * count,
+            "prefix_l1": str(machine.prefix),
+            "prefix_bits": machine.prefix.numerator.bit_length()
+            - machine.prefix.denominator.bit_length() + 1,
+            "non_scalar_sha256": digest(non_scalar),
+            "paid_profile": {str(k): v for k, v in sorted(paid.items())}}
+
+
+def reject(call, name):
+    try:
+        call()
+    except (ValueError, KeyError):
+        return name
+    raise ValueError("negative control accepted: " + name)
+
+
+def controls():
+    f = [["phase", "full_frame_cleanup"],
+         ["scalar", ["add", ["L", 1], ["L", 0], 1, 2], 1],
+         ["scalar", ["add", ["L", 2], ["L", 0], -3, 2], 1]]
+    b = [f[0]] + [["scalar", dual_macro(e[1]), 1] for e in f[1:]]
+    block = forward.plan(f, True)[0]
+    good, _ = compare_boundary(f, b, block)
+    expected = ExactMachine([0, 1, 2, WORK])
+    for event in b[1:]:
+        expected.run(forward.lower_macro(event[1]))
+
+    def check_ops(ops):
+        actual = ExactMachine([0, 1, 2, WORK])
+        actual.run(ops)
+        require(actual.rows == expected.rows, "control endpoint/clearing mismatch")
+        require(actual.prefix < 2**192, "control inherited prefix allowance")
+
+    check_ops(good)
+    cases = [("initial_clear", good[1:]), ("final_clear", good[:-1])]
+    wrong_sign = copy.deepcopy(good)
+    j = next(j for j, op in enumerate(good) if op[0] == "sub" and op[1] == 0)
+    wrong_sign[j][0] = "add"
+    cases.append(("missed_inverse_sign", wrong_sign))
+    cases.append(("omitted_target_write", good[:j] + good[j + 1:]))
+    cases.append(("forward_instead_of_reverse_order", list(reversed(good))))
+    wrong_scale = copy.deepcopy(good)
+    j = next(j for j, op in enumerate(good) if op[0] == "halve")
+    wrong_scale[j][0] = "double"
+    cases.append(("inverted_scale", wrong_scale))
+    growth = good[:-1] + [["double", WORK]] * 200 + [["halve", WORK]] * 200 + good[-1:]
+    cases.append(("endpoint_neutral_excess_growth", growth))
+    result = [reject(lambda ops=ops: check_ops(ops), name) for name, ops in cases]
+    bad = copy.deepcopy(b)
+    bad[2][2] = 0
+    result.append(reject(lambda: compare_boundary(f, bad, block), "frame_crossing"))
+    bad = copy.deepcopy(b)
+    bad[2][1][3] *= -1
+    result.append(reject(lambda: compare_boundary(f, bad, block), "wrong_macro_dual"))
+    result.append(reject(lambda: transpose([["add", 0, WORK]], 0), "forward_source_write"))
+    return result
+
+
+def equal_blocks(events):
+    """Independent grouping for the historical equal-coefficient comparator."""
+    phase, previous, blocks = None, None, []
+    for index, event in enumerate(events):
+        if event[0] == "phase":
+            phase = event[1]
+        current = forward.eligible(event, phase)
+        if current is not None and current == previous:
+            if blocks and blocks[-1]["stop"] == index:
+                blocks[-1]["stop"] += 1
+                blocks[-1]["routes"].append(["shift", 0])
+            else:
+                blocks.append({"start": index - 1, "stop": index + 1,
+                               "routes": [["shift", 0]]})
+        previous = current
+    return blocks
+
+
+def build():
+    programs, native, spec = load_inputs()
+    blocks = forward.plan(programs["forward"], True)
+    certificate = {"schema": "jp8-transposed-private-work/1", "base_commit": BASE,
+                   "assignment_seed": 25, "blocks": []}
+    replacements = {"forward": {}, "backward": {}}
+    for block in blocks:
+        ops, item = compare_boundary(programs["forward"], programs["backward"], block)
+        certificate["blocks"].append(item)
+        replacements["forward"][block["start"]] = (block["stop"],
+            forward.lower_block(programs["forward"], block))
+        replacements["backward"][block["start"]] = (block["stop"], ops)
+    result = {"schema": "jp8-transposed-private-work-replay/1", "base_commit": BASE,
+              "negative_controls": controls(), "orientations": {},
+              "certificate_sha256": digest(certificate), "blocks": len(blocks),
+              "block_input_columns": sum(b["columns"] for b in certificate["blocks"])}
+    for orientation in ("forward", "backward"):
+        events = programs[orientation]
+        row = full_replay(events, native, replacements[orientation], orientation)
+        contract = spec["complete_contract"]["local"][orientation]
+        original = 2 * sum(len(forward.lower_macro(e[1])) for e in events if e[0] == "scalar")
+        require(original == contract["real_component_primitives"], "original count mismatch")
+        require(row["prefix_l1"] == contract["prefix_l1"], "full positive prefix changed")
+        row.update(original_real_primitives=original, saved_real_primitives=original-row["real_primitives"])
+        result["orientations"][orientation] = row
+        print(f"PASS_LITERAL_{orientation.upper()} columns=725 primitives={row['real_primitives']}",
+              flush=True)
+    contract = spec["complete_contract"]
+    maximum = max(r["real_primitives"] for r in result["orientations"].values())
+    result["unintegrated_budget"] = {
+        "adapter_allowances": contract["local_adapters"], "compiler_charge": contract["compiler"],
+        "original_local_bound": contract["local_primitive_bound"],
+        "paired_candidate_local_bound": maximum + contract["local_adapters"] * contract["compiler"],
+        "backward_only_local_bound": result["orientations"]["forward"]["original_real_primitives"]
+        + contract["local_adapters"] * contract["compiler"],
+        "new_kappa_claimed": False,
+    }
+    comparisons = {}
+    for name, baseline in (("equal_magnitude", equal_blocks(programs["forward"])),
+                           ("power_only", forward.plan(programs["forward"], False))):
+        saved = sum(2 * (sum(len(forward.lower_macro(e[1])) for e in
+                            programs["forward"][b["start"]:b["stop"]])
+                         - len(forward.lower_block(programs["forward"], b))) for b in baseline)
+        comparisons[name] = {"blocks": len(baseline), "backward_saved": saved,
+                             "backward_primitives": 257280 - saved}
+    result["comparators"] = comparisons
+    grew = [b for b in certificate["blocks"] if Q(b["isolated_companion_prefix"])
+            > Q(b["isolated_original_prefix"])]
+    worst = max(grew, key=lambda b: Q(b["isolated_companion_prefix"])
+                / Q(b["isolated_original_prefix"]))
+    result["prefix_scope"] = {
+        "isolated_growth_blocks": len(grew), "worst_start": worst["start"],
+        "worst_ratio": str(Q(worst["isolated_companion_prefix"])
+                           / Q(worst["isolated_original_prefix"])),
+        "old_isolated_prefix": worst["isolated_original_prefix"],
+        "new_isolated_prefix": worst["isolated_companion_prefix"],
+    }
+    fixture = [["phase", "full_frame_cleanup"]] + [
+        ["scalar", ["add", ["L", r], ["L", 0], 1, 4], 1] for r in range(1, 5)]
+    dual = [fixture[0]] + [["scalar", dual_macro(e[1]), 1] for e in fixture[1:]]
+    _, witness = compare_boundary(fixture, dual, forward.plan(fixture, True)[0])
+    require((witness["isolated_original_prefix"], witness["isolated_companion_prefix"])
+            == ("2", "4"), "prefix counterexample changed")
+    result["prefix_nonmonotonicity_witness"] = {"sources": 4, "coefficient": "-1/4",
+                                               "old_prefix": "2", "transposed_prefix": "4"}
+    return certificate, result
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument("--write", type=Path)
+    args = parser.parse_args()
+    certificate, result = build()
+    for name, obj in (("CERTIFICATE.json", certificate), ("EXPECTED.json", result)):
+        if args.write:
+            args.write.mkdir(parents=True, exist_ok=True)
+            path = args.write / name
+            require(not path.exists(), "refusing to replace output")
+            if name == "CERTIFICATE.json":
+                # One self-contained literal block per line keeps the review diff small.
+                fields = ["  " + json.dumps(k) + ": " + json.dumps(v)
+                          for k, v in obj.items() if k != "blocks"]
+                fields.append('  "blocks": [\n' + ",\n".join(
+                    "    " + json.dumps(b, separators=(",", ":")) for b in obj["blocks"])
+                    + "\n  ]")
+                path.write_text("{\n" + ",\n".join(fields) + "\n}\n")
+            else:
+                path.write_text(json.dumps(obj, indent=2) + "\n")
+        else:
+            require(json.loads((HERE / name).read_text()) == obj, name + " mismatch")
+    b = result["orientations"]["backward"]
+    print(f"PASS_JP8_TRANSPOSED_WORK backward_saved={b['saved_real_primitives']} "
+          f"blocks={len(certificate['blocks'])} columns=725 controls={len(result['negative_controls'])}")
+
+
+if __name__ == "__main__":
+    main()