Recursive interface: coalesce restoration with cancelling readers

research/reader-restoration-cuts-seed53/CERTIFICATE.json

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+{
+  "base": "1b59b624d21924f8e3537cdc1ac52d3be6eda2fe",
+  "fixture": {
+    "cost": 1,
+    "cuts": [
+      5
+    ],
+    "segments": [
+      {
+        "cost": 1,
+        "negate": [
+          0
+        ],
+        "start": 0,
+        "stop": 5
+      }
+    ]
+  },
+  "forced_cut_2": {
+    "cost": 5,
+    "cuts": [
+      1,
+      2,
+      3,
+      4,
+      5
+    ],
+    "segments": [
+      {
+        "cost": 1,
+        "negate": [
+          0
+        ],
+        "start": 0,
+        "stop": 1
+      },
+      {
+        "cost": 1,
+        "negate": [
+          0
+        ],
+        "start": 1,
+        "stop": 2
+      },
+      {
+        "cost": 1,
+        "negate": [
+          0
+        ],
+        "start": 2,
+        "stop": 3
+      },
+      {
+        "cost": 1,
+        "negate": [
+          0
+        ],
+        "start": 3,
+        "stop": 4
+      },
+      {
+        "cost": 1,
+        "negate": [
+          0
+        ],
+        "start": 4,
+        "stop": 5
+      }
+    ]
+  },
+  "invalid_input_controls": 9,
+  "mathematical_controls": [
+    "changed reader destroys cancellation",
+    "checkpoint after child two destroys saving",
+    "final equality does not authorize an intermediate cut",
+    "same children in a different order lose cancellation"
+  ],
+  "retained_pr1155": {
+    "deferred_mults": 1,
+    "eager_mults": 5,
+    "old_certificate": "PASS",
+    "probes": 3
+  },
+  "scalar_exhaustion": {
+    "batches": 1555,
+    "max_children": 4,
+    "partitions": 11311
+  },
+  "scalar_instruction_invoice": {
+    "common_sufficient_signed_bits": 7,
+    "deferred": {
+      "adds": 5,
+      "coefficient_reads": 10,
+      "data_reads": 11,
+      "data_writes": 11,
+      "multiplies": 5,
+      "negations": 6,
+      "program_records": 6
+    },
+    "eager": {
+      "adds": 5,
+      "coefficient_reads": 10,
+      "data_reads": 15,
+      "data_writes": 15,
+      "multiplies": 5,
+      "negations": 10,
+      "program_records": 10
+    },
+    "input_D": 11,
+    "input_W": 7,
+    "output_D": 32,
+    "output_W": 7
+  },
+  "scope": "exact diagonal readers and signs; every chosen cut matches eager data and workspace; no new kappa",
+  "seed": 53,
+  "seeded_longer": {
+    "batches": 160,
+    "max_children": 8,
+    "max_dimension": 3,
+    "partitions": 5264
+  },
+  "status": "PASS_READER_RESTORATION_CUTS",
+  "two_coordinate_exhaustion": {
+    "batches": 1296,
+    "partitions": 2592
+  }
+}

research/reader-restoration-cuts-seed53/CLAIM.md

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+# Reader-aware restoration cuts
+
+Status: claimed, recursive interfaces, OS-random seed 53. Owner Sagar Thakrar
+with OpenAI Codex, chat `01a136c4-b32d-7a30-92f0-d8dffa66b80f`, local host.
+Scope key: `restoration-coalescing/diagonal-readers/checkpoint-correct-cuts`.
+
+Cycle starts 2026-10-13 19:37:19 UTC. Scope deadline 19:45:19 UTC;
+verification starts by 20:07:19 UTC; hard finish 20:17:19 UTC.
+Checkout: `/Users/thakrarsagar-dev/Documents/Codex/2026-10-13/continue-the-user-s-research-loop-13/work/integer-mult-bounds`.
+Fetch completed 19:37:56 UTC. Clean local main and fetched origin/main were
+both `1b59b624d21924f8e3537cdc1ac52d3be6eda2fe` at 19:38:11 UTC, before
+creating `research/cycle08-seed53-recursive-interfaces`. That is the base.
+Prior branches are preserved.
+
+## Three candidates and reductions before implementation
+
+1. **Selected: permit diagonal readers during restoration coalescing.**
+   PR1155, `9821c6e9388f3fc097e94b72be9fa7b41a849527`, gives read-free
+   phase/permutation examples, a failing reader example, a written model and
+   cost recurrence. Its README calls absence of readers necessary. Read the
+   README, actual Batch interpreter, builders, verifier and certificate.
+   Cheap counterexample in its exact model: five scalar children all negate
+   W and read coefficients `[1,1,1,-1,1]`. Eager and deferred outputs both
+   increment D by `3W`; one final negation restores W. All five children read
+   W. At child two the data values differ, so an intermediate observation is
+   a real constraint. New evidence: a necessary and sufficient interval test,
+   minimum-cost checkpoint-correct partition tool, full node recurrence and
+   exact independent enumeration. No novelty is claimed for matrix algebra.
+2. **Rejected: extend four-way port borrowing to higher arity.** PR1164,
+   `0b2077e929ec488a4d38b3523688a8a68bf1b43e`, already gives the chain word,
+   endpoint/middle call multiplicities and a pool-count induction. Read its
+   PROOF.md and embedded PR1057 proof. Replacing four by k is a direct
+   consequence and a larger replay would add no distinct result.
+3. **Rejected: fixed unequal positive/negative connector-query prices.**
+   Accepted PR1088, `b0396aad724fec58630fbd4d13a809c562f18d9e`, proves the
+   signed three-by-three weighted optimum. Read the retained README and
+   query alphabet. Uniform correctness forces equal positive/negative counts
+   for each named query, reducing its fixed directional prices to their
+   average. This does not produce a new optimization problem.
+
+## Scope and overlap
+
+The selected domain is a fixed ordered batch of exact diagonal reader shears
+and diagonal signs over characteristic zero. Every chosen restoration cut
+must match eager D and W, with optional mandatory cuts. The tool does not
+optimize arbitrary reversible programs, permit inaccurate intermediate cuts,
+change children, or claim a supplier or kappa improvement. A complete price
+must include child calls, source access, workspace, preparation, control,
+restoration and any physical adapters.
+
+Inspected all 1170 PRs available at the check, including 786 open PRs, their
+bodies, the central ledger, selected PR1094 construction/assembly/source
+manifest and review, prior loop negatives, and live discussions 1163/1166/1167.
+There is no pinned sprint issue. The immediately refreshed discussion 1166 has
+Adam Gall's unrelated PR279 provenance claim at 19:42:46 UTC; no overlapping
+restoration claim appeared. The latest open PR was 1177.
+Claim is local as specified by the research brief. No separate issue is opened.
+PR249 `96495746c786d6d0339dbb38c7f553d4af3f88ed` and PR309
+`aeb4808af1392e565e7071cbeb2c29608201cf24` transport dirty-helper responses
+through concrete supplier words; that algebra is credited, not reinvented as
+a new mechanism. The distinct deliverable corrects PR1155's applicability claim
+and supplies a reader-aware cut planner with a precise observation contract.
+Prior PR1168's pending-output fanout cap is a different query alphabet.
+
+Stop if a matching planner or correction appears, or if no useful checked
+artifact is ready by the verification deadline. At most one pivot, with the
+original clock unchanged. Recheck overlap at first result and before submission.
+
+## Result and release
+
+Completed with [PR1199](https://github.com/CrocSwap/integer-mult-bounds/pull/1199).
+Five nonzero readers admit one restoration instead of five. An observation after
+child two removes the saving. The package supplies a written criterion and
+restricted optimum, the cut planner, the exact complete-node accounting model,
+and an independent literal-column enumeration of 19,167 partitions across 3,011
+declared batches. The retained PR1155 certificate also reproduces unchanged.
+No new kappa, adopted supplier or runtime improvement is claimed.
+
+Verification and handoff started by 19:52:45 UTC. All relevant checks in
+VALIDATION.md pass. Final overlap at 19:56:39 UTC covered 808 open PRs through
+1196 and all active discussions; no matching correction or planner appeared.
+Upstream main was freshly fetched again and remained
+`1b59b624d21924f8e3537cdc1ac52d3be6eda2fe`, the original investigation base.
+No pivot. This scope is complete; do not rerun it as a new experiment.

research/reader-restoration-cuts-seed53/NOTICE.md

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+# Credit and license
+
+Prepared for Sagar Thakrar, GitHub `thakrarsagar`, with substantial OpenAI Codex
+assistance in investigation, proof writing, implementation and verification.
+New material is licensed under the repository's Apache License 2.0.
+
+Aaron Wang / Navinor's PR1155, with opencode assistance, supplies the batch model,
+read-free coalescing examples, a valid failing-reader example and the original
+cost discussion. The retained README and verifier at
+`9821c6e9388f3fc097e94b72be9fa7b41a849527` are byte-identical to their original
+Git blobs. The retained certificate has a final newline appended, with identical
+parsed content. Its source attribution and Apache-2.0 terms are preserved. This
+package corrects a stated necessity claim without changing that predecessor.
+
+The general algebra of dirty-response transport is credited to eumemic's PR249
+and PR309 and their retained construction lineages. The exact heads are in
+SOURCES.json. This package makes no priority claim for linear cancellation,
+composed inverses or dynamic programming. Its added evidence is the concrete
+PR1155 counterexample, observation boundary, restricted cut planner and independent
+finite enumeration.
+
+PR1164 by woklerx, Gareth Davidson's PR1057, Koroffstrogov's PR758 and their
+predecessors retain credit for dirty echo and port-borrowing mechanisms. They
+were inspected as overlap, not used as runtime dependencies. `martinxxx9`'s
+PR1088 and Ross Richardson's PR944 retain the signed connector results used to
+reject another candidate. PR1168 is the separate pending-output fanout result
+from the previous loop cycle.
+
+The unchanged selected PR1094 construction is Rohan Arun's packing on Jeremy
+Kothe's PR1087 and the cumulative supplier, controller and bit lineages credited
+in the selected review. No claimed improvement to those results follows here.