Research construction: exact factor schedules save 26 R420 XORs (no new kappa)

research/r420-block7-exact-schedule/CLAIM.md

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+# Local sprint claim
+
+- Track: bit supplier; OS-random assignment seed **17** (`secrets.randbelow(100)`).
+- Base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`.
+- Claimed: 2026-10-13 05:23:34 UTC; expected finish: 05:58 UTC (13:58 Hong Kong).
+- Object: PR543 R420 phase-B block **7 = 17 mod 10**, frame 549,
+  original B[471:512], on the unchanged selected supplier word.
+- Method: split the original 41-XOR block into its upper/lower triangular
+  factors and use an exact subset-state dynamic program to minimize each
+  factor among schedules that finish each destination in one contiguous
+  batch, using only positive additions from the current other wires.
+  Check integer transfer equality, both positive envelopes, literal reverse
+  cleanup, all arbitrary-input columns, frames and complete native bill.
+- Deliverable: frozen circuit, finite-class optimality certificate, verifier,
+  source pins and a scoped research note; no new kappa assumed.
+- Three candidates considered: (1) block-7 factor scheduling; (2) punctured
+  R439 mini-node basis changes (close overlap with #728); (3) full-frame
+  decoder resynthesis (close overlap with #708/#748/#753). Selected (1).
+- Distinct coverage: #670 and #674 optimize blocks 3 and 5, respectively;
+  #691 changes quad 13; #754 synthesizes cleanup windows on at most four
+  wires. This experiment checks all subset states for two twelve-wire
+  factors with a precisely stated schedule class, not another stochastic
+  rewrite seed or a claim of unrestricted XOR optimality. #681 integrates
+  the earlier circuits but does not change block 7.
+- Coordination: browser inspection of the issue listing and API inspection
+  found only issue #650 (request for Discussions), no pinned sprint issue.
+  No reservation issue was created. Initial metadata covered 586 open PRs
+  through #767. The repository review ledger ends at #626; later proposals
+  are not treated as accepted frontiers.
+- Contributor: Yuzhou Wang / @yzhouwang, with substantial OpenAI Codex
+  assistance. Apache-2.0. No delegated workers or independent human review.
+
+## Result update (once)
+
+Completed 2026-10-13 05:39 UTC. Submitted
+[PR #789](https://github.com/CrocSwap/integer-mult-bounds/pull/789).
+The 41-XOR block is replaced by 28 XORs, with exact fixed-factor minima
+14 + 14. Charging the literal reverse cleanup saves 26 native ADDs overall:
+26,916 → 26,890. No new κ. All 660 dirty columns and frame paths pass;
+fresh 5/9/11-stage native checks pass with 96 bridge-omission controls.
+Python 3.9/3.14, optimized/relocated/committed-source replay, source and
+certificate controls and selected-record integrity pass. Scope, source pins,
+proof, inherited obligations and full validation are in README/VALIDATION.
+Final overlap refresh through #786 found no block-7 factor-schedule duplicate.

research/r420-block7-exact-schedule/EXPECTED.json

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+{
+  "columns": 660,
+  "controls": [
+    "omitted local XOR",
+    "F2-neutral positive inflation",
+    "omitted cleanup",
+    "stale scalar-valid cleanup",
+    "omitted dirty compensation",
+    "illegal frame descent",
+    "inflated lower-bound certificate",
+    "unpaid witness gate"
+  ],
+  "factor_audits": [
+    {
+      "feasible_pairs": 20992,
+      "new": 14,
+      "old": 17,
+      "state_destination_pairs": 24576,
+      "states": 4096,
+      "valid_donor_subsets": 20992
+    },
+    {
+      "feasible_pairs": 22528,
+      "new": 14,
+      "old": 24,
+      "state_destination_pairs": 24576,
+      "states": 4096,
+      "valid_donor_subsets": 22528
+    }
+  ],
+  "kappa": "unchanged",
+  "labels": {
+    "frames": 672,
+    "gram_det": 1,
+    "labels": 120,
+    "lattice_rank": 8
+  },
+  "local": {
+    "boolean_inputs": 4096,
+    "integer_columns_each_orientation": 12,
+    "reverse_strictly_smaller_entries": 25
+  },
+  "native": [
+    {
+      "adds": 26916,
+      "copies": 9,
+      "erasures": 9,
+      "forward_prefix": 144651,
+      "paid_histogram": {
+        "1": 1320,
+        "2": 1440,
+        "4": 60,
+        "5": 120,
+        "8": 9
+      },
+      "payload": "998206739997057013039187410944",
+      "payload_bits": 100,
+      "rank_mass": 5112,
+      "records": 29874,
+      "reverse_prefix": 2270064
+    },
+    {
+      "adds": 26890,
+      "copies": 9,
+      "erasures": 9,
+      "forward_prefix": 144651,
+      "paid_histogram": {
+        "1": 1320,
+        "2": 1440,
+        "4": 60,
+        "5": 120,
+        "8": 9
+      },
+      "payload": "942069190649173878911112170496",
+      "payload_bits": 100,
+      "rank_mass": 5112,
+      "records": 29848,
+      "reverse_prefix": 2205308
+    }
+  ],
+  "status": "PASS_R420_BLOCK7_EXACT_SCHEDULE",
+  "word_sha256": [
+    "e597e3885c0a8daa262afa5985ee2930443329126aece76d33159d81882bdcc2",
+    "d8faa85175256c99afbd8f8ac8639c0de9be2e5ff3ad7f64012fe506f58dd43a"
+  ]
+}

research/r420-block7-exact-schedule/NATIVE.json

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+{
+  "status": "PASS_R420_BLOCK7_NATIVE",
+  "words": [
+    {
+      "adds": 26916,
+      "payload_bits": 100,
+      "rank_mass": 5112,
+      "stages": [
+        {
+          "bridge_controls": 6,
+          "columns": 900,
+          "stages": 5
+        },
+        {
+          "bridge_controls": 18,
+          "columns": 1380,
+          "stages": 9
+        },
+        {
+          "bridge_controls": 24,
+          "columns": 1620,
+          "stages": 11
+        }
+      ],
+      "word": "original"
+    },
+    {
+      "adds": 26890,
+      "payload_bits": 100,
+      "rank_mass": 5112,
+      "stages": [
+        {
+          "bridge_controls": 6,
+          "columns": 900,
+          "stages": 5
+        },
+        {
+          "bridge_controls": 18,
+          "columns": 1380,
+          "stages": 9
+        },
+        {
+          "bridge_controls": 24,
+          "columns": 1620,
+          "stages": 11
+        }
+      ],
+      "word": "replacement"
+    }
+  ]
+}