Research construction: save 137 R420 cleanup XORs with exact four-wire synthesis

research/r420-four-wire-cleanup/CLAIM.md

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+# Sprint claim: four-wire cleanup synthesis
+
+- Track: bit supplier. Fresh OS assignment seed **17**, generated with `secrets.randbelow(100)`.
+- Base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`.
+- Object: selected #543 R420 cleanup C, expanded into its ordered binary XORs. Width selected as `2 + (17 % 3) = 4`.
+- Hypothesis: an arbitrary exact CNOT circuit on a contiguous window's same at-most-four registers can shorten that window. This admits noncommuting identities outside #710's completed commuting-deletion class.
+- Method: enumerate GL(k,2), k=2,3,4, with exact shortest-path distances; exhaust contiguous original cleanup windows on at most four registers. If gains exist, choose nonoverlapping windows and check all arbitrary-input maps and complete costs. No extra helpers, copies, source accesses beyond the counted XORs, frame changes or recursive children may be omitted.
+- Deliverable: a verified finite circuit improvement, or a complete scoped shortest-circuit census with rejection controls. No global circuit-optimality or new kappa claim.
+- Finish target: 2026-10-13 05:43 UTC; reserve the last quarter of actual work for independent verification and handoff.
+- Coordination: no pinned issues exist. Open queue initially 566 PRs through #747. Local claim recorded; no reservation issue opened.
+
+## Three candidates considered
+
+1. Frame-zero target-donor sharing in D: already crowded by #713/#716/#733/#742.
+2. Synthesis of B block 7: resembles #670/#674; #688 already supplies composition and global verification.
+3. Exact four-wire cleanup window synthesis: selected because it changes the search class beyond #710's commuting cancellation and specific conjugation.
+
+## Prior work and scope
+
+#710 `d97c8d20a8b885271a24c5a4e8446ac7f5697d5c` closes commuting-only deletion on the fixed cleanup and gives a different two-sandwich rewrite. This experiment does not repeat that search or import its patch. #688 `bb05c40c3e6319aa8c131b06a105e3648ac6c95b` concerns B blocks 3 and 5 and their changed cleanup. The archived R715 commuting-cancellation experiment is a different word and a narrower identity class. The selected result and actual #543 construction were inspected through the current selected certificate and #626 review. Previous contribution #739 is complex-side and independent.
+
+## Result and handoff — 2026-10-13 05:08 UTC
+
+Completed: [PR #754](https://github.com/CrocSwap/integer-mult-bounds/pull/754),
+construction commit `8ee38f714dc80526c05a6653c1ff56a678615d49`.
+All 7,693 original windows were examined; 284 admit shorter circuits.
+The independently certified optimum over disjoint original windows saves
+137 XORs in 101 replacements, taking the complete word from 26,916 to 26,779
+ADDs with unchanged paid paths and no new κ claim. Both words pass native
+legality and 5/9/11-stage replay, including 96 omitted-bridge controls.
+The final part of the sprint was reserved for independent review, relocated
+replay, corruption controls and this handoff. Final queue refresh through
+#753 found distinct concurrent decoder/compensation work, credited in the
+README without an additive or composed saving claim. No pinned coordination
+issue existed, so this is the single local claim update.

research/r420-four-wire-cleanup/EXPECTED.json

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+{
+  "status": "PASS_R420_FOUR_WIRE_CLEANUP",
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "seed": 17,
+  "saving": 137,
+  "selected_windows": 101,
+  "windows_examined": 7693,
+  "independent_distances": {
+    "2": {
+      "states": 6,
+      "directed_edges_checked": 12,
+      "diameter": 3,
+      "layers": {
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+        "1": 2,
+        "2": 2,
+        "3": 1
+      }
+    },
+    "3": {
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+      "directed_edges_checked": 1008,
+      "diameter": 6,
+      "layers": {
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+        "1": 6,
+        "2": 24,
+        "3": 51,
+        "4": 60,
+        "5": 24,
+        "6": 2
+      }
+    },
+    "4": {
+      "states": 20160,
+      "directed_edges_checked": 241920,
+      "diameter": 9,
+      "layers": {
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+        "1": 12,
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+        "3": 542,
+        "4": 2058,
+        "5": 5316,
+        "6": 7530,
+        "7": 4058,
+        "8": 541,
+        "9": 6
+      }
+    }
+  },
+  "independent_windows": {
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+    "boolean_inputs_checked": 1408,
+    "selected_original_gates": 414,
+    "selected_replacement_gates": 277,
+    "saving": 137,
+    "all_replacements_shortest": true,
+    "nonnegative_transfer_columns_checked": 756,
+    "entrywise_dominance_forward_and_reverse": true
+  },
+  "independent_census": {
+    "binary_gates": 2973,
+    "windows_checked": 7693,
+    "shortenable_windows": 284,
+    "maximum_single_window_saving": 3,
+    "optimal_disjoint_original_window_saving": 137,
+    "by_width": {
+      "2": {
+        "windows": 2973,
+        "shortenable": 0,
+        "maximum_length": 1
+      },
+      "3": {
+        "windows": 1927,
+        "shortenable": 45,
+        "maximum_length": 3
+      },
+      "4": {
+        "windows": 2793,
+        "shortenable": 239,
+        "maximum_length": 7
+      }
+    },
+    "saving_histogram": {
+      "0": 7409,
+      "1": 239,
+      "2": 39,
+      "3": 6
+    },
+    "scope": "All original contiguous windows with at most the stated wires; no extra registers, free permutations or iterative rewrites."
+  },
+  "labels": {
+    "labels": 120,
+    "lattice_rank": 8,
+    "gram_det": 1,
+    "frames": 672
+  },
+  "original": {
+    "records": 29874,
+    "adds": 26916,
+    "moves": 2940,
+    "copies": 9,
+    "erasures": 9,
+    "paid_histogram": {
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+      "2": 1440,
+      "4": 60,
+      "5": 120,
+      "8": 9
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+    "rank_mass": 5112,
+    "phase_adds": {
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+      "2": 2376,
+      "3": 568,
+      "4": 1077,
+      "5": 2973
+    },
+    "forward_prefix": 144651,
+    "reverse_prefix": 2270064,
+    "payload_upper": "998206739997057013039187410944",
+    "payload_bits": 100
+  },
+  "rewritten": {
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+    "adds": 26779,
+    "moves": 2940,
+    "copies": 9,
+    "erasures": 9,
+    "paid_histogram": {
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+      "4": 60,
+      "5": 120,
+      "8": 9
+    },
+    "rank_mass": 5112,
+    "phase_adds": {
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+      "2": 2376,
+      "3": 568,
+      "4": 1077,
+      "5": 2836
+    },
+    "forward_prefix": 109173,
+    "reverse_prefix": 2270064,
+    "payload_upper": "429142879200064664450983477248",
+    "payload_bits": 99
+  },
+  "negative_controls": [
+    "dropped replacement XOR",
+    "reversed source and destination",
+    "overlapping windows",
+    "forged distance potential",
+    "missing GL state",
+    "undeclared ancillary register",
+    "unpaid frame rank",
+    "missing cleanup"
+  ],
+  "rewritten_word_canonical_sha256": "5af46aa89fc362cc3fe9541ab833d4bbb26645ca6d767bce7197e995535afa5d",
+  "literal_H4": false,
+  "semantic_H4_prime": true,
+  "all_input_columns": 660
+}