Research: shorten R420 cleanup by two XORs with certified conjugation (no new kappa)

research/r420-cleanup-commutation/CLAIM.json

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+{
+  "track": "bit supplier",
+  "assignment_seed": 10,
+  "seed_source": "Python secrets.randbelow(100), operating-system randomness",
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "object": "PR543 R420 full-frame C cleanup, expanded into binary XORs",
+  "method": "Commute independent transvections and cancel identical pairs; exact symbolic replay on all dirty inputs; seed 10 selects deterministic traversal order",
+  "deliverable": "One finite cleanup circuit improvement or scoped negative certificate, with source pins and negative controls; no kappa claim",
+  "expected_finish_utc": "2026-10-13T04:59:00Z",
+  "coordination": "No pinned issues returned by GitHub GraphQL. Local claim; recent PR metadata checked through PR700 before recording.",
+  "candidate_tasks": [
+    {"task": "phase-A quad synthesis", "decision": "deferred: PR691 already supplies exact synthesis on quad13"},
+    {"task": "frame-zero helper sharing", "decision": "deferred: PR686 and PR694 cover single-pair and matching recipes"},
+    {"task": "full-frame cleanup commutation", "decision": "selected: changes only cleanup, unlike the phase-A/B resynthesis and literal reverse cleanup in PR670/674/688/691"}
+  ],
+  "scope": "No upstream, selected certificate, forward circuit, COPY, frame, helper allocation, or assembly changes. Delegated audit stays within cleanup equivalence and its interface/accounting obligations.",
+  "status": "completed: finite circuit improvement and commutation-only negative result",
+  "finished_utc": "2026-10-13T04:38:41Z",
+  "result": "C 2973->2971; complete native ADDs 26916->26914; 660 symbolic columns and paid frame paths preserved; reverse positive bound 2270064->2266968; no new kappa",
+  "pull_request": "https://github.com/CrocSwap/integer-mult-bounds/pull/710",
+  "final_overlap_refresh": "Open PR metadata through #709 immediately before submission; no pinned issue. #708 changes decoder totals and literal reverse cleanup, while this contribution keeps the forward word fixed and changes cleanup by conjugation. No composition is claimed.",
+  "main_at_final_refresh": "66849e1475c930fd7732b1a0067ffee7e1e95f60"
+}

research/r420-cleanup-commutation/README.md

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+# Two fewer XORs in the R420 cleanup
+
+This finite bit-circuit construction replaces five cleanup XORs by three in
+eumemic's PR543 R420 word. The complete local scalar ADD count falls from
+26,916 to 26,914. Every source and dirty helper is restored, all register frame
+paths are unchanged, and the reverse positive-prefix bound decreases.
+There is no new supplier exponent or integer-multiplication kappa claim.
+
+The accompanying negative result closes commutation-only cancellation on this
+specific cleanup: all 18 repeated-gate pairs have a noncommuting blocker.
+The improvement uses an additional F2 conjugation identity.
+
+## Reproduce
+
+From the repository root, with Python 3.11 or later and its standard library:
+
+```sh
+python3 -B research/r420-cleanup-commutation/verify.py
+```
+
+Expected final line:
+
+```text
+PASS_R420_CLEANUP_CONJUGATION C=2973->2971 native_adds=26916->26914 columns=660 kappa=unchanged
+```
+
+The verifier checks the pinned baseline, reconstructs the candidate from
+`certificate.json`, checks all 672 frames and 660 symbolic input columns, and
+rejects six negative controls. It also exhausts the four-register identity on
+all 16 Boolean inputs. These finite checks do not formalize the multiplication
+theorem. All substantive checks remain active under `python3 -O`.
+
+To export the exact candidate as uncompressed JSON for downstream replay:
+
+```sh
+python3 -B research/r420-cleanup-commutation/verify.py --emit-word /tmp/r420-cleanup.json
+python3 -B research/e8-bit-carrier-skeleton/code/bit/export_records.py /tmp/r420-cleanup.json /tmp/r420-cleanup-records
+```
+
+The existing exporter reports 29,872 records. The candidate's canonical JSON
+SHA-256 is `f87e4574addee69e4c9387f0d7139138454cf24e78e2c1358f239a52546b0abd`.
+Source archive hashes are checked separately from canonical payload equality.
+
+## Object and exact substitution
+
+The base is `66849e1475c930fd7732b1a0067ffee7e1e95f60`. Its accepted conditional
+result is kappa `51110230636731/31250000000000000`, or
+`0.001635527380375392`, from #626 with #543. The construction and its remaining
+assumptions are described in the [selected review](../../docs/research/community-k492-review.md).
+This experiment uses the selected bit word from #543 at
+`2065fe4f72a914d907f1046603f5a36f17018535`, not a larger pending PR claim.
+
+Expand each C gate's source list in its stored order into binary gates
+`(destination, source)`. Indices below are zero-based in that original stream.
+Every C gate has full-frame id 1.
+
+| Gate | Positions | Operation |
+|---|---|---|
+| A1 | 714, 1321 | helper 616 ^= helper 530 |
+| A2 | 718, 1320 | helper 616 ^= helper 540 |
+| B | 1071 | helper 466 ^= helper 616 |
+
+For each Ai, B is the only noncommuting gate strictly between its two copies.
+A1 and A2 commute with each other. Commute the Ai copies toward B to obtain
+`A1 A2 B A2 A1`. On the four registers this sequence leaves 616, 530 and 540
+unchanged and sends 466 to `466 + 616 + 530 + 540` over F2. It therefore equals
+the three gates `B`, `(466,530)`, `(466,540)`.
+
+Delete original offsets 714, 718, 1320 and 1321. Insert the two correction gates
+immediately after original offset 1071. All other binary gates retain their
+order. The certificate records this complete construction. Expanding
+multi-source gates changes the number of JSON records; counts here always
+refer to binary XORs or exported native records, not JSON gate records.
+
+## Interface and complete local cost
+
+The historical [PR543 checker](../e8-bit-r420/code/xcheck.py) requires C to be
+the literal reverse of the non-target gates in A and B. This candidate fails
+that structural H4 check as expected. The original checker is unchanged.
+
+Use H4-prime instead: C touches only source/helper registers at the full frame,
+each such register finishes at the full frame, and C induces the inverse of
+the raw source/helper map U. The [PR543 interface proof](../e8-bit-r420/README.md)
+then applies directly: the raw word sends `(z,y)` to `(Uz,y+Jx+Ks)`, and C sends
+that to `(z,y+Jx+Ks)`. D, J, K and the target map are unchanged. Exact matrix
+equality to the original cleanup establishes H4-prime for arbitrary helper
+contents. The verifier reports the original H4 failure and checks H4-prime;
+it does not weaken or patch the predecessor checker.
+
+| Complete local charge | Original | Candidate |
+|---|---:|---:|
+| Binary cleanup ADDs | 2,973 | 2,971 |
+| All binary ADDs, including D and scatter | 26,916 | 26,914 |
+| MOVE records | 2,940 | 2,940 |
+| COPY / ERASE records | 9 / 9 | 9 / 9 |
+| Helpers | 420 | 420 |
+| Forward positive-prefix row bound | 144,651 | 144,651 |
+| Reverse positive-prefix row bound | 2,270,064 | 2,266,968 |
+
+The four removed reads and two new reads are charged. All helpers already
+exist; there is no scratch allocation, extra copy or omitted restoration.
+Each register has exactly its original sequence of frame transitions,
+including its final climb. Thus all recursive child ranks are unchanged.
+The paid histogram, including the nine rank-8 copies, is
+`{1:1320, 2:1440, 4:60, 5:120, 8:9}`, with rank mass 5,112. MOVE timing can
+change; the claim is equality of each register's path, not of the global event
+order. In the inherited local recurrence the child terms are identical and
+the scalar ADD component of the node toll decreases by two. No new complete
+hosted recurrence or assembled bound is asserted.
+
+The circuit identity is over F2, not equality of positive integer shears.
+The cancellation-free bounds are checked separately on the actual forward
+and reverse streams, including COPY/ERASE. The commutations are also valid
+for the positive envelope, and replacing the joint sandwich lowers the
+envelope on helper 616 while preserving the envelope on 466. This explains
+why fewer gates do not hide a larger local coefficient bill.
+
+## Search coverage and controls
+
+The OS assignment seed was 10. Three candidates and the overlap decision are
+recorded in `CLAIM.json`. The finite scan starts at binary cleanup offset 10
+and wraps once. It covers all 2,973 original gates, all 2,955 distinct binary
+gates, and all 18 duplicate pairs. Each repeated gate occurs twice.
+The stopping rule was one complete census, followed by testing every
+one-blocker pair, with no further circuit resynthesis. Exactly two pairs
+have one blocker and both are used in the joint substitution.
+
+Commuting swaps preserve the relative order of a gate and any noncommuting
+blocker. If a first cancellation were possible after commuting swaps, its
+two equal gates could not still have such a blocker between them. Every
+duplicate pair has one, and none has yet been deleted before the hypothetical
+first cancellation. Hence no first cancellation exists. This is an exact
+obstruction for commuting swaps and pair deletion on the fixed word. It
+does not exclude conjugation, insertion of neutral pairs, general linear
+synthesis, changed forward circuits, or other cleanup words.
+
+Controls omit each correction separately, supply the wrong correction
+source, introduce a second noncommuting blocker, try the uncorrected
+three-to-one deletion, and put cleanup at an illegal lower frame. The first
+two omissions and the uncorrected deletion fail exact F2 equality; the extra
+blocker fails the rewrite premise. These are distinct from a failed search.
+
+## Validation boundary and attribution
+
+`VALIDATION.json` records the checks performed locally. A separate Codex
+worker independently replayed the candidate using no repository optimizer
+or checking imports and obtained the same columns, paths and prefix bounds.
+The existing native exporter was also run on both words; its binary records
+confirm the table above and identical frame/state descriptions.
+
+No C++ cohort replay was run because Boost headers were unavailable in this
+environment. The recursive hosted-parent reconstruction, renamed-instruction
+audits, residual-host checks and full invoice/outer assembly remain integration
+work. The full `make verify` suite and selected cold replay were not run for
+this standalone addition; no selected source, generator, certificate or
+`upstream/` file changes. Uniform/compiler/common-label/mutable-basis,
+hosting, signed-lift, precision/tape and analytic obligations remain inherited.
+
+Credit belongs to eumemic's #543 word and checker, the #493 carrier-skeleton
+lineage, Dylan Crow's record exporter and original bit circuit, Zhihao Chen's
+#425 cleanup interface, the #472 common-label realization, and the cumulative
+contributors credited in those packages. Concurrent #670 by yongjae321,
+#674 by x0hu, #688 by Ali Romman and #691 by mtpsl change forward blocks and
+their literal reverse cleanup. #686 by Mikhail R. Gadelha and #694 by Greer
+Guthrie change frame-zero compensation. Tuomo Kiiskinen's #661 fixes the
+phase-B gate word and excludes cleanup synthesis. This result instead keeps
+D/A/scatter/B fixed and changes only C. None of those pending patches is a
+dependency, and compatibility with their changed cleanup words is untested.
+
+The final refresh through #709 also found [#708](https://github.com/CrocSwap/integer-mult-bounds/pull/708)
+by @thakrarsagar at `2ec522a466ec448f148d4355dc43b51c0ef1e0b5`. It shares
+phase-A decoder operands and regenerates literal reverse cleanup, saving 114
+ADDs while raising the measured prefix bounds. The present result uses a
+different algebraic rewrite, changes C alone and lowers the reverse prefix
+bound. It neither imports nor composes #708. #709 caches geometry checking;
+it changes no circuit and is not used here.
+
+Prepared for GitHub account **@thakrarsagar** with substantial OpenAI Codex
+assistance, including the separate audit worker. No independent human review
+is claimed. New files are under the repository's Apache-2.0 license.

research/r420-cleanup-commutation/VALIDATION.json

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+{
+  "status": "finite circuit improvement; no new kappa",
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "environment": "macOS, Python 3.14.8, standard library only",
+  "reproduce": "python3 -B research/r420-cleanup-commutation/verify.py",
+  "expected_last_line": "PASS_R420_CLEANUP_CONJUGATION C=2973->2971 native_adds=26916->26914 columns=660 kappa=unchanged",
+  "passed": [
+    "Exact source archive/checker and canonical word pins",
+    "All 18 original duplicate pairs and both one-blocker identities",
+    "Complete 660-column cleanup matrix equality",
+    "Pinned E8 lattice and all 672 closed frames",
+    "Complete finite word replay with arbitrary source, target and helper inputs",
+    "All 660 register frame paths, including final climbs",
+    "H1/H2/H3/H5 and all scalar decomposition checks; original H4 intentionally false",
+    "Exact positive forward and reverse COPY/ERASE stream bounds",
+    "Six negative controls and all 16 local Boolean inputs",
+    "python3 -O -B research/r420-cleanup-commutation/verify.py",
+    "make selected-record-check",
+    "Independent Codex-worker scalar, frame-path and prefix replay without project imports",
+    "Existing native exporter on both words, with binary record count/path/prefix comparison",
+    "Minimal relocated Git archive with only this package and its two declared sources, Python -I -B",
+    "Both input Git blobs match GitHub PR543 at the declared source head",
+    "Staged whitespace check and unchanged upstream/selected-certificate paths"
+  ],
+  "native_exporter": {
+    "path": "research/e8-bit-carrier-skeleton/code/bit/export_records.py",
+    "source_sha256": "8a5999601f917dbb492bb9ae9a344611f18cc7cedc53dc3c76496a369e530eb3",
+    "records_before_after": [29874, 29872],
+    "move_records_before_after": [2940, 2940],
+    "copy_records_before_after": [9, 9],
+    "erase_records_before_after": [9, 9],
+    "binary_sha256_before": "f7e6cceaceb04d7bd70918981405dcead988a890cfca03b51e80d935f00a960f",
+    "binary_sha256_after": "13ba784c06ed0a5849a225c9f117613d69016363fa3aa41e95355a99467ceb1b",
+    "frame_and_state_json_equal": true,
+    "per_register_move_paths_equal": true,
+    "copy_erase_subsequence_equal": true
+  },
+  "not_run": [
+    "C++ native cohort replay: Boost headers unavailable",
+    "Hosted-parent reconstruction, renamed-instruction/residual-host audits and complete invoice/assembly",
+    "Repository-wide make verify and unchanged selected cold replay",
+    "Remote CI"
+  ],
+  "review_scope": "Finite computational evidence and written local algebra; not a formal proof of the multiplication theorem or independent human review"
+}

research/r420-cleanup-commutation/certificate.json

@@ -0,0 +1,46 @@
+{
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "source_pr": 543,
+  "source_head": "2065fe4f72a914d907f1046603f5a36f17018535",
+  "source_git_blobs": {
+    "research/e8-bit-r420/data/bitE8mut-r420.json.gz": "0c63ca62759dc63d302be6552aeeee1ede70350f",
+    "research/e8-bit-r420/code/xcheck.py": "fbff8607e4c61e18481f696b99dd38a32444e90d"
+  },
+  "source_sha256": {
+    "research/e8-bit-r420/data/bitE8mut-r420.json.gz": "9721a3819efd68307f66f00d84a952eb28eb7c3ad2ce79653ce295c98a764784",
+    "research/e8-bit-r420/code/xcheck.py": "5737d18a7826fd51e131035f2d7bebba2e2868af91dd7672a5f11208b036d424"
+  },
+  "canonical_input_sha256": "e597e3885c0a8daa262afa5985ee2930443329126aece76d33159d81882bdcc2",
+  "canonical_candidate_sha256": "f87e4574addee69e4c9387f0d7139138454cf24e78e2c1358f239a52546b0abd",
+  "cleanup_binary_adds": [2973, 2971],
+  "native_adds": [26916, 26914],
+  "positive_prefix_before_after": [[144651, 2270064], [144651, 2266968]],
+  "recipe": {
+    "indexing": "Zero-based original C after expanding each source list in its stored order",
+    "delete": [714, 718, 1320, 1321],
+    "insert_after": 1071,
+    "insert_gates": [[466, 530], [466, 540]],
+    "full_frame_id": 1
+  },
+  "duplicate_census_columns": ["first", "second", "gate_dst_src", "blocker_count", "first_blocker"],
+  "duplicate_census": [
+    [634, 1358, [598, 532], 3, 721],
+    [714, 1321, [616, 530], 1, 1071],
+    [718, 1320, [616, 540], 1, 1071],
+    [982, 2375, [658, 335], 8, 1361],
+    [985, 2393, [649, 307], 7, 1381],
+    [986, 2391, [650, 428], 7, 1514],
+    [989, 2385, [653, 417], 8, 1222],
+    [991, 2405, [643, 468], 7, 1269],
+    [992, 2403, [644, 257], 8, 1556],
+    [994, 2399, [646, 478], 8, 1245],
+    [998, 2415, [638, 277], 7, 1164],
+    [1003, 2429, [631, 272], 6, 1376],
+    [1012, 2435, [628, 249], 5, 1209],
+    [1015, 2453, [619, 295], 6, 1377],
+    [1060, 2531, [580, 321], 3, 1234],
+    [1078, 2567, [562, 333], 4, 1489],
+    [1083, 2581, [555, 297], 8, 1567],
+    [1084, 2579, [556, 460], 3, 1608]
+  ]
+}