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

[thakrarsagar]

This finite bit-circuit construction shortens PR543's R420 cleanup from 2,973 to 2,971 binary XORs, with complete local ADDs 26,916 to 26,914. Two repeated XOR sandwiches share one blocker; exact F2 conjugation replaces five gates by three. Status: research construction / finite circuit improvement; no new kappa.

The forward D/A/scatter/B word stays fixed. All 660 symbolic input columns, arbitrary helper restoration, per-register frame paths, 420 helpers, nine copies and rank mass 5,112 are preserved. The forward positive-prefix bound stays 144,651; reverse decreases 2,270,064 to 2,266,968. All four deleted and two added reads are charged.

The interface change is explicit: the original literal-reversal H4 rejects this representation. The package proves and checks H4-prime, legal full-frame cleanup implementing the same inverse U. No predecessor checker is patched. The identity is over F2, and positive integer envelopes are checked separately.

python3 -B research/r420-cleanup-commutation/verify.py

Expected final line: PASS_R420_CLEANUP_CONJUGATION C=2973->2971 native_adds=26916->26914 columns=660 kappa=unchanged.

Exact base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. PR543 dependency: 2065fe4f72a914d907f1046603f5a36f17018535; both input Git blobs match that head. Only research/r420-cleanup-commutation/ is added. The package contains a complete recipe, exact certificate, written proof, reproducible verifier, attribution and validation receipt.

Validation passed on macOS / Python 3.14.8 with standard-library Python only:

• All 672 frames, all 660 symbolic dirty-input columns, inverse map and unchanged paid paths.
• Complete 18-pair duplicate census. Every duplicate pair has a blocker, proving no cancellation is possible using commuting swaps and pair deletion alone on this fixed cleanup. Both one-blocker pairs are used; no general circuit optimality is claimed.
• Six negative controls and the full 16-input local truth table.
• Optimized-Python execution, minimal relocated archive replay and selected-record binding.
• Independent Codex-worker replay without repository checking/optimizer imports.
• Existing native exporter on both words, with binary counts, per-register MOVE paths, COPY/ERASE, frame/state data and positive prefixes compared. Native records decrease 29,874 to 29,872.

OS assignment seed 10, bit-supplier track. No pinned sprint issue exists, so the claim is local. The final metadata refresh through #709 distinguishes this cleanup-only conjugation from #670/#674/#688 phase-B synthesis, #691 phase-A synthesis, #686/#694 frame-zero sharing, and #708 decoder-total sharing. #708 changes the forward totals and raises measured prefix bounds; this result keeps forward gates fixed and lowers the reverse bound. No pending patch is imported or composed.

The C++ cohort replay was not run because Boost headers were unavailable. Recursive hosted-parent reconstruction, renamed-instruction/residual-host audits, full invoice and changed-word outer assembly remain integration work. Repository-wide make verify, unchanged selected cold replay, Lean and remote CI were not run or claimed. All inherited compiler/common-label/mutable-basis/hosting/signed-lift/precision/tape/analytic obligations remain. The selected frontier and upstream/ are unchanged.

Credit remains with eumemic's PR543/493, Dylan Crow, Zhihao Chen, the #472 realization and the cumulative lineage identified in the README. Prepared for Sagar Thakrar (@thakrarsagar) with substantial OpenAI Codex assistance, including a separate audit worker, under Apache-2.0. No independent human review claimed.

[Mister-Meeseeks]

100-PR batch review at d97c8d20a8b885271a24c5a4e8446ac7f5697d5c.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Cleanup commutation saves two XORs with equal complete F2 action and paid paths; semantic inverse cleanup replaces literal H4 syntax.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.