Research: simplify one R420 block from 50 to 33 XOR gates (no new kappa)

[yongjae321]

This reduces one finite circuit in eumemic's PR543 R420 bit word from 50 to 33 binary XOR gates, preserving its exact ordered 12-register F2 map. The matching literal reverse cleanup saves another 17 gates: 34 fewer scalar ADDs overall (26916 to 26882). This is a research contribution / finite local circuit improvement, with no new kappa claim.

Both forward and reversed nonnegative transfer matrices are entrywise dominated by the originals. Every register's physical frame path, the paid histogram, 420 helpers, nine retained copies and rank mass 5112 remain unchanged. The local cohort forward prefix stays 144651; its reverse prefix drops from 2270064 to 2185964. The payload allowance remains 100 bits. No hosted supplier invoice is claimed.

Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. PR543 reviewed dependency: 2065fe4f72a914d907f1046603f5a36f17018535. SOURCES.json pins the input and relevant predecessor files. Only research/r420-block3-xor/ is added; upstream and the selected frontier are unchanged.

python -m pip install -r research/r420-block3-xor/requirements.txt
python -B research/r420-block3-xor/verify.py

Expected: PASS_R420_BLOCK3_XOR gates=50->33 saved_with_cleanup=34 columns=660 rank_mass=5112 controls=10.

Passed on Windows/Python 3.12.14 with SymPy 1.14.0 and mpmath 1.3.0:

• Pinned PR543 labels, frames, all 660 symbolic dirty-input columns, H1-H5 and their decomposed conclusion.
• Independent set-based column replay and reconstruction; exact frame paths, positive transfers and prefix quantities agree.
• Ten corruption controls, including F2-neutral positive-bill inflation and F2-correct cleanup that fails literal-reversal H4.
• Actual inherited record exporter: 29874 to 29840 records, with scalar streams and paid histogram checked exactly.
• Frozen seed-13 search reproduction; Windows launcher; optimized Python; existing-output protection; selected-record binding; staged whitespace check.

Full C++ multistage replay, hosted invoices, frontier assembly, repository-wide make verify, Lean and remote CI were not run for this standalone local addition. No signed endpoint, optimal gate count, practical speedup or complete multiplication theorem is asserted. The README gives the precise positive-prefix argument, inherited assumptions, search domain/stopping rule and remaining integration work.

The assignment seed was 13, selecting block 3 by 13 mod 10. Search stopped after 50 x 400 attempts. No pinned coordination issue existed; CLAIM.json records the narrow local claim and completed result. PR metadata was refreshed through #669. PR661 fixes the scalar gates/order and explicitly excludes resynthesis; PR612 formalizes the original word's literal interfaces; PR589 transports gates in a different punctured word. This contribution is distinct and depends on none of them.

Credit: eumemic's PR543/493 circuit and interface; Dylan Crow, DreamingOfClouds, Zhihao Chen, Chafik Boukhalfa, DaysSky and the cumulative predecessor lineage detailed in the README. Prepared for yongjae321 with substantial OpenAI Codex assistance, including separate nomination, checking and review workers. Apache-2.0; no independent human peer review claimed.

[Mister-Meeseeks]

Integrated at ee055dc44815497927ef618fc985b93c2dcbda2a in the 20-PR pilot, retaining your original commit, source notices and contributor credit.

Block 3: 50→33 XORs, 34 saved including cleanup. Exact F2 endpoint, unchanged paid frame paths, positive-transfer domination and all 660 dirty columns checked; no new exponent.

The scoped package check passed at 2e31fcad7f9ecc8b97039ecb2333e581fd725fa1. This is research integration; the selected κ and inherited theorem assumptions are unchanged. Review, evidence and attribution. Thank you for the contribution.