Research: save 6,934 R420 XORs with a restored helper basis

[g-guthrie]

This reduces the accepted PR543 R420 bit word by 6,934 binary XOR additions: 26,916 → 19,982, about 25.8%. A matching of 201 disjoint helper pairs factors the frame-zero dirt-compensation matrix and restores every helper before phase A. The D stage falls from 19,922 → 12,988 ADDs, including all 201 preparation and 201 restoration operations. Status: finite circuit improvement; no new κ.

The complete per-register frame paths, 420 helpers, nine COPYs and recursive profile remain unchanged; rank mass is 5,112. The measured positive-prefix bounds rise, so they are recomputed rather than inherited: forward 144,651 → 202,699, reverse 2,270,064 → 2,446,788, and the payload allowance 100 → 102 bits, still below the checked 104-bit cap.

Concurrent #686 is credited for the same one-pair mechanism and semantic H2 contract. Its helpers 552/521 occur in this matching. This contribution adds the simultaneous 201-pair construction, complete 6,934-ADD reduction and changed prefix accounting; no priority for the one-pair idea is claimed and no #686 code is imported. #670/#674/#688 modify and compose different phase-B blocks and are not combined here.

The contract change is explicit. The original H2 forbids helper destinations in D, and its literal K extractor cannot read a temporary helper basis; both still reject the new syntax. No predecessor checker is patched. The new prefix contract verifies the full (x,s,y) → (x,s,y+Ks) operator on all 660 independent input columns, restores all 420 arbitrary helper inputs before A, and requires every field outside D to match the pinned source. Actual changed-word frame/scalar replay and all other original interface predicates pass. The written linear-algebra composition proof states the inherited common-label boundary.

python3 -B research/r420-compensation-basis/verify.py
# Add --native for the unchanged C++ legality and 5/9/11-stage checks.

Expected: PASS_R420_COMPENSATION_BASIS D=19922->12988 native_ADDs=26916->19982 saved=6934 rank_mass=5112 payload_bits=102<104.

Local validation on macOS / Python 3.14.6:

• Exact original E8/frame audit, complete forward/inverse 660-column replay, independent set-valued D endpoint, and source-bound canonical interface proof.
• Both original and changed words passed the unchanged record exporter. The independent literal streams, all frame paths, profiles, counts and integer positive-prefix bounds match; records fall from 29,874 → 22,940.
• Seven rejection controls and a positive control that requires an F2-neutral pair to pay its two ADDs.
• Normal and isolated optimized execution; relocated minimal checkout and gzip recompression; corrupted source, recipe and underpriced certificate rejected.
• make selected-record-check, source pins, documentation links and staged whitespace checks pass.

Fresh unchanged C++ checks also passed on macOS and in focused Linux CI on Python 3.11 and 3.14 at commit 9732ac127dd4bef98a846aaa904c20b5a84e72c1: 900/1,380/1,620 formal columns, all 6/18/24 bridge-omission controls, legality and exact prefix/profile agreement. Missing helper restoration is rejected by the C++ endpoint check at every stage count. The local native run used Python 3.14.6, Apple Clang 17.0.0 and temporary Boost 1.85.0 headers; the official archive SHA-256 was 7009fe1faa1697476bdc7027703a2badb84e849b7b0baad5086b087b971f8617. The committed local validation file records the preceding standard-library checks; the linked CI records the completed native checks.

Base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. PR543 source: 2065fe4f72a914d907f1046603f5a36f17018535. Assignment seed 12. All 87,990 helper pairs were scored in both orientations, followed by one seeded greedy disjoint matching; no optimum is claimed. The overlap refresh covered 510 open PRs through #690, followed by an immediate check through #693. No pinned coordination issue existed; the local claim records the scope and concurrent #686 attribution.

Only the new research package and its focused workflow are added. Full hosted-parent reconstruction, whole-owner/finite invoices, integration of the semantic D-prefix contract and a changed-word outer assembly remain separate obligations. No selected-frontier change, practical multiplication speedup, signed integer endpoint equality, repository-wide suite pass or Lean proof is claimed.

Prepared by Greer Guthrie with substantial OpenAI Codex assistance. The package retains credit for eumemic, Dylan Crow/DreamingOfClouds, DaysSky, Zhihao Chen, Boukhalfa, icekylinx, Arun, Mikhail R. Gadelha and the cumulative predecessor lineage. Apache-2.0.