Finite R420 construction: restored helper matching and shared target donors save 8,412 XORs (no new kappa)

[franciscoyne001-svg]

Change and mathematical scope

Contributed by franciscoyne001-svg, with substantial OpenAI Codex assistance in construction, proof, checking and packaging. This is a finite circuit construction on the historical PR543 R420 word, with no new kappa or practical runtime claim.

Add research/r420-two-sided-compensation/: a frozen recipe, proof, exact verifier, separately implemented set-based audit, negative controls and native execution receipts. Combining 202 disjoint helper transvections with 68 target transvections using 47 shared donors reduces D compensation from 19,922 to 11,510 ADDs, including both preparations and restorations. The otherwise unchanged PR543 native word falls from 26,916 to 18,504 ADDs: 8,412 saved. Helpers, COPY/ERASE counts, frame/state files, the complete MOVE/COPY/ERASE subsequence, paid ranks and rank mass are unchanged. The recomputed local payload uses 103 bits under the 104-bit cap.

This extends credited mechanisms, not a priority or best-known claim. In particular:

• Research: compose R420 bases for 8,368 fewer XORs #733 combines helper and disjoint target bases, saving 8,368 on the same baseline; this construction saves 44 more in total, with a different helper basis.
• Research: shared target donors save 3,288 R420 XORs (no new kappa) #742 supplies shared target donors with immutable helpers. Here shared target donors are combined with a restored helper basis.
• Certify fixed-basis R420 target matching and the eight-ADD gap #817 certifies a different fixed-basis disjoint matching at D=11,546, compared with 11,510 here. Its smaller published whole-word count also incorporates separate Consolidate six circuit PRs: 31.72% fewer bit ADDs and 22–25% fewer complex primitives #681 changes and is not directly comparable.
• Research construction: R420 target-basis compensation saves 7,879 XORs within the existing 192-bit nine-stage prefix charge (no new κ) #794 supplies a hosted precision invoice this package does not provide. Later R439 work concerns a different word.

The bounded overlap refresh inspected 200 recent open PRs through #946 and indexed R420-title, donor, two-sided and 8,412 searches. No identical recipe surfaced; this is not an exhaustive novelty guarantee. No pending PR code or frozen recipe is imported. Credit to eumemic, Mikhail R. Gadelha, Greer Guthrie, Scott Woolsey Biggart, Sagar Thakrar, ellipsi2, Dylan Crow, Zhihao Chen, DreamingOfClouds, Chafik Boukhalfa, DaysSky and the upstream lineage is retained in the package.

The original syntactic H2 and literal K extractor intentionally reject this replacement. The package proves and checks the semantic D endpoint (x,s,y) -> (x,s,y+Ks) through M=T K H, restoring arbitrary helpers and targets. Integration still requires acceptance of that interface, full hosted-parent/whole-owner replay, revised global invoices and changed-word outer assembly. Inherited all-size, realization, hosting, residual, precision/tape and analytic assumptions remain. No selected-frontier files are edited.

Validation

python3 -B research/r420-two-sided-compensation/verify.py

Passed again on upstream main 007664306fff84fbb1c9a1b4053b69f439764034:

PASS_R420_TWO_SIDED D=19922->11510 native_ADDs=26916->18504 saved=8412 columns=660 no_new_kappa

Checks include the complete E8 frame audit, 660 arbitrary-input columns, independent set-valued replay, fixed-donor optimal receiver assignments, eight main and three independent negative controls, serialized native paths and prefix accounting, and 5/9/11-stage Python checks. Earlier unchanged-source C++ legality and 5/9/11-stage checks passed for both words; normalized receipts are retained in NATIVE-CPP.json. A missing native D ADD passes legality but is rejected by all three C++ column checks. C++ was not rerun in the final standard-library replay because Boost was not configured for that invocation.

Also checked isolated optimized Python, relocation/recompression, corrupted source/certificate rejection and staged whitespace. Full make verify, current selected-frontier cold replay, hosted-parent/global-invoice integration, remote CI and a Lean proof of the full multiplication theorem are not claimed. These are finite checks and written proof arguments, not independent human review.

[Mister-Meeseeks]

100-PR batch review at 49868be647e52c11a4e305a10d4cedb915a7cf3d.

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

Reviewed scope: Historical R420 two-sided dirty compensation, restoration and independent finite replay; optional native compilation separately identified.

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.