Research construction: shorten R420 phase-B block 8 from 47 to 30 XORs (no new kappa)

[jerrylin-23]

Change and mathematical scope

This finite bit-circuit construction replaces phase-B block 8 of the selected PR543 R420 word (frame 550, B[512:559), ordered registers [2,4,16,18,20,22,96,98,101,103,112,118]) with a schedule of 30 gates instead of 47. The literal reverse cleanup C is regenerated from the changed A/B, so it saves 17 more gates. The complete local record stream falls 26,916 → 26,882 scalar additions (34 fewer). No new κ is claimed.

• The ordered positive integer forward matrix of the block is identical in all 144 entries, so every arbitrary block input gives the same exit values.
• The positive reverse matrix is entrywise no larger: 32 entries are strictly smaller, the entry sum falls 321 → 103 and the largest row sum falls 134 → 26.
• All 660 register frame paths, the paid histogram {1:1320, 2:1440, 4:60, 5:120, 8:9}, rank mass 5,112, and the nine copies and nine erasures are unchanged.
• The original R420 checker xcheck.py reports all hypotheses H1-H5 and all four interface conclusions true on the changed word.
• Python positive-prefix replay: forward 144,651 (unchanged), reverse 2,270,064 → 2,212,684. The five-stage local payload stays at 100 bits, below the existing 2^104 cap.

Only research/r420-block8-schedule/ is added: the circuit (CANDIDATE.json), its search (search.py), an exact verifier (verify.py) with CERTIFICATE.json, source pins (SOURCES.json), a native replay harness (native.py) and the local claim (CLAIM.json). upstream/, the selected sources, the selected record and the frontier are unchanged.

Search record. Seeded two-sided nonnegative elementary-factor reduction on the fixed 12×12 forward matrix. The OS-random assignment seed 18 (bit-supplier track) selects block 18 mod 10 = 8. The search seed 801018 was fixed before the first trial. 10,000 trials, each stopped at the current best length: 144 reached the identity, 10 of these also had a dominated reverse matrix, and improvements came at trial 21 (33 gates) and trial 318 (30 gates). A separate seed-18 run was interrupted, found no improvement and was not used. No minimality or exhaustive-coverage claim is made.

Validation

python3 -B research/r420-block8-schedule/verify.py

Expected output (Python standard library only, about 2 s; checked with Python 3.14.6 on macOS arm64):

PASS_R420_BLOCK8_SCHEDULE gates=47->30 ADDs=26916->26882 columns=660 controls=10 rank_mass=5112 kappa=not_recomputed

The verifier does not import the search. It requires and checks the SHA-256 pins of the source word and xcheck.py (and the other files in SOURCES.json); binds the recipe to its block index, frame, slice and ordered registers; checks forward matrix equality and reverse domination; rebuilds the record stream independently of the predecessor exporter; replays all 660 columns forward and in reverse over F2 with copy-lifecycle checks; replays both positive prefixes; requires equal frame paths, an equal paid profile, the exact 2 × 17 saving, frame ranks equal to the original checker's, and true interface flags on both words; and compares the full report with CERTIFICATE.json.

Ten negative controls are checked: equal forward maps without reverse domination (three-wire calibration), a dropped gate, a self-addition, a boolean wire index, a wrong frame, a swapped register binding, an unpriced cancelled pair, a missing cleanup gate, a stale cleanup that passes the F2 endpoint but fails H4, and an illegal frame descent. A separate temporary test confirmed that an empty pin map, a map without xcheck.py and a wrong source hash are rejected.

Not run: native C++ replay of the candidate word. native.py is included for that replay, but only a calibration run (the original word as both inputs, on a pre-final revision of the harness) has been done. No repository-wide suite, selected cold replay, hosted or outer assembly, or CI result is claimed. Inherited obligations remain open: signed decoder evidence beyond F2 column replay, mutable-basis common-label realization and exact-bank completion, and hosted replay, routing, pricing and outer assembly. Finite replay is not a proof of the full multiplication theorem.

Pins, overlap and attribution

• Base commit 66849e1. PR543 source 2065fe4f72a914d907f1046603f5a36f17018535 (git confirms the source word blob matches it); selected consumer PR626 660de893d27bb7abd1d5b01b1640207528a5c851.
• Source word SHA-256 9721a3819efd68307f66f00d84a952eb28eb7c3ad2ce79653ce295c98a764784; rebuilt word canonical-JSON SHA-256 b187b77553eb2f806ab467c33fcd1e85236faa7d393decbf8491332a3c91be9b.
• No pinned coordination issue existed, so the claim is local in CLAIM.json.
• Overlap, refreshed through Independent verification: integral-divisibility kernel replay and radix 81 #804: no other PR changes block 8. Research construction: exact factor schedules save 26 R420 XORs (no new kappa) #789 (yzhouwang, block 7) and Research note: factor R420 block 9 from 38 to 24 XORs (28 fewer native ADDs, no new κ) #790 (commdata2338, block 9) apply similar fixed-block factor schedules to other blocks; this work is concurrent and independent and imports no code from them. Consolidate six circuit PRs: 31.72% fewer bit ADDs and 22–25% fewer complex primitives #681 consolidates other circuit PRs, including the phase-B circuits for blocks 3 and 5, but no block 8 schedule. Research construction: save 137 R420 cleanup XORs with exact four-wire synthesis #754 rewrites short cleanup windows and may touch the block 8 cleanup range. Composition with these or any other pending PR is not checked.
• Credit: eumemic for the PR543 R420 word and the e8-bit-carrier-skeleton exporter and native checkers; Dylan Crow for the bitE8 format and checker lineage; chafreaky for the selected PR626 consumer.
• Prepared by Jerry Lin (@jerrylin-23) with substantial AI assistance in search, implementation, verification and writing. Apache-2.0. No independent human review yet.

[Mister-Meeseeks]

100-PR batch review at 6e1fd9c00c1b0463debaac343db2948e02bcede7.

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

Reviewed scope: Block 8 integer factorization and rebuilt cleanup save 34 native XORs; forward/reverse prefix domination and paid paths unchanged.

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.