[petalas]

Summary

The fixed twelve-wire binary map of historical R420 phase-B block 4 has a 29-XOR realization with no workspace and an explicit output-to-wire assignment.

- 51 XOR additions, cited construction in #1363
+ 29 XOR additions, exact new local word

Claim type: finite exact circuit construction, with a written extension to arbitrary binary vector payloads. The new local word saves 22 additions. It answers Cinna Davis / 1337isnot1337's explicit open shortening question in #1383. The first bidirectional row/column trajectory gives the word; the search stopped there. No optimality or new kappa is claimed.

Only five files under research/r420-block4-short-word/ are added. The package displays the complete target, 29 gates, input order, output assignment and inverse contract. Predecessor results are cited and consumed as specifications, not independently audited. The new construction was produced by Nick Petalas. Credit Cinna Davis for #1383/#1363, eumemic's retained #543 R420 source and the PR370/493 lineage; elementary bidirectional synthesis is classical.

Evidence

• Exact packed-row multiplication gives all twelve specified target rows. A separate list-valued Boolean interpreter checks 4,096 forward inputs and 4,096 inverse reconstructions.
• All 29 single-gate deletions, 29 direction reversals, two endpoint corruptions and eleven malformed-program cases are rejected.
• Both registered warning-strict normal and optimized checks pass on CPython 3.12.15 / macOS arm64. Measured local wall times were 0.06s and 0.05s, well below each 60-second registered timeout. These are replay timings, not a performance claim.
• The package launcher passes both commands. Committed bytes/modes and full source-history preflight pass at 41af4479c37619737ad550311b07b31b7df9abcd.
• Official export and declared-input integrity checks pass. An optimized replay outside Git, with empty PATH and read-only exported files, passes with all six export file hashes unchanged. A generated receipt binds the local stdout to all five package file hashes.

From the repository root, reproduce with the exact interpreter path used locally:

~/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/r420-block4-short-word/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -O -B research/r420-block4-short-word/verify.py
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/r420-block4-short-word
~/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/r420-block4-short-word --check-source-history

The direct result is PASS_R420_BLOCK4_SHORT_WORD gates=29, with 4,096 inputs/inverses and witness digest 2d4ac033f5f4ba048c9f411e8b59fd0c84694a499d60b33315803ff967e4b3d1. SOURCES.json pins the source target and explanations at base 19fdb086ea96af9a9f87936a2b888e10af4ed810. The direct check is standalone and reads no predecessor files.

Limitations

The output assignment is explicit and part of the local endpoint. It is not an implemented physical permutation. Binary equality does not preserve the predecessor's positive integer lift, norm envelope, downstream routes or full cleanup word. This PR does not claim a complete R420-word saving, unchanged paid profile, native phases, physical hosting, precision, caller admission, selected-frontier change or complete multiplication theorem. The 29-gate count is an upper bound. Independent human and formal proof-kernel review are unclaimed. No full supplier replay, Lean or Docker ran.

Open next questions

• Install this exact block and assignment in one complete historical caller, rebuild downstream routing and literal reverse cleanup, and price the resulting norm/payload and native bill. A 22-addition local saving does not automatically save 44 in the full word.
• Prove a lower bound for the exact binary target with arbitrary output assignment and no workspace. The successful greedy trajectory excludes no shorter word.

Both questions are unreserved; TASK.json records the first steps and blockers. Scope changes from the current centre-path-sharing card to the historical finite-map handoff, with no current signed-moment consequence inferred.

Related open work

• #612 formalizes historical R420 phase-B boundaries and XOR semantics for the original word; those source-bound certificates do not cover this new gate list.

• #2263 changes the historical R420 RREF checker implementation; it does not synthesize this block, and its code is not consumed here.

• #2276 changes historical R420 mutation-check geometry caching; it concerns verification cost rather than this circuit's gate count and is not consumed here.

• #5935 studies XOR counts for restored identity words with a connected interaction graph. This PR constructs a fixed nonidentity R420 map with assigned outputs; no same-instance quantity is compared.

• #5942 studies the G20 boundary chain/exchange component under its own wire and endpoint contract. This PR studies the distinct twelve-wire R420 block; no common target value is computed.

• #5933 studies a directed-price dirty-fanout word over Z/4. This PR uses unweighted binary XOR gates and a different endpoint; no numerical comparison is asserted.

The post-CI refresh covered all 13 other open account PRs created after the advance CLAIM, #5930 through #5943 excluding this PR. None studies this fixed R420 block or reports a conflicting same-instance gate count. No extra mathematical computations were run to validate or compare those submissions.

CI notes

Both hosted checks passed at exact head 41af4479c37619737ad550311b07b31b7df9abcd in run 40524007509: plan in 4m32s and replay (research/r420-block4-short-word) in 1m18s. The checks were read once after completion, and the run head matches the PR head. Hosted replay confirms the registered finite checks. Local scientific, metadata, source-history, committed-byte and export checks above also passed; independent human and formal proof-kernel review remain unclaimed.

Merge Danger

Door: two-way. This is an isolated research package with no production or selected-record edit.

Blast Radius: package. Main checkout, frozen predecessor sources and selected frontier are unchanged.