[petalas]

Summary

Scoped algebraic proof with exact finite corroboration: four inverse cyclic targets diag(g^2,g^2,g^2,g^2) in full F2[C3] require exactly eight positive-g rank-one idempotent calls on exactly four arbitrary dirty roles, even with free interleaved binary scalar mixing.

- Rank and determinant permit a five-call sharing attempt
+ A row-once factorization excludes five; eight coordinate calls are optimal

The proof removes the first factor of a hypothetical five-call word. Four independent binary update directions then force a rank-one projector to have rank four. This is an original changed-alphabet side result for the BIT register-model card. No new kappa or selected-frontier change is claimed.

This contribution was produced by Nick Petalas. Credit eumemic's register framework, Ali Romman's rank-one context, Wentao Li's cyclic 2-power work, Ross Richardson's distinct cubic-caller boundary and the retained finite-group lineage. These are context and cited premises, not predecessor scientific replays. Elementary finite-field, rank and determinant facts are classical; global literature priority is unclaimed.

Evidence

The written proof settles the only count below eight left by rank and determinant. Two exact arithmetic engines corroborate the exclusion: packed F4 rows and separate full F2[C3] coefficient convolution each test all 8,361 normalized complements and find zero five-call hits. The verifier also checks all 65,536 binary matrices for the 120-projector inventory, 4,096 row-once factorization cases, every formal coefficient of the eight-call witness, all twelve faithful regular-action dirty-input columns, eight omitted-call rejections and six semantic controls.

Local CPython 3.12 warning-strict normal and optimized commands pass, approximately 0.30 and 0.31 seconds each, well below the 120-second registered limits. These are check durations, not a performance claim. Registered metadata/replay, committed bytes/source ancestry and official export/input integrity pass. Both modes pass in the read-only Git-free export with unchanged bytes. Three disposable optimized CLI mutations reject wrong field arithmetic, wrong cyclic convolution and omitted witness calls. No generated receipt is handwritten; stdout receipts bind all six package files by SHA256.

Reproduce from the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/c3-four-inverse-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/c3-four-inverse-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/c3-four-inverse-tariff
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/c3-four-inverse-tariff --check-source-history

Use another CPython 3.12 interpreter path elsewhere. Expected markers include PASS_C3_FOUR_INVERSE_TARIFF minimum=8 five_call_hits=0, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

The result fixes exactly four dirty roles, binary rank-one idempotents and only the positive g label. It does not cover extra restored spectators, an independently paid g^2 label, clean allocation, erasure or nonbinary scalar mixing. Physical/native realization, frame and scalar charges, access, ownership, routing, restoration, compiler, precision, stock, tape and complete recursive invoices remain unpriced. There is no complete multiplication theorem or selected-supplier admission. Independent human and formal proof-kernel review are unclaimed. No Lean, Docker, selected cold replay or predecessor scientific suite was run; every registered package check ran locally.

Base is freshly fetched canonical main 19fdb086ea96af9a9f87936a2b888e10af4ed810, priority revision 8. Entry remote overlap covered 341 open PRs and 5,523 Discussion records, supplemented by main-tree and merged/all-state searches and a precommit live refresh. Sparse local notes and unpublished work remain outside the novelty coverage. The advance CLAIM sought a smaller word or a sharp bounded obstruction; the delivered result is the exact four-role eight-call minimum.

Open next questions

Can one additional restored dirty role lower the same four-target tariff? First analyze the initial coupling relative to target and spectator subspaces. The scalar-target proof no longer applies and no five-role search was run. Any physical consequence also needs a fully priced positive-g primitive and complete caller.

Related open work

• PR #5947 uses the same register model: four dirty roles and calls drawn from the 120 rank-one idempotents of M4(F2). It has a different group and target, diag([xyx],[xyx],[e],[e]) in F2[S3]. No numbers are compared.
• PR Construction: three-stage persistent prefix shear chain (no new kappa) #1930 studies a persistent prefix-shear chain; its C3 fanout terminology is a different object from the cyclic group C3 here. No shared numerical minimum is compared.
• PR Prove exact six-call nonunit fanout on a directed cycle (no κ) #5933 studies a directed Z/4 fanout tariff, a different group of gates and endpoint. No shared numerical minimum is compared.

The completion refresh inspected all ten other open PRs from this account created since the advance CLAIM at 2026-10-27T16:08:27Z. None studies this fixed positive-g C3 inverse endpoint or computes its tariff. No additional mathematical computations were run to compare those contributions.

CI notes

Both hosted checks passed at exact contribution head ccea0bf1100b54d3ae84b72e5850cfa491cb3d4c in run 40539516329: plan in 8m58s and replay (research/c3-four-inverse-tariff) in 2m14s. The checks were read once; run metadata confirms that head, completed status and success. Hosted replay certifies the registered normal and optimized package checks. Local metadata, committed-byte/source-history and read-only Git-free export checks also pass. Independent human and formal proof-kernel review remain unclaimed.

Merge Danger

Door: two-way. Blast radius: isolated. Only six files in research/c3-four-inverse-tariff/ are added. Shared code, frozen predecessor inputs and selected records are unchanged.