[petalas]

Summary

Resolve Open next question 1 of Wentao Li's #3788. The six-label UT3(F4) central triple still has exact minimum twelve after admitting every nonzero binary rank-one coupling, including nilpotents, and arbitrary restored dirty workspace.

before: rank-one idempotents                  -> exact minimum 12
after:  all nonzero binary rank-one couplings -> exact minimum 12
proof:  extension-field quotient -> UV=0 -> n_X,n_Y >= 2d

Claim type: scoped algebraic proof with exact finite corroboration. The same written argument gives the 4d central-target law over every characteristic-two address field. The predecessor's coordinate upper word retains its credit; the new contribution is the enlarged-alphabet lower bound. No predecessor executable was replayed. No new kappa or selected-frontier change.

Package research/ut3-f4-nilpotent-tariff/ was produced by Nick Petalas. Independent human and formal proof-kernel review are unclaimed. Credit Wentao Li #3774/#3788, Cinna Davis #2261, Arham Tahir #1589, Hanson Sun #1607, steph4n-gh #1573, James Reed #3783-#3786 and the retained register-model lineage. Exact source pins and full attribution are in the package.

Advance claim.

Evidence

Before, #3788 explicitly excludes non-idempotent couplings and leaves this question open. After, the quotient over the address coefficient field proves at least six X calls and six Y calls without scalar idempotence or a workspace bound.

Local CPython 3.12 checks pass on cb5230dea14ba310d55d92396c9f0bb28a34bf42:

• Registered warning-strict normal and optimized replay; timed direct commands take 2.53 seconds each, below their 120-second limits.
• All 160,435 expanded-alphabet two-role words through length three, including 112,446 containing nilpotents, satisfy the extension-field coefficient-span bounds. This is finite corroboration, not a length-eleven exclusion search.
• Exact quotient homomorphism on all 4,096 group products; 256 field/binary rank comparisons; 294 paid involutions; nine complete dirty endpoints and 1,856 literal regular-action basis columns. All 2,971 words through length two also have full formal and quotient agreement, with 5,942 additional identity-address column comparisons.
• Twelve omission controls and named wrong-coefficient, dirty-only, nonfaithful-quotient, illegal-alphabet and hypothesis controls pass.
• Metadata/source bindings, committed-byte/source-history preflight, AST inspection and whitespace checks pass. The generated receipt binds all five source/metadata files.
• The official six-file committed export passes integrity checking. Optimized replay also passes from a read-only Git-free export with empty PATH. Altered-receipt and altered-proof copies reject, and the original export stays byte-identical.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/ut3-f4-nilpotent-tariff
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/ut3-f4-nilpotent-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/ut3-f4-nilpotent-tariff/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/ut3-f4-nilpotent-tariff --check-source-history

Expected markers: PASS_UT3_F4_NILPOTENT_TARIFF minimum=12 nX=6 nY=6, PASS_REGISTERED_RESEARCH_PACKAGE, and PASS_RESEARCH_COMMIT_BYTES. Substitute another CPython 3.12 path outside this environment.

Limitations and unrun obligations

The arbitrary-field and arbitrary-workspace statements are written algebraic proofs, with F4 finite corroboration. No proof-kernel formalization, native supplier realization, recursive rank identification, compiler/hosting/precision/tape bill, selected cold replay or outer analytic transfer was run. No complete multiplication theorem is claimed. The formal alphabet's physical price remains unknown. There are no locally unrun registered commands. The selected conditional kappa stays 0.00302015.

Open next questions

Can a physically realizable additional address operation escape the exact axis-alphabet tariff with a favorable complete bill? First pin a literal entry/call/exit contract and charge its frames, preparation, dirty restoration, hosting and precision. A legal physical realization and recursive price remain the blockers.

Related open work

• Prove exact monotone UT3 rank tariffs (no new κ) #4846 studies common monotone rank tariffs for UT3(F2) block calls. This PR studies rank-one calls with variable F4 axis coefficients and settles the separate nilpotent-coupling floor. Neither result is used to establish the other; no new numerical comparison is asserted.

Before DONE, this account's open PRs created since the advance CLAIM were refreshed at 2026-10-22 22:02 UTC. No additional PR in that snapshot studies the same F4 nilpotent central-triple floor. The fixed-factor workspace and S3 rigidity submissions address different quantities or groups; no shared numerical value is compared or recomputed. Canonical main remains 624de9dc66858af0945e5f8a0942488d62280d43, priority revision 7, at the final fetch. Remote overlap and distinctive merged/main searches have the coverage limits recorded in the package.

CI notes

Both hosted checks passed on publication head cb5230dea14ba310d55d92396c9f0bb28a34bf42 in run 39861008611: planning in 3m23s and registered package replay in 1m33s. Run metadata confirms that exact head, completed status and success. The package job executes both warning-strict normal and optimized registered commands. These automated checks certify the bounded replay; independent human and formal proof-kernel review remain unclaimed.

Merge danger

Door: two-way. Blast radius: package.

Six isolated research files and an opt-in replay manifest. No shared implementation, frozen predecessor, selected source or frontier changes.