[petalas]

Summary

Scoped mathematical proof with exact finite corroboration, produced by Nick Petalas. The actual cut58 operator families satisfy, for every positive tensor multiplicity f,

P W K P W = -T W K P K
        => chi(K) chi(W) chi(T) chi(P) = 1

Every scalar-fixing extension of either separable tariff test chi_W or chi_T must therefore be odd on the rank-zero ADDRESS swap P. Neither test extends with P even, as the positive native-charge counting argument would require. This resolves part of Open next question 1 in unpaidmoon's merged #4149. Ross Richardson's separable 22f theorem in #4086 remains within its stated alphabet.

Only research/cut58-free-address-character-obstruction/ is added. The new relation and sign obstruction are the contribution; no predecessor census or 22f endpoint is audited or replayed. Credit unpaidmoon for the ADDRESS menu/question; Ross Richardson for the operators, physical tensor conventions and separable proof; icekylinx, Rohan Arun, Chris R / Clear1nsight, ToikY and the retained native lineage. Advance CLAIM.

Evidence

Before, #4149 left extension of the three tariff characters past its ADDRESS menu open. After, a ten-letter exact relation rules out the two stated extensions with P even. The sign remains negative at even f because the global i factors stay outside the tensor powers.

Local CPython 3.12.15 passes PASS_CUT58_FREE_ADDRESS_CHARACTER_OBSTRUCTION in warning-strict normal and optimized modes:

• A new Boolean multilinear polynomial interpreter derives the complete coordinate and phase formula. A separate literal interpreter checks both words on all 4,096 twelve-bit basis columns, including every spectator setting.
• All 16 generator sign assignments are enumerated. Eight satisfy this relation, four with P even. These are assignments unexcluded by one relation, not a count of actual characters.
• Boundary controls distinguish unsigned equality, omitted ADDRESS, discarded global phases, phases incorrectly placed inside even tensor powers, and permitting P to be odd.
• The registered package replay, exact committed bytes and source ancestry pass at abedf047307701d83e40566b06db51e8427815c3. Both registered argv were timed directly, 0.112 and 0.115 seconds, below their 60-second bounds. These are replay-budget measurements.
• The official committed export passes integrity and declared-input coverage. A fresh read-only optimized Git-free replay with empty PATH passes with all eight exported files unchanged. Changing the exported operator input rejects at its SHA256 gate; restoring it restores the original export hashes.

From the repository root, exact reproduction commands:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -I -B research/cut58-free-address-character-obstruction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -I -O -B research/cut58-free-address-character-obstruction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/cut58-free-address-character-obstruction
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/cut58-free-address-character-obstruction --check-source-history

Any CPython 3.12 installation can replace the local interpreter path. Standard library only. Each verifier run prints a generated receipt bound to the six package files and the sole declared source input. No receipt is handwritten.

Limitations and unrun obligations

The relation is a necessary condition for characters and scalar-linear sign twists. It neither classifies the full character group nor establishes extension of chi_KW or any other surviving assignment. The written tensor argument establishes all positive f; the finite computation corroborates its one-coordinate identity.

No cheaper six-bank word, enlarged-alphabet 22f theorem, native supplier, complete paid invoice or new kappa is supplied. ADDRESS's physical costs, BIT compatibility, ownership, compiler, precision/tape and analytic obligations remain. No selected cold replay, Lean, Docker, independent human review or formal proof-kernel acceptance is claimed. The selected frontier is unchanged.

Related open work

No open PR in the checked complete 387-PR snapshot studies the cut58 K/W/T/P character extension. The source #4086 and #4149 are merged predecessors.

The completion refresh read this account's 13 other open PRs created since the advance CLAIM: #6121 through #6134, excluding this PR #6132. They concern defect guards, binary query capacity, centre kernels, sampler services, leaf laws, Gaussian bounds, dyadic gauges, scalar resynthesis and random query compression. None studies this cut58 character extension or reports a competing value for this result. No additional mathematical computation was run for comparison.

Open next questions

• Does chi_KW extend with P even and respect the represented-algebra relations at the required tensor multiplicities? Give a structural formula or a new exact relation that excludes it. This relation alone proves neither existence nor tensor faithfulness.
• Can another invariant or an explicit fully charged word settle the enlarged six-bank cleanup price? Begin with the exact cut58 matrix and an alphabet containing P; losing these two tests supplies no cheaper word.

CI notes

Uncertified by CI. At commit abedf047307701d83e40566b06db51e8427815c3, run 40693575177 failed its plan job in the onboarding helper tests. The package replay was skipped and did not run in CI. The registered scientific commands and publication/source-history/export checks passed locally; this hosted run certified none of the package's scientific checks. A maintainer re-run is needed after main is fixed.

The exact cause is the selected-record/documentation mismatch on current main 721c03822c421b7f7fcbb2598993b947c4d88419. That PR6030 kappa-admission commit was pushed with [skip ci] and updated the selected record without matching the onboarding documents:

• test_research_current_status.CurrentStatusTests.test_current_section_matches_selected_record expects the literal **0.003125804** in docs/research/current-status.md. The document instead bolds 0.003125804 = 781451/250000000 as one expression, so the exact substring assertion fails.
• test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record expects make frontier6030-verify in docs/ci.md, which still describes Raise conditional kappa to 0.00310045 with independent helper splits #5599 and make frontier5599-verify.

The same two failures occur in PR #6120's run 40691776027, confirmed from its failed-job log. The completed PR checks were read once. This contribution changes neither failing document nor the selected record. Its passing local evidence remains separate from the missing CI certification.

Merge Danger

Door: two-way. Blast Radius: isolated.

One opt-in research package is added. Shared code, frozen predecessors and selected-frontier records are unchanged.