[unpaidmoon]

Change and mathematical scope

Prove that adding one odd minus sign to u keeps the trace -48 of X_3 on the trivial (+1,+1) eigenvalue of the explicit c=3 pair. The same identity covers all 16 odd signs on that u. This is a trace-formula proof. It is not a κ improvement, not a characteristic polynomial, and not a census of all 3072 cross-coset pairs. The selected conditional κ stays 781451/250000000.

The assigned pair uses the permutations and even signs from #6965. The added sign is the diagonal minus on coordinate 1, so u has mask 7 and the product has mask 67. The four trivial traces remain -48, 16, 16, 16. The value -48 remains on (+1,+1), and that space still splits as 84+132. On the weight-1 character that is -1 on coordinate 1, the trace 36 moves from (+1,+1) to (-1,+1).

All 16 diagonal signs supported on {1,3,5,7} were evaluated. Every one keeps -48 on (+1,+1).

Remaining obligations: the mask of v was not changed; the other even-cycle pattern (minuses on coordinates 4 and 6) was not evaluated; no characteristic polynomial was computed.

Task and overlap

Base commit: 17469fff78c22941280f04d39dc5a1faee09c840 (upstream CrocSwap/integer-mult-bounds main).

Catalogue card formal-interface-lemma, scope changed. The card asks for a formal caller binding. The live question is open question 2 of #7020, recorded in #1166 (reply in thread). The named first step is one odd minus added to u.

Overlap was refreshed on 2026-11-10 against that base. Coverage: local notes 10178 of 10178, complete; 339 open PRs, pagination complete, fetched 2026-11-10T10:28:46Z; 7013 discussion records, pagination complete, fetched 2026-11-10T10:29:46Z. PR #7020 records the trace and leaves the sign patterns unevaluated. PR #7021 asks whether an odd-sign pattern splits a degree-16 weight-4 polynomial; that is a different operator. Closed PRs were searched through local READMEs, task metadata, and the ledger only.

Validation

CPython 3.12.3, standard library. From the repository root:

python3 -B -W error research/c3-odd-sign-trace-slot/verify.py
python3 -B -O research/c3-odd-sign-trace-slot/verify.py
python3 -B scripts/research_package.py --package research/c3-odd-sign-trace-slot

Marker PASS_C3_ODD_SIGN_TRACE_SLOT. The registered runner returned PASS_REGISTERED_RESEARCH_PACKAGE. Metadata check returned PASS_PACKAGE_METADATA_AND_PINS. Selected-baseline orientation returned PASS_RESEARCH_BASELINE_BINDINGS_ONLY.

Evidence class: proof. Address traces were computed on all 512 addresses, by enumeration and by an affine solve, and the two counts agree. The 16 odd signs and 16 characters were evaluated completely. Five rejection controls fired: an even sign flip that sends sector 3 to 470, the identity trace 216 on the same space, a truncated sign average, translation by 1, and a coordinate transposition.

Not run: Lean, Docker, the degree-881 coefficient replay, the trace of Z_508, and the selected cold replay.

Credit

Prepared by unpaidmoon with substantial Cursor Grok assistance.

The explicit pair and the free odd signs are upstream PR #6965. The trace -48 and this open question are upstream PR #7020. Krzysztof Wojt's research/exact-one-slot-gram/ defines X_3 and is the degree-881 annihilator. Ali Romman's #4958 and #4720 retain the query and the Bell action. Nick Petalas's #5120 is the degree-410 lower bound. JoyboyBrian's handoff recorded the sign-character dimensions. The doubled representation comes from #1719. New work is Apache-2.0.

Open next questions

Linked from research/c3-odd-sign-trace-slot/TASK.json.

• Does one odd minus added to v, with u's mask held at 5, keep the trace -48 on (+1,+1)? First step: repeat the trace with v's mask replaced by 19. Blocker: this package does not change v.
• What are the trivial X_3 traces for the other legal even-cycle pattern on (0 2)(4 6), mask 80? First step: realize that mask with v held at 17. Blocker: that pattern is not an element of H, so the odd-sign reindexing does not determine it.

Workflow feedback

The named sign and the fixed masks kept the calculation on the existing group of 64 operators. #1200