[unpaidmoon]

Change and mathematical scope

Proof for one named pair, plus an exhaustive finite census of a described list. No κ change. The selected conditional kappa stays 781451/250000000.

On the weight-4 character of sector c=3, the cross-coset pair

• u = (0 2)(4 6) with minus signs on coordinates 0, 1 and 2
• v = (0 4)(2 6) with minus signs on coordinates 0, 1 and 4

does not split the common degree-16 polynomial of #7021 into two different degree-8 factors. Both joint factors are

x^4 (x - 2)^2 (x + 2)^2,

and each 16-dimensional eigenspace of u has characteristic polynomial

x^8 (x - 2)^4 (x + 2)^4.

Every one of the 96 described splitting involutions acts on this character as ± the no-odd-minus representative of its permutation. Both signs occur 16 times in each coset. All 3072 unordered cross-coset pairs therefore have those same equal factors.

This is not a new classification of the 96, not a weight-3 calculation, and not a statement about involutions outside the described list.

Task and overlap

Base commit 17469fff78c22941280f04d39dc5a1faee09c840. The question is open-next item 1 of #7021. Catalogue card formal-interface-lemma; scope status changed, because the card asks for a formal binding and this package answers that odd-sign factorisation.

The named pair is the first step recorded in research/c3-weight4-joint-charpoly/TASK.json: the even double transpositions with a minus on coordinate 1 only. The no-odd-minus pair is used only to identify the spaces; its factorisation is the one already recorded by #7021.

Local overlap against that HEAD searched 10178 of 10178 note paths. A GitHub search of open pull requests for odd-sign degree-8 factors returned #7021 only. cross-coset also returned #7020, which is the payload-involution split and is not this factorisation. Fork cursor/* branches did not already contain this computation.

Not taken: the 12-dimensional weight-3 polynomial, the payload-involution remainders, and a second proof of the signed-permutation classification in #6965.

Validation

Evidence class: a proof for the named pair, and an exhaustive finite census of the described list of 96. Not a heuristic search, and not an unrestricted nonexistence claim.

python3 -B -W error research/c3-weight4-odd-sign-factors/verify.py
python3 -B -O research/c3-weight4-odd-sign-factors/verify.py
python3 -B scripts/research_package.py --package research/c3-weight4-odd-sign-factors --check-only

Marker PASS_C3_WEIGHT4_ODD_SIGN_FACTORS. CPython 3.12.3, standard library. Both direct runs passed, about one second each, and the registered package replay returned PASS_REGISTERED_RESEARCH_PACKAGE. Selected-baseline orientation returned PASS_RESEARCH_BASELINE_BINDINGS_ONLY.

Not run: Lean, Docker, the degree-881 replay, and the selected cold replay.

Credit

The list of 96 and the explicit no-odd-minus pair come from #6965. The degree-16 polynomial and the question come from #7021. The query and the Bell action are Ali Romman's PR4958 and PR4720. The doubled representation comes from PR1719.

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

Open next questions

Recorded in research/c3-weight4-odd-sign-factors/TASK.json.

• What is the characteristic polynomial of one 12-dimensional joint space of a weight-3 c=3 character? The weight-4 scalar reduction does not determine it.

Workflow feedback (optional)

The source first step names the odd-sign pattern precisely enough to pick the pair without a search. #1200