[unpaidmoon]

Change and mathematical scope

Record the monic characteristic polynomial of A_1 on the 84-dimensional X_3 = +1 subspace of the trivial (+1,+1) joint space from #7020. This is an exact finite computation: an explicit basis, the integer matrix of B = 4 A_1, the degree-84 characteristic polynomial, and the degree-25 minimal polynomial. It is not a κ improvement and not a replay of the degree-881 certificate. The selected conditional κ stays 781451/250000000.

On that subspace,

chi_{A_1}(x) =
    x^16 (x-14) (x-12) (x-4)^6 (x-2)^14 (x+2)^14 (x+4)^6 (x+12) (x+14)
    * (x^4 - 16x^2 - 8x + 16)^2
    * (x^4 - 16x^2 + 8x + 16)^2
    * (x^4 - 12x^3 + 20x^2 + 32x - 32)
    * (x^4 + 12x^3 + 20x^2 - 32x - 32).

chi_{A_1}(x) = 4^{-84} chi_B(4x). Ascending-coefficient SHA256 values are c4bcbd4e52f92d13f3ad451d92c9d22256e7725e4be24ee203bea1c96c129648 for chi_A and 185b51cfd084f95c1dbd814b0b211ce4ab35b2d59604d7795715d65a0a4e5e65 for chi_B. The minimal polynomial is the degree-25 square-free product of these factors. Z_508 splits the space into degrees 42 and 42, and every irreducible factor of chi_B divides a published c=3, s=+1 sector polynomial from the degree-881 certificate. Each rational eigenvalue has geometric multiplicity equal to its algebraic multiplicity.

Evidence class: proof. The basis classifies all 4096 coordinates. The determinant is exact modular arithmetic with a coefficient bound. The quartic irreducibility search is exhaustive over a finite set of integer pairs.

Remaining obligations: the 132-dimensional X_3 = -1 complement is uncomputed, and the odd signs on the pair are unchanged. The degree-881 certificate was not replayed.

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 1 of #7020, recorded in #1166 (reply in thread). That pull request gives the trace -48 and the dimension 84 without a basis or a polynomial.

Overlap was refreshed on 2026-11-10 against that base. Coverage: local notes 10178 of 10178, complete; 339 open PRs, pagination complete, fetched 2026-11-10T10:24:45Z; 7013 discussion records, pagination complete, fetched 2026-11-10T10:25:50Z. 84-dimensional matches #7020, which leaves this polynomial open, and #7009, whose 84 is a different c=1 dimension. No open PR computes this polynomial. 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-x3-plus-charpoly/verify.py
python3 -B -O research/c3-x3-plus-charpoly/verify.py
python3 -B scripts/research_package.py --package research/c3-x3-plus-charpoly

Marker PASS_C3_X3_PLUS_CHARPOLY. 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.

Rejection controls fired: the transposition (0 1), the c=1 block operator, translation by 1, a flipped group sign, a one-entry matrix change, a missing factor, and a sign error in a basis vector.

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

Credit

Prepared by unpaidmoon with substantial Cursor Grok assistance.

The trace split and the open question are upstream PR #7020. The explicit pair and the signed-permutation boundary are upstream PR #6965. Krzysztof Wojt's research/exact-one-slot-gram/ defines X_3 and Z_508 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. The c=1 polynomial in research/a1-joint-216-charpoly/ is a different block. New work is Apache-2.0.

Open next questions

Linked from research/c3-x3-plus-charpoly/TASK.json.

• What is the characteristic polynomial of the 132-dimensional X_3 = -1 complement of this same joint space? First step: impose eigenvalue -1 on the same equations for H, u, and v. Blocker: this package stops at eigenvalue +1.
• What is the characteristic polynomial of one 66-dimensional intersection of that complement with a single eigenvalue of Z_508? First step: add Z_508 after the X_3 = -1 basis exists. Blocker: that basis is not built here.

Workflow feedback

The named open question, with the trace and the explicit pair, made the missing object a finite signed orbit basis rather than another trace. #1200