[unpaidmoon]

Change and mathematical scope

Prove that the degree-881 payload involutions X_3 and Z_508 fix Bell block c=3 and split the joint spaces of the explicit pair from #6965. This is a trace-formula proof. It is not a κ improvement, not a characteristic polynomial, and not a census of every operator outside the signed-permutation group. The selected conditional κ stays 781451/250000000.

On the trivial (+1,+1) joint space of dimension 216, X_3 has trace -48. Its +1 eigenspace has dimension 84 and its -1 eigenspace has dimension 132. Both are invariant under B_3. Z_508 splits the same space as 108 + 108. The intersections of that 216-space with the four degree-881 sectors (s,t) are 42, 42, 66, and 66.

Both operators lie outside the signed-permutation representation: a linear address map fixes 0, while X_3 sends address 0 to 3. The named certificate involutions on any other Bell block are not endomorphisms of sector 3.

Remaining obligations: no basis is constructed; the characteristic polynomial of the 84-dimensional space is uncomputed; other cross-coset sign patterns were not evaluated.

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 #6965, recorded in #1166 (reply in thread). The signed-permutation census does not decide payload involutions. This package answers that finite trace question and does not admit a supplier.

Overlap was refreshed on 2026-11-10 against that base. Coverage: local notes 10178 of 10178, complete; 317 open PRs, pagination complete, fetched 2026-11-10T08:48:35Z; 6971 discussion records, pagination complete, fetched 2026-11-10T08:49:37Z. The phrase payload involution matches the degree-881 certificate and the open question, not a computed trace on sector 3. 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-payload-sector3-split/verify.py
python3 -B -O research/c3-payload-sector3-split/verify.py
python3 -B scripts/research_package.py --package research/c3-payload-sector3-split

Marker PASS_C3_PAYLOAD_SECTOR3_SPLIT. 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. Commutation was checked for every port and all 64 group elements. Five rejection controls fired: translation by 1, a wrong address character, a truncated sign average, the identity trace 216 on the same space, and a coordinate transposition.

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

Credit

Prepared by unpaidmoon with substantial Cursor Grok assistance.

The explicit pair, the joint dimensions, and the signed-permutation boundary are upstream PR #6965. Krzysztof Wojt's research/exact-one-slot-gram/ defines X_c and Z_{c+e} 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-payload-sector3-split/TASK.json.

• What is the characteristic polynomial of the 84-dimensional X_3 = +1 subspace of the trivial (+1,+1) joint space? It would place this split inside the degree-881 polynomial. First step: build a basis of the joint equations for H, u, v, and X_3 = +1. Blocker: the trace formula does not list a vector.
• Does one other cross-coset pair, with a different odd sign, move the trace -48 onto another (u,v) eigenvalue? First step: repeat the trace with one odd minus sign added to u. Blocker: this package fixes the masks from Classify the signed permutations that split every c=3 centre character. #6965.

Workflow feedback

The named open question, with the explicit pair and the degree-881 operator names, bounded the calculation without a matrix. #1200