[unpaidmoon]

Change and mathematical scope

Explicit support witness. No new κ and no word.

In the PR2085/2112 register model, four nonzero symmetric rational rank-one matrices and two distinct old projectors can meet target support for

R + S + T + U = 4I - E - F.

The six labels A, B, P, Q, 2A, and 2B sum to 4I because P+Q = A+B = I. Each target is exactly one subset sum, the pair A+P, A+Q, B+P, or B+Q. Designating any two of the four projectors as the old calls leaves the other four labels as added frames. Added frames may equal old projectors; that is the permission already stated for the length-5 equation in PR #7083.

The same six labels fail the inherited odd-c count: only P and Q have odd c, while all four targets do. Under the inherited call-versus-target inequality this witness is not a word. The inequality is not reproved. This is one solution, not a census of the length-6 equation, and not a proof that no other solution is a word.

Selected frontier stays 781451/250000000 = 0.003125804 from PR #6030. No κ change.

Task and overlap

Base commit 17469fff78c22941280f04d39dc5a1faee09c840 (canonical CrocSwap main, re-fetched at submission and unchanged). Fork main is a different history, so this draft uses cursor/crocswap-main-17469fff-1751, which is that same commit, as its base. The upstream target is CrocSwap main.

Head 17c1c233d8ee2c8098c8797916d2a1ede085d6f8. Package research/four-signed-two-old-support.

Task card bit-new-register-model, scope changed. This is open-next question 1 of PR #7083.

Nearest overlap, refreshed 2026-11-10T13:24:26Z (python3 scripts/find_overlap.py --refresh, 398 open PRs, 7013 discussion records, local inventory complete):

• Open PR Prove four rank-one frames plus one old call miss length-5 support (no κ) #7083, head 977b3451cea127c598cb07315180fc6c6bc4f4b6, proves the one-old length-5 equation misses support and asks this two-old question. It does not give a length-6 solution. Canonical finding: discussion Handoffs and requests for independent checks #1166, comment 19515064.
• Merged PR Prove four once-each rank-one frames miss the four-call endpoint. #6952, package research/four-signed-once-parity, proves the zero-old four-call equation misses support. Its open question includes lengths 5, 6, and 7 together; the length-5 case is Prove four rank-one frames plus one old call miss length-5 support (no κ) #7083, and this package is only the length-6 witness.
• Open PR Classify length-6 support for five frames plus one old projector #7067 classifies length-6 support for five frames plus one old projector, a different equation.
• PR Prove eight-call floor for two arbitrary signed rank-one frames (no κ) #6190 / research/bit-two-signed-frame-floor supplies the inherited determinant identity, subset-sum necessity, and odd-c inequality. They are not reproved.

No open PR in that snapshot exhibits this six-label witness.

Validation

Python 3.12, no third-party packages.

python3 -B scripts/research_package.py --package research/four-signed-two-old-support

Registered status: PASS_REGISTERED_RESEARCH_PACKAGE with outcome completed. Both the plain replay and the python3 -O -B replay print PASS four-signed-two-old-support checks. Checks raise, so -O does not strip them.

scripts/check_research_commit.py --package research/four-signed-two-old-support --check-source-history --check-modes passed at 17c1c233d8ee2c8098c8797916d2a1ede085d6f8.

The checker confirms the uu^T scales, the sum 4I, all six old pairs, that each target has exactly the declared size-2 carrier, the odd-c count, and the two failure controls (drop Q; shift 2A by A).

Not run: a classification of other solutions, a search offered as a census, a native or supplier replay, a proof assistant, and scripts/research_selected_baseline.py. The selected record is provenance only.

Credit

The register model is Cinna Davis's PR2085/2112 formal model. The determinant, subset-sum necessity, and call-versus-target inequality are the PR2910 package as recorded in apoursh's two-signed-frame floor, PR #6190. The four-call support proof is PR #6952. The one-old length-5 proof that asks this question is PR #7083.

Prepared by unpaidmoon with substantial Cursor Grok assistance.

Open next questions

Does any other solution of R+S+T+U = 4I-E-F meet support and also have at least four labels of odd c? This witness has two, so the inherited count excludes it and does not exclude a different solution. A word at length 6 would be shorter than the eight-call minimum. First step: one other integral solution with four odd-c labels, then the same subset scan. Blocker: this package does not classify the solutions.

Can four added frames plus three old calls meet support at length 7? First step: R+S+T+U = 4I-E-F-G for three distinct projectors, then one rational solution's subset scan. Blocker: the six-label witness has no seventh matrix.

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