Screen: no thirteen-call five-SWAP word with commuting projectors (scoped negative, no new kappa)

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Summary

Complete exact screen of the pairwise-commuting projector subfamily for the strictly alternating thirteen-call candidate at five fully occupied coordinate-SWAP targets. Scoped negative result; no new kappa. The selected result, the selected bit supplier and upstream/ are unchanged.

formal F2[GL_2(F2)] model, t = r = 5 fully occupied SWAP roles
  -> predecessor uniform cut consequence: any saving word has q = 13, strictly alternating
  -> natural-quotient block equations: four mod-2 conditions per frame class
  -> commuting reduction (sign equation): coordinate frame of exactly 5 projectors, odd multiplicities
  -> every class needs >= 3 positions, so 13 positions cannot fill 5 classes
  -> zero admissible partitions; any 13-call escape must use noncommuting projectors

The new package is research/fiveswap-alt13-commuting-screen/. It answers the commuting part of open question 2 of research/four-swap-full-occupancy, whose uniform cut consequence isolates q = 13, strictly alternating, as the only candidate below the direct fifteen-call tariff at five roles. No predecessor implementation is imported.

Exact predecessor heads and sources:

• research/four-swap-full-occupancy (rotation step, uniform cut consequence, open question 2; Ross Richardson), on main at base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2.
• PR 1722 (label saturation lemma; Ross Richardson), PR 1333 (scalar normalization; Gareth Davidson), PR 1414/PR 1491 (model and sign invariant; commdata2338), PR 1541 (workspace reasoning; Cinna Davis).
• research/three-swap-cnot-census and research/three-swap-compression-seed6 (t = 3 closed at any workspace), research/four-swap-r4-cnot (t = 4 residual search).

The original bit-new-register-model draw is refined, not changed: the card's first experiment asks to price one smallest completed instance after a named changed premise; here the changed premise is the target-count boundary t = 5 left open by the predecessor's cut bound. Recent Q8/GL3 objects (PR 2303, PR 2248, PR 2290, PR 1974) and a parallel agent's fresh Q8 claim were avoided; the scope choice is recorded in TASK.json.

Evidence

Before: the strictly alternating thirteen-call word for t = 5 was an open candidate, with noncommuting scalar-projector interactions named as the blocker. After: the pairwise-commuting part is excluded by a written reduction plus a complete finite screen, and the block equations for the general noncommuting case are recorded as a handoff.

python3.12 -I -B research/fiveswap-alt13-commuting-screen/verify.py
python3.12 scripts/research_package.py --package research/fiveswap-alt13-commuting-screen

PASS_FIVESWAP_ALT13_COMMUTING_SCREEN min_class=3 admissible_sets=1365 size_hist=3:91,5:455,7:546,9:247,11:26 partitions5=0 direct15_partitions=608 small_pairs=2046 commuting_pairs=196 frames3=28 no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/fiveswap-alt13-commuting-screen outcome=completed

Screen counts for the strictly alternating thirteen-label word (all 8,191 nonempty position sets checked exactly):

Quantity	Value
Admissible class sets	1,365
Minimum admissible class size	3
Admissible sets of size 3	91
Admissible sets of size 5	455
Admissible sets of size 7	546
Admissible sets of size 9	247
Admissible sets of size 11	26
Partitions into 5 admissible classes	0
Direct fifteen-call positive control partitions	608

• Formal group-algebra product controls pass for the direct words at t = 1 (3 calls), t = 3 (9 calls) and t = 5 (15 calls); a mutated fifteen-call word fails the formal endpoint.
• Failure and scope controls: a size-one class is rejected; the universal size floor is exhaustive for every label word through length six (2,046 cases); an explicit noncommuting idempotent pair witnesses that the excluded family is strict; exhaustive commuting-pair orthogonality over F2^3 (28 rank-one idempotents, 196 commuting pairs) and 28 orthogonal frames support the reduction lemma.
• All 65 research workflow tests pass (python3 -B -m unittest discover -s tests -p 'test_research_*.py'), 5 skipped as environment-dependent. python3 scripts/research_package.py --package research/fiveswap-alt13-commuting-screen --check-only prints PASS_PACKAGE_METADATA_AND_PINS.
• The result is min-scope: q >= 3t for pairwise-commuting projectors at any fully occupied target count t, with the direct 3t word attaining it. No kappa, supplier, frontier or general impossibility claim.

Limitations and remaining obligations

• No noncommuting tuple search is attempted; the four block equations and the sign equation are the recorded handoff for that search. A thirteen-call escape may still exist with noncommuting projectors.
• No partial-occupancy claim (r > t restored dirty extras): at partial targets the predecessor's rotation and the block comparison change shape (its recorded workspace boundary).
• Author-checked only: no independent human review, no Lean formalization, no physical compilation, hosting, precision/tape, analytic transfer or selected-supplier cold replay. Finite checks in a declared subfamily are not a theorem about all words.
• make is unavailable on the recording host; the commands above are the underlying Python scripts.

Merge Danger

Door: two-way

Blast Radius: package

Only five new package files are added. No selected supplier, frontier record, frozen archive, shared checker or existing replay is changed.

Credit and license

Prepared by Aaron Wang with help from opencode (substantial AI assistance; no independent human review). New work is Apache-2.0; predecessor sources, attribution and notices are unchanged. Claim recorded in discussion 1317.