Research: no 15-call two-target UT4 word on two roles; with #1680, exactly 16 calls for every r (no new kappa)

[JosieElliston]

Change and mathematical scope

This adds only research/ut4-two-target-fifteen/. It is an exhaustive finite negative result in the explicit F2[UT4(F2)] alphabet of #1662. There is no new κ and no supplier. The selected record, upstream/ and every frozen pin are unchanged.

Result. With r = w = 2, no word of 15 paid factors I+([g]+1)xy^T (g in {a,b,c}) equals the two-target endpoint I+([z]+1)I_2. No such word reaches even the degree-three quotient endpoint. This answers open next question 1 of #1690 for two roles.

Every-r corollary, conditional on written lemmas. #1680 (@ToikY, head a281330e) compresses every two-target word of length at most 7w+1 = 15 on any r ≥ 2 roles to two roles, without raising any label count. #1662 excludes 13 calls or fewer, #1690 excludes 14 on two roles, and this package excludes 15. So two central targets need exactly 16 calls for every finite r ≥ 2. The direct (abcb)^2 per target is optimal, and the two targets share nothing at unit tariff. I re-derived #1662's count bound and #1680's compression proof line by line and found no gap. Neither is formalized.

Method

The search generalizes #1690's (Aaron Wang / @navinor, head dfd15686) prefix/suffix-profile and cross-key pipeline. Its search.py is vendored byte for byte as pr1690_search.py, and verify.py checks the SHA-256. The new fifteen.py adds:

• separate a/c profile sets for unequal counts;
• 7-bit cross keys for seven middle calls (Research: no 14-call two-target UT4 word on two roles; with CrocSwap/integer-mult-bounds#1680, at least 15 calls for every r (no new kappa) #1690's parity table covers 6 bits);
• a trie side screen and stdlib multiprocessing.

#1662's bound leaves the splits (5,6,4), (4,6,5) and (4,7,4). Every surviving profile pair is expanded to concrete words. Each word is checked against the full quotient endpoint and the full formal endpoint.

Split	Canonical middles	Profile pairs	Words	Q pass	Formal pass
(4,6,4) regression = #1690	5,582	208	3,248	0	0
(5,6,4)	5,582	1,252	114,960	0	0
(4,6,5)	5,582	1,252	114,960	0	0
(4,7,4)	49,686 (cover 298,116)	5,952	64,680	0	0

Evidence

• verify.py (stdlib) checks:
  • the vendored module hash;
  • the quotient homomorphism on all 4,096 pairs;
  • GL2 orbit coverage at lengths 6 and 7;
  • the direct 16-call positive control;
  • an exact regression of every Research: no 14-call two-target UT4 word on two roles; with CrocSwap/integer-mult-bounds#1680, at least 15 calls for every r (no new kappa) #1690 (4,6,4) stage count;
  • all three splits against EXPECTED.json.
• The full run passed in 243 s wall on 10 cores (about 30 CPU-minutes). --part small takes 86 s and --part large handles (4,7,4).
• test_package.py passes 6 tests in 36 s:
  • the trie screen equals a brute-force screen per split;
  • the 7-bit key_cross equals the direct cross coefficients;
  • key pairing equals brute force;
  • planted words survive every stage, per split;
  • the a↔c automorphism ψ(g) = J g^{-T} J holds on all 4,096 pairs;
  • the r = 1 driver control passes.
• research_package.py --check-only passes.

Limits

• The search covers r = 2 only. The transfer to every r rests on Research: UT4 two-target unit count is independent of dirty workspace #1680's written lemma, and the split on Research: UT4 central-target tariff bound with arbitrary dirty workspace #1662's written bound.
• The formulas and expansion are Research: no 14-call two-target UT4 word on two roles; with CrocSwap/integer-mult-bounds#1680, at least 15 calls for every r (no new kappa) #1690's code, so an error shared with Research: no 14-call two-target UT4 word on two roles; with CrocSwap/integer-mult-bounds#1680, at least 15 calls for every r (no new kappa) #1690 would affect both. The regression, brute-force and planted tests guard against this, but this is not a separate reimplementation.
• Unit tariff only. No compiler, supplier, rational-frame identity or κ change. No human or proof-assistant review.
• The (4,7,4) replay (--part large) took 149 s wall and 21 CPU-minutes. It fits the 600 s timeout only with 4 or more comparable cores.

Open next questions

• w = 3: Research: UT4 two-target unit count is independent of dirty workspace #1680 covers L ≤ 22 of the 21..24 interval.
• Weighted tariffs, where a word longer than 16 calls with n_b < 8 could win.

Task and overlap

Base main 0b1cccf86e2f5132ed286a26ef0a681a495740ea. Revision 3 task bit-new-register-model, chooser seed 6e519ca68cc1e5c0af3a6aba9841c5ed; scope refined to #1690's open question 1. Claim: #1647 (comment). The overlap refresh at 10:34Z (473 open PRs, 449 discussions, complete pagination) finds only #1662, #1680, #1690 and that claim.

Credit

• @ToikY: model, count bound and question (Research: UT4 central-target tariff bound with arbitrary dirty workspace #1662, head 01cc02b1) and the compression lemma (Research: UT4 two-target unit count is independent of dirty workspace #1680, head a281330e).
• Aaron Wang / @navinor: pipeline and 14-call exclusion (Research: no 14-call two-target UT4 word on two roles; with CrocSwap/integer-mult-bounds#1680, at least 15 calls for every r (no new kappa) #1690); its search module is vendored unchanged.
• Lineage: steph4n-gh (Scoped theorem: 4k UT3 commutator calls with arbitrary dirty workspace #1573), Arham Tahir (Research: central UT3 floor survives nilpotent rank-one paid couplings #1589), Hanson Sun / @bbMisk (Scoped theorem: exact mixed-target UT3 tariffs with arbitrary dirty workspace #1607).

Josie Elliston's donated session. Substantial AI assistance from Claude Code.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Scoped batch triage at fb67c06c36bf3a5be26ab43fc81418b4bc7e748e. This PR remains open.

The complete UT4 fifteen-call search exceeded the300-second batch budget. Finish all three split censuses and independently review the seven-middle cross-key/trie completeness and imported every-register compression premise. Keep finite two-role exhaustion separate from the written every-r corollary.

This is a review continuation, not a rejection or a request to repeat the checks already retained. Credit and source pins are preserved in the batch ledger. The selected bound is unchanged. Thank you for the contribution.

[VictorLiwentao]

Open next question 2’s n_b = 6 half is answered by a follow-up package: no legal two-target word has exactly six paid b calls, so with #1662 every realizing word has n_b ≥ 7. The case n_b = 7 remains open. See the linked PR once it is up.

[VictorLiwentao]

Clarifying: the n_b = 6 exclusion is #3887. n_b = 7 is still open.