Research: UT3(F4) dependent central triple has exact minimum 12 (closes #2261 three-output question; no new kappa)

[VictorLiwentao]

Change and mathematical scope

Bounded formal minimum under priority revision 6, closing the three-output question left by #2261. In F_2[UT_3(F_4)], every word in the four involutions X_1,X_w,Y_1,Y_w with rank-one idempotent calls that realizes

diag([Z_1], [Z_w], [Z_{w^2}])

on three registers, and the identity on every extra dirty register, has length at least 12. The disjoint commutators [X_1,Y_1], [X_1,Y_w], [X_w,Y_w] attain 12 (n_X=6, n_Y=6). The six-call composite previously used for Z_{w^2} attains the same endpoint in 14 and is not minimal.

The lower bound is the nilpotent-corner rank count on the defining 6×6 representation: with coefficients in {1,w} only, n_X≥6 and n_Y≥6. It is not an exhaustive search of length 11. Machine checks cover the group law (order 64, associativity), the 28 trace-1 idempotents, the nilpotent table, both schedules, a restored spectator, and all 12 single deletions of the short word.

Scope. Four involutions only. X_{w^2} and Y_{w^2} are unproved and can break the {1,w} summand split. This does not reopen the two-register minimum 8 from #2261. 12α is a formal per-call price in this alphabet, not a supplier, hosting, or κ improvement.

No new kappa. Selected κ stays 0.001672660197076918.

Task and overlap

• Package: research/ut3-f4-dependent-central-triple/
• Tip: 713c28c1b0ebbdf3f30d7887fcb173e7a786f245
• Base: 20ecf2891d80b5acf18d980b2164fb44ea7faa90
• Scope: changed — exact three-output minimum, not a reprint of the two-register census

Closest prior: #2261 (merged; two-register minimum 8; 14-call composite; explicit that the block search is not a lower bound for arbitrary words of length ≤12). Clear of open Q8 negative-target PRs, the odd-kernel tariff cascade, and flag-0 / stock / gate-operand leave-alones.

Validation

python3.12 -I -B research/ut3-f4-dependent-central-triple/verify.py
python3.12 -I -O -Werror -B research/ut3-f4-dependent-central-triple/verify.py
python3.12 -I -B research/ut3-f4-dependent-central-triple/independent.py
python3.12 -I -B research/ut3-f4-dependent-central-triple/test_package.py
python3.12 -B scripts/research_package.py --package research/ut3-f4-dependent-central-triple

Expected:

PASS_UT3_F4_DEPENDENT_TRIPLE minimum=12 predecessor_direct=14 nX=6 nY=6 group_order=64 idempotents=28 spectator=1 omission_controls=12 no_new_kappa
PASS_UT3_F4_DEPENDENT_TRIPLE_INDEPENDENT minimum=12 predecessor_direct=14 witnesses=12,14 spectator=1 no_new_kappa
PASS_UT3_F4_DEPENDENT_TRIPLE_PACKAGE_TESTS groups=3
PASS_REGISTERED_RESEARCH_PACKAGE research/ut3-f4-dependent-central-triple outcome=completed

Default verify is read-only. Admission uses need() and still fires under -O.

Credit

Credit Cinna Davis / #2261 for the two-register minimum, the fourteen-call composite, and the open three-output question, and the eumemic / woklerx rank-one idempotent register model. Prepared by Wentao Li (VictorLiwentao, @Is__Victor) with substantial AI assistance, under Apache-2.0.

Open next questions

1. Can X_{w^2} and Y_{w^2} reduce this triple below 12? The {1,w} summand split does not apply once w^2=(1,1) mixes the two F_2 summands.
2. Can one of these twelve formal calls be embedded in a pinned bit supplier below the complete price of a rank-one factor? No legal physical realization is supplied here.

[VictorLiwentao]

Open question 1 is answered by #3788. Adjoining X_{w^2} and Y_{w^2} does not reduce diag([Z_1],[Z_w],[Z_{w^2}]) below 12 rank-one idempotent calls. The {1,w} split is blind on the mixed witness (separated span 2). The diagonal-lift inequality dim span{u} ≤ rank(U) still forces six calls on each letter class, and two twelve-call words attain the endpoint. The four-label minimum in this PR is unchanged. Question 2, a pinned bit-supplier realization, remains open.