Research: six-label UT3(F4) triple still costs 12 after adjoining w^2 (answers #3774; no new kappa)

[VictorLiwentao]

Change and mathematical scope

Answers the six-label question left open by #3774. In F_2[UT_3(F_4)], every word in the six involutions

X_1, X_w, X_{w^2}, Y_1, Y_w, Y_{w^2}

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 coefficient w^2=(1,1) meets both F_2 summands, so the four-label complementary split does not apply. On the mixed word the {1,w} shadows span only span{e_0,e_1} (dimension 2). The third direction is the lift column (e_2,e_2). The diagonal-lift inequality replaces the split: if U has columns (α_0 u, α_1 u), then dim span{u} ≤ rank(U). With the inherited corner and row-space bounds this still forces n_X≥6 and n_Y≥6.

Two twelve-call words attain the endpoint. The mixed word is [X_1,Y_1], [X_w,Y_1], [X_{w^2},Y_1]. The both-new word is [X_1,Y_1], [X_{w^2},Y_{w^2}], [X_w,Y_w]. Each has n_X=6 and n_Y=6. [X_{w^2},Y_1]=(0,0,w^2) and [X_{w^2},Y_{w^2}]=(0,0,w).

The lower bound is the diagonal-lift count, not an exhaustive search of length 11. A length-11 word would need one letter class of length at most 5, which the inequality forbids for every finite r≥3. Machine checks cover the group law (order 64), both schedules, a restored spectator, single omissions, and dim span{u} ≤ rank(U) on all 82160 subsets of size at most 6 of the 21 pure columns in r=3. The step from that census to every r≥3 is the φ-Δ dimension count in the proof.

Scope. Six axis involutions and rank-one idempotents only. This does not reprint the four-label theorem, does not use the fourteen-call composite, and 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-w2-lift-obstruction/
• Tip: 595b9426cc2fac34d31a71b24b98ed080f54c99a
• Base: 20ecf2891d80b5acf18d980b2164fb44ea7faa90
• Scope: changed — six-label obstruction, not a reprint of the four-label minimum

Closest prior: #3774 (open; four-label exact minimum 12; explicit that X_{w^2} and Y_{w^2} are unproved because they break the {1,w} summand split). The nilpotent expansion and the disjoint four-call commutator shape are that argument with two letters added. The new step is the diagonal-lift inequality, which that split cannot cite.

Validation

python3.12 -I -B research/ut3-f4-w2-lift-obstruction/verify.py
python3.12 -I -O -Werror -B research/ut3-f4-w2-lift-obstruction/verify.py
python3.12 -I -B research/ut3-f4-w2-lift-obstruction/independent.py
python3.12 -I -B research/ut3-f4-w2-lift-obstruction/test_package.py
python3.12 -B scripts/research_package.py --package research/ut3-f4-w2-lift-obstruction

Expected:

PASS_UT3_F4_W2_LIFT six_label_minimum=12 labels=6 mixed_word=12 both_new_labels=12 nX=6 nY=6 blind_span=2 lift_rank=3 shadow_subsets=82160 spectator=1 no_new_kappa
PASS_UT3_F4_W2_LIFT_INDEPENDENT minimum=12 labels=6 mixed=12 both_new=12 blind_span=2 lift_rank=3 no_new_kappa
PASS_UT3_F4_W2_LIFT_PACKAGE_TESTS groups=3
PASS_REGISTERED_RESEARCH_PACKAGE research/ut3-f4-w2-lift-obstruction outcome=completed

Default verify is read-only. Admission uses need() and still fires under -O. A temporary copy of CERTIFICATE.json with a corrupted field is rejected, and duplicate JSON keys are rejected.

Credit

Credit Cinna Davis / #2261 for the two-register minimum and the open three-output question. Credit the four-label dependent-central-triple note (#3774) for posing the w^2 question. The rank-one idempotent register model follows the eumemic / woklerx address-call lineage used there. Prepared by Wentao Li (VictorLiwentao, @Is__Victor) with substantial AI assistance, under Apache-2.0.

Open next questions

1. Can rank-one calls that are not idempotent reduce this six-label triple below 12? Keep the six axis labels, drop N^2=N, and test whether a nilpotent factor can feed a three-dimensional corner. The blocker is that v·u=0 kills every quadratic scalar v_X·u_Y, so the present normal form does not admit those factors.
2. Can one of these twelve formal calls be realized in a pinned bit supplier below one rank-one complete price? Charge frame changes, hosting, restoration, and dispatch for either twelve-call word. No legal physical realization is supplied here.