Research: no two-target UT4 central word has n_b=6 (n_b=7 open; no new kappa)

[VictorLiwentao]

Change and mathematical scope

Answers the n_b = 6 half of open next question 2 in #1816. In the explicit F_2[UT_4(F_2)] central-target alphabet of #1662, no legal word realizing two central targets — of any finite length, on any finite restored dirty workspace — has exactly six paid b calls. Combined with #1662’s n_b ≥ 6, every such word has n_b ≥ 7. The case n_b = 7 is left open.

At n_b = 6 the middle quotient forces rank X = rank Y = 2, so every b coupling lies in the active 2-plane. Interior dirty contractions then cannot touch the {a,b} / {c,b} side coefficients or the cross coefficients abc and cba. Those necessary equations reduce to a two-role profile problem with unrestricted outer length.

The census enumerates all 5,582 GL2-minimal zero-sum length-6 middle sequences, keeps the 5,400 with both middle ranks at least two, builds every side-zero transport key, and pairs keys for abc = cba = I_2. Zero pairs survive. All 138,240 identity key pairs that do appear sit on the 182 low-rank middles, so the screen is not empty by accident.

Scope. Quotient-level obstruction for n_b = 6 only. Does not restate the unit-tariff minimum 16 from #1816, does not reopen three-target lengths 21/22, and does not claim a word with n_b = 7. No supplier or κ change.

No new kappa. Selected κ stays 0.001672660197076918.

Task and overlap

• Package: research/ut4-two-target-no-six-b/
• Tip: 1db88eeb57f21689a8a8f025c2535f50b816414d
• Base: 20ecf2891d80b5acf18d980b2164fb44ea7faa90
• Scope: refined — decide n_b = 6 only

Closest prior: #1662 (n_b ≥ 6) and #1816 (unit-tariff exact 16; weighted OQ2). The #1690 profile/key pipeline is reused for the census, but the rank-(2,2) filter and the unbounded dirty reduction newly close n_b = 6 rather than retargeting the unit-tariff result.

Validation

python3.12 -I -B research/ut4-two-target-no-six-b/verify.py
python3.12 -I -O -Werror -B research/ut4-two-target-no-six-b/verify.py
python3.12 -I -B research/ut4-two-target-no-six-b/independent.py
python3.12 -I -B research/ut4-two-target-no-six-b/test_package.py
python3.12 -B scripts/research_package.py --package research/ut4-two-target-no-six-b

Expected:

PASS_UT4_TWO_TARGET_NO_NB6 key_pairs=0 middles=5400
PASS_UT4_TWO_TARGET_NO_NB6_INDEPENDENT key_pairs=0 keys_sum=1568064
PASS_UT4_TWO_TARGET_NO_NB6_PACKAGE_CONTROLS checks=6
PASS_REGISTERED_RESEARCH_PACKAGE research/ut4-two-target-no-six-b outcome=negative-result

Default verify is read-only. Admission uses need() and still fires under -O. A temporary copy of CERTIFICATE.json with corrupted pairs is rejected, and duplicate JSON keys are rejected. independent.py does not import verify.py.

Credit

Credit @ToikY / #1662 for the model and n_b ≥ 6 bound, and #1816 for unit-tariff exactness and the weighted open question. Aaron Wang / @navinor / #1690 for the profile/key pipeline. #1680 for workspace context. Prepared by Wentao Li (VictorLiwentao, @Is__Victor) with substantial AI assistance, under Apache-2.0.

Open next questions

1. Does any two-target word have n_b = 7 at any length and any finite dirty workspace? Extend the unrestricted-profile key census to length-7 middles on rank patterns (2,2), (2,3), (3,2), or find an algebraic obstruction. Blocked by 16^7 profiles and middle excess when a middle rank is 3.

[VictorLiwentao]

The (2,2) slice of open question 1 is answered by #3938: no legal two-target word has n_b=7 with middle ranks (2,2). Patterns (2,3) and (3,2), and unrestricted n_b=7, remain open.