Research: no two-target UT4 word has n_b=7 with middle ranks (2,2) (n_b=7 open; no new kappa)

[VictorLiwentao]

Change and mathematical scope

Narrows the remaining gate left by #3887. 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 seven paid b calls with middle ranks (rank X, rank Y) = (2, 2), under the same unrestricted side-zero profile reduction used there.

At n_b = 7 the #1662 rank bound permits patterns (2,2), (2,3), and (3,2). The (2,2) slice still places every middle coupling in a fixed 2-plane, so interior dirty contractions cannot touch the side or abc/cba coefficients. Those equations reduce to a two-role unrestricted-profile census.

The census enumerates all 49,686 GL2-minimal zero-sum length-7 middle sequences, keeps the 49,140 with ranks exactly (2,2), builds every side-zero transport key via a 3+4 meet-in-the-middle, and pairs keys for abc = cba = I_2. Zero pairs survive. A low-rank control with ranks (1,2) still produces pairs, so the screen is not empty by accident.

Scope. Middle ranks (2,2) only. Patterns (2,3) and (3,2), and unrestricted n_b = 7, are left open. This does not claim that every n_b = 7 word is impossible, does not restate the unit-tariff minimum 16 from #1816, and does not reprint the n_b = 6 census. No supplier or κ change.

No new kappa. Selected κ stays 0.001672660197076918.

Task and overlap

• Package: research/ut4-two-target-nb7-ranks-22/
• Tip: 4995b0ce91c163de821a5d117c6e5a5289c12172
• Base: 20ecf2891d80b5acf18d980b2164fb44ea7faa90
• Scope: refined — decide the (2,2) slice of n_b = 7 only

Closest prior: #3887 (n_b = 6 impossible). The length-7 (2,2) census is the slice that PR left open; it is not forced by the length-6 result, because at n_b = 7 the rank bound newly admits (2,3) and (3,2).

Validation

python3.12 -I -B research/ut4-two-target-nb7-ranks-22/verify.py
python3.12 -I -O -Werror -B research/ut4-two-target-nb7-ranks-22/verify.py
python3.12 -I -B research/ut4-two-target-nb7-ranks-22/independent.py
python3.12 -I -B research/ut4-two-target-nb7-ranks-22/test_package.py
python3.12 -B scripts/research_package.py --package research/ut4-two-target-nb7-ranks-22

Expected:

PASS_UT4_TWO_TARGET_NO_NB7_RANKS_22 key_pairs=0 middles=49140
PASS_UT4_TWO_TARGET_NO_NB7_RANKS_22_INDEPENDENT key_pairs=0 keys_sum=46169088
PASS_UT4_TWO_TARGET_NO_NB7_RANKS_22_PACKAGE_CONTROLS checks=7
PASS_REGISTERED_RESEARCH_PACKAGE research/ut4-two-target-nb7-ranks-22 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 rank bound, #1816 for unit-tariff exactness and the weighted open question, and #3887 for the n_b = 6 exclusion and the unrestricted side-zero reduction. Aaron Wang / @navinor / #1690 for the profile/key pipeline. 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 with middle ranks (2,3) or (3,2)? A middle rank of 3 escapes the fixed 2-plane argument used here.
2. Does any two-target word have unrestricted n_b = 7? That needs a dirty-role argument beyond the side-zero reduction.