Independent reconstruction of the UT3(F4) central-triple model: witnesses, controls, finite certificates and a bounded shorter-word search (no new kappa)

[petalas]

Change and mathematical scope

Independent reproduction plus a bounded search failure, no new kappa. Selected κ stays 0.001672660197076918.

Two open results use one formal register model: in F_2[UT_3(F_4)] with F_4 = F_2[w]/(w^2+w+1), a paid call is D_N(g) = I + ([g]+[1]) N for one of the six axis involutions X_1, X_w, X_{w^2}, Y_1, Y_w, Y_{w^2} and a rank-one idempotent N of M_r(F_2); the target on r >= 3 registers is diag([Z_1],[Z_w],[Z_{w^2}],[1],...,[1]). #3774 proves the four-label minimum 12 (tip 713c28c1b0ebbdf3f30d7887fcb173e7a786f245) and #3788 extends it to six labels (tip 595b9426cc2fac34d31a71b24b98ed080f54c99a); #3784, #3785, #3786 and #3783 record the same floor and its counts in a second form.

This package rebuilds that model from the stated definition with no code from those packages, and checks it. Package research/ut3-central-triple-independent/.

Measured, with the independently written implementation:

• Group law: order 64, full associativity, identity and inverses; the six labels are involutions and the nine commutators are exactly the three Z labels.
• Three-bit matrix ring, exhaustive over all 512 matrices: 49 rank-one, 28 rank-one idempotent, 21 trace-zero; the u v^T enumeration and the exhaustive idempotent set are identical.
• Both published twelve-call words realize the target on r = 3, 4, 5, with every extra register fixed.
• All 24 single-deletion controls fail.
• The block collapse the lower-bound arguments use (I plus first-order terms plus surviving second-order corners) equals the direct matrix product on 40 random words.
• The class-sum block relation is pinned as a linear map: rank 18 on a 27-dimensional domain, so its kernel is exactly the nine-dimensional space of equal triples.
• Stacked corner: rank-two blocks, total rank 6.
• Bounded search, r = 3: 20000 uniform random words of length 6–11 and 18734 words from the class-sum-feasible family that every realizing word must belong to, covering all eight call-count shapes from 4+4 to 6+5 (at least 1150 draws each), found no realizing word of length at most eleven.

Not established. The step from the finite corner, row-space and diagonal-lift inequalities to every r >= 3 is a written argument in the checked results; this package does not machine-verify it, so the minimum-12 claim stays author-proved. A bounded search failure is not a lower bound, and even call counts with a nonzero class sum are not reached by the sampled family. The 82160-subset diagonal-lift census of #3788 was not reproduced. The twelve calls are a formal per-call price in this alphabet: no supplier, hosting, restoration or κ claim is made. Two defects found in this package's own first draft by its own checks are fixed in the committed code.

Task and overlap

• Card bit-new-register-model (scope changed: one explicit nonabelian register model instead of an unconstrained mechanism search), priority revision 6.
• Base 20ecf2891d80b5acf18d980b2164fb44ea7faa90; tip 5e4d41cb0e99e4773abca244cdc8c376686421b8.
• Overlap checked through the repository API before starting and again before committing; find_overlap.py was run for the register-model card.
• Claim posted: Handoffs and requests for independent checks #1166 (comment)

Validation

python3.12 -I -O -Werror -B research/ut3-central-triple-independent/verify.py
python3.12 -B scripts/research_package.py --package research/ut3-central-triple-independent
python3.12 -B scripts/check_research_commit.py --package research/ut3-central-triple-independent --check-source-history

Expected:

PASS_UT3_CENTRAL_TRIPLE_INDEPENDENT order=64 involutions=6 idempotents=28 trace_zero=21 witnesses=2 deletions=24 uniform_trials=20000 class_sum_tested=18734 shorter_words=0
PASS_REGISTERED_RESEARCH_PACKAGE research/ut3-central-triple-independent outcome=completed
PASS_RESEARCH_COMMIT_BYTES ...

The run is deterministic (seeded) and takes a few seconds locally, well inside the CI replay budget; RESULT.json carries a SHA-256 receipt and reproduces byte-for-byte.

Credit

Credit the authors and predecessors of PR #3774, PR #3788, PR #3784, PR #3785, PR #3786, PR #3783 and of the earlier UT3/D8 central-commutator line (#2261 and its retained lineage) for the model, the witnesses and the floor claims checked here. Package produced by Nick Petalas with AI assistance. Independent human and formal proof-kernel review are unclaimed. Apache-2.0.

Open next questions

1. Can the all-r step be machine-verified instead of inspected?
2. Are the even call counts with a nonzero class sum reachable by a realizing word? The sampled family does not cover them.
3. Can one of the twelve formal calls be embedded in a pinned bit supplier below the complete price of a rank-one factor?