Research: exact frame search certifies #741 and #626 providers frame-optimal (rediscovers #741; negative result)

[Rohansguliani]

Update: current selected word (#741). Rebased on main after #741 merged. I gave CP-SAT PR699's SW4 word (d232183f…), using the admissible model with the audit's hpe cone caps and fixed tail frames. In 27 s it returned OPTIMAL with exactly #741's profile {1:2556,2:334,3:430,4:194,5:56,8:8}, so the solver rediscovers #741's gain without guidance. check_sw4.py applies #741's recipe and reproduces its pinned output hash e7804702…. The LP optimum over all frame assignments then equals #741's weight: 18155.950422 linear and 18219.665540 exact-root at #741's complex root, with gaps under 1e-9. So no frame-only change of the selected word can raise κ; PR699's frame slack (6.90 units) is exactly what #741 recovered. Six commuting-reorder restarts on #741, each re-solved by the LP, gave no gain. The receipt is RESULT-SW4.json. This work does not touch the R420/R439 bit or E8 lines (#789/#790/#792, #765/#767/#770).

Negative result, scoped. No κ change. By Rohan Guliani (@Rohansguliani), with substantial AI assistance (an AI coding agent wrote the search code and ran the experiments).

The complex side binds the selected κ. In the 64 jp8 calls, unit climbs make up 160,768 of the 181,659 rank-1 children. Paid children are exactly the frame climbs that hpe_audit_and_transport.py::climb counts, and the scalar program never reads frames. So I treated every A/B/tail gate frame of the selected provider words as a variable and searched all of them exactly.

Result. For #626's jp8 and ck words (content hashes pinned in complex-k492-owner-repair/data/SPEC.json, rebuilt fresh by the package's verify.py), the paid profile is optimal over every frame assignment of the fixed gate sequence:

word	gates	input weight	LP optimum (HiGHS)	CP-SAT
jp8	4471	18475.029780700	18475.029780701	OPTIMAL, same value
ck	4507	18568.580702034	18568.580702035	OPTIMAL, same value

The weight is the linearised moment, Σ d·ln(48/d). Each word's rank mass is fixed by its widths, so at the selected root a* the exact criterion is Σ d·(73/d)^a*. That weight is also LP-optimal at the input (gap below 1e-9). The model relaxes the audit: it omits the hpe cone caps and cold_end conditions and leaves tail frames free. So no frame regroup, coalescence or promotion recipe of any size can raise κ on these words. Lifting every frame to its greatest feasible value costs +3.8%. Annealing over commuting reorders (the build_gate_moves.py rule), followed by an exact frame re-solve, gave no gain in 11 restarts (ORDER-SEARCH.json). That is a sample, not a proof for all orders.

Context for κ > 0.01. Jensen's inequality gives a ≤ −ln p / E_mass[ln(73/r)]. At the current deficit (2560 / 523264), a ≥ 0.01 needs a mass-weighted geometric-mean child rank of at least 44.7; today it is 3.66. At today's child ranks it would instead need a deficit of at least 15,425 (6.0×). Reaching it needs new scalar structure, not frame bookkeeping. About 1,700 of jp8's 2,512 rank-1 climbs are helper unit climbs.

Files. research/frame-lift-search/ contains the model, the LP/CP-SAT solvers, the order search, RESULT.json and ORDER-SEARCH.json. tests/test_frame_lift_search.py checks the lattice model on the in-repo #571 prefix word. There is also a one-line experiment-log link.

Verification. check.py ran on a fresh complex-k492-owner-repair/verify.py rebuild (Linux, Python 3.13) and reports PASS_FRAME_ONLY_LP_OPTIMAL_FOR_SELECTED_PROVIDERS. python3 -m unittest tests.test_frame_lift_search passes. I did not run make verify: no generator, certificate or selected-record file changed.

Limits. HiGHS is a floating-point LP, not an exact rational dual certificate, and CP-SAT's OPTIMAL comes without a proof log. The scalar program, helper count, dispatch, hosting and bit supplier are out of scope.

[Mister-Meeseeks]

100-PR batch review at d8c8b40ab0a95b6d77389e6726d86dddb60fb53c.

This remains open after scoped triage. Next review step: The claimed exact frame optimum relies on floating LP tolerances and solver status without an exact dual/proof certificate. Review/qualify the universal exclusion before incorporation.

This is a specific continuation point, not a rejection or a request to redo already recorded checks. Completed local evidence is retained separately from outstanding whole-head obligations.

Batch review, attribution and evidence. Thank you for the contribution.