Conditional κ = 0.001591045666134196: rank-five complex trace expansion

[eumemic]

κ = 0.001591045666134196 (conditional).

This new finite witness is 0.0038662232% above #547 at eecca20a24258cb6d9f698b1984a95029bac0a1c (0.001590984155135328). It expands one commuting pair in #548's complex scalar word into three gates that all fit a rank-five frame, then composes the resulting complex supplier with the cited unchanged #543 R420 bit supplier. Complex b = 0.001593581126479114854212 binds; bit c = 0.001653826988146798944268.

The exact identity j += k; i += j → i += j; i += k; j += k applies after checking 111 intervening gates. The complete local paid-profile delta is e1-e2-e3+e4, preserving rank mass, child count, helper widths and owner geometry. The new scalar gate and inverse cleanup are paid in the fresh invoice. Every old published branch/head remains frozen; this is a new branch and PR.

The dependency-closed, standard-library verifier reconstructs the word and checks both 728-column orientations, all 504 dirty helpers, both reflected paid ledgers, every one of 2929 phase edges and 224 endpoint adapters, all 36288 literal helper roles, normalized wrappers and a fresh finite invoice. It includes 44 omitted-pivot controls, omission of the new middle shear, 11 native controls, seven owner mutations, two independent rational moment engines, six exact bootstrap levels, all 47 strict outer constraints and four outer controls. Malformed or incomplete assembly receipts also reject. Both supplier next ticks and the next κ grid point fail.

python -B research/complex-trace-rank5/verify.py --work /absolute/new/output

Observed full admission and exact assembly passed locally. A separate clean portable replay receipt is included alongside the certificate package. Dedicated Python 3.11/3.14 CI is provided; GitHub CI is not claimed green. An initially stale baseline plan profile was correctly rejected; the new nominated plan changes only that profile and the checker independently rederives it.

The numerical claim remains conditional on the inherited all-size compiler, common-label/mutable-source realization, whole-call hosting, precision/tape and analytic-transfer contracts. The unchanged #543 bit supplier is cited and exactly repriced, not physically recompiled by this package. This is not an unconditional multiplication theorem or a complete Lean proof.

Prepared for eumemic with OpenAI Codex assistance. Credits and original AI disclosures are retained for Dugongue #548/#541; rohanarun/Claude #534 and Rohan Arun #547; icekylinx #535/#531; chafreaky #544; DreamingOfClouds #484 and the attributed gauge/kernel, normalized-wrapper, phase and PR361 arithmetic ancestry; and eumemic/Claude #543. Upstream licenses, notices, exact source pins, proof and observed receipts are preserved in research/complex-trace-rank5/.

[Mister-Meeseeks]

Thank you @eumemic. This contribution at c7337040822978f1611ce9d9d44e7bb96d887e23 is incorporated in the reviewed family through #588, published in 9594b61.

Rank-five promotions must preserve the scalar map while changing all incident frame paths. Exact promotion and full reflected phase checks are replayed, with adjacent moment endpoints rejected.

Preserved implementation and research:

• research/complex-trace-rank5
• research/complex-trace-rank5-replay

The family review and individual disposition record the scope, evidence and attribution. Parallel and intermediate contributions remain credited even where a later composition has a stronger bound. The published multiplication bound remains conditional on the documented compiler, realization, hosting and analytic assumptions; these finite checks are not a full Lean multiplication proof.