Partition coefficients for generalized peakon transfer matrices

We give a universal factorial formula for every partition coefficient in every positive trace power of the source generalized peakon Lax matrix, for all particle numbers. A curve-conditioned rank-two determinant collapse and the formal trace-log identity prove the formula and both printed families. Lean covers arbitrary coherent complex particle amplitudes and the literal square-root matrices at nonnegative real momenta. Classical determinant and multinomial tools and known families are uncredited as separate outcomes. AI-assisted preprint by Myeongjin Kang, with no affiliation. Internal AI review is not external journal peer review. The accompanying Lean certificate covers the precise scoped main statement; inspect the formal endpoint, assumptions and provenance in the supporting package. Prior-art investigations are documented bounded searches, not worldwide-priority certification. Original prose is offered under CC BY 4.0 and original code/Lean under MIT; third-party rights and existing notices are preserved. The public supporting ZIP contains original proof sources and verification evidence, with bibliographic links and hashes instead of redistributed third-party source manuscripts. Recorded prior-art/source-access limits: Bounded selective semiseparable/full Kostant-Toda/peakon literature and one functioning web engine; no worldwide-priority guarantee; MathDB companion IDs and first-stated-year metadata unreliable and not relied on; No exhaustive subscription-index search