The full automorphism maximum
Abstract
The partition weight is the attained maximum of the full automorphism orders of perfect complex evolution algebras in every positive finite dimension.
Kaygorodov, Martín González and Viruel, arXiv:2610.22248v1, section 2.3, Conjecture 3, pages 27-28, conjecture that the maximum order in dimension n is the maximum over all multiplicities m_l with sum_l l m_l=n of product_l [l(2^l-1)]^(m_l) m_l!. The source states the attaining cyclic examples, and reports the upper bound checked only for n<=4. The assertion is conjectural in that source; the universal proof below is a separate result. No global priority claim is made.
Theorem 1.1 (The universal bound and actual attainment).
Proof. Machine-checked in Lean as D5/S3/QuadraticForms/EvolutionAlgebras/EvolutionAutomorphismMaximum.result (✓ std3). ∎
Resolves. Problems/kaygorodov-martin-gonzalez-viruel-2026-evolution-automorphism-maximum (proved) by D5/S3/QuadraticForms/EvolutionAlgebras/EvolutionAutomorphismMaximum.result.
Source. Repository-derived.
Acknowledgement. Ivan Kaygorodov; Cándido Martín González; Antonio Viruel (2026). Evolution algebras: a survey of results and a list of open problems. DOI: 10.48550/arXiv.2610.22248. URL: https://arxiv.org/abs/2610.22248v1.
Commentary.
For every positive n, every arbitrary complex module of finrank n with any finite natural basis and with all pair products spanning the module has a finite full multiplication-preserving LinearEquiv group of order at most maximumWeight(n). For the same n there exists an actual commutative, perfect bilinear evolution product on Fin(n) to C, whose finrank is exactly n, with a natural basis and full automorphism order equal to maximumWeight(n). Associativity and a unit are never assumed. The bound retains all quotient fibers and their internal kernel. The attaining product sends e_i squared to e_(sigma(i)). Choose the identity for n=1 and every n>=7; one cycle of lengths 2, 3, and 5 in those dimensions; two 2-cycles in dimension 4; three 2-cycles in dimension 6. The exact full-group count follows from the constructed kernel and centralizer lifts. The existing arithmetic theorem gives the values 1, 6, 21, 72, 155, 1296 and n! for n>=7. The witnesses establish a maximum rather than merely an upper bound or a supremum. No realization of every integer partition is needed.