Game Conductors of Finite Groups: Determinantal Torsion from Structured Payoff Probes

Abstract

We attach to a finite group $G$ and a structured payoff probe $φ$ an integer \emph{payoff-difference lattice} $M_φ(G)$ and its \emph{conductor} $C_φ(G)$: the primes at which $M_φ(G)$ loses rank modulo $p$. Our main result is an exact computation: for any CA-group the commuting conductor is rad$(b-1)$, where $b$ is the number of maximal abelian subgroups. In particular, conductor primes need not divide $|G|$: the prime $3$ occurs for a $2$-group of order $64$ with $b=7$. The commuting Smith spectrum is an invariant of the isoclinism class and obeys an exact direct-product law, giving ${\rm C_{comm}}(G\times H) = {\rm C_{comm}}(G) \cup {\rm C_{comm}}(H)$ unconditionally. A Galois-orbit-trace character probe reads a complementary layer: an index-$2$ subgroup forces $2\in {\rm C_{char}}(G)$ while no odd prime is forced, and ${\rm C_{comm}}(D_{2q}) = \{q\}$, ${\rm C_{char}}(D_{2q}) = \{2\}$ for all odd primes $q$. Certified exhaustive computation ($|G|\le128$ commuting, $|G|\le64$ character) and a deformation-family analysis support the general program: classify the Smith torsion of the compressed centralizer-type incidence matrix $B_G$.

Disclosure

“ality and accuracy of all content has been confirmed by the author; every theorem was proved and checked by the author, and every computational claim was verified by the author’s own GAP and Python code; (ii) the AI tools used were Claude (Anthropic), models Opus 4.8 and Fable 5, used for adversarial review of proof exposition, suggestions for computational checks, and editorial revision; all computations reported in this paper were performed and verified by the author’s own GAP and P”

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2026-08-24T00:34:09.244Z

Structural counts