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

Abstract

We attach to a finite group $G$ and a structured payoff probe $\varphi$ an integer \emph{payoff-difference lattice} $M_{\varphi }(G)$ and its \emph{conductor} $C_{\varphi }(G)$: the primes at which $M_{\varphi }(G)$ 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 ${\mathrm{C}}_{\mathrm{comm}}(G\times H)={\mathrm{C}}_{\mathrm{comm}}(G)\cup {\mathrm{C}}_{\mathrm{comm}}(H)$ unconditionally. A Galois-orbit-trace character probe reads a complementary layer: an index-$2$ subgroup forces $2\in {\mathrm{C}}_{\mathrm{char}}(G)$ while no odd prime is forced, and ${\mathrm{C}}_{\mathrm{comm}}(D_{2q})=\{q\}$, ${\mathrm{C}}_{\mathrm{char}}(D_{2q})=\{2\}$ for all odd primes $q$. Certified exhaustive computation ($|G|\le 128$ commuting, $|G|\le 64$ character) and a deformation-family analysis support the general program: classify the Smith torsion of the compressed centralizer-type incidence matrix $B_G$.

17 Pages