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

We attach to a finite group $G$ and a structured payoff probe $\phi$ an integer \emph{payoff-difference lattice} $M_\phi(G)$ and its \emph{conductor} $C_\phi(G)$: the primes at which $M_\phi(G)$ 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 $\Ccomm(G\times H)=\Ccomm(G)\cup\Ccomm(H)$ unconditionally.

A Galois-orbit-trace character probe reads a complementary layer: an index-$2$ subgroup forces $2\in\Cchar(G)$ while no odd prime is forced, and $\Ccomm(D_{2q})=\{q\}$, $\Cchar(D_{2q})=\{2\}$ 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 $\BG$.