{"ID":6024175,"CreatedAt":"2026-07-08T01:00:23.257252134Z","UpdatedAt":"2026-07-09T23:32:41.575780533Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.05698","arxiv_id":"2607.05698","title":"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 $\\Ccomm(G\\times H)=\\Ccomm(G)\\cup\\Ccomm(H)$ 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 $\\BG$.","short_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...","url_abs":"https://arxiv.org/abs/2607.05698","url_pdf":"https://arxiv.org/pdf/2607.05698v1","authors":"[\"Matthew Fried\"]","published":"2026-07-06T23:35:18Z","proceeding":"math.GR","tasks":"[\"math.GR\",\"cs.GT\",\"math.CO\"]","methods":"[]","has_code":false}