arXiv subjects

Matthew Fried

Publications and source records attributed to Matthew Fried.

5 recordsLinked to original sources

arXiv2026

Conjugation invariants determine the metacommutation permutation only up to relabelling

Let $\mathcal{H}$ be the Hurwitz quaternions, $p$ an odd prime, and $Q \in \mathcal{H}$ a prime of norm $q \neq p$. Metacommutation $PQ = Q'P'$ induces a permutation $π_Q$ of the $p+1$ left-associate classes of primes of norm $p$. Cohn and Kumar compute its sign and fixed-point count, and Leite and Machiavelo its full cycle structure, by formulas depending only on conjugation-invariant data of $Q$ (namely $q$ and $\mathrm{tr}\,Q$). We prove this is exactly the boundary of what such invariants can carry: no quantity $I(Q)$ invariant under unit conjugation determines $π_Q$ as a labelled permutation of the intrinsic class set, or even the image of a single specified class. The proof combines an equivariance identity $π_{uQu^{-1}} = ρ_u π_Q ρ_u^{-1}$ with a minimal, fully explicit witness at $(p,q) = (3,5)$: the four primes $2+i$, $2+j$, $2+k$, $2-i$ form a single unit-conjugacy orbit, hence agree under every conjugation-invariant function, yet induce four pairwise distinct $4$-cycles of the same four classes. We further observe that isomorphisms $\mathcal{H}/p\mathcal{H} \to M_2(\mathbb{F}_p)$ form a torsor under $\mathrm{PGL}_2(\mathbb{F}_p)$, so no projective labelling of the classes is canonical, and that by orbit-stabilizer the datum of one destination $π_Q(C)$ is exactly a coset $g_Q G_C$ in $\mathrm{PGL}_2(\mathbb{F}_p)/G_C$. All sixteen refactorizations in the witness are listed in the appendix and have been verified by machine along two independent routes.

cs.GT↗

arXiv2026

What Multichoice Values Cannot See: The Information Content of Anonymous Values for Games with Graded Participation

In a cooperative game with graded participation, each of $n$ players acts at one of $m$ ordered levels; multichoice games, voting with abstention, and graded feature attribution all take this form. We determine exactly what the entire family of linear, player-symmetric values, power indices, and importance measures proposed for this setting, present and future, can and cannot see, for all $m$ at once. The mechanism is short: the functional computing any one player's payoff under an anonymous value is invariant under permutations of the other $n-1$ players, and by the branching rule such invariants exist only in the Specht constituents $(n)$ and $(n-1,1)$ of $(\mathbb{C}^m)^{\otimes n}$. Consequences: the joint information of all anonymous values is a component of dimension polynomial in $n$ against an $m^n$-dimensional game space, with the classical binary theory as the $m=2$ shadow. Three players can hide: for $m\ge3$ the blind space is nonzero already at $n=3$, and the first invisible constituent is not a correlation but a chirality, realized by two distinct monotone abstention-voting rules on three voters that every anonymous power index scores identically. For $n\ge4$ the blind space is spanned by $\pm1$ games on four profiles each. Order-$d$ interaction probes see exactly the partitions with at most $d$ cells outside the first row, with full recovery only at $d=n-\lceil n/m\rceil$. Audit evasion gets easier than in the binary theory: a coalition of $c$ players evades every order-$d$ audit iff $c-\lceil c/m\rceil\ge d+1$, so with three or more levels, trios can restructure invisibly to every value-based payment scheme. Algorithmically, the visible part of a game is a polynomial-size, cheaply estimable sketch, while exact computation of any full-support value requires all $m^n-1$ nonzero queries. All dimension and rank claims are verified computationally.

cs.GT↗

arXiv2026

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

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 $B_G$.

math.GR↗

arXiv2026

Which Wallpaper Groups Arise from Tiled Games?

Which discrete symmetry groups can arise from strategic interaction? We tile the plane with copies of a bimatrix game's support complex, joined by controlled boundary rules, and show that all seventeen wallpaper groups act on the resulting covers: explicit generators, each a machine-verified graph automorphism, every realization certified as the exact toroidal quotient, with types identified by a crystallographic recognizer in exact rational arithmetic and cross-validated in GAP. A three-line lemma turns the classical symmorphic/non-symmorphic distinction into a lattice classification: realizations whose translations contain the full tile lattice exist precisely for the thirteen symmorphic groups, and the four non-symmorphic groups are realized at translation-lattice index exactly two, the minimum possible: the tile is the glide's half-step. Two computational tracks accompany the construction. On the graph track, quotienting a straight cover by its translations recovers the tile exactly, $\beq(M/\calT)=\beq(K)$, and swap boundaries add exactly $\binom m2$, independent of payoffs and of cover size. On the game track, detecting a duplicated-strategy cover is a linear-time payoff scan, one tile solution folds to a full translation orbit of cover equilibria, and the tiled correlated-equilibrium system has dimension exactly $r(d-q)+q$, with expansion impossible. The polymatrix cover then carries the symmetry outright: every wallpaper action, glides included, is a group of genuine game automorphisms, equilibria collapse along any symmetry subgroup to a folded fixed-point problem, and a decorated refinement has game automorphism group exactly the toroidal wallpaper group.

cs.GT↗

arXiv2026

What Semivalues Cannot See: The Information Content of Anonymous Marginal Values

The semivalue family shares a common kernel: games invisible to every anonymous marginal value at once, nonzero from four players (Kleinberg and Weiss, 1985; Amer, Derks and Giménez, 2003). Crisman and Orrison (2015) ask what useful structure this kernel carries; this paper gives a concrete answer. In Harsanyi-dividend coordinates the joint information of all semivalues is exactly each player's total synergy at each coalition size, so the kernel is synergy arranged in closed circuits. We prove: order-$\le d$ mixed-difference audits recover exactly the degree-$\le d$ dividend-slice harmonics, with closed-form dimension at every rung; nonzero blind games fail superadditivity, monotonicity, and core existence, yet distinct convex games with identical values under every semivalue exist from four players, with exact perturbation thresholds; the positive weighted Shapley family attains full information $2^n-1$, so anonymity is the binding axiom within the marginal framework; and a coalition of size $c$ defeats every audit of order $\le d$ precisely when $c\ge2d+2$, within the convex class for small perturbations. An exhaustive census at $n=5$ exhibits non-isomorphic voting rules with identical values under every semivalue power index; no weighted game participates in any collision, prompting a swing-rigidity conjecture. Measured against the theory, classical cooperative games sit at $0.90$ to $1.00$ visibility to the family versus $0.089$ for a random game.

cs.GT↗