Mathematical Form as Invariant-Form Compression: A Finite-System Account of Wigner’s Puzzle

This paper proposes a finite-system account of why physical laws take mathematical form. The central claim is that law-like regularities are those physical regularities that can be compressed into invariant, equivariant, or covariant forms stable across perturbations, coordinate changes, coarsegraining, and finite representational constraints. Raw microstate histories are too large for finite systems to represent. What persists across observers, scales, coordinates, gauges, and perturbations must factor through a compressed sector. In the strict invariant case, q : X → Q, q(Pix) = q(x), while in the equivariant or covariant case, q(Pix) = ρiq(x), F(q) = 0 ⇒ F(ρiq) = 0. A physical law is therefore an operationally stable relation on compressed law-form sectors, rather than a full enumeration of microstates. The “mathematical effectiveness” of physical law is reframed: mathematics is effective because the portable part of physics is the part compressible into invariant-form structures. This account is developed using the language of Finite Distinction Systems (FDS), but the central thesis is independent of the formal framework. The core claim is that finite boundary-maintaining systems must compress stable physical regularities through invariant, equivariant, or covariant representations. The higher-risk philosophical extension is that Wigner’s puzzle is dissolved because persistent physical law and mathematical structure are two perspectives on invariant-form compression.