The Universal Boundary Quotient of an Observed Representation: Maximal Equivariant Structure and an Exact Tripartite Conservation Law

Abstract

AbstractA representation resolves some perturbations of a system and leaves others undetected, and the orbit-stabiliser relation makes that division an exact two-term conservation law. Between the perturbations that move nothing and those that move the output without law lies a third class, moving the output by a single fixed symmetry of the value space. We prove that, once a group A of admissible value symmetries is fixed, one canonical object governs this class. The covariant perturbations form the twisted stabiliser, and their quotient B over the invisible subgroup is universal among group quotients of the law-carrying perturbations that identify the invisible subgroup with the identity. The quotient induces the exact tripartite refinement log|G|= log|Orb G([φ]A)|+ log|B|+ log|Stab|, a noiseless channel of capacity $\log|B|$, and the split H(X|M) = H∂ + H_inv in the uniform finite setting. A converse shows that the identity fixes the sector sizes but neither the boundary homomorphism nor the boundary group's isomorphism type; inequivalent orientations can share the same identity. The classification extends to Haar measure and Lie dimension, and an operational reading separates the boundary's existence from the injectivity of its access map.