The Universal Boundary Quotient of an Observed Representation

Abstract Let G be a perturbation group, φ: X → Y an observed representation, and A the group of allowed symmetries of the value set. We prove that the perturbations which act on φ as a single state-independent symmetry in A form a maximal subgroup K. The strictly invisible perturbations form a normal subgroup H ◁ K, and the quotient B = K/H canonically carries the nontrivial lawful intermediate sector. Every group-valued description compatible with the group operation and insensitive to H factors through B uniquely. The finite case yields an exact tripartite identity and a channel transmitting log |B| zero-error bits per use, and for a uniform perturbation the residual entropy splits exactly into a boundary term and an invisible term. The converse shows that the three sector sizes determine neither the boundary homomorphism nor the isomorphism type of B. We also formulate the classification for closed subgroups with Haar measure and Lie dimension. The classification indicates the type of additional resource required and asserts no general minimal repair, and for a finite boundary torsor the reference-specification cost is log |B| bits.