The Information-Transition Boundary of Representation Systems

A representation of a system hides exactly those transformations that lie in its stabilizer; what it does not hide, it detects. This two-part picture, visible and invisible, is made quantitative by the orbit–stabilizer relation, which we read as an exact conservation law, since the logarithmic size of a structured perturbation group splits without loss or excess into a visible part, the variety of distinguishable appearances, and an invisible part, the stabilizer. The present paper shows that the frontier between the two parts, which is the information-transition boundary of a representation system, is not structureless. Relative to a chosen group A of symmetries of the value space, there is a third, intermediate class of perturbations that do change the representation, but change it lawfully, by a fixed automorphism of the values. We call these the boundary perturbations, and together with the invisible perturbations they form a group, the twisted stabilizer. Sending each boundary perturbation to the automorphism it induces is a group homomorphism, and we identify it with a correspondence subgroup of A×G in the sense of Goursat (1889). Our main result is an exact three-term identity, in which the classical law refines without loss or excess into a detected term that counts appearances modulo lawful dressing, a boundary term equal to the order of the realized automorphism group, and the invisible term. We prove that the boundary behaves as an exact, lossless communication channel, in the sense that an observer learns, about a boundary perturbation, precisely its induced automorphism and nothing else. Boundary information is moreover differential, carried by an ordered before/after pair and by no single observation, and three mobility laws govern it, namely the snapshot lemma, the one-way valve, and the torsor tariff. The identity measures the size of the boundary but not its orientation, and instruments with identical identities can carry inequivalent boundary homomorphisms. A dial without numerals turns out to be a pure boundary instrument, and we close by computing the exact size of the invisible part at the boundary of observational exhaustion, where entropy, dimension, and large-deviation rate coincide.