From Quantum Measurement to Gauge Structure: Minimal Response and Absolute Canonicity

Paper 3 of a four-paper program on generalized quantum measurement, realization geometry, and informational invariance. Generalized quantum measurements admit multiple purification and dilation realizations of the same operational state–measurement data. This paper studies the geometric structure induced by that representation freedom and develops a transport theory on the resulting realization space. On each finite-dimensional regular fixed-rank state–POVM stratum, framed purification amplitudes and framed Naimark realizations are shown to assemble into a smooth principal bundle whose fibers are precisely compact auxiliary gauge orbits. The paper establishes complete-POVM robustness results controlling states, labeled measurement architectures, fixed coarse events, canonical normalization, and smooth operational paths. A framed Hilbert–Schmidt response geometry is then introduced on the realization bundle. The associated minimum-response construction yields a smooth gauge-equivariant principal connection and an induced positive response metric on the operational base manifold. This provides a geometrically distinguished transport law and establishes relative canonicity within the framed response geometry. The manuscript further introduces Quantum Informational Identity (QII) and Dynamical Quantum Informational Identity (DQII) as explicit foundational identity assumptions. Exact gauge transformations are shown to preserve the generated informational profile of a realization, while smooth gauge paths preserve the corresponding dynamical profile history. Under QII and DQII, all quantum-informationally admissible principal connections determine the same represented quantum-physical transport content. Relative to the framed Hilbert–Schmidt response geometry, the minimum-response connection is the unique minimum-response representative of that common content. The paper refers to this combination of: uniqueness of represented quantum-physical transport content, and uniqueness of the framed minimum-response representative, as absolute canonicity within the generated realization framework. The results are restricted to finite-dimensional generated realizations on regular fixed-rank strata with fixed outcome labels. The paper does not claim that the framed response geometry is uniquely derived from informational principles, nor does it extend the transport theory across singular or rank-changing loci.