Rigidity and Conserved Structure in Symmetry–Selection Quantum Channels

We prove a rigidity theorem for symmetry-selection dynamics realized as a random-unitary quantum channel. A collection of histories is weighted by a Gibbs-type rule determined by a positive prior and a rigidity functional, and each history acts on boundary states by unitary conjugation. The resulting spin-amplitude weighted paths (SAWP) channel is completely positive, trace-preserving, and unital. The channel is shown to be covariant under any boundary symmetry that preserves the prior and rigidity functional and acts equivariantly on the history unitaries. In the Heisenberg picture, the fixed-point algebra of the channel is identified exactly with the commutant of the boundary kinematics algebra generated by the history unitaries. Thus the conserved structure is an operator algebra of observables, rather than a single scalar invariant. For finite history sets with a positive rigidity gap separating the minimizing histories, the channel concentrates exponentially onto the minimizing sector as the inverse temperature increases. In this limit, the SAWP channel converges in diamond norm to a canonical equilibrium channel supported on the minimizing histories. The result gives a Noether-type conservation statement for symmetry-selection channels: admissible symmetry and rigidity constraints select an invariant fixed-point algebra and its associated equilibrium sector under the channel dynamics. License note: Distributed under CC BY-NC-ND 4.0.