Symmetry-Forced Fiber Structure in Physical Inverse Design: A Unified Framework

Description: We identify and formalize a structure that pervades physical inverse design problems: when the forward map Phi from generator space, to observable space is equivariant under a discrete symmetry G, an entire subspace of generators collapses onto a single behavioral fiber. All generators within that subspace are observationally indistinguishable. The physically meaningful degree of freedom is the single transversal direction that breaks G-symmetry. Three propositions are proven: G-orbits lie within single fibers; G-preserving directions lie in the kernel of D Phi; G-breaking directions project onto the normal bundle N F (b) under a G-sensitivity condition. A Fiber Stability Theorem establishes that perturbed fibers lie within a tubular neighborhood of radius eps/sigmaₘin, where sigmaₘin is the minimum singular value of D Phi on N F (b). Design stability is governed entirely by sigmaₘin on N F (b), not by the full parameter space. Fiber transitions are classified into three types: continuous symmetry breaking (Type I), singularity of D Phi at exceptional points or quantum critical loci (Type II), and discrete topology jump (Type III). Optimal representative selection within fibers is developed with robustness procedures at Type II singularities and a Pareto front connecting behavioral class to minimum physical implementation cost. The FS-TC algorithm operationalizes the framework for black-box forward maps using Jacobian SVD, Levenberg-Marquardt regularization near Type II loci, and multi-start fiber classification. An analytic benchmark provides exact ground truth with sigmaₘin = 2||theta|| verified to five significant figures and kappa = 1. 00 exactly at the symmetric point. A Z₂ codimension-1 conjecture — fiber transitions generically controlled by a single scalar escape parameter, consistent with the Z₂ pitchfork normal form — is supported by eleven independent physical realizations spanning quantum transport, non-Hermitian topology, quantum criticality, lattice field theory, vacuum entanglement harvesting, hypersonic thermal protection, and topological quantum computing. In every case the normal bundle is 1-dimensional and one scalar parameter controls the fiber transition. The conjecture is grounded in equivariant bifurcation theory and stated explicitly as conjectural pending a complete proof.