Certified Orbit, Witness, and Topology Labels for Equivariant Positive-Dimensional Fiber Benchmarks

Abstract

Numerical methods for positive-dimensional fibers are commonly assessed after root filtering, sampling, meshing, clustering, or topological postprocessing, even when the continuous real component structure is not known independently. We construct an exact-label benchmark oracle for a controlled class of orthogonally equivariant maps. Quotient reduction converts high-dimensional fiber recovery into low-dimensional equations; each regular positive-radius quotient root lifts to an explicit sphere or product-of-spheres orbit component. Beyond component labels, the oracle records the rank-loss mechanism, Betti numbers, algebraic witness data, quantitative root-isolation constants, and scoring rules. We prove determinant factorizations, exact principal-stratum fiber decompositions, a residual-to-distance and perturbation theorem, and a multi-radial extension. For polynomial orbit components, their complexifications are smooth complete intersections whose degree is two raised to the number of radial blocks, and an explicit transverse slice produces the same number of witness points. Monomial, Chebyshev, product-quadric, and exponential-linear families give reproducible instances. The accompanying package regenerates all reported tables and figures and validates symbolic root filtering, structured witness slices, conditioning exponents, independent chain-complex homology, filter collisions, and sampled clustering controls. The construction is not a universal solver benchmark; it is a certified regression testbed for specified real-domain, component-recovery, and singularity-classification failures.