Cos45:A Geometric Boundary for Rank-Admissible Inference Under Noise

This work defines a geometric boundary for rank-based inference under noise using the singular value structure of Hankel matrices. Given an observed Hankel matrix (HW), the decision variable is the third singular value (₃ (HW) ), which equals the exact distance to the set of rank-2 matrices (Eckart–Young–Mirsky theorem). Under an explicit noise assumption (|E|2 ₕ = Cobs h), if (₃ (HW) ₕ), the observation is indistinguishable from a rank-2 structure at the noise scale, and no inference is admissible. If (₃ (HW) ) exceeds a calibrated threshold, deviation from rank-2 structure can be locally certified under spectral gap conditions. The framework does not perform system identification, does not claim optimality, and does not extend Eckart–Young–Mirsky. It defines a boundary induced by noise-scaled indistinguishability within the class of completely monotone signals. A structural counterexample shows that local observables (e. g. , curvature) cannot determine the global rank structure captured by (₃ (HW) ). All results are conditional on explicit assumptions and restricted to the stated model class.