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 (H_W), the decision variable is the third singular value (σ_3(H_W)), which equals the exact distance to the set of rank-2 matrices (Eckart–Young–Mirsky theorem). Under an explicit noise assumption (|E|2 ≤ τ_h = C{obs} σ √h), if (σ_3(H_W) ≤ τ_h), the observation is indistinguishable from a rank-2 structure at the noise scale, and no inference is admissible.If (σ_3(H_W)) 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 (σ_3(H_W)). All results are conditional on explicit assumptions and restricted to the stated model class.