Sharp Minimax Limits and Compatibility Spectra for Critical Near-DFT Index-Only Frequency Estimation

Abstract

An $M$-channel discrete Fourier transform (DFT) channelizer routes an on-grid sinusoid to a single output. When each frame reports only one energy-proportional channel index, signed sub-bin frequency estimation becomes nonregular: dark-channel probability is quadratic in the offset, whereas orientation enters cubically. We study $N$ independent labeled reports under a known uniform-replacement probability ${\epsilon }_N$ and a single deterministic unitary shared by all frequencies and frames, constrained to routing defect $\tau /N$. If $\sqrt{N}{\epsilon }_N\to \lambda <\infty$, we establish an attained global-in-frequency minimax limit at the critical scales $N^{-1/4}$ for frequency offset, $N^{-1/2}$ for unitary perturbation and replacement, and $N^{-1}$ for routing defect. The effective unitary tangent is a symmetric complete-graph edge field modulo one centering nuisance, and its first variation is a radius-dependent weighted divergence. For $\lambda >0$, evaluation at $k$ distinct normalized offset magnitudes yields an exact Fourier-Cauchy nullity spectrum: $\lfloor (M-1)/2\rfloor$ evaluations are necessary and sufficient, for every choice of distinct magnitudes, to certify neutrality at all magnitudes. The terminal nullspace has a greatest-common-divisor dimension formula and positive-definite aggregate curvature. Consequently, exact DFT routing is uniquely minimax within the complete critical tangent class for $M=3$, whereas every positive critical defect budget strictly improves the minimax constant for $M\ge 4$. The analysis also yields a smallest-prime curvature-visibility law and, for $M\ge 5$, discontinuous compatibility geometry but a continuous minimax value at the zero critical replacement floor $\lambda =0$.