Inverse Gaussian Smoothing: Sharp Hölder Exponents and Spectral Boundaries

This preprint studies the recovery of total-variation distances before additional Gaussian convolution, uniformly over unrestricted location mixing laws. On a separable real Hilbert space, the common component covariance is positive, injective and trace class. For nonnegative Hilbert–Schmidt relative noise, the sharp supremal inverse Hölder exponent is determined by the operator norm. Uniform coefficients are controlled by operator and Hilbert–Schmidt norm bounds, and their optimal logarithmic growth is quadratic in the Hilbert–Schmidt bound. Finite active rank admits a determinant correction retaining the relative covariance spectrum, with the critical exponent attained at a simple spectral edge. Infinite-dimensional obstructions yield sharp continuity thresholds for power-law spectra and the first logarithmic boundary. An exact Gaussian contrast reduction also characterizes stability under arbitrarily strong noise through transversality and whitened principal angles. General endpoint attainment and a necessary and sufficient criterion for arbitrary spectra remain open. The files comprise the paper PDF, its matching LaTeX source, and a computation supplement requiring only Python 3.10 or later and the standard library. The supplement verifies exact rational identities and archived results, and provides floating-point illustrations with stated tolerances. Both programs were reproduced in ordinary and optimized modes: 8,775 exact checks and 61 numerical checks per mode. Finite computations support the analytic arguments and do not establish the universal or limiting theorems on their own.