Linear Gaussian SURE Bias for Haar Soft Thresholding under Smooth Noise

We give a fixed, smooth, sub-Gaussian noise distribution for which the bias of the Gaussian Stein unbiased risk estimate does not vanish after division by dimension, even for ordinary Haar soft thresholding. For every dyadic dimension \(d\ge2\), threshold \(1/8\) and signal constant on each finest coordinate pair, the bias is at most \(-273d/1604\). The noise coordinates are independent, identically distributed and standardized; their density is an everywhere positive mixture of two Gaussians. The bound covers both an unchanged and a thresholded constant coefficient. At the zero signal, the normalized bias has an exact absolutely convergent scale series with a strictly negative limit and an explicit truncation bound. The construction uses the positive proportion of localized Haar rows, not independence of transformed coefficients. It concerns fixed-threshold bias and does not contradict strong-log-concavity results or resolve adaptive threshold selection.

The general SURE risk and bias framework is credited to Fathi, Goldstein, Reinert and Saumard. The broader AIM-PROBABILITY-0028 record remains partially resolved; this preprint is not a claim of closing all of its conditional wavelet questions. The package contains the PDF, LaTeX source, exact arithmetic regression program, recorded outputs and verification report. AI-assisted, self-audited and unrefereed; no independent human peer review, formal verification or absolute-priority certification is claimed.