# Verification of the Haar SURE counterexample

The complete analytic result concerns a fixed smooth Gaussian-mixture noise law and Haar soft thresholding at 1/8. In every dyadic dimension, the Gaussian SURE bias is at most -273d/1604 for every pairwise-constant signal. At zero signal the normalized bias has an exact negative scale-series limit. The general AIM question about conditional wavelet adaptivity remains open in this work.

## Analytical audit

The risk is expanded directly; Gaussian integration by parts is not assumed for non-Gaussian noise. Absolute continuity makes the two threshold boundaries null events. The Lipschitz clipped correction is bounded, so the risk expansion and cross terms are integrable. Orthogonality preserves covariance, not independence.

For each finest Haar row, equal independent signs occur with probability 1/2. Conditional on this event, the remaining Gaussian coefficient has variance 1/401. Chebyshev gives a small-ball probability at least 337/802 at threshold 1/8. There are d/2 finest rows; the other probability terms have the correct nonpositive sign, and the cross term is at most 2t per selected row. The resulting bound is exactly -273d/1604. The proof permits arbitrary coarser signal coefficients and either convention for the constant row.

At zero signal, there are d/m rows of block length m, each with the stated binomial-Gaussian marginal. Linearity of expectation gives the scale series. The bound |b_m| <= 2 yields a tail at most 2/d for details-only thresholding, or 4/d including the constant row. Neither scale independence nor a numerical extrapolation is used.

## Reproducible checks

The standard-library program in reproducibility/check_haar_sure.py uses exact arithmetic in Q(sqrt(2)). It checks orthogonality at dimensions 2, 4, 8, 16 and 32; binomial moment normalization; and 1,104 pointwise risk/divergence identities at dimensions 2, 4 and 8, for both threshold conventions and zero and nonzero pairwise-constant signals. Normal and optimized Python modes must give identical results.

The program separately evaluates Gaussian-mixture integrals through normal-moment formulas and checks representative components against piecewise Simpson quadrature. These are floating-point illustrations, not interval-certified numerical bounds. The value near -0.5996787086 at dimension 2048 is not a certified enclosure of the limiting constant. The analytical theorem does not depend on it.

Failure checks retain the bias of an unscaled formula at nonunit variance, the positive second derivative 159999 of the mixture log density at zero, and an atom at the chosen threshold for unsmoothed Rademacher noise at dimension 256. Thus the example does not satisfy strong log-concavity, and discrete derivative conventions are not silently transferred to the smooth proof.

## Scope of assurance

This is an originating-researcher self-audit supported by reproducible finite calculations. It is not independent human peer review or formal verification. The general SURE identities and the existing unrestricted-estimator counterexamples are not claimed as new. The bounded literature comparison and the full mathematical scope are set out in source_and_novelty_review.md. No whole-source closure or absolute-priority certification is claimed.