{
  "schema_version": 1,
  "problem_number": "AIM-PROBABILITY-0028",
  "title": "Linear Gaussian SURE Bias for Haar Soft Thresholding under Smooth Noise",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "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 >= 2, 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.",
  "result_type": "COMPLETE_SCOPED_HAAR_SURE_COUNTEREXAMPLE",
  "categories": [
    "math.ST",
    "math.PR"
  ],
  "keywords": [
    "Gaussian SURE",
    "Haar wavelets",
    "non-Gaussian noise",
    "risk estimation",
    "soft thresholding",
    "counterexample"
  ],
  "manuscript_version_date": "2026-10-11",
  "publication_date": "2026-10-11",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-11",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-probability-0028/",
  "pdf_url": "https://eulersolve.org/papers/aim-probability-0028/paper.pdf?v=de8509dc96bb",
  "doi": "10.5281/zenodo.23299092",
  "zenodo_record_url": "https://zenodo.org/records/23299092",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete explicit counterexample to vanishing dimension-normalized Gaussian SURE bias for fixed-threshold Haar shrinkage under a smooth iid sub-Gaussian unit-variance noise law. The source's broader conditional and adaptive questions remain partially resolved. General SURE identities are prior art; no contradiction of strong-log-concavity theorems is claimed. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "de8509dc96bb70e23c8475be47e7e228e354eaf077c3bc956f2b9b0989304a42"
    },
    "source.zip": {
      "sha256": "2889cf8b1db3e144c4f2a696f927fdb021e7057889a9bf08c2e784705052cac3"
    },
    "verification_report.md": {
      "sha256": "15379e2258aacb7abd1e659a9eff84a71d2901b89ecc8610ef72d5d581364fb4"
    }
  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}