{
  "schema_version": 1,
  "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0271",
  "title": "Sharp Risk Bounds and a Shrinking-Variance Transition for Normal and Gamma Fits",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Suppose lognormal data are fitted by an ordinary normal model and a shape-scale gamma model. If s > 0 is the log variance, the difference Delta(s) between their optimized forward Kullback-Leibler risks satisfies 2s/3 < Delta(s) < s and 2/3 < Delta'(s) < 1; both constants are optimal. We establish the boundary behavior of the maximized gamma-to-normal log-likelihood ratio R_n when s = s_n tends to zero. If n s_n tends to c in (0, infinity), R_n converges in distribution to N(2c/3, 2c/3). If n s_n tends to infinity, gamma is selected with probability tending to one; if n s_n tends to zero, that probability tends to one half. An exact profile identity and uniform sample-moment expansion give the result. Fixed-variance likelihood and compact-prior marginal asymptotics are also included. The KL projection formulas and general misspecification theory are established prior ingredients. This is a scoped response to AIM Singular learning theory Problem 1.3, indexed as AIM-ALGEBRAIC_GEOMETRY-0271 in UnsolvedMath v1.6.0, not a resolution of every interpretation of the source prompt. The preprint is unrefereed and makes no absolute priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.ST",
    "stat.TH"
  ],
  "keywords": [
    "AIM-ALGEBRAIC_GEOMETRY-0271",
    "lognormal distribution",
    "gamma approximation",
    "model selection",
    "Kullback-Leibler divergence",
    "shrinking variance"
  ],
  "manuscript_version_date": "2026-10-16",
  "publication_date": "2026-10-16",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-16",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-algebraic-geometry-0271/",
  "pdf_url": "https://eulersolve.org/papers/aim-algebraic-geometry-0271/paper.pdf?v=cf4e48eeec7f",
  "doi": "10.5281/zenodo.23405918",
  "zenodo_record_url": "https://zenodo.org/records/23405918",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete sharp global risk bounds and a three-regime shrinking-variance likelihood theorem, not an unrestricted boundary Bayes-factor theorem or closure of every interpretation of the AIM source question. Fixed-variance marginal asymptotics require the stated compact-prior hypotheses. Existing KL projection and misspecification results are credited. Unrefereed and AI-assisted; no independent review, formal verification or absolute priority is certified.",
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    "source.zip": {
      "sha256": "aa17d661157f442b84a265c899dc8eef6fe89d4eb813f7eb2b1920016e8d1b9c"
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  "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."
}