{
  "schema_version": 1,
  "problem_number": "AMR-022-7077",
  "title": "Positive Functions Equimeasurable with Their Derivatives: Construction and Rigidity",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Rubel asked for functions whose derivative has the same value distribution as the function (Hayman-Lingham Problem 7.77). We parametrize all strictly positive C1 solutions on a compact interval by continuous Lebesgue-measure-preserving maps satisfying a spectral-radius condition. Twice-differentiable strictly positive solutions are exactly exponentials and an explicitly phased sine family. The tent map gives positive C1 solutions precisely for interval lengths below 2, with exact spectral radius, endpoint value, uniform approximation errors and dense dyadic losses of higher differentiability. At the critical length, a rescaling of the classical Fabius distribution supplies a smooth nonnegative example; that distribution and its smoothness are explicitly credited, not claimed as new. The construction also gives positive functions equimeasurable with any prescribed higher derivative on sufficiently short intervals. This answers the additional-examples request and classifies the strictly positive C1 class. The unrestricted classification of vanishing or sign-changing functions remains open; the whole frozen UnsolvedMath record AMR-022-7077 is not claimed solved. The package contains the five-page proof, exact rational checks and a source/scope audit. AI-assisted self-audited preprint; no independent peer review, formal verification or exhaustive priority certification is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CA",
    "math.DS"
  ],
  "keywords": [
    "AMR-022-7077",
    "Rubel problem",
    "equimeasurable functions",
    "differential rearrangements",
    "tent map",
    "Fabius distribution"
  ],
  "manuscript_version_date": "2026-10-26",
  "publication_date": "2026-10-26",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-26",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-022-7077/",
  "pdf_url": "https://eulersolve.org/papers/amr-022-7077/paper.pdf?v=db3da6713d66",
  "doi": "10.5281/zenodo.23651477",
  "zenodo_record_url": "https://zenodo.org/records/23651477",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "All strictly positive C1 functions equimeasurable with their first derivative on [0,L] are parametrized by a positive initial value and a continuous measure-preserving map T with spectral radius r(K_T)<1. The twice-differentiable subclass consists only of exponentials and a phased sine family. For the tent map r(K_T)=L/2, giving the exact threshold L<2, sharp approximation errors and dense dyadic losses of higher differentiability. A rescaled classical Fabius distribution gives a smooth nonnegative example at L=2. A positive-operator construction supplies functions equimeasurable with every prescribed higher derivative on sufficiently short intervals. The positive C1 class is parametrized and further examples are constructed. The general classification of vanishing or sign-changing functions remains open; the whole source record is not claimed solved. Fabius construction and standard tools are credited. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "db3da6713d667b78400c3b64689dd9a2cec8ea15827bf7fdf61bb14d51d46d5d"
    },
    "source.zip": {
      "sha256": "cb1e206cf9cf3d0447acae9434778e60e6a2567550716979eac5ed9869ff4f42"
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    "verification_report.md": {
      "sha256": "b7dc31da27af71eebe13137be8d6feb6591e197327fe43b2931aaec85466c226"
    }
  },
  "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."
}