{
  "schema_version": 1,
  "problem_number": "AIM-PROBABILITY-0016",
  "title": "Heat Flow Leaves Two-Hidden-Unit Restricted Boltzmann Models",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Independent bit-flip heat flow preserves binary restricted Boltzmann machines with one hidden unit, by their classical representation as mixtures of two product distributions. We give explicit evolving parameters. With two hidden units and at least four visible units, the behavior is different: for every fixed positive vector of flip rates, a generic positive parameter leaves even the algebraic closure of the model immediately. A sixteen-by-sixteen integer Jacobian determinant proves transversality in the four-visible case. A coordinate-permutation argument covers arbitrary positive rates, and conditioning extends the obstruction to every larger visible dimension. The proof distinguishes a rank witness from a pointwise smoothness certificate. For the logarithmic-mean graph transport metric, this yields a generically nonzero residual in projected entropy flow. This is a complete scoped finite-state theorem, not a solution of the broad AIM question about PDE approximation. The preprint is unrefereed and makes no absolute priority claim.",
  "result_type": "COMPLETE_SCOPED_THEOREM",
  "categories": [
    "math.PR",
    "math.AG",
    "stat.ML"
  ],
  "keywords": [
    "restricted Boltzmann machine",
    "heat flow",
    "algebraic statistics",
    "graph transport",
    "AIM-PROBABILITY-0016"
  ],
  "manuscript_version_date": "2026-10-20",
  "publication_date": "2026-10-20",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-20",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-probability-0016/",
  "pdf_url": "https://eulersolve.org/papers/aim-probability-0016/paper.pdf?v=53c1bfe28726",
  "doi": "10.5281/zenodo.23489067",
  "zenodo_record_url": "https://zenodo.org/records/23489067",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete theorem in a specified finite-state setting, not a full solution of AIM-PROBABILITY-0016. The broad PDE-approximation question remains partially solved, and its exact original web wording could not be verified. Generic non-invariance concerns two hidden units and at least four visible units, with fixed positive flip rates; the four-visible case also permits nonzero nonnegative rates. The integer rank witness does not establish its own pointwise smoothness or exit. No continuum limit, approximation error bound or arbitrary-hidden-width classification is claimed. Classical ingredients are credited. AI-assisted, self-audited and unrefereed; no independent review, formal certification, certified novelty or absolute priority claim.",
  "files": {
    "paper.pdf": {
      "sha256": "53c1bfe287263f0558b8d860894591b7ca1172272cc5b069fab0dfb802637780"
    },
    "source.zip": {
      "sha256": "a2d2501f56bb9d58c78d8f90762fc48d06100826288e6464abae16e563c3581c"
    },
    "verification_report.md": {
      "sha256": "ebe60cf9160381acd8e0363851c7fa592238e879b860434bf5240f23deb98234"
    }
  },
  "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."
}