{
  "schema_version": 1,
  "problem_number": "AIM-PROBABILITY-0009",
  "title": "Bidirectional Inference Counts on Forests and Cycles",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a binary restricted Boltzmann machine on a fixed labeled bipartite graph, consider the pair of conditional maximum-a-posteriori maps in both directions, with the same weights used in the two maps and no ties. We count such pairs exactly on forests and on even cycles. On a forest, local threshold functions can share symmetric weights if and only if their monotonicity directions agree on every input edge essential at both endpoints. Positive rescaling gives a constructive proof and an incidence partition function evaluable by a tree recurrence. For paths, the counts satisfy p_1=2, p_2=14, and p_n=10p_(n-1)+6p_(n-2). For even cycles of length n>=4, the count is (5+sqrt(31))^n+(5-sqrt(31))^n-2^(n+1); the subtraction removes exactly two cyclic families of impossible magnitude comparisons. Two trees with identical degree lists in each bipartition class have different bidirectional counts. This invariant is distinct from the classical one-way inference count, which factors over output degrees.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "cs.LG"
  ],
  "keywords": [
    "restricted Boltzmann machine",
    "bidirectional inference",
    "AIM-PROBABILITY-0009",
    "math.CO",
    "cs.LG"
  ],
  "manuscript_version_date": "2026-09-29",
  "publication_date": "2026-09-29",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-29",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-probability-0009/",
  "pdf_url": "https://eulersolve.org/papers/aim-probability-0009/paper.pdf?v=6ccfb4756ef1",
  "doi": "10.5281/zenodo.23045888",
  "zenodo_record_url": "https://zenodo.org/records/23045888",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Explicit bidirectional extension, not closure of the entire AIM record; standard one-way product is prior work. No all-graph higher-degree cyclic formula, new general symmetrization method, or absolute-priority claim. Live AimPL item unavailable; Cosnard-Goles chapter checked at abstract level. Self-audited and unrefereed, with AI assistance; no independent peer review or absolute priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "6ccfb4756ef1d40c1b4e00666e7aaa47249201f7634fa6eb9a4c172225438323"
    },
    "source.zip": {
      "sha256": "5e6522904974c28cc6041794066b2823be35c3549fe8cb7eed1ed1190271a424"
    },
    "verification_report.md": {
      "sha256": "e3734758b0c15dd5e6b6dd1d2dc9d5a5241bf8aac3af3438ba57617941b3a7a9"
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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."
}