{
  "schema_version": 1,
  "problem_number": "AIM-PROBABILITY-0007",
  "title": "Finite Likelihood Traps on Connected Boltzmann Trees",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We construct finite nonglobal local maxima of exact empirical likelihood for a binary restricted Boltzmann machine with one hidden unit and three visible units. All seven weights and biases are free, the data have full support, and the equivalent two-component mixture has distinct components with equal positive mixing weights. We determine the exact local-maximum-to-saddle threshold, including its degenerate case. A conditional likelihood argument joins these examples at a shared visible center: for every k, a connected allowed tree with k hidden vertices has at least 2^k distinct local-maximum visible laws at finite parameters. All but one displayed law are nonglobal, with an explicit likelihood deficit. The extrema are nonstrict and some allowed edge weights vanish. These are complete scoped counterexample theorems about raw-parameter likelihood, not a solution to the general efficient sparse-RBM training question in AIM-PROBABILITY-0007. The source package includes reproducible exact and floating-point supplementary checks. This is an unrefereed preprint; AI assistance is disclosed, and no absolute priority or formal verification is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "stat.ML",
    "math.PR",
    "cs.LG"
  ],
  "keywords": [
    "restricted Boltzmann machines",
    "local maxima",
    "empirical likelihood",
    "tree graphs",
    "latent class models",
    "AIM-PROBABILITY-0007"
  ],
  "manuscript_version_date": "2026-10-17",
  "publication_date": "2026-10-17",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-17",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-probability-0007/",
  "pdf_url": "https://eulersolve.org/papers/aim-probability-0007/paper.pdf?v=9367d1cad0a4",
  "doi": "10.5281/zenodo.23441404",
  "zenodo_record_url": "https://zenodo.org/records/23441404",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete finite three-leaf likelihood threshold and connected allowed-tree counterexample family. The local maxima are nonstrict and some allowed weights vanish. The original AIM efficient-training question remains open. Prior latent-class and degree-two work is credited. AI-assisted, self-audited and unrefereed; no independent review, formal certification or certified absolute priority.",
  "files": {
    "paper.pdf": {
      "sha256": "9367d1cad0a4a88ab440e4d8854e2d0bb56306e1a4e1cddecdd0c32980ecbc22"
    },
    "source.zip": {
      "sha256": "19702b28160ca5f7505faa30a82d905c776dbefbba68ab999d9e7b1984004310"
    },
    "verification_report.md": {
      "sha256": "5c5242911dd086a35bf8669f16f9750740d9b9cf8ca5abe8eb472bac89619e85"
    }
  },
  "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."
}