{
  "schema_version": "1.1.0",
  "volume_id": "union_closed",
  "title": "The Element in Half the Worlds",
  "subtitle": "Frankl’s conjecture and the structure of union-closed families",
  "status": "Open conjecture; full-tier documentary exposition; no proof claim",
  "claim_boundary": "This is a source-normalized illustrated documentary. It does not prove Frankl's conjecture, establish a new universal frequency bound, certify unreviewed proof claims, or make a novelty, priority, or public-release claim.",
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    "archival_role": "enhancement_only",
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  },
  "palette": {
    "navy": "#071A36",
    "parchment": "#FAF5E8",
    "gold": "#B78935",
    "ink": "#1D2940"
  },
  "plates": [
    {"id":"cover","label":"Cover","title":"The Element in Half the Worlds","asset":"assets/documentaries/union_closed/cover.svg","alt":"An illuminated navy, cream, and gold cover showing overlapping set-worlds, element nodes, and the exact half-frequency conjecture.","authority":"pedagogical_orientation_only"},
    {"id":"garden","label":"Plate I","title":"The Garden That Closes","asset":"assets/documentaries/union_closed/plate_garden.svg","alt":"A branching garden of finite sets flows upward to their pairwise unions, contrasting local closure with the global search for an abundant element.","authority":"pedagogical_orientation_only"},
    {"id":"frequency","label":"Plate II","title":"The Half-Way Balance","asset":"assets/documentaries/union_closed/plate_frequency.svg","alt":"A brass balance compares one element’s frequency with half the number of sets, above the incidence-counting identity linking all frequencies to total set size.","authority":"pedagogical_orientation_only"},
    {"id":"entropy","label":"Plate III","title":"The Entropy Bridge","asset":"assets/documentaries/union_closed/plate_entropy.svg","alt":"A uniform random set passes through coordinate entropies and coupled unions to a dimension-free constant lower bound that still falls short of one half.","authority":"pedagogical_orientation_only"},
    {"id":"lattice","label":"Plate IV","title":"The Lattice Cathedral","asset":"assets/documentaries/union_closed/plate_lattice.svg","alt":"A finite join-semilattice rises from bottom to top through atoms, joins, and irreducible elements, illustrating the lattice formulation and deletion routes.","authority":"pedagogical_orientation_only"},
    {"id":"certificate","label":"Plate V","title":"The Certificate Forge","asset":"assets/documentaries/union_closed/plate_certificate.svg","alt":"A formal forge separates exact finite replay, Lean theorems, imported results, hybrid packages, source audits, and the universal conjecture’s claim boundary.","authority":"pedagogical_orientation_only"},
    {"id":"frontier","label":"Plate VI","title":"Islands of Theorem","asset":"assets/documentaries/union_closed/plate_frontier.svg","alt":"Islands representing special families, rigorous constant bounds, and formal lattice results approach but do not reach the open one-half threshold.","authority":"pedagogical_orientation_only"}
  ],
  "chapters": [
    {"id":"reader-note","title":"A Note to the Reader","source_heading":"A Note to the Reader","claim_classes":["exposition","guardrail","open"]},
    {"id":"garden","title":"The Garden That Closes Under Union","source_heading":"The Garden That Closes Under Union","claim_classes":["exposition","established","open"]},
    {"id":"half-way","title":"The Half-Way Balance","source_heading":"The Half-Way Balance","claim_classes":["established","open","guardrail"]},
    {"id":"small-worlds","title":"Small Worlds, Exact Ledgers","source_heading":"Small Worlds, Exact Ledgers","claim_classes":["formal_bounded","guardrail"]},
    {"id":"mirrors-lattices","title":"Mirrors, Intersections, and Lattices","source_heading":"Mirrors: Complements, Intersections, and Lattices","claim_classes":["established","formal_bounded","guardrail"]},
    {"id":"average-size","title":"The Average-Size Door","source_heading":"The Average-Size Door","claim_classes":["imported","guardrail"]},
    {"id":"entropy","title":"Entropy Enters the Garden","source_heading":"Entropy Enters the Garden","claim_classes":["imported","conditional","open","guardrail"]},
    {"id":"theorem-frontier","title":"The Theorem Frontier","source_heading":"The Theorem Frontier","claim_classes":["established","imported","formal_bounded","open","guardrail"]},
    {"id":"certificate-forge","title":"The Certificate Forge","source_heading":"The Certificate Forge","claim_classes":["formal_bounded","guardrail","open"]}
  ],
  "appendices": [
    {"id":"appendix-definitions","title":"Precise Definitions and Normalizations","source_heading":"Precise Definitions and Normalizations","claim_classes":["established"]},
    {"id":"appendix-elementary","title":"Elementary Proof Spine","source_heading":"Elementary Proof Spine","claim_classes":["established","formal_bounded"]},
    {"id":"appendix-entropy","title":"Entropy Method Skeleton","source_heading":"Entropy Method Skeleton","claim_classes":["imported","guardrail"]},
    {"id":"appendix-lattice","title":"Lattice Correspondence Notes","source_heading":"Lattice Correspondence Notes","claim_classes":["established","formal_bounded","guardrail"]},
    {"id":"appendix-formal","title":"Formal and Computational Inventory","source_heading":"Formal and Computational Inventory","claim_classes":["formal_bounded","guardrail"]},
    {"id":"appendix-sources","title":"Source Audit Ledger","source_heading":"Source Audit Ledger","claim_classes":["imported","conditional","guardrail"]},
    {"id":"appendix-trust","title":"Claim-Level Trust Matrix","source_heading":"Claim-Level Trust Matrix","claim_classes":["established","imported","formal_bounded","open","guardrail"]}
  ],
  "sources": [
    "https://doi.org/10.1017/S0963548302005230",
    "https://arxiv.org/abs/2211.09055",
    "https://arxiv.org/abs/2211.11504",
    "https://arxiv.org/abs/2212.00658",
    "https://arxiv.org/abs/2212.12500",
    "https://doi.org/10.37236/12232",
    "https://arxiv.org/abs/2306.08824",
    "https://arxiv.org/abs/2503.00277",
    "https://arxiv.org/abs/2504.13454"
  ]
}