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  "volume_id": "hodge",
  "title": "The Geometry of Hidden Harmony",
  "subtitle": "A guided journey through cohomology, Hodge decomposition, and algebraic cycles",
  "status": "Open Millennium Prize Problem; rational projective statement; no algebraicity proof claim",
  "claim_boundary": "Browser-native, source-normalized exposition of the rational Hodge conjecture for smooth projective complex varieties. Low-dimensional cases, numerical coincidence, integral classes, arbitrary compact Kähler manifolds, Hodge loci, Abel–Jacobi invariants, the Tate conjecture, Mumford–Tate theory, and motivic heuristics are not promoted to the general algebraicity statement.",
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      "id": "cover",
      "label": "Cover",
      "title": "The Geometry of Hidden Harmony",
      "asset": "assets/documentaries/hodge/cover.svg",
      "alt": "An illuminated complex variety is overlaid by a Hodge diamond and golden algebraic cycles.",
      "authority": "pedagogical_orientation_only"
    },
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      "id": "memory",
      "label": "Plate I",
      "title": "How a shape remembers",
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      "alt": "Cycles thread through a complex geometric world while cohomology records their global traces.",
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      "label": "Plate II",
      "title": "The Hodge decomposition",
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      "alt": "A luminous Hodge diamond separates cohomology into conjugate bidegrees.",
      "authority": "pedagogical_orientation_only"
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      "id": "cycles",
      "label": "Plate III",
      "title": "From subvarieties to classes",
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      "alt": "Algebraic subvarieties map through the cycle-class map into rational Hodge classes.",
      "authority": "pedagogical_orientation_only"
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    {
      "id": "frontier",
      "label": "Plate IV",
      "title": "The exact rational frontier",
      "asset": "assets/documentaries/hodge/frontier.svg",
      "alt": "Known divisor and low-dimensional terrain is separated from false integral and unrestricted Kähler variants.",
      "authority": "pedagogical_orientation_only"
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    {
      "id": "reader-note",
      "title": "How to Read the Harmony",
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      "id": "cohomology",
      "title": "How a Shape Remembers",
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      "id": "decomposition",
      "title": "The Hodge Decomposition",
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      "id": "cycles",
      "title": "Algebraic Cycles and Their Classes",
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    {
      "id": "conjecture",
      "title": "The Exact Rational Conjecture",
      "source_heading": "The Hodge Conjecture, Exactly",
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      "id": "terrain",
      "title": "Known Islands and False Extensions",
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  "appendices": [
    {
      "id": "appendix-categories",
      "title": "Category and Coefficient Restrictions",
      "source_heading": "Technical Appendix: Categories and Coefficients",
      "claim_classes": [
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      "id": "appendix-cycle-map",
      "title": "The Cycle-Class Map",
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      "id": "appendix-low-dimensional",
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      "id": "appendix-trust",
      "title": "Hodge Loci, Motives, and Trust Matrix",
      "source_heading": "Technical Appendix: Adjacent Conjectures and Trust Matrix",
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  "sources": [
    "https://www.claymath.org/millennium/hodge-conjecture/",
    "https://www.claymath.org/wp-content/uploads/2022/05/hodge.pdf",
    "https://doi.org/10.1007/978-3-662-03080-7",
    "https://doi.org/10.1007/BF02684695",
    "https://doi.org/10.1007/s00222-003-0329-0"
  ]
}