{
  "schema_version": "26.0-internal",
  "paper_id": "PFT:THE_HODGE_CONJECTURE_A_LATTICE_COHERENT_RESOLUTION_THREE_FORMULATIONS_VIA_COHOMOLOGY_EVENT",
  "entity_type": "paper",
  "slug": "the-hodge-conjecture-a-lattice-coherent-resolution-three-formulations-via-cohomology-event",
  "title": "The Hodge Conjecture: A Lattice-Coherent Resolution - Three Formulations via Cohomology, Event Cascades, and Unified Field Structure on the Allen Orbital Lattice",
  "visibility": {
    "approved_public": false,
    "scope": "internal",
    "approval_state": "unreviewed"
  },
  "publication": {
    "published": null,
    "verification_status": "unchecked",
    "last_checked": null,
    "locations": [
      {
        "provider": "patternfieldtheory",
        "url": "https://patternfieldtheory.com/papers/the-hodge-conjecture-a-lattice-coherent-resolution-three-formulations-via-cohomology-event/",
        "verification_status": "discovered-unverified"
      },
      {
        "provider": "zenodo",
        "url": "https://zenodo.org/communities/pft",
        "verification_status": "discovered-unverified"
      },
      {
        "provider": "academia",
        "url": "https://independent.academia.edu/JamesAllen375",
        "verification_status": "discovered-unverified"
      }
    ]
  },
  "representations": {
    "html": {
      "available": true,
      "path": "index.html",
      "canonical": true
    },
    "json": {
      "available": true,
      "path": "paper.json"
    },
    "pdf": {
      "available": true,
      "path": "paper.pdf"
    },
    "latex": {
      "available": false,
      "path": null,
      "original_filename": "millenium-4-hodge-conjectureGOOD.tex"
    },
    "timestamp": {
      "available": false,
      "path": null
    }
  },
  "sections": [
    {
      "level": 1,
      "title": "The Hodge Conjecture: A Lattice-Coherent Resolution - Three Formulations via Cohomology, Event Cascades, and Unified Field Structure on the Allen Orbital Lattice",
      "anchor": null
    },
    {
      "level": 2,
      "title": "Repository Files",
      "anchor": null
    },
    {
      "level": 1,
      "title": "The Hodge Conjecture: A Lattice-Coherent Resolution - Three Formulations via Cohomology, Event Cascades, and Unified Field Structure on the Allen Orbital Lattice",
      "anchor": null
    },
    {
      "level": 1,
      "title": "Introduction",
      "anchor": "#introduction"
    },
    {
      "level": 1,
      "title": "AOL Structure and Higher-Dimensional PAL Forms",
      "anchor": "#aol-structure-and-higher-dimensional-pal-forms"
    },
    {
      "level": 2,
      "title": "Allen Orbital Lattice",
      "anchor": "#allen-orbital-lattice"
    },
    {
      "level": 1,
      "title": "Formulation A: PAL-Cohomology and Hodge \\((p,p)\\) Classes",
      "anchor": "#formulation-a-pal-cohomology-and-hodge-pp-classes"
    },
    {
      "level": 2,
      "title": "Hodge classes as PAL-stable harmonic regions",
      "anchor": "#hodge-classes-as-pal-stable-harmonic-regions"
    },
    {
      "level": 2,
      "title": "Algebraic representability",
      "anchor": "#algebraic-representability"
    },
    {
      "level": 1,
      "title": "Formulation B: Event Cascades and Algebraic Cycles",
      "anchor": "#formulation-b-event-cascades-and-algebraic-cycles"
    },
    {
      "level": 2,
      "title": "Algebraic cycles as cascade families",
      "anchor": "#algebraic-cycles-as-cascade-families"
    },
    {
      "level": 2,
      "title": "No non-algebraic PAL cascades",
      "anchor": "#no-non-algebraic-pal-cascades"
    },
    {
      "level": 1,
      "title": "Formulation C: Unified Spectral Field Structure",
      "anchor": "#formulation-c-unified-spectral-field-structure"
    },
    {
      "level": 2,
      "title": "AOL harmonic operator",
      "anchor": "#aol-harmonic-operator"
    },
    {
      "level": 2,
      "title": "Spectral decomposition",
      "anchor": "#spectral-decomposition"
    },
    {
      "level": 1,
      "title": "Main Theorem: Structural Resolution of the Hodge Conjecture",
      "anchor": "#main-theorem-structural-resolution-of-the-hodge-conjecture"
    },
    {
      "level": 1,
      "title": "References",
      "anchor": "#references"
    }
  ],
  "external_locations": [
    {
      "provider": "patternfieldtheory",
      "url": "https://patternfieldtheory.com/papers/the-hodge-conjecture-a-lattice-coherent-resolution-three-formulations-via-cohomology-event/",
      "verification_status": "discovered-unverified"
    },
    {
      "provider": "zenodo",
      "url": "https://zenodo.org/communities/pft",
      "verification_status": "discovered-unverified"
    },
    {
      "provider": "academia",
      "url": "https://independent.academia.edu/JamesAllen375",
      "verification_status": "discovered-unverified"
    }
  ],
  "software": [],
  "data_files": [],
  "legacy_metadata": {
    "title": "The Hodge Conjecture: A Lattice-Coherent Resolution - Three Formulations via Cohomology, Event Cascades, and Unified Field Structure on the Allen Orbital Lattice",
    "slug": "the-hodge-conjecture-a-lattice-coherent-resolution-three-formulations-via-cohomology-event",
    "canonical_html": "https://patternfieldtheory.com/papers/the-hodge-conjecture-a-lattice-coherent-resolution-three-formulations-via-cohomology-event/",
    "pdf": "https://patternfieldtheory.com/papers/the-hodge-conjecture-a-lattice-coherent-resolution-three-formulations-via-cohomology-event/paper.pdf",
    "ots": null,
    "ots_file_date": null,
    "original_folder": "millennium-04-Hodge-Conjecture",
    "original_tex_filename": "millenium-4-hodge-conjectureGOOD.tex",
    "original_pdf_filename": "millenium-4-hodge-conjectureGOOD.pdf",
    "original_ots_filename": null,
    "images_folder_copied": [
      "images"
    ],
    "csv_files_copied": [],
    "pandoc_status": "success",
    "source_last_write_utc": "2025-11-14T18:04:28Z",
    "generated_utc": "2026-05-08T11:01:52Z",
    "status": "canonical_html_export"
  },
  "migration": {
    "source": "papers(1).zip",
    "content_policy": "existing scientific HTML preserved; v26 shell and resource controls injected",
    "formula_status": "not rewritten during structural migration",
    "requires_source_formula_audit": true,
    "source_html_sha256": "01ea02a1e13833dfa0b765cc5cb7b9b170e1072cf906c192218bae65761fb110"
  }
}