{
  "schema_version": "26.0-internal",
  "paper_id": "PFT:THE_ALLEN_ORBITAL_LATTICE_IS_COMPLETION_EQUIVALENCE_BACKGROUND_INDEPENDENCE_VIA_ZERO_GEOME_2",
  "entity_type": "paper",
  "slug": "the-allen-orbital-lattice-is-completion-equivalence-background-independence-via-zero-geome-2",
  "title": "The Allen Orbital Lattice is Completion Equivalence: - Background Independence via Zero–Geometry Determination",
  "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-allen-orbital-lattice-is-completion-equivalence-background-independence-via-zero-geome-2/",
        "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": "aol_gauge.tex"
    },
    "timestamp": {
      "available": false,
      "path": null
    }
  },
  "sections": [
    {
      "level": 1,
      "title": "The Allen Orbital Lattice is Completion Equivalence: - Background Independence via Zero–Geometry Determination",
      "anchor": null
    },
    {
      "level": 2,
      "title": "Repository Files",
      "anchor": null
    },
    {
      "level": 1,
      "title": "The Allen Orbital Lattice is Completion Equivalence: - Background Independence via Zero–Geometry Determination",
      "anchor": null
    },
    {
      "level": 1,
      "title": "Introduction",
      "anchor": "#sec:intro"
    },
    {
      "level": 1,
      "title": "Background Dependence in Existing Frameworks",
      "anchor": "#sec:hist"
    },
    {
      "level": 2,
      "title": "General relativity",
      "anchor": "#general-relativity"
    },
    {
      "level": 2,
      "title": "Quantum field theory",
      "anchor": "#quantum-field-theory"
    },
    {
      "level": 2,
      "title": "Lattice and discrete approaches",
      "anchor": "#lattice-and-discrete-approaches"
    },
    {
      "level": 2,
      "title": "Quantum gravity proposals",
      "anchor": "#quantum-gravity-proposals"
    },
    {
      "level": 1,
      "title": "Allen Orbital Lattice Structure",
      "anchor": "#sec:aol"
    },
    {
      "level": 2,
      "title": "Sites, edges and faces",
      "anchor": "#sites-edges-and-faces"
    },
    {
      "level": 2,
      "title": "Phase Alignment Lock",
      "anchor": "#phase-alignment-lock"
    },
    {
      "level": 2,
      "title": "Event cascades and operators",
      "anchor": "#event-cascades-and-operators"
    },
    {
      "level": 1,
      "title": "Zero–Geometry Determination",
      "anchor": "#sec:zerogeom"
    },
    {
      "level": 2,
      "title": "Physical observables",
      "anchor": "#physical-observables"
    },
    {
      "level": 2,
      "title": "Zero–geometry configurations",
      "anchor": "#zerogeometry-configurations"
    },
    {
      "level": 1,
      "title": "AOL Completions and PAL–Diffeomorphisms",
      "anchor": "#sec:completions"
    },
    {
      "level": 2,
      "title": "Completions",
      "anchor": "#completions"
    },
    {
      "level": 2,
      "title": "PAL–diffeomorphisms",
      "anchor": "#paldiffeomorphisms"
    },
    {
      "level": 1,
      "title": "Main Completion–Equivalence Theorem",
      "anchor": "#sec:mainthm"
    },
    {
      "level": 1,
      "title": "Lagrange–Hex Ghost Kernel",
      "anchor": "#sec:ghostkernel"
    },
    {
      "level": 2,
      "title": "Lagrange–hex projection",
      "anchor": "#lagrangehex-projection"
    },
    {
      "level": 2,
      "title": "Definition of the ghost kernel",
      "anchor": "#definition-of-the-ghost-kernel"
    },
    {
      "level": 2,
      "title": "Invariance under PAL–diffeomorphisms",
      "anchor": "#invariance-under-paldiffeomorphisms"
    },
    {
      "level": 1,
      "title": "Consequences for GR, QFT and Unification",
      "anchor": "#sec:conseq"
    },
    {
      "level": 2,
      "title": "Background independence in PFT",
      "anchor": "#background-independence-in-pft"
    },
    {
      "level": 2,
      "title": "GR as PAL–induced metric sector",
      "anchor": "#gr-as-palinduced-metric-sector"
    },
    {
      "level": 2,
      "title": "QFT as PAL–constrained cascade sector",
      "anchor": "#qft-as-palconstrained-cascade-sector"
    },
    {
      "level": 2,
      "title": "Unified gauge structure",
      "anchor": "#unified-gauge-structure"
    },
    {
      "level": 1,
      "title": "Comparison with Other Approaches",
      "anchor": "#sec:compare"
    },
    {
      "level": 2,
      "title": "Continuum GR",
      "anchor": "#continuum-gr"
    },
    {
      "level": 2,
      "title": "Lattice gauge theory",
      "anchor": "#lattice-gauge-theory"
    },
    {
      "level": 2,
      "title": "Loop quantum gravity and spin networks",
      "anchor": "#loop-quantum-gravity-and-spin-networks"
    },
    {
      "level": 2,
      "title": "String theory and related models",
      "anchor": "#string-theory-and-related-models"
    },
    {
      "level": 1,
      "title": "Discussion and Outlook",
      "anchor": "#sec:outlook"
    },
    {
      "level": 1,
      "title": "Appendix A — Glossary of Terms and Acronyms",
      "anchor": "#appendix-a-glossary-of-terms-and-acronyms"
    },
    {
      "level": 1,
      "title": "Appendix B — PFT Internal Bibliography",
      "anchor": "#appendix-b-pft-internal-bibliography"
    },
    {
      "level": 1,
      "title": "Appendix C — Notes on PAL–Diffeomorphism Structure",
      "anchor": "#appendix-c-notes-on-paldiffeomorphism-structure"
    },
    {
      "level": 2,
      "title": "C.1 Group–like properties",
      "anchor": "#c.1-grouplike-properties"
    },
    {
      "level": 2,
      "title": "C.2 Orbits of completions",
      "anchor": "#c.2-orbits-of-completions"
    },
    {
      "level": 2,
      "title": "C.3 Relation to continuum diffeomorphisms",
      "anchor": "#c.3-relation-to-continuum-diffeomorphisms"
    }
  ],
  "external_locations": [
    {
      "provider": "patternfieldtheory",
      "url": "https://patternfieldtheory.com/papers/the-allen-orbital-lattice-is-completion-equivalence-background-independence-via-zero-geome-2/",
      "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 Allen Orbital Lattice is Completion Equivalence: - Background Independence via Zero–Geometry Determination",
    "slug": "the-allen-orbital-lattice-is-completion-equivalence-background-independence-via-zero-geome-2",
    "canonical_html": "https://patternfieldtheory.com/papers/the-allen-orbital-lattice-is-completion-equivalence-background-independence-via-zero-geome-2/",
    "pdf": "https://patternfieldtheory.com/papers/the-allen-orbital-lattice-is-completion-equivalence-background-independence-via-zero-geome-2/paper.pdf",
    "ots": null,
    "ots_file_date": null,
    "original_folder": "old copies nov 2025",
    "original_tex_filename": "aol_gauge.tex",
    "original_pdf_filename": "aol_gauge.pdf",
    "original_ots_filename": null,
    "images_folder_copied": [
      "images"
    ],
    "csv_files_copied": [],
    "pandoc_status": "success",
    "source_last_write_utc": "2025-11-17T06:46:03Z",
    "generated_utc": "2026-05-08T11:02:19Z",
    "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": "aa57ebbe96e96ee4d4583ad35780d99c889fe02a6c32cad3721218d0af99f538"
  }
}