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      "title": "The PFT Algebra is Closed: - Operator Closure Under PAL–Coherence on the Allen Orbital Lattice",
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      "title": "Introduction",
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      "level": 1,
      "title": "PFT Operator Hierarchy",
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      "level": 2,
      "title": "Configuration space",
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      "level": 2,
      "title": "Operator set",
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      "level": 2,
      "title": "Hierarchy indices",
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    {
      "level": 1,
      "title": "PAL–Coherent States and Operator Domain",
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      "level": 2,
      "title": "Phase Alignment Lock",
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      "level": 2,
      "title": "PAL–coherent subspace",
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      "level": 2,
      "title": "Bracket definition",
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    {
      "level": 1,
      "title": "Closure Theorem",
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    {
      "level": 1,
      "title": "Structure Constants and Prime Harmonics",
      "anchor": "#sec:constants"
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    {
      "level": 2,
      "title": "Prime–indexed harmonic sums",
      "anchor": "#primeindexed-harmonic-sums"
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    {
      "level": 2,
      "title": "Spectral boundedness",
      "anchor": "#spectral-boundedness"
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    {
      "level": 2,
      "title": "Example: \\([\\mathcal{R},\\mathcal{N}]\\) to \\(\\mathcal{G}\\)",
      "anchor": "#example-mathcalrmathcaln-to-mathcalg"
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    {
      "level": 1,
      "title": "Lagrange–Hex Ghost Kernel and Invariance",
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    {
      "level": 2,
      "title": "Ghost kernel",
      "anchor": "#ghost-kernel"
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    {
      "level": 2,
      "title": "Operator action on the kernel",
      "anchor": "#operator-action-on-the-kernel"
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    {
      "level": 2,
      "title": "Invariance under closure",
      "anchor": "#invariance-under-closure"
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    {
      "level": 1,
      "title": "Consequences for Representation Theory and Unification",
      "anchor": "#sec:conseq"
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    {
      "level": 2,
      "title": "Finite–dimensional Lie–type structure",
      "anchor": "#finitedimensional-lietype-structure"
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    {
      "level": 2,
      "title": "Representations as physical sectors",
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      "level": 2,
      "title": "Unification",
      "anchor": "#unification"
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      "title": "Discussion and Outlook",
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      "title": "Appendix A — Glossary of Terms",
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      "level": 1,
      "title": "Appendix B — PFT Internal Bibliography",
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      "title": "C.1 Jacobi identity",
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