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      "level": 1,
      "title": "PFT Hierarchy as Renormalization Group: - Multi–Scale Flow on the Allen Orbital Lattice",
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      "level": 1,
      "title": "Introduction",
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    {
      "level": 1,
      "title": "PFT Hierarchy and Indices",
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    {
      "level": 2,
      "title": "Scale index \\(r\\)",
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      "level": 2,
      "title": "Network indices \\(m\\) and \\(n\\)",
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      "level": 2,
      "title": "Recursion index \\(a\\)",
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      "level": 2,
      "title": "Cascade index \\(k\\)",
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      "title": "RG Interpretation of the PFT Hierarchy",
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      "level": 2,
      "title": "Effective couplings on the AOL",
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      "level": 2,
      "title": "Coarse–graining and prime–composite merging",
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    {
      "level": 2,
      "title": "RG flow on the PFT hierarchy",
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    {
      "level": 1,
      "title": "Beta Function and Fixed Points",
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    {
      "level": 2,
      "title": "Definition of the beta function",
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    {
      "level": 2,
      "title": "UV and IR behaviour",
      "anchor": "#uv-and-ir-behaviour"
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    {
      "level": 1,
      "title": "Lagrange–Hex Coarse–Graining Unit",
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      "level": 2,
      "title": "Ghost layers and displacement classes",
      "anchor": "#ghost-layers-and-displacement-classes"
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    {
      "level": 2,
      "title": "Unit cell for RG transformations",
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    {
      "level": 2,
      "title": "Iterated coarse–graining",
      "anchor": "#iterated-coarsegraining"
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    {
      "level": 1,
      "title": "Collapse Taxonomy",
      "anchor": "#sec:collapse"
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    {
      "level": 2,
      "title": "Subcritical, critical and ultra–sparse regimes",
      "anchor": "#subcritical-critical-and-ultrasparse-regimes"
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    {
      "level": 2,
      "title": "Collapse and universality",
      "anchor": "#collapse-and-universality"
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    {
      "level": 1,
      "title": "UV and IR Limits: Connection to GR and QFT",
      "anchor": "#sec:uv_ir"
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    {
      "level": 2,
      "title": "UV fixed point and conservation–dominated regime",
      "anchor": "#uv-fixed-point-and-conservationdominated-regime"
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    {
      "level": 2,
      "title": "IR limits and emergent GR",
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    {
      "level": 2,
      "title": "IR limits and emergent QFT",
      "anchor": "#ir-limits-and-emergent-qft"
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    {
      "level": 1,
      "title": "Comparison with Wilsonian RG",
      "anchor": "#sec:compare"
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    {
      "level": 2,
      "title": "Degrees of freedom",
      "anchor": "#degrees-of-freedom"
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    {
      "level": 2,
      "title": "Coupling space",
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      "title": "Fixed points and universality",
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      "level": 2,
      "title": "Background independence",
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      "title": "Appendix A — Glossary of Terms and Indices",
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      "level": 1,
      "title": "Appendix B — PFT Internal Bibliography",
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      "title": "Appendix C — Notes on RG Flow Structure in PFT",
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      "level": 2,
      "title": "C.1 Completion equivalence and RG",
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