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      "level": 1,
      "title": "A Lattice-Consistent Reconstruction of Lagrange's Four-Square Geometry - on the Allen Orbital Lattice",
      "anchor": null
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      "level": 2,
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      "level": 1,
      "title": "A Lattice-Consistent Reconstruction of Lagrange's Four-Square Geometry - on the Allen Orbital Lattice",
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      "level": 1,
      "title": "Introduction",
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    {
      "level": 1,
      "title": "Canonical Double-Cube Geometry and \\(\\sqrt{1}\\) – \\(\\sqrt{6}\\)",
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      "level": 1,
      "title": "Correction of the \\(\\sqrt{3}\\) Placement",
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      "title": "AOL Projection and Ghost-State Representation",
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      "level": 1,
      "title": "Prime and Composite Reservation on the Allen Orbital Lattice",
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      "level": 2,
      "title": "Prime Reservation Channels",
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      "level": 2,
      "title": "Composite Reservation from Prime Channels",
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      "level": 2,
      "title": "Odd–Even Structural Split",
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    {
      "level": 2,
      "title": "Relation to \\(\\pi^2/6\\) and the \\(\\zeta(2)\\) Partial Sums",
      "anchor": "#relation-to-pi26-and-the-zeta2-partial-sums"
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    {
      "level": 1,
      "title": "AOL Hex-Footprint of the \\(\\sqrt{1}\\) – \\(\\sqrt{6}\\) System",
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      "level": 1,
      "title": "Figures: AOL Grid and Footprint",
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      "level": 2,
      "title": "TikZ Hex-Lattice Diagram",
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    {
      "level": 2,
      "title": "PNG Placeholder Diagram",
      "anchor": "#png-placeholder-diagram"
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      "level": 1,
      "title": "Lagrange’s Four-Square Decomposition on the AOL",
      "anchor": "#lagranges-four-square-decomposition-on-the-aol"
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    {
      "level": 2,
      "title": "Geometric Module",
      "anchor": "#geometric-module"
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    {
      "level": 2,
      "title": "AOL Projection of the Four-Square Structure",
      "anchor": "#aol-projection-of-the-four-square-structure"
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    {
      "level": 2,
      "title": "Connection to \\(\\pi^2/6\\)",
      "anchor": "#connection-to-pi26"
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    {
      "level": 2,
      "title": "Height Functions on the Hexagonal AOL",
      "anchor": "#height-functions-on-the-hexagonal-aol"
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    {
      "level": 2,
      "title": "Relation to Gibbs Measures and Delocalization",
      "anchor": "#relation-to-gibbs-measures-and-delocalization"
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    {
      "level": 2,
      "title": "Dominion Fixing and Symmetry Breaking in the AOL",
      "anchor": "#dominion-fixing-and-symmetry-breaking-in-the-aol"
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    {
      "level": 2,
      "title": "Interpretation in the Context of the Lagrange Projection",
      "anchor": "#interpretation-in-the-context-of-the-lagrange-projection"
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    {
      "level": 1,
      "title": "Conclusion",
      "anchor": "#conclusion"
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    {
      "level": 1,
      "title": "Dominion Spokes and Prime-Indexed Scaffolds",
      "anchor": "#sec:dominion-spokes"
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    {
      "level": 2,
      "title": "Prime-indexed dominion spokes",
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    {
      "level": 2,
      "title": "Charge as dominion spoke imbalance",
      "anchor": "#charge-as-dominion-spoke-imbalance"
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      "level": 1,
      "title": "The Dominion Sequence",
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      "level": 2,
      "title": "Step 1: AOL Sheet (Base Hex Lattice)",
      "anchor": "#sec:seq-step1"
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      "level": 2,
      "title": "Step 2: Ring Structure (n-Rings)",
      "anchor": "#sec:seq-step2"
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      "level": 2,
      "title": "Step 3: Dominion Spokes (Six Radial Sectors)",
      "anchor": "#sec:seq-step3"
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    {
      "level": 2,
      "title": "Step 4: Dominion Shells (3D Elevation)",
      "anchor": "#sec:seq-step4"
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      "level": 2,
      "title": "Step 5: Dominion Trajectories (Prime-Controlled Flow Lines)",
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      "level": 2,
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      "level": 1,
      "title": "Riemann Duplex Elevation in the Dominion Framework",
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      "level": 1,
      "title": "Implications for Physics",
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      "level": 2,
      "title": "1. Electrons Are Not Particles",
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      "level": 2,
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      "title": "3. Unification with Cosmological Structure",
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      "level": 2,
      "title": "4. Riemann Structure as Physical Geometry",
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      "level": 1,
      "title": "Dominion Spokes on the Allen Orbital Lattice",
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