{
  "schema": "co-emergent-trinary-algebra / phi-solid relation lattice",
  "description": "The twelve corpus vantages placed on the vertices of an icosahedron (coordinates (0, ±1, ±φ), unit radius). Each relation is a mathematical identity shared by two vantages. Vertex assignment maximizes relations lying on true icosahedron edges: 18 of 21. Generated from coemergent_phisolid.html.",
  "seed_identity": "α ≡ Ω ≡ X ≡ φ ≡ e^(iπ) ≡ 1/φ − φ",
  "edge_length_unit_solid": 1.051462,
  "counts": {
    "relations": 21,
    "on_edge": 18,
    "chord": 3,
    "antipodal_axes": 6
  },
  "vertices": [
    {
      "index": 1,
      "title": "The Trinary Seed",
      "coordinates": [
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        -0.850651
      ]
    },
    {
      "index": 2,
      "title": "Prismatic Inversion",
      "coordinates": [
        0,
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      ]
    },
    {
      "index": 3,
      "title": "The Zero Relation",
      "coordinates": [
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        0
      ]
    },
    {
      "index": 4,
      "title": "Coupling as Origin",
      "coordinates": [
        -0.525731,
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        0
      ]
    },
    {
      "index": 5,
      "title": "The Machine 𝔐₀",
      "coordinates": [
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      ]
    },
    {
      "index": 6,
      "title": "The Co-Emergent Axiom",
      "coordinates": [
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      ]
    },
    {
      "index": 7,
      "title": "The Prism of Seven",
      "coordinates": [
        0.850651,
        0,
        -0.525731
      ]
    },
    {
      "index": 8,
      "title": "The Octave Return",
      "coordinates": [
        0.850651,
        0,
        0.525731
      ]
    },
    {
      "index": 9,
      "title": "The Dₙ(r) Lattice",
      "coordinates": [
        0.525731,
        -0.850651,
        0
      ]
    },
    {
      "index": 10,
      "title": "The Machine Spec",
      "coordinates": [
        -0.525731,
        -0.850651,
        0
      ]
    },
    {
      "index": 11,
      "title": "The Listing",
      "coordinates": [
        0,
        -0.525731,
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      ]
    },
    {
      "index": 12,
      "title": "Traditional Algebra",
      "coordinates": [
        0,
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      ]
    }
  ],
  "relations": [
    {
      "from": {
        "index": 1,
        "title": "The Trinary Seed",
        "coordinates": [
          0,
          -0.525731,
          -0.850651
        ]
      },
      "to": {
        "index": 3,
        "title": "The Zero Relation",
        "coordinates": [
          0.525731,
          0.850651,
          0
        ]
      },
      "shared_identity": "−1 + 1 = 0 becomes the closure X + 1 = 0",
      "type": "zero-relation / X",
      "vertex_distance": 1.701302,
      "on_icosahedron_edge": false
    },
    {
      "from": {
        "index": 1,
        "title": "The Trinary Seed",
        "coordinates": [
          0,
          -0.525731,
          -0.850651
        ]
      },
      "to": {
        "index": 5,
        "title": "The Machine 𝔐₀",
        "coordinates": [
          -0.850651,
          0,
          -0.525731
        ]
      },
      "shared_identity": "𝒯 is the alphabet of 𝔐₀ = (𝒯, 𝓘, Fix)",
      "type": "machine / spec",
      "vertex_distance": 1.051462,
      "on_icosahedron_edge": true
    },
    {
      "from": {
        "index": 1,
        "title": "The Trinary Seed",
        "coordinates": [
          0,
          -0.525731,
          -0.850651
        ]
      },
      "to": {
        "index": 7,
        "title": "The Prism of Seven",
        "coordinates": [
          0.850651,
          0,
          -0.525731
        ]
      },
      "shared_identity": "3 + 3 + 1 = 7 — the prism built from 𝒯",
      "type": "zero-relation / X",
      "vertex_distance": 1.051462,
      "on_icosahedron_edge": true
    },
    {
      "from": {
        "index": 1,
        "title": "The Trinary Seed",
        "coordinates": [
          0,
          -0.525731,
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        ]
      },
      "to": {
        "index": 9,
        "title": "The Dₙ(r) Lattice",
        "coordinates": [
          0.525731,
          -0.850651,
          0
        ]
      },
      "shared_identity": "φⁿ = Fibₙφ + Fibₙ₋₁ feeds the lattice magnitude",
      "type": "golden / φ-ladder",
      "vertex_distance": 1.051462,
      "on_icosahedron_edge": true
    },
    {
      "from": {
        "index": 2,
        "title": "Prismatic Inversion",
        "coordinates": [
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        ]
      },
      "to": {
        "index": 6,
        "title": "The Co-Emergent Axiom",
        "coordinates": [
          -0.850651,
          0,
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        ]
      },
      "shared_identity": "the pipeline enumerates the projections of 𝔄",
      "type": "inversion / axiom",
      "vertex_distance": 1.701302,
      "on_icosahedron_edge": false
    },
    {
      "from": {
        "index": 2,
        "title": "Prismatic Inversion",
        "coordinates": [
          0,
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        ]
      },
      "to": {
        "index": 5,
        "title": "The Machine 𝔐₀",
        "coordinates": [
          -0.850651,
          0,
          -0.525731
        ]
      },
      "shared_identity": "𝓘(x) = −x is the transition function of 𝔐₀",
      "type": "inversion / axiom",
      "vertex_distance": 1.051462,
      "on_icosahedron_edge": true
    },
    {
      "from": {
        "index": 2,
        "title": "Prismatic Inversion",
        "coordinates": [
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        ]
      },
      "to": {
        "index": 3,
        "title": "The Zero Relation",
        "coordinates": [
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          0
        ]
      },
      "shared_identity": "stage one of the pipeline: seed → identities",
      "type": "inversion / axiom",
      "vertex_distance": 1.051462,
      "on_icosahedron_edge": true
    },
    {
      "from": {
        "index": 2,
        "title": "Prismatic Inversion",
        "coordinates": [
          0,
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          -0.850651
        ]
      },
      "to": {
        "index": 7,
        "title": "The Prism of Seven",
        "coordinates": [
          0.850651,
          0,
          -0.525731
        ]
      },
      "shared_identity": "the inversion stage is the pairing 𝒯 ∘ (−𝒯)",
      "type": "inversion / axiom",
      "vertex_distance": 1.051462,
      "on_icosahedron_edge": true
    },
    {
      "from": {
        "index": 3,
        "title": "The Zero Relation",
        "coordinates": [
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          0
        ]
      },
      "to": {
        "index": 4,
        "title": "Coupling as Origin",
        "coordinates": [
          -0.525731,
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          0
        ]
      },
      "shared_identity": "X + 1 = 0 as the shadow of ΩC² ≡ e^{iπ}",
      "type": "coupling / Ω-phase",
      "vertex_distance": 1.051462,
      "on_icosahedron_edge": true
    },
    {
      "from": {
        "index": 3,
        "title": "The Zero Relation",
        "coordinates": [
          0.525731,
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          0
        ]
      },
      "to": {
        "index": 12,
        "title": "Traditional Algebra",
        "coordinates": [
          0,
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        ]
      },
      "shared_identity": "x² − x − 1 = 0, solved by the quadratic formula",
      "type": "zero-relation / X",
      "vertex_distance": 1.051462,
      "on_icosahedron_edge": true
    },
    {
      "from": {
        "index": 4,
        "title": "Coupling as Origin",
        "coordinates": [
          -0.525731,
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          0
        ]
      },
      "to": {
        "index": 12,
        "title": "Traditional Algebra",
        "coordinates": [
          0,
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        ]
      },
      "shared_identity": "e^{iπ} = −1, recovered by Taylor partial sums",
      "type": "coupling / Ω-phase",
      "vertex_distance": 1.051462,
      "on_icosahedron_edge": true
    },
    {
      "from": {
        "index": 4,
        "title": "Coupling as Origin",
        "coordinates": [
          -0.525731,
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          0
        ]
      },
      "to": {
        "index": 6,
        "title": "The Co-Emergent Axiom",
        "coordinates": [
          -0.850651,
          0,
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        ]
      },
      "shared_identity": "α ≡ Ω is the coherence primitive of 𝔄",
      "type": "inversion / axiom",
      "vertex_distance": 1.051462,
      "on_icosahedron_edge": true
    },
    {
      "from": {
        "index": 6,
        "title": "The Co-Emergent Axiom",
        "coordinates": [
          -0.850651,
          0,
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        ]
      },
      "to": {
        "index": 8,
        "title": "The Octave Return",
        "coordinates": [
          0.850651,
          0,
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        ]
      },
      "shared_identity": "𝒞(n+1) = 𝒞(𝒞n): the axiom under recursion",
      "type": "inversion / axiom",
      "vertex_distance": 1.701302,
      "on_icosahedron_edge": false
    },
    {
      "from": {
        "index": 6,
        "title": "The Co-Emergent Axiom",
        "coordinates": [
          -0.850651,
          0,
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        ]
      },
      "to": {
        "index": 11,
        "title": "The Listing",
        "coordinates": [
          0,
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        ]
      },
      "shared_identity": "the listing enumerates 𝔄 line by line",
      "type": "inversion / axiom",
      "vertex_distance": 1.051462,
      "on_icosahedron_edge": true
    },
    {
      "from": {
        "index": 7,
        "title": "The Prism of Seven",
        "coordinates": [
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          0,
          -0.525731
        ]
      },
      "to": {
        "index": 8,
        "title": "The Octave Return",
        "coordinates": [
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          0,
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        ]
      },
      "shared_identity": "7 → 8: completed closure returns at the next scale",
      "type": "golden / φ-ladder",
      "vertex_distance": 1.051462,
      "on_icosahedron_edge": true
    },
    {
      "from": {
        "index": 8,
        "title": "The Octave Return",
        "coordinates": [
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          0,
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        ]
      },
      "to": {
        "index": 9,
        "title": "The Dₙ(r) Lattice",
        "coordinates": [
          0.525731,
          -0.850651,
          0
        ]
      },
      "shared_identity": "the φ-power ladder drives 𝒜ₙ = φ·Fibₙ·2ⁿ·Primeₙ",
      "type": "golden / φ-ladder",
      "vertex_distance": 1.051462,
      "on_icosahedron_edge": true
    },
    {
      "from": {
        "index": 8,
        "title": "The Octave Return",
        "coordinates": [
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          0,
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        ]
      },
      "to": {
        "index": 12,
        "title": "Traditional Algebra",
        "coordinates": [
          0,
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        ]
      },
      "shared_identity": "φ² = φ + 1 in ordinary algebra",
      "type": "golden / φ-ladder",
      "vertex_distance": 1.051462,
      "on_icosahedron_edge": true
    },
    {
      "from": {
        "index": 9,
        "title": "The Dₙ(r) Lattice",
        "coordinates": [
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        ]
      },
      "to": {
        "index": 10,
        "title": "The Machine Spec",
        "coordinates": [
          -0.525731,
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          0
        ]
      },
      "shared_identity": "Dₙ(r) emitted as the lattice block of the spec",
      "type": "machine / spec",
      "vertex_distance": 1.051462,
      "on_icosahedron_edge": true
    },
    {
      "from": {
        "index": 10,
        "title": "The Machine Spec",
        "coordinates": [
          -0.525731,
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        ]
      },
      "to": {
        "index": 5,
        "title": "The Machine 𝔐₀",
        "coordinates": [
          -0.850651,
          0,
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        ]
      },
      "shared_identity": "𝔐₀ formalized as the executable machine spec",
      "type": "machine / spec",
      "vertex_distance": 1.051462,
      "on_icosahedron_edge": true
    },
    {
      "from": {
        "index": 10,
        "title": "The Machine Spec",
        "coordinates": [
          -0.525731,
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          0
        ]
      },
      "to": {
        "index": 11,
        "title": "The Listing",
        "coordinates": [
          0,
          -0.525731,
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        ]
      },
      "shared_identity": "the spec compresses to the numbered listing",
      "type": "machine / spec",
      "vertex_distance": 1.051462,
      "on_icosahedron_edge": true
    },
    {
      "from": {
        "index": 11,
        "title": "The Listing",
        "coordinates": [
          0,
          -0.525731,
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        ]
      },
      "to": {
        "index": 12,
        "title": "Traditional Algebra",
        "coordinates": [
          0,
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        ]
      },
      "shared_identity": "listing identities checked in float64 arithmetic",
      "type": "machine / spec",
      "vertex_distance": 1.051462,
      "on_icosahedron_edge": true
    }
  ],
  "antipodal_axes": [
    {
      "pair": [
        {
          "index": 1,
          "title": "The Trinary Seed",
          "coordinates": [
            0,
            -0.525731,
            -0.850651
          ]
        },
        {
          "index": 12,
          "title": "Traditional Algebra",
          "coordinates": [
            0,
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          ]
        }
      ],
      "closure": "1 + 12 = 13"
    },
    {
      "pair": [
        {
          "index": 2,
          "title": "Prismatic Inversion",
          "coordinates": [
            0,
            0.525731,
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          ]
        },
        {
          "index": 11,
          "title": "The Listing",
          "coordinates": [
            0,
            -0.525731,
            0.850651
          ]
        }
      ],
      "closure": "2 + 11 = 13"
    },
    {
      "pair": [
        {
          "index": 3,
          "title": "The Zero Relation",
          "coordinates": [
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            0
          ]
        },
        {
          "index": 10,
          "title": "The Machine Spec",
          "coordinates": [
            -0.525731,
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            0
          ]
        }
      ],
      "closure": "3 + 10 = 13"
    },
    {
      "pair": [
        {
          "index": 4,
          "title": "Coupling as Origin",
          "coordinates": [
            -0.525731,
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            0
          ]
        },
        {
          "index": 9,
          "title": "The Dₙ(r) Lattice",
          "coordinates": [
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            -0.850651,
            0
          ]
        }
      ],
      "closure": "4 + 9 = 13"
    },
    {
      "pair": [
        {
          "index": 5,
          "title": "The Machine 𝔐₀",
          "coordinates": [
            -0.850651,
            0,
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          ]
        },
        {
          "index": 8,
          "title": "The Octave Return",
          "coordinates": [
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            0,
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          ]
        }
      ],
      "closure": "5 + 8 = 13"
    },
    {
      "pair": [
        {
          "index": 6,
          "title": "The Co-Emergent Axiom",
          "coordinates": [
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        },
        {
          "index": 7,
          "title": "The Prism of Seven",
          "coordinates": [
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            0,
            -0.525731
          ]
        }
      ],
      "closure": "6 + 7 = 13"
    }
  ]
}