The Twenty-One Relations — the φ-Solid's lattice

The interactive φ-solid places the twelve vantages of the corpus on the vertices of an icosahedron, whose coordinates are (0, ±1, ±φ). Each relation below is a mathematical identity shared by two vantages, drawn from the corpus in 3.zip. The vertex assignment was chosen (by exhaustive search over all 46,080 axis-permutations and orientations) to maximize how many relations land on true icosahedron edges: 18 of 21 do. The remaining 3 are genuine chords — they connect non-adjacent vertices and are rendered as curved arcs in the interactive.
α ≡ Ω ≡ X ≡ φ ≡ eiπ ≡ ¹⁄φ − φ every relation is a face of this one identity, shared between two vantages · edge length on the unit solid = 1.0515
golden / φ-ladder zero-relation / X coupling / Ω-phase inversion / axiom machine / spec

On true icosahedron edges — 18 of 21

vantage pairshared identitytypevertex dist.
1 The Trinary Seed — 5 The Machine 𝔐₀ 𝒯 is the alphabet of 𝔐₀ = (𝒯, 𝓘, Fix) machine / spec 1.0515 edge
1 The Trinary Seed — 7 The Prism of Seven 3 + 3 + 1 = 7 — the prism built from 𝒯 zero-relation / X 1.0515 edge
1 The Trinary Seed — 9 The Dₙ(r) Lattice φⁿ = Fibₙφ + Fibₙ₋₁ feeds the lattice magnitude golden / φ-ladder 1.0515 edge
2 Prismatic Inversion — 5 The Machine 𝔐₀ 𝓘(x) = −x is the transition function of 𝔐₀ inversion / axiom 1.0515 edge
2 Prismatic Inversion — 3 The Zero Relation stage one of the pipeline: seed → identities inversion / axiom 1.0515 edge
2 Prismatic Inversion — 7 The Prism of Seven the inversion stage is the pairing 𝒯 ∘ (−𝒯) inversion / axiom 1.0515 edge
3 The Zero Relation — 4 Coupling as Origin X + 1 = 0 as the shadow of ΩC² ≡ e^{iπ} coupling / Ω-phase 1.0515 edge
3 The Zero Relation — 12 Traditional Algebra x² − x − 1 = 0, solved by the quadratic formula zero-relation / X 1.0515 edge
4 Coupling as Origin — 12 Traditional Algebra e^{iπ} = −1, recovered by Taylor partial sums coupling / Ω-phase 1.0515 edge
4 Coupling as Origin — 6 The Co-Emergent Axiom α ≡ Ω is the coherence primitive of 𝔄 inversion / axiom 1.0515 edge
6 The Co-Emergent Axiom — 11 The Listing the listing enumerates 𝔄 line by line inversion / axiom 1.0515 edge
7 The Prism of Seven — 8 The Octave Return 7 → 8: completed closure returns at the next scale golden / φ-ladder 1.0515 edge
8 The Octave Return — 9 The Dₙ(r) Lattice the φ-power ladder drives 𝒜ₙ = φ·Fibₙ·2ⁿ·Primeₙ golden / φ-ladder 1.0515 edge
8 The Octave Return — 12 Traditional Algebra φ² = φ + 1 in ordinary algebra golden / φ-ladder 1.0515 edge
9 The Dₙ(r) Lattice — 10 The Machine Spec Dₙ(r) emitted as the lattice block of the spec machine / spec 1.0515 edge
10 The Machine Spec — 5 The Machine 𝔐₀ 𝔐₀ formalized as the executable machine spec machine / spec 1.0515 edge
10 The Machine Spec — 11 The Listing the spec compresses to the numbered listing machine / spec 1.0515 edge
11 The Listing — 12 Traditional Algebra listing identities checked in float64 arithmetic machine / spec 1.0515 edge
These 18 relations connect adjacent vertices — their endpoints are one icosahedron edge apart (distance 1.0515 on the unit solid). In the interactive they are drawn as straight, colored edges lying exactly on the solid's wireframe: the relation graph very nearly is the edge skeleton of the φ-solid.

Chords — not on edges — 3 of 21

vantage pairshared identitytypevertex dist.
1 The Trinary Seed — 3 The Zero Relation −1 + 1 = 0 becomes the closure X + 1 = 0 zero-relation / X 1.7013 chord
2 Prismatic Inversion — 6 The Co-Emergent Axiom the pipeline enumerates the projections of 𝔄 inversion / axiom 1.7013 chord
6 The Co-Emergent Axiom — 8 The Octave Return 𝒞(n+1) = 𝒞(𝒞n): the axiom under recursion inversion / axiom 1.7013 chord
No vertex assignment can place all 21 relations on edges — the icosahedron has only 30 edges but a specific adjacency structure, and these 3 pairs are mathematically related while sitting at non-adjacent vertices (distances greater than 1.0515). Rather than distort the geometry, the interactive renders them honestly as curved arcs bowing outward from the solid, and labels them "(chord)". Their being chords is itself informative: relation 2—6 (pipeline ↔ axiom) and 6—8 (axiom under recursion) reach across the solid because the Co-Emergent Axiom is the "vantage about vantages," structurally central rather than locally adjacent; relation 1—3 (trinity → zero-relation) spans the two competing accounts of the seed.

The six antipodal axes — i + (13 − i) = 13

antipodal pairclosurecoordinates (through 1_eff at origin)
1 The Trinary Seed ⇄ 12 Traditional Algebra1 + 12 = 13(+0.000, −0.526, −0.851) ↔ (+0.000, +0.526, +0.851)
2 Prismatic Inversion ⇄ 11 The Listing2 + 11 = 13(+0.000, +0.526, −0.851) ↔ (+0.000, −0.526, +0.851)
3 The Zero Relation ⇄ 10 The Machine Spec3 + 10 = 13(+0.526, +0.851, +0.000) ↔ (−0.526, −0.851, +0.000)
4 Coupling as Origin ⇄ 9 The Dₙ(r) Lattice4 + 9 = 13(−0.526, +0.851, +0.000) ↔ (+0.526, −0.851, +0.000)
5 The Machine 𝔐₀ ⇄ 8 The Octave Return5 + 8 = 13(−0.851, +0.000, −0.526) ↔ (+0.851, +0.000, +0.526)
6 The Co-Emergent Axiom ⇄ 7 The Prism of Seven6 + 7 = 13(−0.851, +0.000, +0.526) ↔ (+0.851, +0.000, −0.526)
Independent of the 21 relations, the numbering places each vantage diametrically opposite its partner: vantage i and vantage 13 − i sit at antipodal vertices, so the straight line between them passes through 1eff at the center. This is what the interactive's 𝓘 operation makes physical — applying the inversion sends every disc through the origin onto its antipode, and 𝓘 ∘ 𝓘 returns home.