We derive the geometric foundations of physics from a single axiom: "I AM I" — self-reference that closes. The derivation proceeds directly from closure to dimension to icosahedral geometry, establishing the icosahedron as the central mathematical object. From its structure emerge the dimensionless constants α (fine structure) and γ (Euler-Mascheroni). The key insight is the structure ratio e⁵ = e^(E/D!) = e^(30/6), where E = 30 edges and D! = 6 closure terms. This geometrically-determined exponent appears in the fine structure constant α⁻¹ = e⁵ − 6√3 − 1 + 1/66 (0.0σ match) and in nucleation theory as ΔG*/Q* = e⁵/1640. The same icosahedral geometry governs materials science: the glass transition occurs at 1/41 = 2.439% free volume, where 41 = V + E − 1 is the interior threshold. An independent arithmetic derivation through prime numbers (Appendix A) arrives at the same structure ratio e⁵, strengthening confidence through convergence. Spatial dimension D = 3 is derived via the unique identity D(D+1) = 2×D!, justified by the Borwein theorem. One constant (e). Zero free parameters. Everything derived.