The Architecture — Pentagon Physics

σ = 1/(1+σ)

The axiom

↓

Discriminant 5 → ℚ(√5) → φ, σ

Two trunks

Trunk 1: partition

{σ², σ, 1, φ, 2}/√5

Bridge positions (no α needed)

↔ 5 Galois-invariant irreps

↓

Three geometries

φ coheres · π closes · e grows

Trunk 2: state space

2I → character table → McKay → E₈

Galois split: 94 invariant / 26 active

60 fermions = 60 bosons (exact)

↓

Gauge structure from E₈

4 EW bosons · 3 colours · 8 gluons

3+1 doublets · 3 generations (D₄)

Y = Z₅/dim(ρ₄) = 1/3

Both trunks required

Bypass α

sin²θ₁₃ = 1/45
sin²θ₂₃ = 3/(2φ²)
sin²θ_W → σ³
α_s = σ³/2

α⁻¹ = 137.036

Exchange rate: algebra → physics

E₈ propagator at z = φ: ratios 2, φ, 0

↓

Bridge line: depth = α⁻¹ × R + σ²

↓

G = α¹⁸×12/7

18 = cage bars
R = σ/√5 ↔ ρ₆

H₀

R = 1/√5 ↔ ρ₇

R = 2/√5 ↔ ρ₉

FDM mass

49.9 nHz

↓

Impedance cage

3 colours × 3 gen × 2 chiral = 18
gravity = EM behind the cage

Mass / gravity duality

mₚ/mπ = 7×(12²−1)/12²
7/12 inward · 12/7 outward

mₑ, mₚ, mᵥ, nuclei

integers = irrep dims
6, 2, 5, 3, 7, 12, 360