This paper presents the algebraic skeleton underlying Pentagon Physics. The discriminant 5 of σ = 1/(1+σ) generates the field ℚ(√5). The field ℚ(√5) has class number 1 (unique factorisation). The unique automorphism √5 → −√5 acts as the fundamental symmetry operation. The pentagon, with its diagonal-to-side ratio φ, is the geometric realisation. The binary icosahedral group 2I is the symmetry group of the unique regular polytope in four dimensions that realises this structure. This is the algebraic chain that connects the axiom to the Standard Model.