The Bridges of Pentagon Physics

Pentagon Physics previously derived all constants of the Standard Model from the single self-referential axiom σ = 1/(1+σ), which has discriminant 5 and positive root φ = (1+√5)/2. Previous papers established the individual derivations. This paper establishes further connection between them.

The partition identity 1 = φ⁻¹ + φ⁻² forces four bridge ratios:

R_Λ = 2/√5,

R_H = 1/√5,

R_G = φ⁻¹/√5,

R_floor = φ⁻²/√5.

These are not independent numbers. They are four faces of a single algebraic object, and they generate three exact identities: R_Λ = 2R_H, R_G + R_floor = R_H, and R_Λ − R_G = 1/φ. From these identities, the cosmological constant, the gravitational constant, the Hubble rate, the U(1) hypercharge coupling, and the Higgs quartic coupling are connected as nodes in one partition rather than five separately derived results.

The paper proves the D2 diagonal identity g′² = λ_H = 2φ/25 exactly, confirms it survives one-loop RGE running from the geometric scale to M_Z, and organises all 26 derived constants into a depth taxonomy. The bridge algebra answers a question that constant-by-constant derivations cannot: why these constants sit at these scales relative to each other.

Keywords: fine-structure constant, cosmological constant, gravitational constant, Hubble constant, Higgs quartic coupling, golden ratio, self-referential axiom, Pentagon Physics