α⁻¹ IS THE UNIVERSAL CONVERSION RATE OF Λ, G, H₀

α⁻¹ IS THE UNIVERSAL CONVERSION RATE OF Λ, G, H Three cosmological and gravitational measurements recover the fine structure constant as their common slope, with no electromagnetic input Three of the most precisely measured numbers in physics have never been explained. The cosmological constant Λ tells us how fast the universe is accelerating its expansion. The gravitational constant G sets the strength of every gravitational interaction. The Hubble rate H₀ describes how fast the universe is currently expanding. Each is measured by a completely different experimental programme — satellite surveys of the cosmic microwave background, laboratory torsion balances, and supernova distance ladders respectively. Each is treated in standard physics as a free parameter: measured, inserted, and unexplained. This paper shows they are not independent. They are three points on a single straight line. The horizontal axis of that line is pure algebra, derived from the self-referential equation σ = 1/(1+σ). This equation has one fixed point, φ = (1+√5)/2, which generates a partition identity φ⁻¹ + φ⁻² = 1. This partition produces exactly four geometric coordinates — called bridge ratios — one for each structurally distinct way to traverse a two-part budget: a round trip, a one-way trip, an escape, and a stored floor. These four numbers are fixed by algebra alone. No measurement enters their derivation. The vertical axis is physical measurement: how many orders of magnitude below the Planck scale each constant sits. Λ sits at depth 122.95. G at depth 38.23. H₀ at depth 61.70. These come from three entirely separate experimental communities. Plot the four points. They fall on a straight line. The slope is 137.036. That number is α⁻¹ — the inverse fine structure constant, measured independently by atomic recoil experiments to eleven significant figures. It does not enter the derivation of the bridge ratios. It does not enter the depth measurements. It is simply what the slope comes out to be when algebra meets observation. Four consequences follow, all with zero free parameters: α⁻¹ is overdetermined. It is now recoverable from three measurements that contain no electromagnetic physics whatsoever. No electrons. No atoms. No QED. Three sectors — cosmology, gravity, expansion — all recover the same slope. This is not a consistency check. It is an independent determination of the fine structure constant from the large-scale structure of the universe. The Dirac large numbers are resolved. Since Λ, G, and H₀ all sit on the same line, their ratios are fixed by the partition geometry. log(Λ/G) = −α⁻¹/φ exactly. log(H₀/G) = −α⁻¹φ⁻² exactly. The mysterious large numbers that Dirac noticed in 1937 and spent years trying to explain with time-varying constants are not mysterious at all. They are the lengths of intervals on a straight line. The Friedmann equation is constrained. When the bridge formulas for Λ, G, and H₀ are substituted into the Friedmann equation, the α⁻¹ terms cancel exactly — forced by the partition identity R_Λ = 2R_H. The dark energy fraction Ω_Λ is a pure function of φ, not a free parameter set by initial conditions. The universe is converging toward Ω_Λ = φ⁻¹ = 0.618 from its current observed value of 0.685. The Hubble tension is a diagnostic, not a crisis. Because Λ is H₀ traversing a return trip (R_Λ = 2R_H exactly), they are not two independent constants. Any analysis that treats Λ as independent and fits H₀ from it will systematically underestimate H₀. The CMB infers 67.4 km/s/Mpc by exactly this method. Local measurements give 73.0. Pentagon Physics predicts 71.7 km/s/Mpc. The gap is not noise. It is the imprint of an inverted causal arrow. The paper also identifies the bridge line as one face of the Galois boundary that separates the free sector of physics (coupling constants, cosmological scales) from the confined sector (mass eigenvalues, mixing angles). The Breit-Wheeler pair production threshold — the energy at which light becomes matter, confirmed by the STAR experiment in 2021 — is the same boundary expressed in physical units. Three descriptions, one object. Kill conditions are stated explicitly throughout. The framework makes specific numerical predictions for DESI DR2 (w₀ ≈ −0.97), for next-generation gravitational constant measurements (α⁻¹ recovered from G to <50 ppm), and for the hard upper bound that no cosmological constant weaker than Λ can exist in this family. If any of these fail, the paper fails with them. This paper is part of the Pentagon Physics programme, which has derived all 26 free parameters of the Standard Model from the single axiom σ = 1/(1+σ), with zero free parameters throughout. Over 70 papers are published on Zenodo. Full documentation, interactive explanations, and collaboration invitations are at https://www.pentagonphysics.com. Published April 1, 2026. The author stands by the epigraph. Keywords: fine structure constant, cosmological constant, gravitational constant, Hubble constant, bridge partition, golden ratio, Pentagon Physics, Galois boundary, Breit-Wheeler, dark energy, Hubble tension, Dirac large numbers, overdetermination, zero free parameters