The Universal Equilibrium Constant (I = 4πGσ / c⁴)

We extend previous work on the photogravitational ratio Φ = L / (M g) by introducing a new invariant quantity that reveals a fundamental constant of nature: I = (Φ β²) / (T⁴ R²) = 4 π G σ / c⁴ = 5.888 × 10⁻⁵¹ m⁻¹ s⁻¹ K⁻⁴ Building upon the established photogravitational framework, where Φ quantifies the balance between radiation and gravity, we show that this combination yields a universal constant equal to 4 π G σ / c⁴. This invariant links radiative equilibrium, Newtonian gravity, and relativistic compactness, demonstrating that all thermal, optically thick photospheres—from main-sequence stars to neutron stars and formally to black holes—obey the same equilibrium condition. The discovery of this universal constant reduces the independent stellar observables (L, M, R, T) from four to three, revealing that classical equilibrium laws—Stefan–Boltzmann, Eddington, Kelvin–Helmholtz, and Hawking—are not independent relations but algebraic projections of a single identity. The constant bridges microscopic physical constants (G, σ, c) and macroscopic observables, establishing photospheres as geometric–thermodynamic boundaries of spacetime. Empirical verification across 23 orders of magnitude in mass—from the Sun, Sirius A, Vega, and Wolf–Rayet 124 to theoretical compact objects—confirms invariance within measurement uncertainties. In Wolf–Rayet stars, the photogravitational parameter Φ clusters by subtype (±10 %), allowing direct mass determination from (L, R) alone and revealing a systematic ≈43 % mass deficit relative to evolutionary models. This equilibrium constant provides a unified, falsifiable framework linking gravity, radiation, and thermodynamics across nature.