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

We present a universal framework linking gravity, radiation, and thermodynamics through a single invariant identity. The relation I = Φβ²/(T⁴R²) = 4πGσ/c⁴ = 5.888 × 10⁻⁵¹ m⁻¹s⁻¹K⁻⁴ remains constant across more than twenty orders of magnitude in mass, from stellar photospheres and compact objects to the observable universe. This invariance unifies classical equilibrium laws such as those of Stefan-Boltzmann, Eddington, Einstein-Schwarzschild, Chandrasekhar, Hawking, and the cosmic balance, all within one photogravitational framework. The work provides both theoretical derivation and empirical verification showing that the photogravitational constant acts as a universal equilibrium law connecting radiation pressure, gravity, and heat flow. Eight discrete cases: Stefan-Boltzmann, Eddington, Chandrasekhar, Schwarzschild, Carnot, Hawking, Planck, and cosmic equilibrium—demonstrate numerical invariance of I to within 0.1% precision. The results establish this invariant as a bridge between microphysical constants and macroscopic phenomena, suggesting that the balance between gravity, radiation, and thermodynamics represents an intrinsic property of spacetime itself.