Horizon thermodynamics involves several dimensionless coefficients across independent theoretical frameworks: the proportionality between acceleration and temperature in the Unruh effect, the coefficient in the saturated Bekenstein entropy bound and the coefficient in the horizon area–entropy relation. While their mutual consistency is often assumed, it is seldom addressed explicitly. Here the three coefficients are treated as a priori free parameters. Consistency is imposed by requiring agreement across three independent expressions for the entropy–mass derivative (dS/dM), evaluated at a common horizon scale. Requiring equality of the Unruh First Law route, the saturated Bekenstein bound and a general area–entropy relation generates two independent equations defining the relationship between these coefficients: CuCb=1 and Cu=1/8πCa. These ’bridge relations’ identify an algebraic structure that is otherwise only implicit in the literature. We demonstrate that within the family R = εGM/c² and assuming inverse-square scaling, only R = 2GM/c² allows for the simultaneous consistency between the Unruh temperature, the Bekenstein bound and the First Law. The same approach is extended to leading quantum (logarithmic) corrections, providing a compact diagnostic for the internal consistency of semiclassical gravity models.
Russell Borland (Fri,) studied this question.