A Geometric Counting Law at the Horizon: Local Étendue and the Geometric Origin of Hawking Entropy

The local étendue constant is introduced as the limiting form of the optogeometric factor. While global étendue is a standard concept in radiometry, it is shown that for an infinitesimal surface element the local density converges to the universal value of pi. This provides a direct counting rule for spatial–angular modes per unit area. Applied to black hole horizons, the rule implies that the number of modes is proportional to the horizon area divided by the square of the wavelength, leading to a geometric derivation of the Hawking–radiation entropy flux. The result takes the compact form of a dimensionless coefficient multiplied by the horizon area and the cube of the Hawking temperature. This coefficient encapsulates greybody transmission and spin multiplicities. Accordingly, apart from universal quantum statistics and transmission effects, the entropy of black–hole radiation is determined purely by geometry (the horizon area) and surface gravity (through the Hawking temperature). The derivation isolates the universal scaling and treats all model-dependent details as a dimensionless prefactor. It assumes local hemispherical acceptance and massless fields on a stationary background; deviations primarily affect the coefficient but not the fundamental area–temperature scaling.