Bekenstein–Hawking Entropy from Gaussian Unit Group Order Without Free Parameters

The Bekenstein–Hawking entropy formula S = A/4 contains the denominator 4 as an empirical constant with no standard structural derivation. Loop Quantum Gravity requires a free parameter (the Barbero–Immirzi parameter γ) tuned to reproduce this factor. We propose that the denominator 4 is |Z4| = |ℤ[i]*| = |{1, i, −1, −i}|, the order of the Gaussian unit group—the group of invertible elements in the Gaussian integers. This identification eliminates the Immirzi parameter, replacing it with a topological invariant of the underlying arithmetic. The formula becomes S = A/|ℤ[i]*|, with the denominator fixed by the discrete rotational symmetry of the Gaussian lattice. We show this agrees with string-theoretic microstate counting and predicts a minimum of eight horizon punctures at the Bit+ radiation level.