We derive the exact Bekenstein-Hawking entropy formula S = A/ (4lP²) from the first principles of a Planck-scale discrete computational geometry, the Universal Processing Law (UPL). The derivation requires no quantum field theory on curved backgrounds, no statistical mechanics, and no free parameters. The Cascade Freeze Mechanism When the UPL processing yield = N₋₎₂₀₋/C_ drops to zero at the event horizon, boundary pixels become computationally frozen, each locking exactly one Planck bit of state data. The total information on the boundary is A/lP². The 2D Projection and the Factor of 1/4 An external observer probes this boundary through data packets (photons) arriving from a specific direction and can only access the two-dimensional projection of the three-dimensional frozen surface. By Cauchy's surface projection theorem, the average projected area of any convex body equals exactly one quarter of its total surface area: S = 14 AlP² = A4\, lP² Computational Verification We verify this result computationally on discrete voxel spheres of radii R = 3 to R = 50 Planck lengths, confirming convergence to the exact 1/4 factor in the macroscopic limit. Unique Prediction A unique, falsifiable prediction of UPL is that microscopic black holes exhibit a finite-size lattice correction to the Bekenstein-Hawking formula of order O (lP/R). This derivation complements the companion result deriving the cosmological constant to 97. 3% accuracy from the same UPL framework. All theoretical concepts, derivations, and original ideas are the sole intellectual work of Ahmed Lahmidi. Contact: ahmed. lahmidi. contact@gmail. com
Ahmed Lahmidi (Thu,) studied this question.