Universal Processing Law (UPL) Track 4: Derivation of the Bekenstein-Hawking Entropy from Discrete Pixel Geometry via the Universal Processing Law

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 & 2D Projection The central mechanism is the Cascade Freeze: 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 therefore A / lP². However, 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 (1/4) of its total surface area. The thermodynamic entropy, defined as the externally accessible information, is therefore precisely S = A / (4lP²). Computational Verification & Unique Predictions 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. 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 exact same UPL framework. This result builds upon the foundational UPL preprint (DOI: 10. 5281/zenodo. 19226622). Authors: Ahmed Lahmidi (Independent Researcher) Contact: ahmed. lahmidi. contact@gmail. com