Universal Processing Law (UPL) Track 6: Resolution of the Ultraviolet Catastrophe from Discrete Click Statistics

We derive the functional form of the Planck blackbody spectrum from the discrete computational axioms of the Universal Processing Law (UPL) without assuming energy quantization (E = h). The Click Budget Mechanism In UPL, each pixel of the Planck-scale hardware graph executes at most C_ irreducible state-updates (Clicks) per hardware cycle. Rendering a radiation mode of frequency costs one Click per oscillation cycle, so higher-frequency modes consume more Clicks per hardware cycle. In thermal equilibrium, the hardware distributes its finite Click budget among competing modes. Because Clicks are indivisible, mode occupation numbers are restricted to non-negative integers. The Planck Spectrum from Click Statistics Maximizing the number of microstates (Click allocation configurations) subject to the finite per-pixel budget yields the Bose-Einstein distribution: n () = 1 () - 1 producing the exact Planck spectrum shape: ³ () - 1 with no free parameters beyond the Lagrange multiplier (conjugate to the Click budget, identified with h/ (kT) ). Resolution of the Ultraviolet Catastrophe The ultraviolet catastrophe is resolved because high-frequency modes are combinatorially suppressed: they cost too many indivisible Clicks per quantum for the finite pixel budget to sustain. Planck had to assume energy comes in packets. UPL explains why: because the hardware updates come in indivisible Clicks. This derivation complements companion results deriving the cosmological constant (97. 3% accuracy), the Bekenstein-Hawking entropy (S = A/ (4lP²) ), and the objective wavefunction collapse time (= / (8 EG) ) from the same UPL master equation. All theoretical concepts, derivations, and original ideas are the sole intellectual work of Ahmed Lahmidi. Contact: ahmed. lahmidi. contact@gmail. com