Dimensional Analysis over Finite Holographic Substrate

The paper derives the fundamental physical units in a universe defined over a finite holographic substrate: a finite totality whose capacity bounds the state count of every embedded observer. The units are anchored at the substrate’s four horizons: the Planck length, momentum, time and energy. The constants c, ℏ, G, kB follow uniquely from this quartet, over realisations stated with their falsifiers: c the ratio across the two Fourier pairs, ℏ the product within each conjugate pair, kB the mixed product, G the normalisation to the totality. The quartet admits one cancellation identity and no further independent relation; on an admissible substrate the constants are exact residues read as unit-face magnitudes. Beneath the quartet, dimensional analysis is a graded modular domain algebra on the framed shell, recovering the classical calculus exactly inside the sub-capacity window. The domain of ℏ is the unit flag; its torsion-free surrogate is the classical mass dimension, temperature inherits the acceleration domain, and the flag cancels in every count-valued comparison. Two predictions are staked: a resolution floor on single coherent registration and the exact constancy of the constants’ dimensionless ratios. Every theorem of the paper is numerically verified and formalized in Lean 4.