Dimensional Analysis in Finite Ring Cosmology

The paper explores the provenance of the fundamental physical units in a physical universe defined over a finite relational arithmetic substrate. The units are identified as the four horizons of the substrate: the Planck length, momentum, time, and energy, one per representation domain of the framed shell. The four physical constants \( c \), \( \hbar \), \( G \), \( k_B \) follow uniquely from this quartet, over declared identifications whose epistemic status is tagged throughout. The speed of light is the ratio across the two Fourier pairs of the domain square. The Planck constant is the product within each conjugate pair. The Boltzmann constant is the mixed product. The gravitational constant is the normalisation of the quartet to the totality. The quartet admits exactly one internal cancellation identity. No further algebraically independent relation exists within its lattice. On an admissible substrate the four constants instantiate as exact residues; their unit-face magnitudes are their observer-facing manifestation. Beneath the quartet, dimensional analysis is developed as a graded modular domain algebra over the framed shell. Classical dimensional analysis is recovered exactly inside the sub-capacity window. The domain of the Planck constant is the unit flag: the generator of the unique order-four torsion subgroup that the domain lattice itself carries. The flag marks the crossing between the observer's counts and the substrate-anchored measures. Its torsion-free surrogate is the classical mass dimension. Temperature inherits the acceleration domain. The flag cancels in every count-valued comparison.