Unit Primacy and Ordinal Superiority: the Mathematical Foundations of Scale-Invariant Physical Constants

We develop a mathematical framework for understanding physical constants through renor-

malization group theory, percolation phenomena, and categorical invariants. A carefully cir-

cumscribed set of dimensionless invariants measured at the unit scalethe RG plateau where

|∂φ/∂log L|≈0behave as true mathematical constants, while observations at quantum and

cosmic scales represent these constants modulo systematic obstructions. The framework syn-

thesizes the Chicago school's categorical universality, the Russian tradition of ergodic theory,

and the French geometric approach to information theory. We establish identity verication

protocols for Feigenbaum, Ising, and KPZ universality classes, introduce the organizer eld

construction for percolation, and formalize ordinal superiority as a category-theoretic criterion

for hypothesis comparison.