This work develops a rigorous framework for constructing physical laws from primitive invariants and sparse geometric structure. Dimensional analysis identifies the admissible invariant space but does not uniquely determine atomic balances or residual dynamical structure. This paper proves that primitive circuits form the canonical building blocks of admissible invariants and introduces a coordinate system that yields an exact decomposition of physical laws into structural, dynamical, and residual components. In a thermally active horizon setting, the framework reduces the problem to a single fundamental clock variable. A canonical phase realization theorem shows that this clock is uniquely determined by the geometry of a smooth closed Euclidean cycle. Evaluating this canonical structure at the fundamental endpoint fixes the normalization and reproduces the Hawking temperature for black holes. The Euclidean cycle is then interpreted physically as a thermal equilibrium period. The result is a fully theorem-based construction of the temperature law from invariant structure and geometric closure alone.
Akiva Shmalo (Wed,) studied this question.