On the Foundation of Physics Admissibility

This work establishes admissibility as a necessary condition for well-defined comparison, defined by invariance under relabeling, refinement, composition, finite propagation, and closure. From these constraints, relational structure and compositional consistency are forced, yielding a unique scalar comparison form. All alternatives introduce dependence on representation, decomposition, or descriptive scale, and are therefore excluded. The result is model-independent and shows that the quadratic invariant structure appearing in both general relativity and quantum theory is not assumed, but required. Any theory admitting well-defined scalar comparison must conform to this constraint.