Lawhood Necessity: A Structural Inevitability Theorem for the Architecture of Physical Law

This paper establishes a necessity result for the structural form of physical law. Building on The Tier-0 Framework: A Law-Level Closure and Selection Principle for Physics (Rodgers, 2026), which introduced a canonical lawhood architecture based on boundary normalization, admissibility filtering, and closure, the present work addresses the remaining foundational gap: Are these structures merely sufficient, or are they unavoidable? The paper proves that any viable notion of physical law under minimal and widely accepted requirements must realize three irreducible functional roles: removal of presentation-dependent artifacts (invariance under representation), enforcement of admissibility and persistence (robust solution selection), completion under composition and extension (global closure of the law space). It then shows that these roles force a unique structural architecture, consisting of: a quotient operation (eliminating representational redundancy), an admissibility filter (selecting stable, physically meaningful structures), a closure operator (ensuring compositional completeness). A final minimal principle, the Universal Certifier Principle is introduced and formalized to complete the bridge from role inevitability to full operator inevitability. The main result is that: Any system that functions as a physically meaningful law must implement this architecture, or an equivalent structure. This converts the Tier-0 framework from a constructive proposal into a necessity theorem for lawhood itself, significantly strengthening its foundational status. Related paper The Tier-0 Framework: A Law-Level Closure and Selection Principle for Physics DOI: https://doi.org/10.5281/zenodo.19382264