Tier-1 Structural Closure_ Formal Rules, Failure Modes, and Consequences of Violation

This paper formalizes Tier-1 structural closure as a regime-specific forcing phenomenon. A Tier-1 system is defined as a quadruple T = (P, A, S, C), where premises P, admissibility constraints A, derived structure S, and closure condition C together determine whether a system admits no remaining independent structural degrees of freedom. The paper establishes a minimal rule set (R1–R8) required for Tier-1 validity, proves that violation of each rule produces identifiable structural failure modes, and introduces a diagnostic procedure for evaluating candidate theories. A regime dependence theorem shows that all Tier-1 results are necessarily bounded to their defining premise–admissibility pair and cannot be universally extended across problem classes. Cross-domain instances demonstrate that distinct premise sets yield distinct forced structures while preserving the same underlying rule logic. The result is a formal framework for identifying, validating, and diagnosing structurally closed systems without invoking ontological or empirical claims.