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

Abstract This work introduces a formal framework for Tier-1 structural systems, defined by the tuple T = (P, A, S, C), and establishes the necessary and sufficient conditions under which structural closure is achieved. A complete rule set (R1–R8) is derived and shown to be required for any system claiming Tier-1 validity. Each rule is paired with a corresponding failure mode, providing a full classification of structural breakdowns when constraints are violated. The paper further presents a diagnostic procedure for evaluating arbitrary theories and proves a regime dependence theorem demonstrating that all Tier-1 results are bounded to their defining premises and admissibility constraints. Cross-domain analysis confirms that classical mechanics, relativity, and information theory each instantiate Tier-1 closure within distinct regimes. The framework is purely structural and makes no ontological or empirical claims.