Axiom Zero: Structural Conditions for Interpretive Stability in Bounded Systems

Abstract Axiom Zero is a formal specification of the structural conditions required for interpretive stability in bounded systems. It defines a constraint grammar governing how systems mediate structural pressure through interpretation, maintain internal coherence, accumulate misalignment, and undergo structural transition when stability limits are exceeded. The specification operates at the grammar layer, defining admissibility conditions for coherent constraint-mediated state transition rather than predicting empirical outcomes. From these axioms, the specification derives an invariant state sequence governing system evolution: constraint, interpretation, alignment, misalignment, threshold, collapse, revision, renewal, and recursion. This sequence applies recursively across scales, including cognitive, institutional, and distributed social systems. Extending this grammar, the Cognitive Manifold Model (CMM) provides a geometric reconstruction framework for representing interpretive structure. CMM formalizes cognition and interpretation as positions within a latent high-dimensional manifold that cannot be directly observed but can be reconstructed through structured projection and measurement. This allows interpretive systems to be represented as coordinates within a continuous structural space rather than categorical classifications. From the grammar and manifold reconstruction principles, a set of operational operators is necessarily derived as measurable consequences of admissibility boundaries, stability gradients, and structural stress accumulation. These include the Logical Seam Stress-Test (LSST), which probes admissibility boundaries; the Least-Resistance Test (LRT), which maps stability gradients and revision resistance; Manifold Stress Topography (MST), which aggregates structural stress distribution across interpretive ecosystems; and Temporal Manifold Mapping (TMM), which models trajectory evolution and rupture likelihood under constraint across time. Together, the constraint grammar and manifold reconstruction framework form a unified structural system for analyzing interpretive stability, structural stress accumulation, and constraint-driven transition across cognitive and institutional domains. Specification Version: 1.0Author: Devin RushFull formal specification to be published as a versioned continuation of this record.