Axiom 0: The Octomorphic Constraint Foundations of Admissible Structure

Axiom 0: The Octomorphic Constraint is a foundational specification defining a single admissibility law governing which structures can exist on a finite, non-associative octal lattice. Rather than proposing a dynamical model or domain-specific theory, this work formalizes existence itself as a constraint problem. A configuration is admissible if and only if it can close its phase loops without exceeding a fixed curvature capacity intrinsic to the lattice. This condition is expressed through a global action functional that accounts for geometric (metric), topological (phase), and contextual curvature contributions. The document establishes: The discrete octal substrate and the returnability requirement Axiom 0 as a binary admissibility law A fixed lattice capacity (Ω = 25.0), derived rather than fitted A normative execution procedure (the Octomorphic Universal Evaluator, OUE) A rigorously audited null result showing that connected structure is forbidden under frozen topology A second operational regime permitting curvature trade, enabling connected matter without violating the constraint Physics, chemistry, biology, and information theory are treated as domain projections of the same admissibility accounting, not as independent rule systems. This specification is normative and auditable. All constants declared as locked are non-tunable, and all claims are structured to be computationally verifiable. The document serves as the constitutional reference for all subsequent implementations, evaluations, and applied work within the Octomorphic Framework. Keywords admissibility constraint, discrete lattice, non-associative geometry, topology, curvature accounting, foundational physics, formal specification, octal lattice, structural invariants License Octomorphic Open License v1.1