From Relational Axioms to Global Hyperbolicity: A Rigorous Analysis of the Logical Gap

This paper presents a rigorous analysis of the central open problem identified in "Time as Constraint, Not Dimension" (DOI: 10.5281/zenodo.19039979): whether the relational axioms (A1)–(A7), combined with the monotonicity of the causal relational change functional Φ, mathematically entail global hyperbolicity. The analysis proceeds in three stages. First, it demonstrates that axioms (A1)–(A7), having established Φ = τ via the uniqueness theorem, impose no global topological constraints by themselves — proper time is a path functional, not a manifold function, and is well-defined even in spacetimes admitting closed timelike curves. Second, it introduces a Global Coherence Axiom (A8′) formalizing the framework's commitment that cumulative causal change produces genuinely distinct states. Combined with axiom (A7)'s universality applied to metric perturbations, this yields the paper's strongest novel result: a proof that the spacetime must be stably causal, guaranteeing a smooth global time function and the structural exclusion of closed timelike curves. The derivation of stable causality from the universality axiom provides genuine mathematical content beyond reformulation. Third, the analysis identifies the irreducible gap between stable causality and full global hyperbolicity — the compactness of causal diamonds — and shows this requires one additional physically motivated input: either an explicit Causal Completeness axiom or invocation of the Choquet-Bruhat–Geroch theorem (physically realistic spacetimes arising from initial data are automatically globally hyperbolic). The paper contains four numbered theorems with complete proofs, an alternative approach via anti-Lipschitz temporal functions, and an honest assessment of every gap. The complete logical chain is: (A1)–(A7) → Φ = τ → (with A8′) causality → (with A7 universality) stable causality → (with A9 or Choquet-Bruhat–Geroch) global hyperbolicity. Acknowledgment: This paper was developed with the assistance of AI language models (Claude by Anthropic). The research question, framework, and all foundational commitments originated with the author. The AI tools contributed mathematical analysis, proof construction, and formal exposition. The author takes full responsibility for the claims presented.