Structural Admissibility Framework: A Geometric Theory of Internal Time

This white paper introduces the Structural Admissibility Framework, a novel geometric theory positing that a system’s internal time is not an independent parameter but an induced invariant of its structural organization. The core principle—the Admissibility Law—states that intrinsic relaxation timescales are strictly constrained by the geometry of structural compatibility. [Key Concepts] • Ontological Reordering: The framework proposes a fundamental hierarchy: Structure → Time → Energy/Information. It argues that classical energetic laws and speed limits (quantum, thermodynamic, or control-theoretic) are measurable "projections" or shadows of deeper structural geometric constraints. • Mathematical Foundation: Utilizing operator class geometry, the theory defines an admissible flow manifold (F_{adm}) where stability topology and spectral continuity are preserved. It derives a Lipschitz bound on intrinsic timescales relative to a canonical structural metric. • Operational Projections: The theory unifies disparate phenomena across domains, including: • Energetic: Proven lower bounds on structural work (W) relative to the rate of change in internal time (\tau). • Informational: Predicted complexity costs (\Delta I \ge C_{struct}) for structural deformation. • Topological: Predicted temporal mode-switching at Morse bifurcations. • Universality Hypothesis: The paper outlines a "Universality Pathway," suggesting that if structural geometry predicts temporal constraints consistently across at least three independent coordinate systems (energetic, informational, and biological), the framework serves as a candidate universal law for adaptive organization. [Significance] By shifting the focus from microscopic mechanisms to operator-level geometry, this framework provides a domain-independent criterion for the feasibility of temporal acceleration and structural adaptation. It offers testable falsification criteria and a programmatic roadmap for experimental validation in physics, biology, and machine learning.