Structural Axioms and Theorems of Coherence, Drift, and Directionality for Long-Horizon Adaptive Systems

This release (v1.4) refines and extends a compact axiomatic and theorem-based foundation for long-horizon adaptive systems operating under structural coherence constraints. The Concept DOI refers to the complete evolving body of work; the Version DOI uniquely identifies this release. Information is treated not as stored data or retrievable memory, but as a persistent structural constraint shaping admissible future evolution. The framework formalizes coherent structural closure, residual-driven reactivation, inevitable drift, internal time as a bounded viability horizon, and directionality (Will) as a primary ontological operator rather than a goal-, reward-, or optimization-derived construct. Building on the phase-related structural cost formalized in version 1.3, version 1.4 explicitly incorporates semantic stabilization and regime-level admissibility as necessary structural conditions for long-horizon viability. The framework now distinguishes context-level interpretation from meaning-level commitment, and conditions irreversible regime transitions on stabilization within internal time rather than on instantaneous plausibility, correctness, or local consistency. These refinements clarify why systems may preserve admissible behavior and stable performance while undergoing irreversible identity degradation. Regime transitions are shown to impose irreducible structural burden independent of action-level correctness, and semantic commitment is treated as a source of irreversible load rather than a reversible contextual adjustment. The resulting axioms and theorems establish invariant principles governing coherence preservation, identity continuity, and controlled adaptation under prolonged interaction with a changing environment. Drift is shown to be unavoidable, inertial propagation to be energetically dominant, residual structure to be the sole carrier of relevance, and internal time to bound structural validity independently of instantaneous error, reward, or performance. These axioms and structural theorems constitute a minimal but non-trivial theoretical core applicable to coherence-preserving adaptive architectures, including systems such as PETRONUS, in which spin, drift, residual, internal time, directionality, and regime-level semantics are realized as explicit operational variables, without restricting the framework to any specific implementation, algorithm, or platform.