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

This record contains the canonical v1.7 release of a unified axiomatic and theorem-based framework for long-horizon adaptive systems operating under irreversible structural constraints.The work formalizes a class of architectures in which viability, identity continuity, and meaningful behavior cannot be reduced to causal control, optimization, or performance-driven regulation. The framework treats information not as stored data or retrievable state, but as a persistent structural constraint delimiting the space of admissible future evolution. Coherence, drift, internal time, and directionality are established as invariant architectural properties governing long-horizon viability. A central result is the strict separation between causal dynamics and non-causal admissibility: causal evolution determines what may occur, while admissibility - derived from accumulated structural state - determines what is permitted to occur. These influences act concurrently but are irreducible to one another. Version 1.7 completes the axiomatic closure of admissibility regulation by establishing a set of structural impossibility results. Admissibility is shown to be non-derivable from instantaneous or augmented state representations, incompatible with optimization-based or penalty-based formulations, and non-restorable once violated without a change of structural class. Any exposure of admissibility-relevant or commitment-related constraints as causally actionable feedback is proven to induce reflexive optimization and collapse long-horizon viability. These results necessitate a reclassification of the cybernetic regime suitable for long-horizon adaptive systems. Classical second-order cybernetics permits observation to enter causal regulation and is insufficient to prevent silent structural degradation. Third-order cybernetics allows reflexive self-modification and optimization against structural limits, which is shown here to be structurally unsafe. The framework therefore defines a discrete intermediate architectural class - termed cybernetics of order 2.5 - which preserves reflexive visibility of structural viability conditions while categorically prohibiting their causal actionability. The axioms, lemmas, and theorems presented do not constitute a metaphysical ontology or an implementation prescription. They define an admissibility calculus governing what may be stabilized, committed to, or preserved within an adaptive system operating under irreversible interaction and finite internal time. Within this context, ONTOΣ I–IV and UTAM specify admissible structural conditions for adaptive existence rather than descriptive claims about the external world. This release serves as a canonical theoretical anchor for PETRONUS™ and related architectures and establishes a defensible prior-art foundation for a distinct school of cybernetics concerned with identity preservation and viability over long horizons.