Navigational Cybernetics 2.5 — Structural Axioms, Lemmas and Theorems of Coherence, Drift, and Directionality for Long-Horizon Adaptive Systems (v2.1)

A closed axiomatic framework for long-horizon adaptive systems in which drift, rather than equilibrium, is the primary structural medium of irreversible evolution. The framework establishes formal conditions under which identity continuity, admissibility, and internal-time viability can persist under sustained directed interaction. NC2.5 introduces an admissibility-first architectural class — Cybernetics of Order 2.5 — characterized by reflexive visibility of structural constraints combined with categorical prohibition of their causal exploitation. Admissibility is not a reward, penalty, or optimization constraint; it is a non-causal structural predicate that excludes forbidden effect-classes before any evaluation, planning, or action selection occurs. Version 2.1 extends the structurally closed core of v2.0 (60 axioms, 62 theorems, 17 lemmas) with structural pressure as a new architectural primitive, regime depth classification, multi-agent coordination enforcement, extreme identity regime analysis, and the gate bifurcation theorem proving Cybernetics 2.5 as the unique structurally stable intermediate within the admissibility-first class. A structural hardening pass adds proof-status stratification, a unified Lyapunov doctrine collecting four canonical monotone witnesses, five canonical falsification witness forms, dependency graphs for load-bearing theorems, and a minimal counterexample protocol. Three specification layers close deployment-blocking gaps: effect-class specification template, dual semantics for internal time, and a unified threat model for non-reconstructibility. 170 pages. 61 axioms. 69 theorems. 21 lemmas. Tags: adaptive systems, structural cybernetics, admissibility, drift, identity continuity, long-horizon viability, architectural theory, formal axioms, non-causal control, structural pressure License: All Rights Reserved — Copyright © 2025–2026 Maksim Barziankou Category: Engineering / Systems Engineering Author: Maksim Barziankou (MxBv), PETRONUS