Event-Triggered Invariant-Gated Governed Evolution: A Minimal Theorem for Long-Horizon Stability in Adaptive Systems

Why This Matters Modern adaptive systems fail not because they lack intelligence, but because they lack governance at the moment it actually matters. Continuous constraints suppress learning, while unconstrained adaptation leads to drift, instability, and loss of identity. This work shows that stability and adaptability are not opposites: by enforcing invariants only during critical coherence events, systems can evolve freely and remain stable over arbitrarily long horizons. The result reframes safety, alignment, and long-term autonomy as problems of when to govern, not how much to govern—opening a new design space for resilient adaptive systems. We present a minimal, inspectable formal framework for long-horizon stability in adaptive and self-modifying systems based on invariant-gated governed evolution. The core result establishes that unconstrained adaptive updates can be safely admitted if and only if they pass an explicit event-triggered admissibility gate enforcing preservation of a defined invariant set (e.g., mass, entropy, phase, or identity functional). The framework is deliberately non-ontological and domain-agnostic: it does not assume semantic goals, reward optimization, or teleological criteria. Instead, it separates proposal dynamics from governance, yielding a piecewise evolution rule in which candidate updates are either accepted, modified, or rejected based on invariant violation thresholds. A Lyapunov-style admissibility condition is used to guarantee bounded drift and non-explosive behavior over arbitrarily long horizons. We prove a minimal theorem showing that, under mild regularity conditions, invariant-gated evolution ensures persistence and prevents runaway adaptation without requiring global optimization or continuous supervision. This establishes a formal distinction between adaptive capability and governed admissibility, clarifying how systems may evolve structurally while remaining stable and auditable. The result applies broadly to adaptive control, autonomous agents, recursive algorithms, and long-running socio-technical systems. This release is intended as a public-safe, minimal publishable core, suitable for inspection, critique, and extension, and serves as a foundation for richer governed-evolution architectures.