κ / κ Irreversibility and the Sign-Flip Law

This work introduces a foundational law governing irreversibility at the level of continuity, rather than energy, entropy, or dynamics. Many physical, biological, computational, and institutional systems exhibit failures that cannot be reversed even when external conditions are restored. Existing frameworks describe irreversibility statistically or dynamically but do not identify when the ability to continue has been structurally exhausted. The paper defines a continuity functional, κ, over admissible transition composition, and a sign-valued directional indicator, κ̇, which tracks whether executed transitions preserve or contract admissible continuation. A cumulative structural exposure parameter, γ, records irreversible commitments imposed by executed transitions. The central result is a sign-flip law: there exists a critical threshold γ* such that once γ ≥ γ*, κ̇ flips from +1 to −1 and cannot return under any admissible reversal of controls, boundary conditions, or initial states. Beyond this threshold, continuity contraction is structurally irreversible. The irreversibility identified here does not arise from entropy increase, dissipation, stochasticity, or coarse-graining. It results from exhaustion of admissible compositional structure: executed transitions permanently remove legal future continuations. The law is substrate-independent and scale-independent, applying equally to physical, biological, informational, and institutional systems. The framework is minimal and negative-law in character. No optimization principles, predictive models, control strategies, or policy claims are introduced. The result defines a structural boundary on continuation itself, identifying when recovery within a given regime becomes impossible by construction.