Universal Stability Law: Coherence, Separation, and Representation (The Spectral-Anchor Framework)

We present a universal framework for the stability of dynamical, physical, and cognitive systems. Classical stability theory often fails to distinguish between internal state coherence and external representation fidelity, leading to hidden failure modes where a system appears stable but is actively drifting from its true structural reference. To resolve this, we introduce the Spectral-Anchor Framework, a universal dual-control architecture demonstrating that true, long-term stability requires the simultaneous satisfaction of two independent mechanisms: the spectral suppression of orthogonal noise (coherence) and the geometric restoration of an invariant reference anchor (truth). At the extreme thermodynamic limit—defined here as the SigmaZeta limit—this framework proves that systems completely suppress entropy and amplify Fisher information, forcing the collapse of the system into a single-mode, structurally aligned, and completely coherent state. The theoretical core is fortified by rigorous operator-theoretic and quantum field-theoretic proofs, including non-perturbative resolvent bounds, the Birman-Schwinger principle, and path integral localization. Furthermore, the paper demonstrates the universality of this dual-control law by mapping its direct realizations across multiple domains, including machine learning (the elimination of catastrophic forgetting), quantum systems (divergence suppression), gravitational systems (singularity evasion), and cognitive epistemology. Ultimately, this framework proves mathematically that stability across all admissible systems is not merely persistence, but an inevitable, forced convergence to truth.