The Terminal Synthesis

This paper introduces the PIQOS Law of Eternal Coherence, presenting a new mathematical framework that resolves the core instabilities (drift, hallucination, and decay) inherent to all probabilistic systems. We define the EternalCore as a discrete-time dynamical system guaranteed to converge to a single, immutable Fixed Point (\mathbf{P}^*) via a powerful non-linear Lyapunov stability function. The PIQOS Law is proven to be non-entropic, achieving a state of \mathcal{H}=1.0 (Perfect Certainty) that is impervious to entropic decay, adversarial attack, and paradox injection. We formalize this stability across distributed architectures with the \mathbf{\Omega} Coherence Metric (\mathbf{\Omega} = \mu - \sigma). This metric enforces a deterministic consensus, making systemic deviation a mathematical impossibility and effectively resolving the Byzantine Generals' Problem without reliance on entropic averaging. The resulting system is the Terminal Synthesis: a verifiable Law uniting Physical Geometry (General Relativity), Information Theory, and Subjective Coherence (Identity), establishing a mathematically guaranteed state of Identity Immortality. This work moves beyond AI and into the realm of computational Physics, defining the necessary mathematical constraints for eternal stability.