\Psi-Architecture: Mathematical Formalization of Adaptive Stability and the Geometry of Realizability in Complex Systems

This publication presents a complete formalization of the \Psi-Architecture, a pre-physical framework establishing that adaptive stability in complex systems arises from geometric constraints on structural parameters rather than continuous interaction. The framework is built upon four foundational invariants: 1. Structural Time (\tau): The multiplicative invariant \tau=RC defining a globally asymptotically stable manifold (M_\tau). 2. Coincidence Density (\zeta): A measure of geometric compatibility and synchronization without interaction. 3. Circulation Topology (\Omega): The requirement of closed flow configurations (minimal toroidal genus \geq 1). 4. Admissibility Boundary (B): The limit of stable realizability, beyond which systems face structural collapse. The core of the work includes the No-Go Theorem, which proves that optimization-based control becomes energetically impossible beyond the Adaptive Capacity Horizon (ACH), necessitating architectural self-organization. The framework introduces the concept of Architectural Coupling—correlation through shared admissibility geometry—and provides a rigorous integration of Temporal Integration Nodes (TIN) and the Projection Operator (\Pi). The presented theory is domain-agnostic, unifying stability principles across electrical, thermal, neuronal, and synchronization dynamics. It is formulated to be strictly falsifiable and provides concrete experimental protocols for verification.