Adaptive Admissibility and Tactical Geometry: Constraint Evolution, Borwein Plateau-Collapse, and the Unified Variational Principle

This manuscript introduces the framework of Adaptive Geometric Dynamics, establishing that the admissible manifold of a physical or strategic state space is not fixed, but dynamically contracts in response to input non-coherence. By elevating constraints from static boundaries to evolving operator-theoretic parameters, the paper proves that increased non-coherence strictly tightens the forbiddenness functional, shrinking the space of permitted motion and forcing systemic adaptation. The manuscript maps this constraint evolution across multiple distinct physical and tactical regimes. It identifies the Spiral of Theodorus and Borwein integrals as the exact discrete and spectral prototypes for overlapping constraint accumulation and plateau-collapse. Extending the Omega-Sigma constraint mechanism to fluid measure transport and tactical interception, the paper proves that multi-disciplinary mastery is not a stylistic choice, but a rigorous geometric expansion of the admissible trajectory set under the constrained Hamilton-Jacobi-Bellman equation. A central contribution is the Drivetrain Recognition Principle, which classifies all dynamical systems by their operator-theoretic mechanisms of propulsion: FWD (projected constraint), RWD (unrestricted generator), and AWD (hyperoperational spectral redistribution). The paper concludes that precise power is mathematically defined by bounded spectral redistribution, and that ultimate systemic stability is achieved solely through the AWD-class Omega-Sigma spectral governor. The terminal phase of systemic failure is characterized as "Quantum Twilight"—the razor-thin geometric margin preceding complete spectral collapse. Finally, the manuscript culminates in a 12-part Master Synthesis Theorem. It proves that the Omega-Sigma admissibility mechanism survives infinite-dimensional scaling, non-linear semigroup flows, stochastic noise, pseudodifferential microlocalization, and fractional dissipation. It establishes the Strict Obstruction Law, proving that ultraviolet cascades and systemic ruptures are dynamically arrested if and only if coercive dissipation strictly dominates the nonlinear transfer rate. Under this framework, systemic collapse is never an accident of complexity; it is the strict, localized failure of coercive dominance.