The Strategic Operator Framework Unified Control, Absorbing-Boundary Survival, and Epistemic Phase Transitions

We establish a unified operator-theoretic framework that formalizes competitive strategic decision-making across disparate physical and epistemic scales. By mapping strategic behaviors onto controlled stochastic differential equations and Partially Observable Markov Decision Processes (POMDPs), we isolate three fundamental control regimes: tempo-dominant state contraction (the Musashi limit), entropy-gated feasibility projection (the Obama limit), and adversarial variance injection (the Nixon limit). We prove that these are not behavioral heuristics, but exact asymptotic limits optimizing distinct topological cost functionals. Using regularized Fredholm determinants (det₂), we prove the Musashi operator forces opponent spectral collapse, inducing a strict resolvent blow-up that scales linearly with the enforced temporal phase gap (Δ). Conversely, we prove the Obama operator minimizes epistemic regret through delayed projection, governed by a continuous-time optimal stopping boundary where Fisher information balances the temporal penalty. The Nixon limit is mathematically formalized as adversarial measure deformation via the Girsanov Theorem. Furthermore, we resolve the apparent tension between these approaches by deriving the Diffusion-Deliberation Tradeoff, demonstrating that absorbing-boundary survival and epistemic decision-optimality occupy strictly partitioned regimes in stochastic phase space. Monte Carlo validations and a unified phase diagram confirm the exact analytic transition boundaries governing continuous strategic survival. This framework provides a complete, mathematically proven operating system for absolute stability under arbitrary environmental uncertainty, bridging quantum operator mechanics, stochastic calculus, and executive strategic theory.