Blind-Spot Decomposition and the Geometry of System Collapse∗

We propose a geometric framework for detecting and controlling structural instability in coupled dynamical systems via energy-topology equivalence. A Blind-Spot Decomposition Theory (BSDT) energy functional is shown to be topologically equivalent to the system potential through a Morse-index equivalence at every critical point, with two-sided gradient bounds and homotopy equivalence of sublevel sets. This recasts instability detection as a control problem on a critical manifold where adaptive friction achieves marginal stability. Two physics-based System Mode engines operationalise the theory with Lyapunov-certified convergence, fused topological scoring, and distribution-free conformal calibration. The framework is validated across three domains: (1) FDIC banking data (6-quarter GFC lead, AUROC = 0.867, zero false alarms), (2) ERCOT power-grid dispatch (+72 h lead, AUC = 0.9997), and (3) three-dimensional incompressible Navier–Stokes via a 34-run GPU stress test (N up to 256³, Re up to 62,832, NVIDIA RTX PRO 6000 Blackwell, 532 min wall time). All runs complete without blow-up: adaptive viscosity reduces peak vorticity by 1.77× at Re ≈ 1,257; the BKM integral remains finite at Re ≈ 62,832; twelve random-phase initial conditions confirm IC-independence; five distinct feedback laws produce identical dynamics (spread < 0.04%); and production–dissipation cross-correlations exceed 0.85 across the full laminar-to-turbulent range. Nine named theorems are proved, including an Energy-Topology Equivalence, Entropy Descent Lemma, Phase Transition Theorem, Conditional NS Regularity, and Strengthened Regularity via topological Betti-number monitoring. The paper includes comparison to LES/Smagorinsky models, hyperviscosity, and prior DNS blow-up studies (Kerr 1993, Hou & Li 2006).