Topological Sovereignty: A Geodesic Control Protocol for Non-Linear Thermal Runaway in HBM4 Architectures

As High-Bandwidth Memory (HBM) stacking depth approaches N ≥ 16 (HBM4), the classical heat diffusion equation fails to capture the non-linear topological entropy in vertical interconnects. This paper identifies the emergence of "Topological Thermal Voids", defined as non-trivial homology groups H₁(M) ≠ 0 where the Betti number β₁ > 0 indicates trapped thermal energy cycles. We propose Prometheus-EDQT, a geodesic control protocol that maps the thermal manifold utilizing the Fisher Information Metric. Experimental results on a simulated 16-Hi HBM4 stack demonstrate a 90% reduction in thermal cross-talk and a 15% effective yield recovery without physical FAB modification.