Overcoming the Curse of Dimensionality in 10D Ricci Flow: Exponential Speedup via Adaptive Sparse Hash and Spectral Engines

We present two complementary numerical methods for computing the Ricci flow of 10-dimensional symmetric metrics that bypass the O(N^{10}) exponential wall of standard dense-grid solvers. Unlike traditional State-of-the-Art (SOTA) tensor methods that require intractable memory for high-dimensional manifolds, our engines restructure the computational topology to deliver exponential speedups and memory reduction. Method I (spectral worldsheet engine) evolves a 10x10 metric field on a 32x32 periodic surface using Fourier-Galerkin spectral derivatives, Shannon-truncated tensor fields, and ETDRK4 / Euler exponential time integration. Method II (adaptive sparse hash engine) discretizes the full 10-dimensional space via a dynamic hash map, computing Christoffel symbols and the Ricci tensor with O(dx^2) finite differences. Wavefront expansion and thermodynamic pruning dynamically manage only the active geometric perturbation, avoiding the calculation of empty space. As a proof of this algorithmic breakthrough, both methods execute entirely on a low-resource mobile ARM processor with peak memory below 12 MB. Method I achieves spectral-accuracy convergence toward Ricci-flat geometry, while Method II dynamically scales up to 89,745 active nodes before converging exactly to flat Minkowski space under Richardson extrapolation. Deployed on enterprise-grade hardware, this framework offers a scalable, Green AI paradigm for non-perturbative 10D gravity, string theory vacua analysis, and high-dimensional optimization.