Emergent Low-Dimensional Topological Dynamics Dictate Generalization and Escape Mechanics in Billion-Parameter Neural Networks

arXiv:2611.08502v1 Announce Type: new Abstract: The remarkable generalization of modern over-parameterized neural networks defies classical statistical learning bounds. Here we establish that optimization trajectories in billion-parameter models do not sample the ambient dimension $\mathbb{R}^N$ homogeneously; instead, they are dynamically constrained to a low-dimensional execution submanifold $\mathcal{M}_{\text{exec}}$ (with empirically observed effective dimension $d_{\text{eff}} \le 12$). By equipping the parameter space with a regularized Hessian metric $g_{ij} = \nabla_i \nabla_j L + \epsilon \delta_{ij}$ and analyzing stochastic Langevin diffusion on loss sublevel sets, we derive a rigorous Topological Generalization Bound grounded in projected subspace persistent homology.