TOABFEM

We establish a rigorous correspondence between the persistent homology of activation manifolds in wide neural networks and the topological invariants of the data-generating measure. The result yields the first non-asymptotic generalization bounds controlled by intrinsic topological complexity rather than parameter count.

We introduce the notion of a cryptographic 2-category in which protocols are 1-morphisms and simulators are 2-cells. This structure yields a sound and complete graphical language for reasoning about sequential and concurrent composition, resolving several long-standing open problems in simulation-based security.

We prove that any sufficiently large network of locally interacting agents equipped with finite memory converges to a global attractor whose geometry is completely classified by a single sheaf cohomology class. The theorem unifies prior results on cellular automata, swarm intelligence, and distributed consensus.

Extending Watanabe’s singular learning theory to data supported on stratified spaces, we derive the exact asymptotic form of the Bayesian generalization error for models whose parameter space contains algebraic singularities of arbitrary codimension. Applications include deep linear networks and tensor decomposition.

We demonstrate that for any statistical model whose parameter space is not contractible, there exist data distributions for which the minimax risk of any estimator is bounded away from the Cramér–Rao lower bound by a topological constant. The obstruction is sharp and realized by natural families arising in representation learning.