Cohomological Characterization of Non-Solvable Neural Trajectories: Automorphic Rigidity and the Inverse Galois Problem in Recursive Synaptic Field Extensions

Mathematical Applications of Science Fiction We present a rigorous algebraic formulation of neural network dynamics viewed through the lensof arithmetic geometry and Galois cohomology. By treating the synaptic weight space not as atrivial manifold over R, but as a tower of recursive field extensions K ⊂ L1 ⊂ · · · ⊂ Ln, we map theconvergence properties of deep learning algorithms to the solvability of the associated Galois groupsGal(Ln/K). We establish the Synaptic Inverse Galois Theorem, demonstrating that for any finitegroup G, there exists a neural architecture whose learning trajectory is governed by G. The centralcontribution of this monograph is the Cohomological Rigidity Theorem, which asserts that non-solvabletrajectories (where the loss landscape possesses non-trivial topological obstructions) are characterizedby the non-vanishing of the second cohomology group H2( Gal(N ), W×). Furthermore, we introducethe concept of Automorphic Synaptic Forms, linking the spectral decomposition of the neural operatorto the Langlands correspondence, thereby providing a unified theory for the "unlearnable" regions ofthe parameter space.