Non-Abelian Ramification and Automorphic Rigidity in the Inverse Limits of Stochastic Neural Pathing: Cohomological Obstructions within the Absolute Galois Group of Synthetic Weight Fields

Mathematical Applications of Science Fiction We present a rigorous arithmetic–geometric framework for analyzing the convergence properties of deep stochastic neural networks viewed as dynamical systems over non-Archimedean fields. By constructing the Synthetic Weight Field as a tower of Galois extensions governed by the stochastic gradient descent operator, we identify the limit of neural pathing with the inverse limit of a projective system of Artinian rings. We prove that the stability of the loss landscape minima is equivalent to the automorphic rigidity of a specific Galois representation from the Galois group of the synthetic field to a general linear group. Our main result shows that non-Abelian ramification at the wild primes of the weight field produces non-trivial classes in the second Galois cohomology group, acting as topological obstructions to global optimization. Furthermore, these obstructions correspond to singularities in the automorphic L-functions associated with the network architecture, implying a fundamental limit to learnability in the presence of high-dimensional arithmetic chaos.