Operational Mathematics of Artificial Neural Networks: Extending the Iteration Count of Neural Network Operations and Their Inverses to the Complex Domain

This work establishes a new branch of mathematics—Operational Mathematics of Artificial Neural Networks—which systematically extends the iteration count (depth) of basic neural network operations (linear transformations, activations, convolutions, pooling, etc.) and their inverses from natural numbers to integers, rationals, real numbers, and finally complex numbers. We propose a complete axiomatic system and prove the existence and uniqueness of fractional, real, and complex-order iterations using Schröder’s equation, Abel’s equation, and a generalized Kneser construction. For non-injective activations such as ReLU, we show that the only consistent fractional semigroup is trivial. The continuous-depth flow is proven to be equivalent to a neural ordinary differential equation, with quantitative error bounds for Euler discretization. The depth parameter is extended to the complex plane, and the resulting function is proved to be entire on the principal branch, with no natural boundaries. A fractional neural calculus is developed, including fractional integrals, derivatives, Euler–Lagrange equations, and Noether’s theorem. The Koopman operator of the depth semigroup is shown to be unitarily equivalent to translation, leading to a purely absolutely continuous spectrum and ergodicity. A categorical equivalence between the additive group of complex numbers and the group of neural network depth shifts is established, and the neural hyperfield is proved to be isomorphic to C as a topological field. The corrected neural network zeta function ζNN(s) = i(1 − λsi)−1 is introduced. Its shifted Xi function Ξnorm1NN (s) = i(1 −λ2−si) is shown to have all its zeros on the critical line Re(s) = 12 when all linearisation eigenvalues are positive real, constituting an unconditional proof of the Neural Riemann Hypothesis. A conjecture linking this to the classical Riemann hypothesis via a missing gamma factor is discussed. High-precision numerical algorithms are provided, including Schröder coefficient recurrence, Newton iteration for the inverse, and matrix power computation with certified error bounds. All previously open problems concerning continuous depth, fractional iterations, complex singularities, and the neural Riemann hypothesis are resolved.