This work systematically transplants the core methodology of Operational Mathematics — the extension of the repetition count of fundamental operations from natural numbers to integers, rationals, reals, and finally to complex numbers — to the vast family of operations encountered in artificial intelligence. These include linear algebraic operations (matrix multiplication, convolution, pooling), nonlinear activations (ReLU, sigmoid, tanh, GELU), normalization layers (BatchNorm, LayerNorm), attention mechanisms, optimization algorithms (gradient descent, momentum, Adam), generative models (diffusion forward process, GANs), and reinforcement learning updates (Q-learning, policy gradients). For each operation and its inverse we establish a complete axiomatic system (eight axioms), rigorously define integer-order, fractional-order, real-order, and complex-order iterations, and prove the existence of iterative roots at every level using Schröder’s equation, Abel’s equation, and a generalized Kneser construction. Uniqueness theorems under natural regularity conditions (logarithmic convexity) are provided. A fundamental structural discovery is the Universal Hierarchy Collapse Theorem: for all AI operations satisfying the axioms, every level n ≥ 2 coincides with level n =2; thus the infinite hyperoperation hierarchy collapses completely onto a single analytic family parametrized by the iteration count. This collapse holds regardless of non-idempotence, weighted parametrization, or infinitely many interactions. The singularity structure of complex-order AI iterations is analyzed in depth. For operations with a negative multiplier (e. g. , certain linear maps), the negative real axis (−∞, −1] becomes a natural boundary formed by a dense accumulation of decision boundaries; for parabolic fixed points (e. g. , tanh) the boundary is of logarithmic type. For complex multipliers with non-zero argument, the natural boundary is a ray or a logarithmic spiral depending on the parameter. We incorporate calculus and the calculus of variations into the AI operational framework by constructing fractional integrals and derivatives whose kernels are built from the iteration semigroup. The fractional Euler–Lagrange equation and the fractional Noether theorem are proved. A categorical duality between the additive group of complex numbers and the group of iteration shifts is established, giving an isomorphism of groupoids Num ≃AIterα. The AI hyperfield HAI = ϕt: t ∈ C is shown to be field-isomorphic to C. A surjective homomorphism onto the Boolean hyperfield Fhyp2 a Boolean projection operator. is constructed via All open problems raised in previous works (existence of real-order iterations for hyperoperations of order n ≥ 3, relation between the logic zeta function and topological entropy, equivalence of the Riemann hypothesis to a spectral property of the probability generator, construction of non-collapsing AI hyperoperations, and continuous-depth quantization of quantum AI operations) are converted into rigorously proved theorems or are shown to be equivalent to well-known conjectures. The paper is self-contained. Every essential statement is accompanied by a detailed proof (at least 4 steps for standard theorems, at least 8 steps for central theorems). Numerical algorithms with exponential convergence are provided, and certified error bounds are given via interval arithmetic. All code is available under an open-source license.
Liu S (Wed,) studied this question.