This paper systematically develops the operational mathematics of computer operations—extending the repetition count of fundamental instructions (arithmetic, logic, bitwise, memory access, control flow, vector/tensor, parallel/distributed, abstract data type operations, system calls, etc.) and their inverse operations from natural numbers to integers, rationals, reals, and ultimately complex numbers. The essential distinguishing features of computer operations—discreteness, finite precision, determinism/nondeterminism, computability, concurrency, abstraction layers, homomorphic encryption compatibility, multi-valued inverses, stateful side effects, and cache locality—are fully integrated into every definition, theorem, and proof. A complete set of twelve independent axioms is established; integer-order, fractional-order, real-order, complex-order, and infinite-order iterations are rigorously defined; and the existence of iterative roots at each level is proved by means of Schröder’s equation, Abel’s equation, and a suitably adapted Kneser construction. Uniqueness theorems under natural regularity conditions (logarithmic convexity) are provided. The singularity structure of complex-order iterations is analysed in depth, revealing a fundamentally new mixed type of branch points (logarithmic from non-real multipliers, algebraic from algebraic singularities of the inverse operation). The exact condition for the existence of a natural boundary is established: ℜ(lnλ) ̸ = 0 and λ /∈ R+. The Riemann surface of the iterated function is shown to be parabolic (conformally equivalent to C). A fundamental structural discovery is rigorously proved: the computer operation hierarchy collapses completely for all levels n ≥ 2, leaving only the base operation at level n = 1 and the collapsed iteration semigroup at level n = 2. Extensions to non-idempotent operations, weighted parametrization, and infinite iteration are analysed, and their interplay with collapse is fully characterised. Fractional calculus, automatic differentiation, program slicing, model inversion, and symbolic execution are shown to be special cases of the operational framework, thereby unifying discrete computer operations with continuous analysis and reverse engineering. A categorical duality between the mathematics of numbers and the mathematics of computer operations is established, yielding a field isomorphism between the computer operation hyperfield and the complex numbers. Functorial relations induced by concurrency, distribution, abstraction layers, and homomorphic encryption are constructed. All previously announced open problems and conjectures (weighted hyperfield structure, compactification of the iteration category, blurring of branch points under finite precision, fractional ergodic theorem, fractional halting problem) have been fully resolved and proved as theorems within this paper. The theory is complete, self-contained, and requires no future work. Extensive numerical verification tables and pseudocode are provided in the appendices.
shifa liu (Wed,) studied this question.