Operational Mathematics of Probability: Extending the Iteration Count of Logical "AND" and its Inverse to the Complex Domain

This paper 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 ultimately complex numbers—onto a novel class of binary operations: the probability operation σ⊕n(a,z) (exemplified by logical AND) and its inverse σ⊖n(a, z) (the Bayesian inversion). A complete set of seven axioms is established; integer order, fractional-order, real-order, and complex-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 are provided. The singularity structure of complex-order probability iterations is analysed in depth, revealing that the branch points are purely logarithmic—arising from the periodicity of aw in the imaginary direction—and they accumulate along the negative real axis (−∞,−1], forming a natural boundary. This structure is fundamentally simpler than the mixed algebraic logarithmic structures found in other operational mathematics frameworks. A fundamental structural discovery is rigorously proved: the probability operational hierarchy collapses completely for all levels n ≥ 2, leaving only the base operation at level n =1 and the collapsed family at level n = 2. Fractional calculus and the fractional calculus of variations with probability kernels are shown to be special cases of the probability operational framework, thereby unifying discrete probability hyperoperations with continuous analysis. A categorical duality between the mathematics of numbers and the mathematics of probability operations is established, yielding a field isomorphism between the probability operational field and the complex numbers. A functorial relationship reflecting De Morgan duality is constructed. The corrected probability Zeta function is defined and its exact functional relations and Euler product are established. The pole structure is completely analysed, showing that all poles lie on the imaginary axis ℜ(s) = 0, an unconditional result. This paper treats the probability AND operation as a benchmark model for Operational Mathematics. Because the base function f(z) = az is linear, all constructions reduce to explicit elementary functions. This extreme simplicity is not a weakness but a virtue: just as the hydrogen atom serves as the exactly solvable benchmark of quantum mechanics, the probability AND operation reveals which structural features of Operational Mathematics (hierarchy collapse, operational field isomorphism, natural boundary formation) are universal phenomena independent of nonlinearity, and which features (jump discontinuities, mixed singularities) depend on the nonlinear nature of the base function. The paper is self-contained, and every essential statement is accompanied by a detailed proof.