Operational Mathematics of Logic Operations: Extending the Iteration Count of NAND 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, rational numbers, real numbers, and ultimately complex numbers -- onto a new class of binary operations: the logical NAND operation ₍^ (a, b) and its inverse ₍^ (a, b). Because classical logic operations are defined on the finite Boolean domain B=\0, 1\, a continuous extension is mandatory in order to accommodate continuous iteration counts. We first expand the basic operation to the unit interval I=0, 1 via product fuzzy logic and erect a complete axiom system on this continuous framework. Within this extension we rigorously define integer-order, fractional-order, real-order, and complex-order iterations, and prove the existence of iterative roots at each level by means of an explicitly solvable Schr\"oder equation. Uniqueness theorems under natural regularity conditions are provided. We analyze in depth the singularity structure of complex-order logic iterations and uncover a fundamentally new phenomenon determined jointly by the finiteness of the truth table, the idempotent law a a = a, and the negative multiplier = -a: non-integer-order iterations produce continuous values in the complex plane, yet logic operations must ultimately return Boolean values \0, 1\. Hence we introduce a compulsory Boolean projection operator that collapses the complex result back onto the Boolean domain. This projection endows non-integer iterations with an intrinsic discontinuity whose decision boundaries constitute a dense set in the parameter space and turn the negative real axis into a natural boundary. We further prove that the logic operation spectrum collapses completely for all levels n 2: the continuous extension of all higher-level logic operations degenerates into a single family of functions because of the idempotent law. Fractional calculus with a logic kernel is incorporated into the logic operational framework; using Abel-coordinate conjugation we prove rigorously the semigroup property of the fractional integral and derive the fractional Euler--Lagrange equation, the fractional Legendre condition, and the fractional Noether theorem, thereby unifying discrete logic hyperoperations with continuous analysis. A categorical duality between the mathematics of numbers and the mathematics of logic operations is established; we construct the logic hyperfield and prove that it admits a surjective homomorphism onto the Boolean hyperfield. Moreover, all sixteen binary Boolean functions are unified under the umbrella of the continuous operation spectrum and are completely classified according to their dynamical character (attracting, repelling, parabolic, degenerate). The theory is further applied to neural networks, quantum computation, and combinatorics; we propose an adaptive-depth training algorithm for continuously deep logic networks, construct a quantum channel that simulates logic iteration, and reveal combinatorial links between the iteration coefficients and Stirling numbers of the first kind. Finally, by means of Domain theory and the Scott topology we build a non-metric continuous extension for G\"odel logic and prove rigorously its information-collapse theorem; ukasiewicz logic is incorporated into the framework of parabolic iteration, and logic iteration is applied to fixed-point acceleration in automated reasoning. The paper is self-contained, and every essential statement is accompanied by a rigorous and detailed proof (always at least 4 steps, and at least 8 steps for theorems of central importance).