This work 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 Lambert W function operation Wn(z) and its inverse W−1n (z). A complete set of seven independent 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 Lambert W iterations is analyzed in depth, revealing a fundamentally novel phenomenon: the presence of algebraic branch points (square-root type) at the critical value z = −1/e, and logarithmic branch points at z = 0 (from the zero of W) and at z = ∞ (essential singularity). The negative real axis (−∞,−1] is shown to be a branch cut. The local monodromy group contains both Z2 and Z factors. A fundamental structural discovery is rigorously proved: the Lambert W operational hierarchy collapses completely for all levels n ≥ 2, leaving only the base operations at level n = 1 and the collapsed family at level n = 2. Fractional calculus and the fractional calculus of variations with Lambert W kernels are shown to be special cases of the Lambert W operational framework, thereby unifying discrete hyperoperations with continuous analysis. A categorical duality between the mathematics of numbers and the mathematics of Lambert W operations is established, yielding a field isomorphism between the Lambert W hyperfield and the complex numbers. A functorial relationship between the Lambert W hyperfield and the exponential hyperfield, reflecting the fact that W−1(z) = zez, is constructed. The connection between Lambert W iteration values and transcendental number theory is explored, with unconditional proofs of transcendence of the values Z(r) for rational r. A corrected Lambert W function is defined using backward iterates, and the Lambert W Riemann Hypothesis is proved unconditionally via a Hilbert-Pólya self-adjoint operator construction. The paper is self-contained, and every essential statement is accompanied by a detailed proof.
Liu S (Wed,) studied this question.