This work presents a completely rigorous and self-contained extension of the full apparatus of Meta-Operational Mathematics to the Lambert W function W and its multi-valued compositional inverse Λ(z) = zez. The central philosophical principle — that operations upon operations constitute meta-operations — is established with complete mathematical precision through a four-level hierarchical framework: Level 0 (elements of a base space), Level 1 (operations as mappings on the base space), Level 2 (meta-operations as mappings on operations), and Level 3 (meta-meta-operations acting on meta-operations). Within this framework, the Lambert Wfunction W and its inverse Λ are shown to admit canonical lifts to meta operations via composition, and these meta-operations interact with one another through composition, pointwise addition, pointwise multiplication, differentiation, exponentiation, logarithm, and branch swapping in arbitrarily many iterations —integer, fractional, real, and complex. A fundamental distinction from the zeta and hypergeometric cases is established: the Lambert W function is characterized by the transcendental equation W(z)eW(z) = z, the Lagrange inversion series, and the branch point at z = −1/e. This leads to the Lambert W Duality Axiom (Axiom 2.18), in which the branch swapping group Z2 acts non-trivially, and the intertwining relations W ◦ Mez = ι and Σ◦W0 =W−1◦Σreplace the elliptic quotient group C/Λ or the zeta reflection. The seven fundamental meta-operations generating the whole Lambert W operad are composition, pointwise addition, pointwise multiplication, differentiation, the branch-swapping meta-operation Σ, the identity operation ι, and the Lambert W operation W. The three essential features of the Lambert W function — its defining transcendental equation, its Lagrange inversion series, and its deep connection to exponential and algebraic operations — are systematically elevated to the meta-operational level as algebraic axioms, analytic tools, and geometric objects, constructing a selfcontained Lambert W Meta-Operational Mathematics. All conjectures and open problems originally stated have either been resolved as theorems within the body of this paper or are precisely formulated as remaining open problems with partial progress indicated. Every theorem is proved in full detail, with each major proof containing at least eight explicit steps. As a conceptual innovation, we introduce the structure of an anti-bialgebra — a coassociative coalgebra with an anti-involution that relaxes the antipode axiom — which naturally accommodates the nonlinear Lambert W operations W and Λ that fail to satisfy the classical Hopf antipode condition.This new algebraic framework is shown to be fully compatible with the Connes–Kreimer renormalization Hopf algebra and serves as a universal template for meta-operational structures beyond the Lambert W case.
shifa liu (Wed,) studied this question.