Critical Partition Function Theory IX: From Algebraic Critical Endpoints to Lambert-W Zero Ladders

We connect an algebraic endpoint of a positive critical measure to the complex zeros generated by finite atomicization. The starting point is the Wright interior of Critical Partition Function Theory (CPFT) III, where the inverse mean map develops a finite backward endpoint aₘᵢₙ. We compute the local Puiseux expansion of the log-Laplace transform and obtain an explicit positive $3/2$ coefficient. A Bingham--Doney type Tauberian theorem then gives an endpoint-tilted tail s-3/2 at the measure level, and, under a monotone-density hypothesis, a density tail pc(s)~ C_*e^aₘᵢₙss-5/2. After lattice atomicization this becomes a summable coefficient law cⱼ~ Cd j-5/2. The mathematical core is a deterministic finite-section theorem independent of CPFT. Let FN(z)=∑ⱼ₌₀Ncⱼzʲ, cⱼ≥0, cⱼ~ Cj-α-1, A:=∑j≥0cⱼ∈(0,∞), with $C>0$ and α>0. In the endpoint scaling z=exp(u/N), every fixed nonprincipal Lambert branch produces a unique zero uN,k=-Wₖ\!(C/AN^α)+o(1). Consequently the local zero ladder satisfies Im uN,m±→±(2m-1)π, while the nearest-zero distance obeys minFN(z)=0|z-1|~ αlog N/N. For a CPFT atomicization with cutoff TN=Nd, this yields aN,m±=a_*+1TNW± m\!(Dd/TN^α)+o(TN⁻¹). In the CPFT III endpoint sector α=3/2, hence dist(aₘᵢₙ, ZN⁽ᵃ⁾)~ 32log TN/TN. A fully explicit CPFT III benchmark, η=2, ω=1/2, k₂=1, numerically supports the s-5/2 tail, identifies the first two Lambert-W zero-ladder branches, and exhibits the predicted asymptotic trend under point and bin atomicization. No disorder, random Hamiltonian, or random-matrix assumption is used.