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 $a_{\min}$. 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

$p_c(s)\sim C_*e^{a_{\min}s}s^{-5/2}.$

After lattice atomicization this becomes a summable coefficient law $c_j\sim C_d j^{-5/2}$.

The mathematical core is a deterministic finite-section theorem independent of CPFT. Let

$F_N(z)=\sum_{j=0}^{N}c_jz^j,$

$\qquad c_j\ge0,$

$\qquad c_j\sim Cj^{-\alpha-1},$

$\qquad A:=\sum_{j\ge0}c_j\in(0,\infty),$

with $C>0$ and $\alpha>0$. In the endpoint scaling $z=\exp(u/N)$, every fixed nonprincipal Lambert branch produces a unique zero

$u_{N,k}=-W_k\!\left(\frac{C}{AN^\alpha}\right)+o(1).$

Consequently the local zero ladder satisfies $Im u_{N,m}^{\pm}\to\pm(2m-1)\pi$, while the nearest-zero distance obeys

$\min_{F_N(z)=0}|z-1|\sim \alpha\frac{\log N}{N}.$

For a CPFT atomicization with cutoff $T_N=Nd$, this yields

$a_{N,m}^{\pm}=a_*+\frac1{T_N}W_{\pm m}\!\left(\frac{D_d}{T_N^\alpha}\right)+o(T_N^{-1}).$

In the CPFT III endpoint sector $\alpha=3/2$, hence

$dist(a_{\min},\mathcal Z_N^{(a)})\sim \frac32\frac{\log T_N}{T_N}.$

A fully explicit CPFT III benchmark, $\eta=2$, $\omega=1/2$, $k_2=1$, 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.