Critical Level-Spacing Statistics of the Three Wigner--Dyson Anderson Classes in a Wright--Lambert Family

Critical Partition-Function Theory (CPFT) and the Anderson transition start from different microscopic descriptions, so no direct identification of their critical exponents is assumed. We instead use the critical nearest-neighbor level-spacing distribution Pc(s) as a shared observable. A Lambert-W boundary of the CPFT Wright--Lambert positive-measure family is represented by z(1+a/R+klog z)=1, log Lc(a)=R[log z+k(z-1)], where Lc(a)=∫₀^∞ e⁻ᵃˢ Pc(s)\,d s. Its small-spacing asymptotics satisfy Pc(s)~ A sR-1 with A=e-RkRR/Γ(R). We test this common family in three dimensions using representative orthogonal, unitary, and symplectic Anderson models. A realization-resolved production run contains 290, 215, and 115 disorder realizations, respectively. Using the conventional critical repulsion powers q1,2,4 as asymptotic anchors, hence $R=2,3,5$, one-parameter fits to five low-a Laplace values give kO=0.335±0.024, kU=-0.003±0.022, kS=-0.395±0.028, with realization-level bootstrap uncertainties. The values reproduce an independent pilot calculation and retain the sign pattern kO>0, kU0, kS<0. Direct inverse-Laplace distributions track the numerical histograms, and size-resolved estimates at $L=6,8,10$ show no statistically required drift of k at the present precision. Covariance-aware fitting, pooled-normalization checks, histogram-binning tests, and inverse-Laplace stability checks do not remove the three-class pattern. We also compare the same production data with Nishigaki's deformed critical random-matrix theory. KS-optimal parameters are aN,O=2.965, aN,U=3.577, and aN,S=3.350. Although point estimates place CPFT and Nishigaki at slightly different distances from the numerical distributions, realization-level bootstrap resampling does not statistically resolve the difference in the orthogonal or unitary samples. The present symplectic sample conditionally favors the constrained CPFT curve, but refitting uncertainty and finite-size/systematic effects are not included, so no thermodynamic-limit ranking is claimed. The visible unitary peak shift is traced to the explicit CPFT constraint $R=3$: releasing R moves the CPFT bulk fit toward the Anderson/Nishigaki peak at the cost of changing the strict s² edge exponent. Thus the present data support a common CPFT coordinate description $(R,k)$ of the three critical Wigner--Dyson spacing distributions, while neither uniqueness of the representation nor universality of k is claimed.