Critical Partition Function Theory III

We develop the distributional-flow sector of critical partition-function theory without assuming disorder, a random Hamiltonian, a random-matrix ensemble, or any stochastic microscopic mechanism. A normalized nonnegative critical measure is evolved by exponential tilting followed by rescaling to unit mean. With intrinsic coordinate τ=-log m(a) and normalized variance g(τ)=V(a)/m(a)², the complete normalized law is reconstructed from the single function g through a(τ)=∫₀^τ ᵗ/g(t)\, t, log L(a(τ))=-∫₀^τ t/g(t). We connect this exact semigroup to the weighted rank-one critical geometry of Paper II and obtain a positive Wright--Lambert source family, with the Paper-I family as the ω=1 one-mode boundary and Lambert W as the confluent ω=0 boundary. The central result is an exact finite-dimensional closure of the mean-normalized Esscher orbit. For normalized cumulant ratios $A,B,H$, let D₄=B-A(2A-1), D₅=H-(7A-3)B+A(4A-1)(2A-1). On the regular branch D₄>0, orbit membership is dynamically characterized by constancy of I(τ)=2A(τ)-D₅(τ)/D₄(τ), and equivalently the inverse normalized variance $q=1/g$ satisfies a constant-coefficient second-order linear equation. Generic Wright orbits have two distinct exponential modes, the Lambert boundary is their repeated-root Jordan limit, and the Paper-I boundary is a one-mode degeneration. Four low cumulants reconstruct a generic orbit point, with the effective dimension dropping from four to three, two, and one on the Lambert, Paper-I, and Gamma strata. The closure admits equivalent differential-Hankel, all-order cumulant, and mean-generating-function formulations. A positive orbit coordinate yields finite polynomials with strictly positive coefficients, proving monotone fixed-order relaxation to the Paper-I boundary. The inverse mean map produces a normalized positive kernel that gives exact coefficient extraction and a uniform Wright--Lambert completion theorem. It also resolves the nonuniform confluent hierarchy through =ωlog n, gives a negative-binomial refinement of the compact completion layer, and separates Gaussian correction-count fluctuations from heavy-tail mesoscopic mass. The Wright interior has a positive-stable mesoscopic boundary with a Wright-function profile; the confluent critical edge has, after exact centering, a universal Landau-type α=1 stable local profile. The same kernel determines the fixed-rank branch point and square-root amplitude, giving the universal (n-1)-3/2 factor in the high-order rank-two asymptotics. Thus finite-order completion and fixed-rank large-order growth arise as distinct, noncircular consequences of one inverse-mean structure. No identification of this semigroup with a model-specific renormalization-group flow is assumed.