The Log-Spectral-Prime (LSP) framework constructs a geometric phase spacefor prime number distribution independently of the Riemann zeta function ζ(s)and its known zero structure. This paper extends the two-dimensional constructionof Paper I and the Mellin–Tauberian analysis of Paper II to a three-dimensionalRiemannian manifold MLSP with coordinates (ρ,θ,τ), where ρ = loglogp is thelogarithmic radial scale, θ ∈ S1 is the phase coordinate, and τ is the growth-rateparameter.The LSP metric is isometric to the Poincare half-plane H2 in the (ρ,θ)-sector,with constant Gaussian curvature K = −1 (Theorem 3.10) and scalar curvatureR = −2 (Theorem 3.12). The critical scale ρ = 1/2 emerges from three independent geometric principles (Corollary 3.16): shell energy maximum (Theorem 3.15),Gammareflection antisymmetry (Theorem 2.30), and phase balance from Paper I. Aseven-component hybrid functional Ψ(X,t) is assembled from the Li2 energy kernel,Gaussian localization, holonomy corrections, and Gamma modulation. Numericalvalidation against the first ten nontrivial Riemann zeros yields mean relative proximity 0.15% at cutoff X = 104.Scope of claims: The LSP functional produces approximate local phase zeros inthe vicinity of Riemann zero ordinates. These are critical points of a geometric phasefunction derived from prime distribution in LSP space, not Riemann zeros of ζ(s).The rigorous conditions under which LSP critical points converge to Riemann zerosare deferred to subsequent work
Okur et al. (Sun,) studied this question.