The LSP Space, Geometric Refinement, and Links Between the Imaginary Axis, Complex Numbers, and Prime Numbers

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