Mellin-Kernel Analysis and Error Structure in the Log-Spectral-Prime Space

This appendix accompanies the paper A Geometric Phase Space Construction for Prime Number Distribution: The Log-Spectral-Prime (LSP) Space with Branch Structure and Dual Number Extensions. It provides supplementary material and detailed derivations for the Log-Spectral-Prime (LSP) counting function: πLSP(X)=∑p≤Xp⋅Li2(1/p),πLSP(X) = ∑p ≤ X p · Li_2(1/p),πLSP(X)=p≤X∑p⋅Li2(1/p), where the asymptotic structure is rigorously established via Mellin–Tauberian analysis. Building on the geometric constructions of the main paper, we prove: πLSP(X)=π(X)+14log⁡log⁡X+O(1),πLSP(X) = π(X) + 1/4 log log X + O(1),πLSP(X)=π(X)+41loglogX+O(1), without assuming the Riemann Hypothesis. The analysis includes: Dirichlet series decomposition A(s)=P(s)+14P(s+1)+H(s)A(s) = P(s) + 1/4P(s+1) + H(s)A(s)=P(s)+41P(s+1)+H(s) with explicit coefficient bounds. Tauberian theory via Wiener–Ikehara and Karamata slowly varying functions. Contour integration with explicit growth estimates and residue analysis. Numerical validation over X∈[103,107]X ∈ [10^3,10^7]X∈[103,107] confirming the bound ∣πLSP(X)−π(X)−14log⁡log⁡X∣≤5|πLSP(X) - π(X) - 1/4 loglog X| ≤ 5∣πLSP(X)−π(X)−41loglogX∣≤5. The dilogarithm kernel Li2(1/p)Li_2(1/p)Li2(1/p) appears both geometrically (via Parseval theorem) and analytically (via coefficient structure). This appendix serves as a direct supplement to the main paper, providing additional derivations, proofs, and numerical data.