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), ₋ₒ₏ (X) = ₏ ₗ p Li₂ (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), ₋ₒ₏ (X) = (X) + 14 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) + 14P (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, 107X 10³, 10⁷X∈103, 107 confirming the bound ∣πLSP (X) −π (X) −14log⁡log⁡X∣≤5|₋ₒ₏ (X) - (X) - 14 X| 5∣πLSP (X) −π (X) −41loglogX∣≤5. The dilogarithm kernel Li2 (1/p) Li₂ (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.