This paper develops the ninth volume of Arithmetic Power Geometry (APG) by proving the source-concentration component of the APG spectral-conductor program. Earlier APG volumes introduced local entropy-controlled closure defects, information-geometric stabilization, integrated closure-defect functionals, scale-corrected invariants, discrete prime-chain regularization, conditional APG–Arakelov height coupling, discrepancy-source compatibility, and the Spectral–Green Bridge. The unresolved source-side obstruction was whether the deformation-induced density ρAPG on the compactified modular curve X = X₀ (N) * (ℂ) could develop non-integrable mass concentration, especially near cusp neighborhoods. This paper gives a clean APG IX resolution by formulating the APG source as a Radon–Nikodym density on a cusp-regularized compact modular surface, proving a non-circular Hilbert-space decomposition ρAPG = 1/V + αAPGσ, constructing a Hilbert–Schmidt transport kernel for the variance-defect transfer, proving the entropy-amplitude inequality αAPG ≤ C₁H (W), and deriving the cusp damping factor Ωₛtack (q) = |q|² = e⁻²ˢ from the APG cusp-regularized metric. The main theorem proves ‖ρAPG − 1/Vol (X) ‖²₂ ≤ C₁H (W) ². Consequently, ρAPG ∈ L² (X), ‖ρAPG‖₂ < ∞, including cusp neighborhoods. The result removes the source-blow-up obstruction and reduces Open Problem 15. 1 to the remaining spectral-gap and conductor-growth estimate λ₁ (ΔAPG) ⁻¹H (W) ² ≤ C deg (π) log N. The paper does not claim an independent proof of Fermat’s Last Theorem, the abc conjecture, the Szpiro conjecture, or the full APG–Arakelov Projection Theorem.
Md. Amir Khusru Akhtar (Thu,) studied this question.