Foundations of Arithmetic Power Geometry IX: Source Concentration Theory and Entropy-Controlled APG Density Bounds

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 ρ_APG 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 ρ_APG = 1/V + α_APGσ, constructing a Hilbert–Schmidt transport kernel for the variance-defect transfer, proving the entropy-amplitude inequality α_APG ≤ C₁H(W), and deriving the cusp damping factor Ω_stack(q) = |q|² = e⁻²ˢ from the APG cusp-regularized metric. The main theorem proves ‖ρ_APG − 1/Vol(X)‖²₂ ≤ C₁H(W)². Consequently, ρ_APG ∈ L²(X), ‖ρ_APG‖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(Δ_APG)⁻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.