Foundations of Arithmetic Power Geometry VIII: A Spectral-Green Bridge for APG Energy and the Boundary-Control Program

This paper develops the eighth volume of Arithmetic Power Geometry (APG) by proving an unconditional analytic bridge between APG source distributions, Poisson potentials, Green energy, and spectral control on compactified modular curves. Earlier APG volumes introduced local entropy-governed closure defects, information-geometric stabilization, integrated defect energy, scale-corrected invariants, discrete prime-chain regularization, and a conditional APG–Arakelov projection program. The remaining difficulty identified in APG VI–VII is the conversion of continuous APG deformation energy into conductor-scale Arakelov intersection control. This paper proves a Spectral–Green Bridge Theorem: on the compact Riemann surface X = X₀(N)*(ℂ), the APG Dirichlet energy of the mean-zero Poisson potential satisfies E_APG ≤ λ₁(Δ_APG)⁻¹ ‖ρ_APG − 1/Vol(X)‖²_{L²(X)}, where λ₁(Δ_APG) > 0 is the first nonzero eigenvalue of the positive APG Laplacian. The theorem is unconditional once the APG metric is fixed as a smooth compact Arakelov-compatible metric. We then formulate the Spectral Conductor Criterion, showing that the Green-energy distortion estimate required in APG VII reduces to an explicit spectral-conductor inequality. The paper does not claim a proof of Fermat’s Last Theorem, the abc conjecture, the Szpiro conjecture, or the full APG–Arakelov Projection Theorem. Its contribution is a rigorous analytic reduction: the APG projection wall is transformed into a precise spectral bound involving the first Laplace eigenvalue and the L²-size of the APG source.