Foundations of Arithmetic Power Geometry XI: Spectral Gap Stability Under Canonical Bounded Green-Renormalized Metrics

Arithmetic Power Geometry (APG) is a deformation-theoretic framework in which algebraic closure relations are studied under continuous and regularized exponent deformation. Previous volumes established entropy-governed local closure defects, information-geometric stabilization, integrated defect functionals, scale-corrected invariants, discrete prime-chain regularization, conditional APG–Arakelov height coupling, discrepancy–source compatibility, the Spectral–Green Bridge, source concentration, and entropy-controlled spectral energy. The remaining analytic obstruction in the APG spectral-conductor program is the possible collapse of the first positive eigenvalue of the APG Laplacian on compactified modular curves. This paper develops APG XI by introducing a canonical bounded Green–renormalized APG metric and proving polynomial spectral gap stability for that canonical metric. If X = X₀(N)*(ℂ), g_APG = e^(2φ_APG)g_std, where g_std is a compact Arakelov-compatible reference metric and φ_APG is the bounded Green-renormalized APG conformal factor, then osc_X(φ_APG) ≤ C log log N. Consequently, λ₁(Δ_APG) ≥ c / (log N)^A whenever the reference spectral gap satisfies a conductor-polynomial lower bound. In the congruence modular-curve setting, this is compatible with the classical Selberg spectral-gap input for the standard hyperbolic model. The paper does not claim a proof of Fermat’s Last Theorem, the abc conjecture, the Szpiro conjecture, the Birch and Swinnerton-Dyer conjecture, Open Problem 15.1, or the full APG–Arakelov Projection Theorem. Its contribution is the spectral-gap stability component of the APG IX–XIII roadmap under a canonical APG metric normalization.