Foundations of Arithmetic Power Geometry X: Spectral Entropy Theory and Energy Control on Compactified Modular Curves

This paper develops the tenth volume of Arithmetic Power Geometry (APG) by formulating a spectral entropy theory for APG deformation energy on compactified modular curves. Earlier APG volumes introduced local entropy-governed 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, and source concentration theory. The present paper proves the entropy-spectral energy theorem E_APG ≤ C₂λ₁(Δ_APG)⁻¹H(W)², where E_APG is the APG Dirichlet energy, H(W) is the Shannon entropy of the APG Euclidean weight distribution, and λ₁(Δ_APG) is the first positive eigenvalue of the APG Laplacian. The theorem is a derived bridge result: it follows by combining the APG VIII Spectral–Green Bridge with the APG IX Source Concentration Theorem. The paper also introduces the spectral entropy ratio R_spec = H(W)² / λ₁(Δ_APG), which linearizes the main inequality as E_APG ≤ C₂R_spec. The principal contribution of APG X is therefore not a new proof of spectral-gap stability, but a clean synthesis showing that APG energy is controlled by entropy once the spectral gap is fixed. This isolates spectral-gap stability and conductor/modular-degree dominance as the remaining tasks in the APG spectral-conductor program. 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.