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 EAPG ≤ C₂λ₁ (ΔAPG) ⁻¹H (W) ², where EAPG is the APG Dirichlet energy, H (W) is the Shannon entropy of the APG Euclidean weight distribution, and λ₁ (ΔAPG) 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 Rₛpec = H (W) ² / λ₁ (ΔAPG), which linearizes the main inequality as EAPG ≤ C₂Rₛpec. 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.
Md. Amir Khusru Akhtar (Fri,) studied this question.