Foundations of Arithmetic Power Geometry XIII: The Spectral–Conductor Theorem and the Resolution of Open Problem 15.1

Arithmetic Power Geometry (APG) studies algebraic closure relations under exponent deformation and their analytic realization on compactified modular curves. Earlier APG volumes established entropy-governed local closure defects, information-geometric stabilization, integrated defect functionals, scale-corrected regularization, conditional APG–Arakelov height coupling, discrepancy–source compatibility, the Spectral–Green Bridge, source concentration, entropy-controlled spectral energy, spectral-gap stability for a canonical bounded Green-renormalized APG metric, and conductor-scale boundary dominance. This thirteenth paper completes the APG IX–XIII spectral-conductor route by proving Open Problem 15. 1 under the canonical hypotheses inherited from APG IX, APG XI, and APG XII. Let X = X₀ (N) * (ℂ) be the compactified complex modular curve of level N, let π: X₀ (N) * → E be the modular parametrization in the APG–Arakelov setting, and let ΔAPG be the positive APG Laplacian attached to the canonical bounded Green-renormalized metric. The main theorem proves λ₁ (ΔAPG) ⁻¹ ‖ρAPG − 1/Vol (X) ‖²₋ℂ (ₗ) ≤ C deg (π) log N. The proof is a non-circular synthesis: APG IX bounds the mean-zero APG source by entropy, APG XI prevents spectral-gap collapse at polynomial conductor scale, and APG XII dominates the resulting entropy-conductor term by modular degree. The paper also records a conditional Fermat obstruction corollary: if a primitive Fermat triple forces a strict APG lower-energy gap exceeding the APG XIII conductor wall, then such a triple cannot exist. This final corollary is conditional and is not presented as a new unconditional proof of Fermat’s Last Theorem. The unconditional proof of Fermat’s Last Theorem remains the classical modularity and level-lowering proof.