Foundations of Arithmetic Power Geometry VI: A Conditional APG-Arakelov Height Coupling Program

This paper develops a conditional APG–Arakelov height coupling program connecting Arithmetic Power Geometry (APG) deformation functionals with Arakelov intersection invariants associated with Frey curves. Earlier APG volumes introduced local entropy-controlled closure deformation, stabilized information-geometric formulations, integrated closure-defect functionals, coordinate-dependent damping, and discrete prime-space regularization. The remaining obstruction is the passage from APG deformation energy on a compactified modular curve X₀(N)* to a height lower bound for the associated Frey elliptic curve Eₐ,ᵦ,ₚ. This paper corrects the weight notation by distinguishing the original Euclidean APG weights from Fermat-normalized weights, formulates the APG potential on a compact Riemann surface, proves smooth solvability of the APG Poisson problem, proves positivity and quadratic Dirichlet-energy scaling, and derives a conditional APG-to-Faltings lower bound under an explicit APG projection hypothesis. The main novel addition is the APG modular-efficiency invariant ε_APG(E) = [D*ₚ(a,b)H(W̃)]² / deg(π), which measures APG deformation energy per unit modular complexity. The paper proves that, assuming the projection hypothesis, the Faltings height is bounded below by this modular-efficiency invariant up to a logarithmic conductor error. The paper does not claim an independent proof of Fermat’s Last Theorem. Instead, it gives a precise conditional reduction and identifies the exact open projection and modular-efficiency problems needed for a future unconditional APG height-obstruction program.