Foundations of Arithmetic Power Geometry XII: Conductor-Modular Degree Dominance and Boundary-Control Growth Estimates

Arithmetic Power Geometry (APG) studies algebraic closure relations under exponent deformation and their analytic realization on compactified modular curves. Earlier volumes established entropy-governed local closure defects, stabilized information geometry, integrated defect functionals, scale correction, discrete arithmetic regularization, conditional APG–Arakelov height coupling, discrepancy-source compatibility, the Spectral–Green Bridge, source concentration, entropy-controlled spectral energy, and spectral-gap stability for a canonical bounded Green-renormalized APG metric. This paper develops the twelfth volume of the APG program. Its purpose is to connect APG analytic size to conductor and modular degree growth before the final spectral-conductor synthesis of APG XIII. Let X = X₀ (N) * (ℂ) and let π: X₀ (N) * → E be the modular parametrization attached to the APG–Arakelov setting. We introduce the canonical APG boundary package BAPG (N, π) = Bcusp + Bᵥert + BGreen, where the three terms measure cusp-width complexity, vertical arithmetic complexity, and Green-energy metric distortion. The main unconditional result of this manuscript is the conductor-scale upper dominance theorem Bcusp + Bᵥert + BGreen ≤ C deg (π) log N, under the canonical normalization inherited from APG VI–XI. We also prove a direct non-circular entropy-to-boundary lower comparison in the logarithmic case A = 1, and formulate the general Spectral Entropy Boundary Realization Problem for A ≥ 1. If this realization property holds, then APG XII yields the desired conductor-modular degree dominance estimate H (W) ² (log N) A ≤ C₃ deg (π) log N. Consequently, APG XII isolates the precise final arithmetic-growth input required by APG XIII. The paper does not claim a proof of Fermat’s Last Theorem, the abc conjecture, the Szpiro conjecture, the Birch and Swinnerton-Dyer conjecture, or the full APG–Arakelov Projection Theorem. Its contribution is a rigorous boundary-growth theorem and a precise reduction of the remaining APG spectral-conductor obstruction to an entropy-to-boundary realization theorem.