Structural Inertia as a Detector of Higher-Order Additive Structure: The IDT-AP Programme

We introduce a spectral framework, IDT-AP, that detects k-term additive structure in subsets of Z_N through a generalised eigenvalue problem (GEP) attached to a Gowers-norm potential. MAIN RESULT Define the AP-invariant C_AP,k(p) := N^(k-2) * T_k(p) - 1, where T_k(p) is the k-AP count for density p on Z_N. We show that whenever |C_AP,k| >= epsilon on Z_N prime, the balanced indicator g = 1_A - alpha satisfies ||g||_{U^(k-1)} >= c_k * epsilon, and the resulting potential Phi_k has a Hessian which is not proportional to the Fisher metric G, forcing the structural inertia M_k > 1. NUMERICAL EXPERIMENTS AT N=101 Two distinct detectors realise a double dissociation between linear and quadratic additive complexity: * Fourier-IDT detector: for the interval set, z(M_4) grows from 10 sigma (N=47) to 84 sigma (N=101). Quadratic phases give z ~ 0. * Gowers-IDT detector: matched-q quadratic phases at N=101 give z ~ 650 sigma for q=3 and z ~ 590 sigma for q=7. The interval set gives only z ~ 2. This implies that additive complexity is vector-valued: no single scalar inertia detects all higher-order structure. Each Gowers level U^(k-1) requires its own information lens. STATUS This is the IDT-AP foundational paper, part of the broader Information-Dynamic Theory programme (Master DOI 10.5281/zenodo.18296688). The result is unconditional in the direction "AP-bias implies inertia"; the converse fails (counterexample provided in the paper). WHAT IS NOT PROVEN The Erdos-Turan conjecture is not closed. The prefactor beta in Phi_k is not derived from first principles (it is calibration-robust over beta in [0.40, 0.79], see companion Q01 closure). No direct physical observable for M_k is yet identified. These are the three honest open problems.