A Hybrid AI-Spectral Solver for the Prime Gap Inverse Problem: Overcoming Ill-Conditioning via Anderson Localization

The inverse spectral problem for Jacobi matrices—recovering the exact matrix entries strictly from its eigenvalues—is fundamentally ill-posed due to a deficit in degrees of freedom. In this paper, we present a hybrid computational framework that overcomes this classical limitation for the Prime-Gap Jacobi Matrix, D_N. By demonstrating that the prime gap sequence induces a strict O(1) Anderson-type topological localization, we utilize participation-weighted eigenvectors to spatially reorder the bare spectrum. We then apply a lightweight 1D-Convolutional Neural Network (1D-CNN) constrained by a novel differentiable Diophantine penalty, sin^2(π x/2), to enforce even-integer predictions. Tested on 50,000 spectral windows sampled from the first 148,931 prime gaps, the model achieves a coefficient of determination R^2 = 0.317, strictly breaking the theoretical diagonal-correlation ceiling of r^2 ≈ 0.23. The algorithm achieves an exact even-integer sequence match for 11.3% of the gaps. This open-source framework serves as a rigorous, verifiable toy model for overcoming severe ill-conditioning in discrete inverse problems.