Generalisation of Geometric Fourier Extension II: Non-Weyl Spectral Geometries and the Nonlinearity–Non-Markovianity Duality

This paper extends the Geometric Fourier Extension (GFE) framework to non-Weyl spectral geometries, proving a completeness theorem that characterizes all physically admissible causal memory kernels as completely monotone functions representable via arbitrary positive Borel spectral measures. Explicit spectral measures, admissibility proofs, and sum-of-exponentials (SOE) approximation rates are derived for fractals (spectral dimension ds ∉ N), quantum graphs (with oscillatory corrections), and sub-Riemannian structures. Additionally, a rigorous nonlinearity–non-Markovianity duality is established: nonlinear Markovian systems on a full Hilbert space project via Mori–Zwanzig reduction to linear non-Markovian systems on reduced subspaces, with the memory kernel encoding the discarded nonlinearity; the SOE embedding provides the converse lifting. The effective memory dimension Deff serves as a stability diagnostic, implying controlled Sobolev growth when bounded and singularity formation when diverging. Applications include anomalous diffusion on the Sierpiński gasket, signal propagation on quantum graph networks, and turbulent cascade dynamics, with numerical recipes provided.