We construct a real, even spectral triple (A, H, D) that couples a four-dimensional spin manifold to a discrete arithmetic space defined over the prime-gap lattice ₚ. We investigate the topological stability of this coupling via the McKean-Singer index formula. We prove that the requirement of a vanishing total Fredholm index, Ind (D) = 0, induces a topological quantization on the generation number Ng of the finite module. By mapping the arithmetic spectral flow to a dynamical system on the torus, we show that KAM (Kolmogorov-Arnold-Moser) stability against chaotic resonance uniquely selects the Lambert-W fixed point ₄₅₅ = W (1/4) 0. 225209973392 for the geometric-arithmetic coupling constant.
August Tudor (Mon,) studied this question.