Higher-Order Jet Geometry of the Zeta Kernel and Multiplicity-Preserving Spectral Realization: A Conditional Construction

We investigate the higher-order derivative structure of a de Branges-type kernel associated with the Riemann xi function at its nontrivial zeros. Our first main theorem establishes that, at a zero of multiplicity greater than one, all normalized derivative kernels below a specific order vanish identically. Starting from that order, however, every finite collection of consecutive derivative kernels is linearly independent. In particular, the first collection containing as many nonvanishing derivative kernels as the multiplicity of the zero has exactly that dimension. This result resolves the analytic degeneracy of lower-order derivatives identified in our previous work. We then identify two remaining operator-theoretic challenges. The first is to construct a closed operator whose generalized eigenspaces reproduce the multiplicities of the nontrivial zeros. The second is to establish a correspondence between these eigenspaces and their Krein signatures. We also derive a recurrence relation describing how multiplication by the complex variable acts on the derivative kernels. Throughout the paper, we distinguish established analytic results from operator-theoretic statements that remain to be proved. Our analysis does not assume that the nontrivial zeros lie on the critical line, nor does it claim to prove the Riemann hypothesis.