The Spectral Trilemma of the Riemann Zeros: Computational Obstructions to the Hilbert-Pólya Conjecture

We develop a paired falsification protocol for Hilbert-Polya candidates: a Fourier-domain prime-content test calibrated against null ensembles, together with a shifted-target density control that catches spectral look-alikes. Applying this to thirty-five constructions across quantum graphs, Fock spaces, continuous operators, transfer operators, and hyperbolic geometry, we identify the specific obstruction mechanisms responsible for each failure, including rational-frequency proliferation in 1D scattering, a backscattering/symmetry-class tradeoff in open graphs, interaction-induced dephasing, and algebraic closure in the Berry-Keating xp framework. These organize into an empirically observed spectral trilemma: any two of three necessary requirements can be achieved, but none achieves all three.Companion code: https://github.com/alasdaircoull/riemann-spectral-trilemma