A Half-Completed Proof of the Riemann Hypothesis: On the Spectral Realization of the Riemann Xi-Function — Dilation Generators, Haar Multipliers, and Spectral Scaling Obstructions

The Riemann Hypothesis (RH) asserts that all nontrivial zeros of the Riemann zeta function ζ(s) satisfy Re(s) = 1 2. In this work we will deliver half of the proof required to solve RH. Within the Hilbert–Pólya framework, one seeks a self-adjoint operator ˆH = ˆH∗whose discrete spectrum coincides with the zeros of the shifted completed xi-function Ξ(z) := ξ( 1 2 + iz). We will construct a mathematically coherent operator-theoretic framework on the Haar-weighted space H = L2(R+, d×x) with d×x = x−1dx, resolving previous dimensional and parity mismatches. We define the intrinsic dilation generator ˆH0 = −ix d dx and establish its essential self-adjointness and exact parity-anticommutation ˆJ ˆH0 ˆJ−1 = −ˆH0 under multiplicative inversion ( ˆJψ)(x) = ψ(x−1). We prove two fundamental structural negative results that delimit the Hilbert–Pólya program:

@article{deceuster2026_66-a-half-completed-,
  title = {A Half-Completed Proof of the Riemann Hypothesis: On the Spectral Realization of the Riemann Xi-Function — Dilation Generators, Haar Multipliers, and Spectral Scaling Obstructions},
  author = {De Ceuster, Peter},
  year = {2026},
  month = {02},
  institution = {SIG Labs},
  doi = {10.5281/zenodo.21969688},
  url = {https://peterdeceuster.uk/papers/66-a-half-completed-proof-of-the-riemann-hypothesis},
  note = {Full text available at https://peterdeceuster.uk/articenter/work/Rxf.pdf}
}