Toward a Spectral Proof of the Riemann Hypothesis: Spectral Unitarity, Hecke Algebras, and Automorphic Scattering in Adelic Langlands Manifolds

. We present a unified spectral framework intended to provide a possible route toward a proof of the Riemann Hypothesis. By embedding the completed Riemann zeta function into the automorphic spectral geometry of the adelic quotient space, we develop a chain of constructions linking automorphic scattering, Hilbert–P´olya-type operators, and Eisenstein series. The resulting framework identifies several key mathematical statements whose establishment would imply the Riemann Hypothesis. We prove a number of founda- tional results, formulate the remaining critical steps explicitly, and discuss computational architectures capable of validating the proposed program.

@article{deceuster2026_84-toward-a-spectral,
  title = {Toward a Spectral Proof of the Riemann Hypothesis: Spectral Unitarity, Hecke Algebras, and Automorphic Scattering in Adelic Langlands Manifolds},
  author = {De Ceuster, Peter},
  year = {2026},
  month = {02},
  institution = {SIG Labs},
  doi = {10.5281/zenodo.21757195},
  url = {https://peterdeceuster.uk/papers/84-toward-a-spectral-proof-of-the-riemann-hypothesis},
  note = {Full text available at https://peterdeceuster.uk/articenter/work/openq (1).pdf}
}