Complete-tail projections and finite multiscale experiments for a discrete Riemann hypothesis criterion

This preprint studies finite fractional-part approximants in a weighted Hilbert space associated with a discrete formulation of the Nyman–Beurling criterion for the Riemann hypothesis.

We assemble exact scaling, joint projection and transport identities that preserve the complete infinite tail. We derive a uniform comparison between an admissible two-packet family and an auxiliary projected loss, a determinant-weighted representation using two readings, and a Mellin representation retaining the full projection gain.

The computational evidence comprises two distinct campaigns: five endpoints through q = 10,000 and twelve fixed-grid endpoints between q = 3,360 and q = 6,720. Their certification methods, inherited reference enclosures, adverse results and reproducibility limitations are stated separately. Supporting records include rational coefficient maps, archived numerical enclosures, source code and provenance hashes.

The unresolved step is an arithmetic estimate controlling the complete residual along an unbounded sequence. No such estimate, improved unconditional power exponent, or proof of the Riemann hypothesis is claimed. Internal checks do not constitute independent replication or external peer review.

AI assistance: ChatGPT Astra (OpenAI), alongside Codex-assisted tools, supported mathematical exploration, symbolic derivations, literature searches, software implementation and computational orchestration, numerical analysis, internal consistency checks, interpretation, and manuscript drafting and editing. Numerical enclosures were produced by the documented arithmetic software. The author is responsible for the selection, interpretation and presentation of the results.