The Riemann Hypothesis: Echo-Silence and Coprime-Diagonal Reduction Program

Author: William Goodfellow

Public Status: Research framework, not a proof of RH

Latest Public Audit: April 2026

Subject: Mathematics - Number Theory (math.NT)

Status correction: this page now presents the Riemann work as a reduction program and research map. The former downloadable preprints used stronger proof-facing language than the current audit supports, so public downloads are paused pending revision.

Current Mathematical Shape

The useful part of the project is the attempt to express RH through mirror or echo-silence functionals and then attack the resulting vanishing problem with coprime-diagonal and dispersion estimates.

The current bottleneck is not presentation. It is the analytic endpoint: closing the required uniform vanishing without importing an assumption as strong as the target. In the local notes this is tracked as a direct endpoint / real-part barrier.

What Remains Useful

Echo-Silence Criterion

A mirror-functional formulation may still be a useful diagnostic language for RH-type obstructions, provided every equivalence and uniformity claim is kept explicit.

Coprime-Diagonal Analysis

The CDH route remains valuable as a way to identify exactly which bilinear or dispersion estimates would be needed to force the desired vanishing.

Barrier Map

The main productive output is a clearer map of where the route fails: endpoint control and avoiding circular use of RH-strength input.

Downloads Paused

The previous PDF and TeX files have been moved out of the public download surface. They are retained privately for audit history, but they should not be cited as a completed proof.

Review Invitation

Feedback is welcome on the reduction architecture, especially on whether the mirror-functional criterion can be made non-circular and whether the coprime-diagonal endpoint can be separated from RH-strength assumptions.

Initial public version: July 2025

Current public status: audited research framework / reduction program