The Zeta Project
Eight papers on the geometry of the Riemann zeta function, an interactive explorer for the object itself, and one refereed paper. The programme set out to find exact conditions forbidding zeros off the critical line. It did not find them — and then measured, precisely, why the search failed.
A question, and an honest answer to a different one
The Riemann zeta function has zeros. Some of them are known to lie on a single vertical line in the complex plane, and the Riemann Hypothesis says that all of the interesting ones do. Nobody has proved it. This project asked a narrower question: what kind of mathematical object would forbid a zero from sitting off that line?
Over eight papers it built about seventy exact objects and scored each one against seven requirements fixed in advance. Every single one failed the same requirement, and the reason turned out to be structural rather than accidental:
The objects that separate ζ from its known counterexamples are inequalities carried by the local prime data. Every exact object this programme built is an equality inherited from the functional equation. And the functional equation is exactly what ζ and the counterexamples share.
That is not a proof of anything, and it is not progress toward one. It is a checkable statement about why a particular search cannot work — which is worth writing down, because the search is a natural one and other people will try it.
Two objects of the right shape are already known: the classical zero-free region, approaching the critical line from one side, and one of this programme's own lemmas, approaching from the other. Both stall. What is missing is not the existence of such an object but the rate at which one reaches the line.
What this project does not claim
Nothing here decides the location of any zero of ζ. No result is progress toward a proof of the Riemann Hypothesis, and none should be read that way. Every count on this site certifies a finite window above a finite detection floor and forbids nothing.
Zeta Function Explorer
The instrument, not a picture of it. Partial sums of ζ are drawn as they accumulate — the spiral, the saddle packets Paper 1 is named after, and the critical line running through the middle of all of it. Built for a desktop screen; the camera stays where you put it.
Published papers
All eight, in full — main text, figures with their commentary, and the supplementary audit layer. Nothing is held back: the refuted conjectures, the priority concessions and the errata are printed with the same weight as the results.
One paper was held to a different standard
A 98-page selection went through fourteen blind referee rounds and is the version prepared for arXiv. It carries 28 of the programme's 62 claims — the ones that survived that process. The other 34 live on this site and nowhere else.
Eight papers, and the machinery underneath
The programme ran as a long sequence of pre-registered measurement rounds. A round names its statistic, its window, its bar and its falsifier before it runs; a gate that cannot fail is thrown out and reported as thrown out; a result that survives is written into a ledger with the run that produced it. Roughly 150 such rounds stand behind these papers, along with several hundred certified computations and a deposit of 171 data and code files.
It was also, from the first line to the last, run with AI. Models designed the probes, executed them, adjudicated the results, drafted the text and refereed it blind. The author directed the programme and made the decisions; he did not independently verify every proof, and the papers say so in their own front matter. That disclosure is on the about page in full.
What the arc looks like from the outside
| Papers | What happened |
|---|---|
| 1–2 | The geometry: partial sums travel in packets, the packet centroids miss ζ by a computable amount, and the error term is identified to a parameter-free prediction. Ends on a structural obstruction. |
| 3–4 | The counterexample is put under the same instruments as ζ. A zero set is shown to read back its own coefficient arithmetic. The first construction that separates the two by using multiplicativity at step one. |
| 5–6 | An exact per-event law of the critical line — and a controlled experiment showing it holds identically on a function that violates the hypothesis. Two of the programme's own conclusions are corrected here. |
| 7–8 | The inequality side and the symmetry side of the same sentence, both priced out with numbers on both sides. |
Read this as a workbench
This site is a record of a workbench, not a record of finished results. Rigorous standards were applied to the arXiv paper alone. The eight papers include measurements that were later corrected, conjectures that were refuted, and observations that have never been checked against the literature. Each is marked as what it is — the marks are defined here.