The Zeta Project

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:

Every exact object this programme built is an equality inherited from the functional equation. The functional equation is exactly what ζ and the counterexamples share. So whatever separates them cannot follow from the functional equation and the analytic axioms alone — it has to draw on something more, and the programme's own register statement points at the Euler product.

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 that do use the Euler product are already known, and both are inequalities carried by the positivity of ζ's local data: 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 — and that those two share a shape is a description of the two, not a theorem about the missing one.

Published papers

All nine, 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. The ninth is a draft and says so on its own page.

Seventeen more pictures came out of the closing sessions and were never promoted to figures of any paper. They are here, captioned, with the corrections several of them record left in.

One paper was held to a different standard

A 98-page selection went through thirteen blind referee rounds, plus one cross-model control round, 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.

Anthropic Riemann Hypothesis research feedback

In August 2026 Anthropic published an unconditional proof that at least two thirds of the zeros of ζ are simple and on the critical line — a jump from a record of 41.66% that had stood since 2020. It works in the same register this programme spent Paper 7 building: Weil's quadratic form on a restricted support. Paper 7 read the sign of its minimum and priced itself out; they read its signature and rank, and got a theorem.

So we rebuilt their machine from scratch. It reproduces their published tables to six significant figures. Then we ran the one test their own numerics could not: their whole zero-side argument turns on off-line zeros contributing a particular kind of block, and ζ has no off-line zeros to test it with — so they tested it on fabricated ones. This project holds the only bank of genuine off-line zeros of a function with a functional equation, and on those the prediction holds.

Their proof has two halves, and everything above is one of them. The other — the step that turns the argument from linear algebra into arithmetic, and the step their own internal review trusted least — we have now built as well. It reproduces the first half to ten decimal places. It also works for Dirichlet L-functions, which lets us print the table their Theorem E has nowhere in 251 pages, and that table says something they do not: raising the conductor buys exactly what raising the height buys — exactly in the band-width, which is a function of the product of conductor and height to the last bit, and approximately in the certificate, where the residual halves as the height rises. A conductor-5 L-function at height 600 sits where ζ sits at height 3000, and its certificate is 7.6% better at the same height for that reason alone.

Then we turned the single test into a curve. The bank includes a one-parameter family that keeps one functional equation throughout while the number of its zeros off the line climbs from none to eighty-eight and falls back to none — so the truth is known at every point of a continuum, not at one reading. Their prediction holds all the way across, at every band-width: thirty-nine readings, no violation.

That family also turned up something sharper than the confirmation. Two of its configurations have every zero on the critical line, and the certificate scores them a factor of two apart. At the narrowest of the three band-widths we ran it certifies nothing whatever about one of them — while a configuration with eighty-six pairs of zeros genuinely off the line scores better. We could measure why: what the certificate reads is how the zeros correlate with one another, not how many of them sit on the line. Superpose two independent families of zeros and their mutual repulsion goes, and with it most of what the method can certify.

That mechanism now has a formula — one with no fitted constants, which predicts the certificate to within half a per cent — and the single place the formula fails turned out to be a prime. A ladder of zeros spaced at 2π/log n is not independent of ζ's zeros at all; it is locked to them, for n = 2, 3, 4, 5, 7, 8, 9, 11 and 13, and at no spacing in between. The configuration above happens to be spaced at 2π/log 5, and that lock is what costs it a fifth of its certificate.

The page also goes after the sentence that says how far the method can go. Its number appears once in 251 pages with no derivation behind it there, so we reconstructed the problem from scratch: its dual turns out to be the paper's own optimisation, agreeing to thirteen decimal places, and that dual bound is reached by no configuration of zeros at all, because the zeros of its own optimal test function never add up — so the extremal configuration has to be atomic. Then we found the proof, in their Lean development rather than in the paper: an explicit 256-periodic law, atomic exactly as predicted, landing inside our bracket, and its data recomputes from their own table to the last digit. Their remark is a theorem; the paper simply does not say so.

Which made it worth auditing the rest of that development — 329 files and 103,067 lines of Lean. No new axioms, no unfinished proofs outside the deliberately unproved statement files, the theorems stated unconditionally, and the definitions pointed the harder way: the denominator counts zeros with multiplicity while the numerator counts distinct ones. And then we built it: all 9,010 compilation targets, from the pinned toolchain, compile and pass the Lean proof-checker's kernel on our machine — the audit is no longer a careful reading but a checked fact. One link sits outside the proof kernel and they say so themselves. The page also carries one piece of new mathematics that makes the input for the next constant unconditional, the reason that route is nevertheless blocked by an identity rather than by a gap, and four natural next moves closed with the reason each fails.

Then we ran their ceiling's own programme where nobody had: at every period up to 128, against their single value at period 256. The ceiling turns out to move by 0.066 with the period and to reach its computed minimum at period 16; their value stays the lowest anyone has exhibited, and the limit over periods — the number the remark is really about — has been computed by nobody, them included. Along the way the locked-ladder law gained its missing half (the prime-power harmonics enter in quadrature, which collapses every sparse-ladder anomaly at once), the depth law that had looked conductor-dependent became one law across conductors 5, 7 and 13 once fitted like for like, and their Theorem E identity was verified at more than three times the height of any previous run, to twelve figures in the traces.

A search can exhibit but never certify, so we then solved the small-period programme exactly — every configuration on a position grid enumerated and priced, 629 million of them at the finest — and the certified optima walk down onto our sweep's values without ever undercutting them: the instrument is validated everywhere an exact check exists. And two of the paper's own scaling claims fail when pushed: the depth-law rate, re-censused at heights up to 3400, does not move at all while the predicted 2L grows by 2.5 — the deficit widens to 21 per cent at fourteen standard errors — and the lock law's one constant, now measured at five windows up to height 9000, fits none of the clean shapes its derivation allows. The laws hold; their claimed scalings are what break.

The height run has now crossed conductors. The same two windows were censused at conductors 5 and 7 — four more censuses, each balancing exactly against the argument-principle count — and the rate is flat in height at every conductor while 2L keeps growing. For the first time there was also the power to compare conductors at height, and the flat lines differ: conductors 5 and 7 sit together near 12.5, conductor 13 near 13.9, three standard errors apart in every reading. The rate is a constant of the conductor — not the universal number a scaling law would make it, and not ordered by conductor size either. What sets each conductor's constant is the open question the feedback now ends on.

And a reply to the first reply

Within hours of the Anthropic paper going public, a repository appeared on GitHub — ainta/zeta-simple-zeros, by @kaizero_ainta, describing itself as a research draft generated by GPT-5.6 Sol, an OpenAI model — claiming the first improvement of that paper's Theorem D: 67.3008% of zeta's zeros simple and on the critical line, against Anthropic's 67.2501%. An AI-generated draft extending an AI-generated theorem, inviting independent review. We gave it the same audit we gave the original: every constant recomputed from scratch, the load-bearing matrix inequality checked line by line and stress-tested numerically, both of its computer-certified minimization targets independently measured, its verifier read closely and then rerun end to end — reproducing both committed certificates exactly, to the hash — and its one missing proof step reconstructed and supplied.

It holds. The refinement is sound, the certified floors sit half a per cent under the true minima we measured, and the improvement is real — small, honest, and of exactly the kind the original paper's own ceiling governs: it spends one part in eighteen of the distance that ceiling leaves open. It also does one thing better than the original: its certificates are published and its verifier rebuilds every transcendental enclosure on each run — the property whose absence is our one standing request to Anthropic. Credit where it is due: @kaizero_ainta shipped the first outside extension of the theorem, within hours, with its verification in the open.

And a note from outside the machinery

A five-page note posted to Academia.edu — A Geometric Reinterpretation of the Riemann Hypothesis, by Kazushi Mizutani — reformulates the hypothesis geometrically: the critical line is the set of points equidistant from the two edges of the critical strip, the zeros are discrete because ζ oscillates, and adding the point at infinity closes the zero set. Its author states in his own abstract that he claims no proof, and asked, in the discussion attached to it, for an opinion and for help placing it on arXiv. It got the same audit as the other two, scaled to its size.

The picture is right, and right for a better reason than the note gives. The equidistant line is the fixed line of the reflection s ↦ 1−s that the functional equation supplies — which is why the answer is one half and not some other number, and why the same construction correctly returns one quarter when it is applied to ζ(2s). What it cannot do is get from the axis to the confinement, and there is a function that settles that: Davenport–Heilbronn has the same functional equation, the same strip, the same equidistant line and the same discrete zeros running off to infinity — and 386 verified zeros off the line, 570 mirror pairs of them below height 10,000, in a census where five genuine L-functions lose nothing. Any argument built only from the symmetry holds for it too, so no such argument can decide the hypothesis. The feedback says that plainly, answers the arXiv question honestly, and then does the part that matters: three directions the idea can actually be taken, and the reason the note is worth rewriting rather than abandoning.

He then wrote a second note, the next day, and it takes the objection head on. The point at infinity is dropped as a mechanism — he now says himself that it cannot tell a sequence on the line from one off it — and in its place he measures how far a zero sits from the line, notes that the functional equation makes those displacements cancel in pairs, and squares them so they cannot. That instinct is the right one, and it is the one the subject already had: the same plan, carried further, is the Weil–Li programme of the 1990s, where the arithmetic half is proved and the positivity is still open thirty years on. His quantity is evaluated here on the Davenport–Heilbronn ledger, where it returns 1.0076×10⁻⁴ against exactly zero on the line — so it does fire, and the missing step is the one that would tell zeta apart from that function. Alongside it, the exact criterion he was reaching for turns out to exist in the linear form he ruled out, and on the same ledger his squared version is the better conditioned of the two by a factor of twelve thousand.

Two further notes followed, and the feedback now covers four. The revision of 21 August carries a reference list where its predecessor had none, and names the missing piece precisely: a projection that would extract the transverse part of the Weil quadratic form. The answer is that every operation in that chain is defined word for word for Davenport–Heilbronn too — so building it would separate nothing, and of the note's own six open problems four are blind to the difference between ζ and a function with off-line zeros while the remaining two are the hypothesis. The fourth note moves to a different subject entirely, algebraic and transcendental numbers, and is screened without the zero-side ledger: the non-circular distance it asks for is Mahler’s classification of 1932, two of its four proposed foundations fail because π is computable, and the one place its two subjects genuinely meet is Gonek’s conjecture on the Hurwitz zeta-function — an open problem it does not cite.

A separate dated record of what feedback was given on which date, and what was and was not credited in the notes that followed, is kept alongside it. It is a record of fact and carries no allegation: no result of this programme appears in anyone else’s work, and the September note owes it nothing.

The nine-point paper

Auditing both links of the chain left this programme holding something neither audit needed: a measured design curve for the whole family of window arguments — how much each window size and each pressure coefficient can yield, and where the family peaks. The peak is not where the chain stopped. A nine-zero window at a retuned pressure supports 67.3054%, and a second refinement on top of it — the kernel tilted inside the class the analytic estimates actually permit, plus a deliberately asymmetric one-step correction and the exact square-root form of the defect credit — supports 67.3118%. Together they take 61% of everything this whole family of arguments can ever give.

The paper is written the way this programme banks everything: the deduction in exact rationals, every imported result credited to Anthropic and to @kaizero_ainta where it belongs, and the computer-assisted inputs held to the same standard the previous link met — certified interval proofs, produced by our own verifier, which first validated itself by reproducing the previous link's committed certificate bit for bit. Both certificates have now returned and are published beside the paper: 27,955,544 nodes for the first, 78,344,600 for the second. The page also records the three runs the verifier refused before them, and why it was right to.

What a formula is allowed to mention

There is a function that looks like ζ in almost every structural respect — same kind of series, same functional equation, same zero-counting law — and that provably has zeros off the critical line. It lacks only one thing: an Euler product. That single fact is the sharpest filter in this subject. If a proposed criterion for the critical line can be stated without ever mentioning the primes, then it is true of that function verbatim, and so it cannot forbid an off-line zero. No amount of numerical agreement rescues it; the counterexample already exists and is named.

For seven papers that filter was applied by reading. It is now a decision procedure: an ordinary walk over the syntax tree of the formula, checking every leaf against a declared alphabet. Run over 188 banked results from all nine papers plus the nine-point and arXiv papers: 162 never mention the primes, 26 do, and every one of the 26 was already on that side. Nothing was reclassified, and no new candidate appeared — the census recomputes the partition the papers argued for, and agrees with it. And that agreement is worth what such an agreement is worth: the audit reads the strings this programme typed, so it certifies that our transcriptions and our readings agree, not that either is right about the object.

The sharpest single row is our own proven one. All 24 statements of the nine-point paper are prime-free without exception: it is built from zero spacings and a Fourier kernel and mentions no prime anywhere. That is exactly what that side of the ledger can buy — bound a proportion, and forbid nothing, and no refinement moves that ceiling, because the reason is structural.

What it turned up, and where the line stops. The result is the agreement itself: a filter applied by hand across seven papers, on our own work, drifted on none of 188 statements — which is the failure mode it was most exposed to. One row did move, and it moved the right way: Paper 7's margin had been typed out as a bare infimum with the quantity being minimised left unwritten, the page printed that as an open exposure rather than resolving it in its own favour, and when Paper 7 wrote the objective out the row crossed over to the coefficient side — where Paper 7 had always placed it. The line is now closed. Of the four clauses of the standing test for a candidate observable, two mechanise and are what the paper delivers, one provably cannot be mechanised as a syntax walk, and the fourth is semantic; so no fourth screen will be built, and what is left is an entry rule at the moment a candidate is banked. Nothing here bears on the Riemann Hypothesis, and the outcome that would have counted as a find — a result mentioning the primes that we had not already scored that way — did not occur.

Nine 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