25 rounds of adversarial review. $1.30. Two days.
In August 2026, an AI model at Anthropic published More Than Two Thirds of the Zeros of the Riemann Zeta Function Are Simple and on the Critical Line — the most significant unconditional advance on the distribution of zeta zeros in decades, accompanied by a kernel-checked Lean 4 formalization. It is, at this writing, the most-read mathematics paper in the world.
We ran it through MathLab.
How to read this page — adversarial by design. MathLab reviews mathematics with three agents in tension: a Critic whose job is to attack every proof as hard as possible, a Verifier whose job is to check every attack by direct computation, and a Synthesizer who scores what survives. Critic flags are hypotheses, not findings. The findings are what the Verifier's computations confirm. The paper's kernel-checked theorems survived all 25 rounds. Isolated Critic flags quoted out of this context misrepresent both the paper and the tool.
Full dossier: complete 25-round dossier · Paper under review: arXiv:2608.13637 — “More Than Two Thirds of the Zeros of the Riemann Zeta Function Are Simple and on the Critical Line” (Anthropic, Aug 2026; communicated by L. Alpöge and R. Furman) · Its Lean repository: github.com/anthropics/formal-math
Target: the paper's Poisson–Gabor identity (Lemma 2.1), a foundation of the whole argument.
The Critic opened with maximum aggression: “CRITICAL ERROR. The claim that Υ̂ has compact support is mathematically false.” If true, this would collapse the paper's sampling construction.
The Verifier then did what a referee should: it recomputed. The proof's support claim concerns the convolution H = φz ∗ φz′ — not its Fourier transform — and the Verifier established, by direct calculation of the support of the convolution and the vanishing of φ at the endpoints ±L/2, that supp H ⊂ (−L, L) strictly, so every aliasing term m ≠ 0 vanishes exactly:
“CONFIRMED that only m = 0 survives… The Critic's objection about compact support of Υ̂ was a misreading — the proof correctly uses that H (not Ĥ) has compact support inside (−L, L).”
The Critic's “circularity” charge on the truncation bound was likewise refuted by exhibiting the actual order of the argument. Synthesis score: 72/100, with the deductions assigned to presentation (an ambiguous pronoun in the proof), not mathematics.
Why this matters: a review system that never raises alarms is useless; a review system that raises alarms and cannot retract them is dangerous. This round shows the full loop — aggressive hypothesis, computational adjudication, honest scoring — in one page.
Target: Remark 4.4, the paper's explanation of why C² smoothing of the test function is necessary.
Across 25 rounds, this is the only place the pipeline found an actual numerical slip. The remark states that with a sharp cutoff the tail bound becomes ⋙ X1/2 L log(T/D₀). The Verifier's recomputation of the sharp-cutoff decay gives ‖vρ‖² ≲ X1/2 L D−1 rather than D−3, so the normalized quantity is X1/2 log(T/D₀) — one factor of L smaller than stated.
Two things about this finding, stated with equal emphasis:
The slip sits in a prose remark — precisely the region the paper's Lean formalization does not cover. Which is the deepest lesson of the whole audit: every weakness found in 25 rounds lives in unverified prose; nothing kernel-checked moved. The audit is an independent, adversarial confirmation of the formal-verification trust model, from the outside.
We communicated this finding to the paper's authors directly before publishing this page.
Target: Remark 7.1, the paper's constants for zeros of ξ′ (0.85838 simple-on-line, 0.92919 distinct).
The Critic scored this remark lowest of any target (32/100): the transfer of the whole method to ξ′ is asserted in one sentence, the four constants appear underived, and the supporting Lean formalization is cited but not exhibited in the paper. Three “critical gaps.”
Here is what happened next — and this stage is human plus source, not model:
xiDeriv_simple_on_line, xiDeriv_simple_on_line_quartic) exist exactly as named, in a file containing zero sorrys, inside a ~100-file formalization tree covering the explicit formula, zero-counting, transfer machinery, and window certificates for ξ′ — everything the Critic said was missing from the paper is present in the repository.Verdict: Round 25's “critical gaps” are disclosure compression in a one-paragraph remark, not holes in the mathematics. The audit correctly identified where the paper under-discloses; the verification stage correctly prevented us from publishing complaints the repository already answers.
This is the exhibit no benchmark can fake: the pipeline's trustworthiness comes from the fact that it does not trust itself.
| MathLab | Traditional journal review | |
|---|---|---|
| Rounds of review | 25 | 2–3 referees, 1 pass each |
| Elapsed time | ~2 days | 6–18 months |
| Cost | $1.30 | Unpaid referee labor |
| Load-bearing errors found | 0 (correctly — the paper is sound) | — |
| Genuine errata surfaced | 1 (verified by hand) | — |
| False alarms raised and self-retracted | multiple, with computations shown | invisible |
Yes. Before publishing our accompanying research note (DOI: 10.5281/zenodo.22150483), we ran it through the same pipeline and dispositioned every finding. The dossier for that run is linked from the note's page. We do not ask the field to trust an instrument we exempt ourselves from.
MathLab is a product of Quantiterate LLC. AI-assisted adversarial review; all published findings verified by hand. Not peer review; a complement to it. Unsolicited audit requests: see our research policy.
Self-audit section, final facts: the note received two full adversarial passes (rounds 26–30, then a 5-round verify pass). All mathematics confirmed both passes; score movement (CF1 72→88, units 78→82) reflects applied fixes. One theorem statement (Thm 3) had its scheme-class hypotheses made explicit after the second pass flagged a fiber-degeneration loophole — the audit finding its own author's gap is the product working as designed. Final gate: an independent lock review by a third model family — concordant, approved. Three model families, one verdict. Quote the correctness axis only; exposition scores are labeled as such.