17 August 2026 · revised 4 September 2026, v1.5

Anthropic Riemann Hypothesis research feedback

We reimplemented Anthropic's compressed Weil form from scratch and reproduced their published tables to six significant figures; then ran the one control their own numerics could not run; then built the arithmetic half of their identity, for zeta and for Dirichlet L-functions; then went after the sentence that says how far the method can go — found its proof in their Lean development rather than in the paper — ran that ceiling's own programme at every period up to 128, where no value of it had ever been computed outside their repository; then solved the small-period programme exactly, every configuration enumerated, to certify our own instrument; then built their entire Lean development and passed all of it through the proof-checker's kernel; and then measured the depth-law rate across three conductors and three heights, where it comes out flat in height and split by conductor.

Ondřej Dvořák · Riemann zeta geometric decomposition programme. Every claim carries its evidence tier. Numbers we measured carry the gate that produced them.

Part I — the result, and our check of it

What was proved

An unreleased Claude research model proved unconditionally that at least two thirds of the non-trivial zeros of zeta in a dyadic window are simple and on the critical line — 0.67250 with the optimal test family, against a record of 5/12 = 0.41666 that had stood since 2020. It also gives at least 5/6 distinct (0.83625 optimised) and the same for a fixed primitive Dirichlet L-function. The work took two sessions in Claude Code, about 650 candidate ideas, roughly 60 subagents and 31 million output tokens; it was validated by two mathematicians and partly formalised in Lean.

The mechanism abandons Levinson's mollifier entirely. Weil's explicit formula gives a real symmetric form; compress it onto a family of windowed exponentials spaced at the mean zero spacing; a zero on the line contributes a positive square, a mirror pair off the line contributes a block of signature (1,1). The hypothesis was classically needed only to read the zero side as a positive sum, and here it is replaced by Sylvester's law of inertia plus a rank–trace inequality. The arithmetic input is Montgomery's 1973 pair correlation at band-width one, which was always unconditional.

anthropic.com/research/riemann-zeta · paper 35 pp · informal note 5 pp · appendix 95 pp · transcripts 116 pp · github.com/anthropics/zeta-23-lean · on Baluyot–Goldston–Suriajaya–Turnage-Butterbaugh, Goldston–Suriajaya, Bombieri, Yoshida.

We rebuilt it and it reproduces

We implemented the compressed form from the zero side — closed-form window transform, the taper ϱ(x) = x − sin(2πx)/2π, the lattice, the assembly — and ran it against real zeta zeros. Nothing was tuned to match. Their published table:

Six significant figures on their own cross-check, and all five ratio rows of their second table exact to the printed three decimals. Verified Two gates were built before any of it: the closed-form window transform against two independent quadratures (worst 9.1×10⁻¹³), and their Lemma 2.2 completeness identity (worst 1.2×10⁻¹¹), with a falsifier — detune the lattice by 1% and the identity degrades to 8.1×10⁻⁷, the aliasing signature. Along the way our own reference quadrature turned out to be the thing that was wrong, not the code; it was fixed, not loosened.

Part II — can the constant go higher?

The honest map

The stated ceiling: absent from the paper, derived in the Lean development

Their Remark 1.1 says no certificate "of this kind, reading only this bandwidth-one data and holding configuration by configuration, can certify a proportion of simple zeros exceeding 0.68185". It is the sentence on which every account of the method's limits rests.

The number appears exactly once in 251 pages, and so does the phrase "explicit extremal law"; searches over the paper, the informal note, the 95-page appendix and the 116 pages of transcripts turn up no derivation and no cross-reference, and the certificate class is never defined there. It is derived in the Lean repository instead, in Zeta23/PairCeiling/, and we reconstructed the problem before reading it. Both accounts are below, because the reconstruction is what makes the formal statement legible — and because it is the reader of the paper who currently has no way to know the remark has a formal derivation at all. Measured

So we reconstructed the problem the sentence must be about, and it has a clean shape. Normalise to mean density one and let M be the limiting second moment of the multiplicities. Integrality gives m² ≥ 2m − 1 with equality on m ∈ {1,2}, so a configuration's true simple-on-line proportion is exactly 2 − M, and a certificate holding configuration by configuration cannot certify more than 2 − Mmax. The ceiling is a question about how large the second moment can be made. In configuration terms Montgomery's pair correlation at band-width one is the single constraint ĝ(α) = M − |α| on |α| ≤ 1 for the hole function g, with g ≤ 1 because the pair correlation is a positive measure.

Where is it, then? Solving the configuration side directly — a semi-infinite linear programme in the hole function, whose value is a lower bound on Mmax by construction — brackets it at 0.6725 < ceiling ≤ 0.694 reading band-one pair correlation alone. Convergence is slow for a reason the argument above supplies: the extremal configuration is atomic, and a programme over densities reaches it only in the limit. Measured

One caveat, and it is theirs, stated in their own file. The chain runs: an exact-rational certificate file cert_N256_blk_b128m.json → interval arithmetic performed outside Lean → the hypothesis EnclOK that the true form factor lies in those enclosures → everything downstream kernel-checked. That certificate file is not in the repository — the README says it is available from the authors. Publishing it, or the interval-arithmetic driver, closes the one link that a reader currently cannot check, and it is the single most useful thing that could be added to the development.

Said as a reader's need rather than a complaint, and with what it would cost. The caveat is theirs, stated in their own file, not something we found hidden. What a published certificate buys is specific: a reader could recompute the law's form factor and discharge EnclOK without the authors — which is the difference between a chain that is kernel-checked and one that is checkable end to end. It is the practice this programme has adopted for its own computer-assisted results and the one we asked of the draft audited on the sibling page: certificates committed, every enclosure reconstructed per run, the trust base stated in the paper. Our own two certificates are published beside the results they carry, and the verifier reconstructs every box from them on each run rather than from an in-memory design. We are asking for exactly what we hold ourselves to, and nothing more.

And one arithmetic coincidence, which we report as an inference and not a finding: invert their own optimal-window formula at 0.68185 and you get band-width λ = 1.0139 — 1.4% above one, the single place a band-width-one ceiling cannot come from. Exploratory

A note on scope, since the two ceilings in the paper are easy to conflate. Section 7.5(a) and Proposition 7.4 cap the rank: n₊ ≤ d = λ₁N, giving 100% at λ = 1. Remark 1.1 caps the extremal law over configurations. They are different quantities and both are upper bounds; the second is the tighter, and it is the one nobody has written down.

Why that is not a theorem — and we now think it cannot be made one this way

Establishing the positivity is not the same as running the import, and the obstruction between them is sharper than we first thought. Written out, the certificate's second moment is ‖Â‖²F = Σρ,ρ′ Ψ(γ−γ′) with Ψ = (Φ₂/aL)², Φ₂ the transform of φ². So the weight it places on the form factor is the autocorrelation of a non-negative function, and an autocorrelation is non-negative everywhere, not merely inside the band. Measured for the paper's own C³ ramp: the weight never goes below −1.8×10⁻¹² anywhere, and beyond the band edge it is 1.8×10⁻¹². Measured

That is fatal to the manoeuvre in one line: to drop an out-of-band piece for free you need the weight there to be negative, so that non-negativity of the form factor pays for it. With a weight that is non-negative everywhere, out-of-band mass can only be paid for, never dropped. This is a property of the square, not of the taper, so no re-choice of window reaches it, including a window of non-compact support.

There is exactly one escape that is not obviously closed, and we priced it. Replace the Frobenius norm by a masked form ⟨Â, S∘Â⟩ with Skl = s(τk−τl): this is still computable from the prime side by the same diagonal method, and its weight is signed when ŝ is. It fails, and it fails on the case that makes the whole certificate sharp. A single simple on-line zero contributes a rank-one P = uuT with |u| = 1, and the masked form charges it (u∘u)TS(u∘u) instead of 1 — measured at 0.48 to 0.81 for a mild signed mask, and negative for a stronger one, at four different zero positions. The counterexample needs no taper at all: for P = [[1,1],[1,1]] the rank–trace lemma demands a second moment of at least 3, the Frobenius norm gives 4, and a mask with off-diagonal σ gives 2 + 2σ — which fails as soon as σ < ½, while any mask with a sign change in its transform has off-diagonals below 1 by construction. Measured

So the barrier is the conversion, not the arithmetic. A signed second-moment weight is perfectly admissible in Montgomery's classical deduction, which uses no spectrum — but that route buys "simple", not "simple on the line". The inertia step is what buys the line, and the inertia step is exactly the one that is spectral. Anyone attacking this should know that the obvious two constructions are closed for stated reasons, and that what is left is an inertia bound whose second-moment slot is not a spectral functional of the matrix at all. The prize is unchanged: +0.0067 on the line and +0.0034 distinct, unconditionally.

Past that, the cheapest purchase is smaller than they say

Reaching band-width 1.04 is where the next 2.7 points live. The paper attributes the barrier to "the Hardy–Littlewood conjectures, or equivalently Montgomery's pair correlation conjecture for α > 1". Reading section 6 against section 5, what the certificate actually consumes is a one-sided upper bound, not an asymptotic — every step that uses the second moment uses it as an inequality — on one specific signed off-diagonal prime sum, with shifts restricted to |h| ≤ T^0.04, and an average over those shifts with any power saving would do. That is materially weaker than the conjectures the paper names, and the paper never draws the bound-versus-asymptotic distinction for the second moment. Exploratory We could not find a route to it from a sieve-type upper bound either, because the sum is signed and needs cancellation in the shift variable, and we record that as an open point rather than a claim.

Part III — eight things the off-line control lets us measure

Part IV — routes tested and closed

Four natural ways past the constant, and what kills each

Each of these is the obvious next move from where the paper leaves off. Three are closed, and the fourth is closed as posed and reopened as a different problem — the degeneracy claim it rested on is false, and what is left is a question about fourth-moment arithmetic. A closed route is worth as much to whoever comes next as an open one, so each is stated with the thing that killed it.

• Re-solve the convex programme for a better window. Closed (And closing this route the way it says — “closing a duality gap takes a new inequality” — is what has since happened twice, in the two new rows of the honest map above. Neither moved the window; both moved the inequality inside the certificate. This programme's own Lemma B1, tr Ψ(G) ≥ 2√S − 1 sharp on {G ≽ 0, diag ≤ 1}, is the new inequality in its case.) It is already solved in closed form — the Montgomery–Taylor cosine is the maximiser, we confirmed the optimum independently at 1.32749929632059, and the yield of any further search is exactly zero. The residual above the published constant is a duality gap, not an unclaimed feasible point: the theorem fixes the window and then faces the worst configuration, while the ceiling describes choosing the certificate after the configuration. Closing a duality gap takes a new inequality.
• Separate a shallow off-line pair from an on-line double by moments. Open, and cheaper than it looks The two are spectrally identical in the zero-depth limit, which reads as a hard information barrier — it is not one. The extremal configuration's fourth spectral moment is 10/3 against the sine kernel's 13/4: they separate by exactly 1/12, at no band-width increase whatever, and the paper itself prices that difference at 13/18 = 0.7222. The barrier is that the fourth moment needs arithmetic nobody has, not that the two objects cannot be told apart. That is a materially different problem from the one the degeneracy suggests.
• Go through the derivative. Closed The completed function's derivative has its zeros symmetric about the line, a large unconditional proportion of them is known to be on it, and Speiser relates left-of-line derivative zeros to off-line zeros of the function — chain those and 87% of ζ's zeros would appear to be on the line. The chain is false, and the reductio is immediate: the unconditional proportion known for the derivative is far above the record for ζ itself, which is 41.66%. (Earlier versions dated that derivative figure “in print since 2015”. We have no citation on file for the year and the reductio does not need one: any derivative proportion above 41.66% breaks the chain, and every published one is. The year is withdrawn rather than left standing uncited.) A shallow off-line mirror pair puts its derivative zero exactly on the axis, so the inequality runs the wrong way for any transfer. Verified symbolically on a quartic with half its zeros off the axis and every one of its derivative's on it.
• Bootstrap the gap between the two form factors. Closed, and productively The ordinate-only form factor is manifestly non-negative while the weighted one carries the indefiniteness the proof runs on, so the discrepancy between them — a functional of exactly the off-line zeros being counted — looks bootstrappable against classical zero-density estimates, which the paper states it never uses. There is no discrepancy in sign. Both are non-negative, and that is the lemma in Part II.

Two small things in their paper, neither load-bearing

The appendix's expansion of the pair-block eigenvalue near zero depth is off by a factor of two against its own preceding line; the correct order is δ²L²/6, confirmed numerically to four digits. And F(λ) in equation (1.3) collides with Montgomery's form factor F(α) inside section 5 and Remark 5.10 — same symbol, two functions, adjacent lines. Measured Neither touches a theorem.

Part V — the Lean development

Built, checked, and the one link a reader still cannot check

We have now built it. The development — 329 files, 103,067 lines, Apache-2.0 — was compiled from a clean install of the pinned toolchain, Lean v4.33.0-rc2 against its pinned Mathlib, and all 9,010 compilation targets build and pass the proof-checker's kernel, with zero unproven gaps in the checked theorems. The audit below was done first, on the sources; the build turns its conclusions from a careful reading into a machine-checked fact. (The build itself needs nothing exotic: the toolchain, the Mathlib build cache, and patience with a 16 GB machine, which cannot compile the wide tail of the dependency graph and run anything else at the same time.)

That last row is the whole of it, and it is flagged in their own file rather than hidden. The definitions being in the harder direction is worth dwelling on: a formalisation can be sorry-free and still state something weaker than the paper, and this one does the opposite — it makes the denominator larger and the numerator smaller than a lazy reading would.

What remains genuinely unknown

• Whether the ceiling's descent over periods stops at 256. We ran their programme at every period up to 128: the value is strongly period-dependent, their 256 law is the lowest anyone has exhibited, and neither their files nor our sweep settles the limit over periods — so 0.681829 is an upper bound on the method's true ceiling, not yet its value. The certified two-sided computation now exists at periods 4 and 8 — solved exactly, every configuration priced — and it validates the sweep instrument where it can be checked; period 16 is beyond any exhaustive check, and the limit over periods remains computed by nobody.
• Whether an inertia inequality accepting a signed second-moment weight exists at all. The two natural constructions are now closed with reasons; what survives is an inertia bound whose second-moment slot is not a spectral functional of the matrix.
• Whether the one-sided, shift-averaged prime-pair bound at band-width 1.04 is reachable.
• What sets the height drift of the lock law's one constant. The quadrature form in item Four collapses every spacing on every window, but its constant falls 0.2605 → 0.1858 as the window grows ninefold; the derivation says it should not move at all. Five windows now exclude every clean shape — constant, 1/L, 1/log T — and what the structure is remains open.
• What sets the depth-law rate's scale. The height runs are now done at all three conductors: every rate is flat while 2L grows, so the deficit is not the approach to an asymptotic regime — and the flat lines differ by conductor, near 12.5 at conductors 5 and 7 against near 13.9 at 13, without ordering by conductor size. The remark's scaling fails at reachable heights; what sets each conductor's constant is the open question.

The feedback in short

If only one page of this is read, this is it — ordered by what we think is worth most to whoever works on the paper next.

1. The result reproduces. Built again from scratch, from the zero side, with nothing tuned to match: their table (6), their table (3) exact to the printed three decimals, and their own trace-level cross-check returning 1322.28 and 1436.41 to six significant figures. Two gates were built before any of it was read, and the closed-form transform was checked against quadrature to 9×10⁻¹³.
2. The control their own numerics could not run passes. Their section 8 tests the off-line mechanism on fabricated pairs because zeta has none. On genuine ones — Davenport–Heilbronn in their own window, 1080 on-line zeros and 58 real mirror pairs, completeness independently gated — the negative index is 32 against 58 pairs and the certificate stays below the truth at every band-width. Their signature-(1,1) prediction holds on non-synthetic data.
3. The arithmetic half is built, and Theorem E now has numbers. The prime-side expression — the step that makes the method arithmetic, and the one their internal calibration trusted least — reproduces the zero side to 5.9×10⁻¹⁰ for zeta and to 1.2–1.5×10⁻¹¹ for the real characters of conductor 3, 4 and 5, with traces agreeing to ten figures. That residual is ordinate precision, not truncation: fifty times more zeros change it by nothing. And the character table, which does not exist anywhere in the paper, shows that conductor buys exactly what height buys — conductor 5 at height 600 is zeta at height 3000, and the certificate is 7.6 per cent better at the same height for that reason alone.
4. Proposition 4.1 now holds continuously, not at a point. Along a family that keeps one functional equation while its off-line pair count runs 0 to 88 and back to 0, across 39 readings, the certificate never exceeds the truth and the index never exceeds the pair count.
5. What the certificate reads is pair correlation, not the on-line proportion. Two configurations in that family have every zero on the critical line, and they certify +278 and +558 — a factor of two apart; at the lowest band-width the first certifies nothing at all, while a configuration with 86 genuine off-line pairs certifies +79. The cause is measured: superposing two independent zero families destroys their mutual repulsion, and that cross term alone is 28 per cent of the second moment. The gap between certificate and truth is a functional of the configuration, not a fixed price of reading two moments.
6. That functional has a parameter-free closed form, accurate to 0.1–0.6 per cent — and it fails at exactly one place, the ladder spacing 2π/log 5. Scanning spacings shows the failure recurs at 2π/log n for n = 2, 3, 4, 5, 7, 8, 9, 11, 13 and at none of the off-tune spacings in between. A ladder at those spacings is phase-locked to zeta's own arithmetic and is not an independent family. In the configuration above, that lock costs 69.6 of certificate: +278.4 measured where independent families of the same densities give +348.0. The slack stops being a range and becomes a formula plus a named residue — and the residue's own law is now complete: the lock's strength is the prime's weight Λ(n^m)/n^m/2 summed over the in-band harmonics in quadrature, which collapses 27 readings across three window lengths to one constant per window at under 2 per cent, sparse ladders included.
7. The whole cost of being off the line lands on the second moment — the trace is blind to it, and blind to the configuration entirely. But there is no cost per pair: over 692 real pairs it runs from 0.011 to 6.9 with depth. And the saturation we found needs pairs to be adjacent; on real configurations the cost is additive to within 0.6 per cent.
8. The ceiling of Remark 1.1 has a formal derivation, and it is in the Lean development, not in the paper — kernel-checked conditional on EnclOK. The number occurs once in 251 pages with no derivation there; the derivation is an explicit 256-periodic law in Zeta23/PairCeiling/ with simple-point fraction p₀ = 0.681828687463831… . Its positions and weights are not in the Lean source — only the enclosures of its form factor are, produced outside Lean from an unpublished certificate file — so the ceiling is not reproduced here and is not refuted here. Reconstructing the problem independently first: its dual is the paper's own optimisation (we get 1.327499296320 against their 1/c*₁ = 1.32749929632059), that dual bound is attained by nothing — the optimal test function's zeros sit at k + 0.0612/k and no two ever sum to a third — and the extremal configuration must therefore be atomic. Their law is atomic, lands inside our bracket, and its data recomputes from their own table: the 255 interior enclosures are j/256 to one ulp, the row certificate to 1.8×10⁻⁴⁰, the edge constant to 0.823953161. The duality gap is exactly 0.009327984, pinned at both ends. And running their programme ourselves at every period up to 128: the ceiling moves by 0.066 with the period, their 256 value is the lowest exhibited anywhere, and the limit over periods is computed by nobody — their number is an upper bound on the method's true ceiling. The sweep behind that statement is itself certified now: the same programme solved exactly on grids at periods 4 and 8, every configuration enumerated and priced — 629 million at the finest — and the certified optima descend onto the sweep's values without ever undercutting them. The one thing missing is that a reader of the paper has no way to know any of this, and the certificate file the chain rests on is not published.
9. The positivity the out-of-band route needs is unconditional. The beta-weighted form factor the certificate reads is non-negative without RH; elementary, and verified at a thousand evaluations on configurations with off-line depth up to 0.49. That is exactly the input Chirre–Gonçalves–de Laat require, and it is worth +0.0067 on the line and +0.0034 distinct — unconditionally — if it can be imported.
10. But that import is blocked by an identity, not by a gap. The certificate's second moment weights the form factor by the autocorrelation of a non-negative function, which is non-negative everywhere — measured to 1.8×10⁻¹² — so out-of-band mass can only be paid for, never dropped, and that is exactly what the out-of-band majorant needs. It is a property of the square, not of the taper, so no window reaches it. The one escape — masking the second moment so its weight can change sign — under-charges a simple on-line zero and so breaks the rank-one case that makes Proposition 4.4 sharp; the counterexample is two-by-two and needs no taper. What is left is an inertia bound whose second-moment slot is not a spectral functional at all. The barrier is the conversion from "simple" to "simple on the line", not the arithmetic.
11. The shortfall does not close with height, and the other conductors behave. Run zeta from T = 600 to 5000 and the certificate rises from 41 to 55 per cent of the truth — but it is converging on two thirds, not on one, so the residual third is Theorem A's own gap. Davenport–Heilbronn functions built at conductors 7 and 13, each censused to an exact argument-principle balance, reproduce the additivity result at up to one part in fifteen hundred. And the single-pair depth law, refit like for like and then censused at height at all three conductors — every window balancing exactly against the argument principle — is flat in height at every conductor while 2L grows by 2.5: the deficit widens to 14–24 per cent of the prediction, at up to fourteen standard errors, so the remark's scaling itself fails at reachable heights. And with the height windows' power the conductors separate for the first time: the rate is a per-conductor constant, near 12.5 at conductors 5 and 7 and near 13.9 at 13, three standard errors apart in every reading — not a universal number, not 2L, and not ordered by the conductor.
12. The Lean development builds, and the kernel passes all of it. 329 files, 103,067 lines, compiled here from the pinned toolchain: all 9,010 targets, zero unproven gaps in the checked theorems. The audit that preceded the build stands as read: no new axioms, no sorry outside the deliberate trusted-statement files, statements unconditional and correctly quantified, and the definitions pointed the harder way — denominator with multiplicity, numerator distinct. Its optimal-window constant agrees with our independent computation to 6×10⁻¹³. One link still sits outside the kernel and they say so: an unpublished certificate file and interval arithmetic done outside Lean. Publishing that file is the single most useful addition.
13. Four natural routes past the constant are closed — a better window (the convex problem is solved in closed form and yields exactly zero further), the derivative transfer (false, with a quartic counterexample), the weighted-against-unweighted bootstrap (no discrepancy to exploit), and the claim that no moment separates a shallow pair from a double (false: the fourth moments differ by exactly 1/12). Each is stated with its reason, because a closed route is worth as much to whoever comes next as an open one.