A new ceiling for Λ: the de Bruijn–Newman constant is at most 0.1787854

The primes 2, 3, 5, 7, 11, 13, … are the atoms of arithmetic: every whole number factors into primes in exactly one way, so facts about primes become facts about all numbers. Individually they are irregular — no known rule produces the next prime from the ones before it. Counted in bulk, they obey a law: the number of primes up to x stays close to a single smooth curve (the prime number theorem, proved in 1896). The open question is the size of the error — how far the true count can stray from the curve. That error term is what the Riemann hypothesis governs, and it is why RH matters: sharpen the error term and you sharpen hundreds of results in number theory that depend on it.

In 1859 Bernhard Riemann explained where that hidden order comes from. He took Euler's identity, which connects the primes to a single function of one complex variable,

extended it to the whole complex plane, and discovered that the wobble of the prime count around its smooth curve is governed — exactly, via an explicit formula — by the locations of the zeros of this function. Each zero contributes one wave to the error; the zero's height sets the wave's frequency and, crucially, its horizontal position sets the wave's amplitude. Riemann observed that every zero he could examine sat on one vertical line, Re s = ½, now called the critical line — the position giving the smallest possible amplitude — and remarked it was “very probable” all of them do. That remark is the Riemann hypothesis. Its concrete content: the prime-count error up to x never exceeds roughly √x, the same size as the wobble of a fair coin flipped x times. The primes are allowed to look random; RH says they are never allowed to drift with a bias.

The wave description is an actual formula, and you can run it below. The slate staircase counts prime powers (a cousin of the staircase above, weighted so the mathematics is exact), and the vermillion curve is Riemann's formula built from the smooth trend plus one wave per zeta zero. Drag the slider and watch thirty zeros carve the primes:

It has now been open for 167 years. It is part of the eighth of Hilbert's problems (1900) and is one of the Clay Millennium Prize problems today; hundreds of theorems across number theory and beyond are proved conditionally, “assuming RH.” Its zeros have been checked by computer into the trillions — every one on the line — but a check is not a proof, and no proof is in sight.

Λ (defined properly in the next chapter) repackages the Riemann hypothesis as a statement about one real number: RH holds if and only if Λ ≤ 0 (a proved equivalence, established in Chapter 1). That reformulation has three consequences. First, progress becomes measurable: a yes/no conjecture has no partial credit, but an upper bound on Λ can shrink: ½ → 0.22 → 0.2 → and now 0.1787854. Second, since 2018 we know Λ ≥ 0, so Λ is confined to the interval from 0 to the current ceiling, and RH is the statement that Λ sits at the left endpoint; every improvement to the ceiling is measured distance toward the answer. Third, bounds on Λ are unconditional — nothing in them assumes RH itself. Lowering Λ is one of the few rigorous, quantifiable ways to make progress on the Riemann hypothesis.

Start with Riemann's xi function , a repackaging of the zeta function: $\xi(s)=\tfrac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$. The Riemann hypothesis says all its zeros lie on the critical line $\mathrm{Re}\,s=\tfrac12$. Rotate coordinates so that line becomes the real axis (this proof uses the Polymath 15 normalization $H_0(z)=\tfrac18\,\xi\!\left(\tfrac12+\tfrac{iz}{2}\right)$) and RH becomes a single sentence:

In 1950 de Bruijn had the idea of deforming this function with a one-parameter flow — mathematically, running the heat equation on it:

where $\Phi$ is the fixed super-exponentially decaying kernel with $H_0=\tfrac18\xi$. Positive t smooths the function and, as we'll see, herds its zeros toward the real axis; negative t roughens it and pushes zeros off. De Bruijn proved that once all zeros are real they stay real at every later time. So there is a single threshold, made precise by Newman in 1976:

That threshold is the de Bruijn–Newman constant, and it converts the Riemann hypothesis from a statement about infinitely many zeros into a statement about one real number:

One direction is now settled. Newman conjectured Λ ≥ 0, famously adding that if RH is true, it is “only barely so” — and Rodgers and Tao proved this in 2018. So Λ is confined: $0\le\Lambda$, and progress can now come only from the upper side. This proof moves the ceiling to $\Lambda\le 0.1787854$. With the floor at 0, this removes just over 10% of the interval that remained.

De Bruijn proved $\Lambda\le\tfrac12$ in 1950. Fifty-nine years later, Ki, Kim and Lee improved it to the strict inequality $\Lambda<\tfrac12$. The current method dates from 2018–2019, when Terence Tao's Polymath 15 project built an effective theory of the heat flow: explicit, computable approximations to $H_t$ with rigorous error bounds, and a criterion (their Theorem 1.2) that converts three finite, checkable statements into an upper bound on Λ. Polymath 15 instantiated it to get Λ ≤ 0.22; plugging in Platt–Trudgian's 2020 record-height RH verification pushed the same machinery to 0.2.

The squeeze on Λ has two independent sides, and they work by completely different mechanisms. Both live in the upper half-plane: since $H_t$ takes real values on the real axis, every off-axis zero at $x+iy$ has a mirror twin at $x-iy$, so it is enough to track the region above the axis — position x running along the axis, height y measuring how far a zero still has to fall. RH is the statement that this upper region is empty.

1. The floor: why Λ ≥ 0 (Rodgers–Tao, 2018)

   Suppose Λ were negative. Then by time 0 the zeros would already have spent a positive stretch of time evolving as a gas of real particles, and that gas has a strong smoothing property: the flow drives neighboring gaps toward equal spacing. Rodgers and Tao showed the actual spacings of zeta zeros fluctuate more than any such relaxed configuration allows — the statistics, read off the explicit formula, are incompatible with having flowed for positive time. Contradiction; so Λ ≥ 0. The argument locates no zero off the line; it shows only that the observed spacings are incompatible with earlier relaxation.

2. The ceiling machine: Tao and Polymath 15's barrier method (2019)

   The upper bound must rule out zeros in the canopy — Polymath's name for the region at height $y_0$ and above at the final time $t_0$, stretching along the entire infinite axis. A theorem of de Bruijn says an empty canopy forces $\Lambda\le t_0+y_0^2/2$. Their method empties it in three zones. Left of a chosen abscissa X: the computer-verified stretch of RH keeps the region clean at t = 0, and a continuity analysis of each zero's velocity keeps it clean afterward — provided nothing crosses in from the right. Blocking that crossing is the barrier construction: a rectangle one unit wide at X, meshed along its boundary, where the effective approximation is evaluated at a ladder of time slices; certified bounds on how fast the values can move between mesh points and between slices, plus the argument principle (winding number 0), keep it zero-free at every intermediate time. Right of the barrier, the canopy is checked at the final time only, using the approximation $H_t\approx B_t f_t$ and Euler-product mollifiers. Polymath 15 ran this at $t_0=0.2$, $y_0=0.2$, $X=6\times10^{10}+83952-0.5$, with the barrier parked at the limit of Platt's 2011 verification, the fractional shift chosen to make the small-prime Euler product large at the barrier — and obtained Λ ≤ 0.22. Their bottleneck was the final-time check (ii).

3. Platt–Trudgian's lever (2020–21)

   The criterion consumes exactly one external computation, the verified height: the barrier must stand below it, and a barrier further out allows better $(t_0, y_0)$. Platt and Trudgian raised the rigorously verified height a hundredfold, to exactly 3,000,175,332,800, itself an interval-arithmetic computation. Feeding it to the same criterion gives Λ ≤ 0.2, noted in their own paper.

4. What I changed (2026)

   Same criterion, same verified height; the differences are in the parameters and in how the result is certified. I parked the barrier 175,239,886.5 below the exact Platt–Trudgian height (0.006% from the edge), re-optimized the row in exact rationals to the criterion's boundary, and built the heavier certification the thinner row demands: mollifiers up to the prime 11 where the floors are thinnest, the window-freeze, native-binding and Dini-transfer lemmas, an all-N tail theorem in place of sampling, and a 883-prism barrier certificate that includes t = 0 exactly. That takes 0.2 to 0.1787854, about $2\times10^{-7}$ below the most this criterion can give at this height. The rest of this post is the detail.

In numbers, my row is $X=6{,}000{,}000{,}185{,}827$, $t_0=\tfrac{129}{800}$, and $y_0^2=\tfrac{87677}{2500000}$, supported by roughly 3.15 million interval-arithmetic certificates and an audit trail built so a reviewer can rebuild every piece from source. The next chapter explains what the flow does to zeros; then the three checks, one at a time.

Under the flow, the zeros of $H_t$ move like interacting particles: real zeros repel each other along the axis, and each complex-conjugate pair gets pulled toward the axis. For an isolated pair at height y (that is, at $z=x\pm iy$), the dynamics are governed, up to smaller corrections, by

so the squared height falls at a constant rate 2, and a pair starting at height y touches down at time about $y^2/2$. The final bound has the same form: it will be $t_0+y_0^2/2$: “run the flow to time $t_0$, show nothing flies higher than $y_0$, then wait $y_0^2/2$ longer for the stragglers to land.” Polymath 15's Theorem 1.2 is the rigorous, effective version of that sentence.

Two structural facts make the schedule provable. First, zeros move continuously in t: a zero cannot appear at a point without traveling there. Second, a zero's vertical speed is bounded by the dynamics, so a region that is empty stays empty unless a zero crosses its boundary. Together these reduce the problem to fencing a region and checking the fence — which is what the criterion asks.

Pick three numbers: a barrier location $X$, a final time $t_0$, and a landing height $y_0$. Polymath 15's Theorem 1.2 says: if the three hypotheses below hold, then $\Lambda\le t_0+y_0^2/2$. (A theorem's hypotheses are its if-part: supply them, and the conclusion is yours. They are the three “checks” of the chapter titles ahead.) The colors below match the map and the chapter headings that follow: slate for (i), brass for (ii), vermillion for (iii). I chose the exact row

chosen so that $t_0+\tfrac{y_0^2}{2}=\tfrac{893927}{5000000}=0.1787854$ exactly — every parameter is a rational number, and the final arithmetic is exact, not floating-point.

Why these three numbers and not others? Try it yourself — the bound, the criterion's validity condition, and the feasibility of actually certifying a row all move as you drag:

The three hypotheses are supplied by three completely different technologies — a cited world-record computation, a 3.15-million-row certified scan with a tail theorem, and an argument-principle certificate. The next four chapters take them in order.

The first hypothesis is the one part I didn't prove myself but licensed from the literature: Platt and Trudgian's record verification (2020) that every nontrivial zeta zero up to height

lies exactly on the critical line. The criterion at our row consumes height $X/2=3{,}000{,}000{,}092{,}913.5$, which sits inside the verified range with an exact margin of $175{,}239{,}886.5$. The zero-height endpoint $T=0$ is closed classically: on the real segment $0<\sigma<1$ a two-line alternating-series argument shows $\zeta(\sigma)<0$ (strictly negative, so no zero hides at the bottom), and at $s=1$ zeta has a pole, not a zero. Nothing conjectural enters: RH is used only as a finite, machine-verified fact up to a specific height, plus the functional equation to map $H_0$-zeros to zeta zeros by the exact change of variables $x=2T$.

Hypothesis (ii) demands that at the final time $t_0$, the function $H_{t_0}$ has no zeros to the right of the barrier in the band $y_0\le y\le\sqrt{1-2t_0}$, an infinite region. Polymath 15's Theorem 1.3 makes this tractable: it supplies an explicit approximation

where $B_t$ is an explicit nonvanishing factor, together with hard error bounds whose certified total we call $E_{\max}$. So it suffices to prove $|f_{t_0}|>E_{\max}$ everywhere: if the computable sum always beats the worst possible approximation error, the true function cannot vanish — a strict inequality between computable quantities. The region is cut into natural “Riemann–Siegel windows” $W_N=[x_N,x_{N+1})$ with $x_N=4\pi(N^2-t_0/16)$, indexed by the length N of the finite sum — the natural coordinate system of the Riemann–Siegel approximation. The scan runs from the window containing the barrier, $N=690{,}988$, to the tail cutoff $N=3{,}840{,}000$: one certified row per window, 3,149,013 rows — each a strict interval-arithmetic inequality — with no gaps, duplicates, or uncertain records.

Most of this proof consists of certificates, so the word needs a precise meaning. A certificate is not an approximate computation. It is a guaranteed enclosure: an interval the true value provably cannot escape, which narrows with more work. A claim is proved only when the whole interval lands on the required side. Try it yourself:

Three ideas make the scan rigorous rather than merely numerical:

1. Freeze conservatively, prove monotonicity

   Within each window, the certificate evaluates the bound at the window's left edge and proves — by elementary monotonicity in x, including through the kink of a positive-part term — that this frozen value is conservative across the whole window (the Window Freeze theorem). Exact rational Machin brackets for π and exact square-root brackets pin the very first window: $x_{690988}<X+\sqrt{1-y_0^2}<x_{690989}$ with millions to spare on both sides.

2. Mollify with small primes

   Raw $|f_t|$ can dip close to zero. Multiplying by a short Euler product $E$ (auxiliary primes 2, 3, 5, 7, 11 near the barrier, tapering to 2, 3 far out) flattens the dips — a damping factor of this kind is called a mollifier, from the Latin for “to soften”. The native binding lemma proves by exact Dirichlet convolution that the stored floors bound the paper's normalized $|f_t|$ directly — the certified numbers are in the right units, with no extra conversion factor left to trust.

3. Certify one height, transfer to all heights

   Each row certifies a floor at one height; a separate Dini-derivative theorem proves the relevant mass is monotone in y — with the scan's thinnest certified ratio, $0.99999860767275095<1$, so the floor transfers to the entire band $y_0\le y\le\sqrt{1-2t_0}$. The margin is thin, but it is a proven strict inequality, checked at two precisions that agree to 17 digits.

The payoff: the binding floor across all 3,149,013 rows is $|f_{t_0}|\ge 7.91366\times10^{-7}$, against a certified total error $E_{\max}\le 2.33495\times10^{-7}$: nonvanishing with a 3.4× cushion.

Past the cutoff $N_*=3{,}840{,}000$ no finite list of certificates can help — there are infinitely many windows left. This infinite remainder is called the tail, and the tail lemma handles all of it at once. Its engine is a contraction: the mollified sum is written as its leading term 1 plus a correction mass $D$, and every ingredient of D is bounded by quantities that decrease as N grows. A capping lemma — each ingredient is bounded by its value at the cutoff, which is its worst case — plus exact finite Dirichlet convolution reduce all $N\ge N_*$ to one interval computation at the cutoff itself:

after subtracting all error terms — uniformly for every window past the cutoff, every height in the band (in fact an extended band), and a full closed box of times around $t_0$ — with no sampling in N and no assumed monotonicity in t. The computation is a single standalone FLINT/Arb program, run at 256-bit and 512-bit precision (agreeing), with a separate Python interval implementation as corroboration.

The barrier is where the proof must control every intermediate time, not just the endpoints: no zero of $H_t$ may enter the closed box

which strictly contains the curved wall Theorem 1.2 actually requires (the floor clearance is exact: $y_0^2-0.1809^2=\tfrac{234599}{10^8}>0$). The tool is the argument principle, the classical way to count zeros inside a region: walk the boundary of R, watch the function's value wind around the origin; winding number zero means no zeros inside. The certificate makes this rigorous under the flow by slicing time into 883 consecutive closed prisms and, on each prism, gating a strict interval inequality:

In words: the boundary values stay so far from zero that neither the gaps between mesh points, nor the motion of the function during the prism's time slice, nor the approximation error of using $f_t$ for $H_t/B_t$, could possibly let a zero slip through. The certified total approximation error, $3.565\times10^{-4}$, fits the 0.00125 allowance 3.5 times over; the worst prism margin is 0.5198.

The function's values along the rectangle's boundary trace a closed loop in the complex plane, and a zero inside the rectangle forces that loop to wrap around the origin. The certificate proves the loop stays far away:

The corners were the risky parts, so I gave each one its own lemma. The $t=0$ endpoint is included exactly — the first prism's seam is required to be literally zero, where the boundary modulus is 4.278, and the derivative majorants provably do not blow up as $t\to0$. The derivative box lemma proves the uniform bounds $D_z,D_t$ on whole prisms (never sampled at a single endpoint). The 7,688 stored Taylor-series coefficient components were independently regenerated — all contained in their stored balls — and the truncated Taylor tail is bounded by $1.95\times10^{-22}$, carried explicitly through every value. Every gate is fail-closed: an interval that straddles it counts as failure.

By design, nothing new happens in the final step. A fail-closed assembly checker confirms that hypothesis (i) is licensed by the cited Platt–Trudgian height (with its exact margin of 175,239,886.5), that the finite lane and the tail jointly cover hypothesis (ii) with an overlapped seam, that the 883-prism certificate covers hypothesis (iii) down to exactly $t=0$ and up to an endpoint enclosing $129/800$, and then performs the one line of mathematics left, in exact rational arithmetic:

None of these margins is a floating-point comparison. Every gate is a directed interval inequality: outward-rounded enclosures on both sides, strict comparison, overlap treated as failure. The three cited inputs that code cannot prove (the two Polymath theorems and Platt–Trudgian's verification) are printed by the checker as cited theorem inputs and left ungated, marking the boundary between what the machine certifies and what the literature supplies. The audit also confirmed there is no circularity: the three hypotheses are logically independent, and no certificate consumes a value derived from the bound it proves.

A proof with this much computation is only as good as its audit trail, so I engineered the trail as carefully as the mathematics. Four layers, in escalating order of independence:

1. Checksums and fail-closed checkers

   Every artifact — certificates, logs, sources, references — is pinned by SHA-256 (362 files) and re-attested by verify.sh. The parsers are fail-closed: 54 strict checks on the barrier alone; any gap, duplicate, overlap, or indeterminate interval anywhere is a hard failure.

2. Cross-implementation and cross-toolchain replays

   Every interval computation the bound rests on — the finite scan's error budget, the tail contraction, and the 883-prism barrier — is certified by FLINT/Arb ball arithmetic, with Python interval code as cross-checks. The finite error budget and the margin above it are certified by a standalone C program run at 256-bit and 512-bit precision inside the pinned container, gated as prerequisite P17 of the 40-gate fail-closed assembly (the original Python interval computation is retained as same-answer corroboration); the tail agrees across 256 and 512 bits plus a separate Python implementation; and the barrier replays end-to-end on two different OS/compiler/library stacks (Linux/FLINT 3.0.1 and macOS/FLINT 3.6.0), emitting identical verdicts on all 883 prisms.

3. An adversarial review, then a full recompute

   I then set a four-agent adversarial AI review panel loose on each proof leg separately (July 2026), with instructions to break it. Verdict: no fatal or bound-invalidating defect; it surfaced — and fixed — one real gap (the thinnest inequality, the Dini transfer, wasn't wired into the assembly gate) and flagged the precise-height citation subtlety of Chapter 5. Every heavy computation was then rebuilt fresh from source: all 7,688 coefficients, all 883 prisms, all 3,149,013 finite rows regenerated and matched line-for-line.

4. Independent human review: Dan Romik

   Dan Romik (author of the standard modern exposition of the ξ function's orthogonal-polynomial theory) reviewed my proof independently: he worked through the full package, verified the cited literature against the arXiv sources, re-proved every analytic lemma, and rewrote the mathematics as two journal-grade manuscripts — an expository account and a research paper — with the computer-assisted components isolated as seven standalone, self-contained verification programs.

This bound will not be improved by re-running the same machinery with more computing time: I chose the row at the edge of what this certificate lane admits — an early companion analysis placed the wall for these parameters about $2\times10^{-7}$ below 0.1787854. Later research in the repository showed that wall belongs to the lane, not to the verified height: the mollifier is tuned to $t_0$ (lowering the time while holding it fixed fails structurally, at any precision), and a lower-time lane at the same height reaches 0.1782354 — 5.5×10⁻⁴ lower, its arithmetic and archive machine-checked though its mathematics is not yet independently reviewed. Getting below that needs a mollifier retuned to the lower time, not a bigger dial.

In the long run the binding lever is verified height — and it pays inverse-logarithmically: the achievable bound shrinks roughly like 1/log(height). Concretely, halving the bound requires roughly squaring the verified height, from 3×10¹² to about 10²⁵ (far beyond any conceivable computation), and driving the bound all the way to 0 would require verifying RH to infinite height, which is just RH again. Together the two bounds say the following about $0\le\Lambda\le0.1787854$: Rodgers–Tao's floor says RH has no safety margin; the wall says brute force alone will never finish the job. Closing the remaining interval will take new mathematical ideas, most likely alongside heavier computation — each idea still has to be instantiated and certified. Working the bound downward is itself one way those ideas get found: every improvement forces a sharper structural understanding of how the zeros move under the flow, and of what the primes do and do not permit.