How Significant Was The LZ Event?

A dark matter detection experiment, Lux Zepplin (LZ) has seen one event that could be dark matter. 

How likely is it that it could just be an extreme outlier of the background?

A new preprint attempts to quantify that better than the original note reporting the detection. The paper doesn't reach a final conclusion which is highly model dependent, but points the way to determining if the LZ event could be just a fluke of ordinary physics rather than a singular dark matter detection, and points the way to how to clarify the matter.

This is mandatory reading for anyone proposing a beyond the Standard Model explanation of this result.

We reconstruct the continuum electron-recoil contribution to the LZ background expectation from the collaboration's released simulation configuration. The released model specifies a definite response tail, but the published calibration information supplies no bound on how far the physical tail may depart from it at the probabilities the expectation depends on. 

Distributions reproducing the executable model's mean, variance, and standardised skewness exactly, and its event-conditioned 10, 50, and 90% (N_e) quantiles to about one per cent, differ in the relevant tail probability by more than five orders of magnitude. 

The ladder is short: a tail agreeing with that central characterisation to 0.14% in the quantiles lifts the electron-recoil contribution out of the negligible range, and one agreeing to 0.47% makes it comparable to the entire background quoted for the comparison region. We go on to identify the measurements that would settle it.

Triparno Bandyopadhyay, Rajarajeswari Ramasamy, Nishitha TP, "The Electron-Recoil Response Tail and the LZ High-Energy Event Background" arXiv:2610.27105 (October 19, 2026).

Thoughts On The Riemann Hypothesis

AI has purported to prove two new and improved versions of the quasi-Riemann hypothesis, as discussed at Quanta magazine and a Twitter thread discussing additional proofs (by humans) that flow from the new quasi-Riemann hypothesis proof.

What is the Riemann Hypothesis?

The Riemann hypothesis can be explained as follows (from the first link):

The Riemann hypothesis is about which values of s will make the zeta function equal zero.

But it is also about far more than how this particular function behaves. As Bernhard Riemann realized when he first made the conjecture in 1859, the function relates to patterns in the prime numbers — integers divisible only by 1 and themselves that many mathematicians call the “atoms of arithmetic.”

It’s long been known that as you move along the number line, the primes thin out. Riemann showed that exactly how the primes thin out is, in a sense, governed by the zeta function — in particular by the places where it equals zero.

It’s straightforward to prove that if s is a negative even number (−2, −4, −6, …), then ζ(s) = 0. These are called trivial zeros. The Riemann hypothesis says that the only other zeros of the zeta function, called nontrivial zeros, occur when s has a real part of 1/2. That is, they should all be located on a vertical line:

If any nontrivial zero were to be found somewhere away from that vertical line, it would imply that the primes must form an extremely surprising pattern, one that has never been observed to date.

Just narrowing the area where the nontrivial zeros can appear has had huge ramifications. Riemann himself proved that nontrivial zeros of the zeta function can only be found in a vertical strip between zero and 1. Then, in 1896, two mathematicians showed that zeros could not lie on the edges of the strip. As a direct consequence, they were able to prove an important result called the prime number theorem, a formula for how quickly, on average, the primes thin out.

If the Riemann hypothesis holds, then mathematicians can guarantee that any deviations from the average given by the prime number theorem will always be small.

Mathematicians have numerically verified(opens a new tab) that the first 12 trillion nontrivial zeros are on the 1/2 line. But they can’t be sure the hypothesis holds for all the zeros, and even one solitary outlier would land like an earthquake. As Enrico Bombieri(opens a new tab) wrote in the paper(opens a new tab) laying out the Millennium Prize formulation of the problem: “The failure of the Riemann hypothesis would create havoc in the distribution of prime numbers.”

The connection between the zeta function and the primes was known before Riemann. In 1737, Leonhard Euler showed that for any complex number s, the zeta function is equivalent to another function that can be written entirely in terms of primes:

This brings us to the quasi-Riemann hypothesis. Where the Riemann hypothesis posits that all the nontrivial zeros are on the vertical line at 1/2, the “quasi” version just asks one to prove that they are all contained in a vertical strip that’s narrower than the known strip between zero and 1. The narrower the strip, the more one can say about how the primes are distributed.

What did AI recently prove? 

OpenAI announced two different versions of their quasi-Riemann hypothesis proof. The first version, dated September 30, 2026(opens a new tab), claimed to show that all nontrivial zeros must be between 1/8 and 7/8, and was formally verified using the Lean programming language. The second version, which was completed almost a week later(opens a new tab) (and is not yet formally verified), claimed to prove a weaker result: that the zeros must have a real part between 1/12 and 11/12. But while the first paper is 199 pages long, the second weighs in at a comparatively modest 49 pages.

More background

The Euler version makes it more obvious, but the zeta function is undefined and at s=0, and approaches a limit of infinity close to s=0.

Even this simple explanation for a relatively simple hypothesis (all the good ones are) begs the question of what a complex exponent of an integer even means (and I won't try to explain that here). Suffice it is say that this isn't trivially intuitive in the way that an integer exponent is, and that it is less intuitive than a non-integer real exponent is.

Commentary on the AI proof

First off, consider that the new proofs are less impressive than they seem. We already knew that the real part of the exponent s in a zeta function zero has to be more than 0 and less than  1. We knew that in 1896, one hundred and thirty years ago. Narrowing that range to greater than 1/12 and less than 11/12, or greater than 1/8 and less than 7/8 improves on that result, but it isn't that Earth shaking of an improvement over the status quo.

Ideas for breaking down the Riemann Hypothesis into subparts

Next, lets break this down and consider some points that would help prove the overall Riemann hypothesis. Some of them may already have been proven. All of them must be true if the Riemann Hypothesis is correct, which numerical methods strongly suggest.

* It would be nice to prove that for any given value of the imaginary part of s that there is only one zero.

* It would be nice to prove that for any given value of the real part of s that there are either zero or an infinite number of zeros.

* It would be nice to prove that, to produce a zero, the real part of s must be a rational number (a subset of the real numbers).

* It would be nice to prove that, to produce a zero, the imaginary part of s must be a whole number (i.e. a positive, non-zero, integer), or failing that, that it must be a positive rational number, or in between, a positive rational number related by some rule to the rational number that is the real part of s, or better still, a whole number related by some rule to the rational number that is the real part of s.

* It would be nice to prove that the imaginary part of s must be an odd number to produce a zero.

* It would be nice to prove that all values of s with a real part of 1/2 and an imaginary part which is a prime number of zeros. Keep in mind that the Riemann Hypothesis is an if and only if conjecture: it says that non-trivial zeros have a real part of 1/2 and an imaginary part that is prime, and it also says that there are no other non-trivial zeros. Proving the affirmative part standing alone would be a big deal, even without a proof that there are no other non-trivial zeros, but requires consideration of far fewer possibilities.

* Proof of the affirmative part of the Riemann Hypothesis would likely require that the term in the Euler infinite series definition of the zeta function where integer in the imaginary part of s is equal to the number that it is the exponent of is special somehow, because it cancels out all of the other terms of the infinite series (I don't think that a single term of either infinite series definitions of the zeta function can be zero by itself, but proving that would itself be a step).

* Proof of the negative part of the Riemann Hypothesis might start with a determination that the imaginary part of s can't produce a zero if it is even. 

* Proof of the negative part of the Riemann Hypothesis might establish that the imaginary part of s can't produce a zero if it has exactly two, or better yet, an even number, of prime factors.

* Proof the the negative part of the Riemann Hypothesis might establish that the imaginary part of s can't produce a zero if it has an odd number of prime factors greater than one.

* Proof of inability to produce a zero, in a given case, amounts to showing there are one or more terms of the infinite series that don't cancel out at that value.

* It would be nice to tie in the conclusion about trivial zeros of the zeta function to the conclusion about non-trivial zeros of the zeta function somehow.

Post Script

Of course, I'm not going to prove (or disprove) the Riemann Hypothesis. Complex analysis (i.e. the calculus of imaginary numbers) was my weakest subject as a math major in college, and I never formally studied number theory at all. 

And, I only scratch the surface of the most obvious ideas in a deeply reached and explored corner of mathematics.

But I had a long dentist appointment this morning and needed a distraction, so I thought about it then, and that's why I'm writing about it now.

A New Anomaly

There are four sets of valence quarks that produce kaons, two of which are charged and two of which are electromagnetically neutral: 

K⁺ (positive charge: up quark + anti-strange quark)

K⁻ (negative charge: anti-up quark + strange quark)

K⁰ (neutral: down quark + anti-strange quark)

K̄⁰ (neutral antiparticle: anti-down quark + strange quark)

In certain heavier atom collisions, there are more charged kaons than expected relative to the number of neutral kaons observed, which means that there are more up quarks (and anti-up quarks) relative to down quarks (and anti-down quarks) than expected.

Isospin symmetry is the concept that particles with up quarks substituted for down quarks (and vice versa) are identical in strong force interactions (in part, due to their similar mass and the indifference of the strong force to electromagnetic or weak force charge).

For some reason, however, more charged kaons are seen in argon-scandium collisions and also in some other particle collisions. The paper below fails to come up with a good explanation for why this is the case and calls for further study. The introduction to the paper explains that:

This paper focuses on hadron–nucleus and nucleus–nucleus collisions at center-of-mass (cms) energies of the order of 10 GeV per nucleon pair. Such processes are governed by (mostly) non-perturbative strong interactions, and result in the production of a large number– tens, hundreds– of new particles in the final state (multiparticle production). 

Isospin symmetry (IS) which can be considered as a remnant of the QCD flavor symmetry between u and d quarks, remains an essential even if approximate feature of the production process. 

At energies equal (or higher) to those specified above, the bulk features of multiparticle production (inclusive spectra, total multiplicities, etc) are widely believed to obey IS up to a very good accuracy, of the order of 3−4%. Nevertheless, the simultaneous measurement of inclusive spectra and yields of charged (K+, K−) and neutral (K0 S) mesons reported by the NA61/SHINE experiment in central Ar+Sc collisions at √ sNN = 11.9 GeV [1] reveals an unexpectedly large IS violation (18 ±6% at kaon mid-rapidity). 

As will be discussed below, measurements from other experiments over a wide range of collision energy appear consistent with this observation. At the present moment (Oct., 2026), the origin of this effect remains unclear.

The paper and its abstract are as follows: 

The NA61/SHINE experiment at the CERN SPS reports an unexpectedly large excess in the inclusive production of charged over neutral kaons in 0-10% central Ar+Sc collisions at sNN‾‾‾‾√=8.8 and 11.9 GeV. This effect corresponds to an unexpectedly large violation of isospin symmetry, that cannot be explained by known mechanisms contained in the present theoretical models. 

The comparison of the NA61/SHINE results to data from other experiments reveals, within sizeable uncertainties, a consistent isospin symmetry-violating excess of charged kaon production in the energy regime 2.6<sNN‾‾‾‾√<200 GeV. At LHC energies (0.9<sNN‾‾‾‾√<13 TeV) the charged-over-neutral kaon ratio appears consistent with unity, but the uncertainties remain sizeable. 

A similar excess of charged over neutral kaon yields is observed by NA61/SHINE in minimum bias π−+C reactions at sNN‾‾‾‾√=17.3 and 25.7 GeV, also in disagreement with model predictions. 

No clear theoretical explanation of these phenomena has been provided so far. An unexpectedly large violation of isospin symmetry brings potential heavy consequences for our understanding of particle production, and calls for a concentrated experimental and theoretical effort.

Andrzej Rybicki for the NA61/SHINE Collaboration, "Unexpectedly large violation of isospin symmetry in nucleus-nucleus collisions at CERN SPS energies" arXiv:2610.23498 (October 16, 2026) (18th International Workshop on Meson Physics (MESON 2026), Kraków, Poland).

I'm personally confident that ultimately an explanation consistent with Standard Model physics will be found. But it is a fairly recent anomaly that hasn't gotten a lot of attention yet, so it deserves a post here.