Yes. Running government like a business has always been a terrible idea.

I haven't seen anything as stupid as the WeWork IPO document in a very long time - that is, until Elon Musk filed to take SpaceX public. WeWork was a joke. SpaceX is a threat. And if Musk and his bankers have their way, you are going to be their bagholder.

Lots of the top-line details leaked long before the S-1 filing itself became public. There's the rumored valuation of more than $1 trillion. That's despite the nearly $5 billion in losses last year. The total addressable market (TAM) for SpaceX - the amount of revenue SpaceX thinks it could make if won over what it thinks is its entire customer base - was listed as $28.5 trillion. By way …

Read the full story at The Verge.

Just a few years ago, Ben Shapiro was the defining voice of right-wing media. His podcast sat near the top of the charts. Posts from the Daily Wire, his media company, routinely dominated the competition on Facebook. His team was even coming for Hollywood, putting out “anti-woke” comedies and an epic fantasy series that cost millions per episode.

All that feels like a distant memory now. Shapiro’s social media traffic has collapsed, as the Washington Post’s Drew Harwell recently reported; the Daily Wire has gone through multiple rounds of layoffs since 2025. The epic fantasy series flopped. Shapiro’s struggle to stay relevant is clear on his YouTube page, where you can find painfully forced videos of the pundit reacting to trending culture.

So what happened? Ryan Broderick, a longtime internet culture reporter who publishes the Garbage Day newsletter, has a succinct explanation: “The age-old problem with working at the racism factory! They eventually make a new racism that includes you,” he wrote in May.

To learn more about the Daily Wire’s decline, Today, Explained co-host Noel King spoke with Broderick about how Charlie Kirk’s murder precipitated a MAGA vibe shift that has left Shapiro out in the cold, the new media figures rising to replace him, and whether we will miss Shapiro once he’s gone. (We very likely will.)

 Below is an excerpt of their conversation, edited for length and clarity. There’s much more in the full podcast, so listen to Today, Explained wherever you get podcasts, including Apple Podcasts, Pandora, and Spotify.

Explain your “racism factory” line, please. 

It was a pithy way to describe what I think is happening to Ben Shapiro right now, which is that he’s found himself on the wrong side of a far-right vibe shift that’s happening. 

The question of “Should American conservatives support Israel?” I think, has quickly become the deciding factor in canonizing the new wave of MAGA, or even post-MAGA conservatism in America. There’s a lot of creators on one side who say we should not be involved with Israel. They say that largely for antisemitic purposes, but also because they’re xenophobic and isolationists, but they know that this is a red line that they can go across. 

Ben Shapiro cannot follow them there because he is an Orthodox Jew who supports Israel and is a fairly standard conservative, all things considered. And so this is among the many other problems that Shapiro is having right now in trying to hold his digital media empire together.

Alright, so Ben Shapiro’s on one side. As you said, he is unlikely to ever turn his back on Israel. On the other side are people who are going hard at Israel and have been since approximately, I don’t know, October 8, 2023. Who are they? Who are the players here?

The biggest one is Nick Fuentes. He is the de facto leader of this far-right splinter cell movement, the “Groypers.” He’s got a live stream that he’s on every single day, and he’s just the most vile kind of far-right personality you could imagine. But you also have more and more creators, I think, sensing this vibe shift and moving towards him. 

Candace Owens was going so far as to even claim that Charlie Kirk was killed by Mossad. You also have Tucker Carlson, Megyn Kelly — a lot of these people I would sort of put in the camp of pretty run-of-the-mill conservative commentators who understand that Trump is not popular, and so they’re trying to feel out new territory there. And then you also have “manosphere” guys like Tim Dillon who have even started to kind of go against Israel. 

It is this thing that is happening, and social media, I think, always prioritizes the newest, most taboo idea. And so this would be a new taboo that has been discovered by far-right commentators.

So in that camp of people, you have critics of Israel that run the gamut from Candace Owens, who seems kind of nutty, to Megyn Kelly, who often seems pretty straight. What do they all have in common? Is it just their criticism of Israel?

No, my read on this is that it all stems from Charlie Kirk, actually.

The MAGA movement is not one movement. It is not one ideology. The 2024 winning coalition was this weird mismatch of far-right live streamers, manosphere podcasters, neoconservatives and the TPUSA/Charlie Kirk kind of middle-of-the-road MAGA people. I think Charlie Kirk was very instrumental in holding a lot of this together, if only because it seemed like — to them at least — he was possibly a replacement for Trump. 

I’ve read into it as the MAGA movement was trying to home-grow their own version of Trump. Charlie Kirk may have been that figure. He dies, and the whole thing starts to fall apart. And I have to give, unfortunately, some credit to Nick Fuentes here, who has always hated Charlie Kirk.

So Charlie Kirk is killed, and then these alliances form and they fracture and they reform and they refracture. What events of the last, say, eight months do we place in the post-Charlie Kirk’s assassination moment?

It’s a lot of reading the tea leaves of online discourse, I would say. But you know when the movement is working and when they’re all falling in lockstep with one another.

Sydney Sweeney’s jeans would be a good example of [that], or Cracker Barrel. They’ve been able to get this talking point to surface out of their DMs and into the general consciousness. And if you look back at the months immediately after Charlie Kirk’s murder, that hasn’t really been happening the same way. They’re not really working together. They’re fighting with each other a lot, and they’re also telling on each other. 

These people are very messy. Even as we speak, Ashley St. Clair is on TikTok sharing secrets from inside the MAGA movement and going on Hasan Piker’s stream. All these guys are unfollowing each other and fighting with each other. And it’s a lot of right wingers who are super dependent on internet attention and monetizing internet attention, and they’re really, really nervous about the internet landscape the same way all digital media publishers are. I think that’s having a negative impact on the stuffiest of the digital media-era people. And Ben Shapiro is the stuffiest.

There is something else that I’ve been thinking about a lot, which is: Ben Shapiro, when he started out, he was so young, and it was like this young man that appealed to people who were much older because he was super well-spoken and he was pugnacious. 

Now he just sort of seems old. He seems like he doesn’t really know what he should be doing on TikTok. He seems like he doesn’t really know who in the culture is relevant anymore. You could make the same argument about Tucker Carlson, even though he’s surviving, but he openly seems scared of Nick Fuentes. 

Do you think that the guys that we were used to are now the old guys and they know it, and the young guys that are coming after them are worse?

I would say that Ben Shapiro from the very beginning was much better at talking to old people than talking to young people. And it seems like what he was doing was creating a digital media company that looked hip and cool to old people, who would then give him money and he would spend that money on advertising and sort of dominate Facebook and create this flywheel that allowed him to grow pretty quickly. 

A lot of the weird preoccupations the Daily Wire has had with dominating Hollywood, for instance, feel very old to me. It feels like an 80-year-old conservative’s fever dream of what the internet could be. Just very strange stuff. 

I think it’s only gotten stranger in the last year or two, because it also feels like the Trump movement has kind of moved beyond the need for someone like Ben Shapiro. In the era of DOGE and Project 2025 and ICE occupations [and] JD Vance/AI stuff, none of it feels like Ben Shapiro is really in the mix anymore.

Do you think we’re going to look back in a few years and miss Ben Shapiro for his sort of sobriety?

Yes. I think that when digital publishers on the right, in the early 2010s, began to really lean into the internet, they inadvertently connected American conservatism and by extension global conservatism with the sea changes and tides of internet discourse. And that’s always going to go towards the thing that feels the most dangerous and the most taboo, because that’s what’s most exciting on social media. 

If you have every major conservative figure in America making money directly from the internet, there’s no real incentive for them to become more moderate. They’re going to be hitting themselves in the face with hammers and smoking meth and attacking people on the street and going full white nationalist, race-science Substack nonsense. We’re already seeing this. The days of Prager University or the Daily Wire trying to do a sensible conservative’s reaction to Cardi B’s “WAP” or whatever are just not going to come back.

The dwarf planet Ceres has a surface that seems to get more perplexing with each new study. A recent paper presented at EGU26 in Vienna only adds to its mystery.

In mid-May, OpenAI announced that an internal AI model had disproved the Erdős unit distance conjecture, a famous problem in discrete geometry that had stumped human mathematicians for the last 80 years.

OpenAI gave several mathematicians early access to the result and published their reactions. Tim Gowers—who won the Fields Medal, the most prestigious prize in mathematics—wrote that “there is no doubt that the solution to the unit-distance problem is a milestone in AI mathematics.”

University of Toronto professor Daniel Litt wrote that “this is the first example of a result produced autonomously by an AI that I find exciting in itself, as opposed to as a leading indicator.”

It’s arguably the first time that an AI system has found a proof resolving a major open conjecture. That’s impressive, but I don’t view it as a radical break from the previous trajectory of AI progress in mathematics.

Three years ago, LLMs struggled to solve arithmetic problems. It was only last year that LLMs started acing high school mathematics competitions.

When I attended the Joint Mathematics Meetings—the largest annual mathematics conference in the world—in January, I learned that AI systems were starting to contribute to mathematical research, but only in constrained settings. It took significant human interpretation to turn an AI output into a publishable theorem.

OpenAI’s new result is the next step in this progression. The AI model cleverly applied existing ideas drawn from several subfields of mathematics to create a full proof. But it didn’t pioneer any genuinely new techniques. The result has since been cleaned up and extended by human mathematicians.

This points to a medium-term future where human mathematicians and AI models complement each other: AIs have a broader knowledge of past work than any human alive and much more willingness to grind through tedious proof strategies that aren’t likely to work. But humans can still think more deeply about any one problem and ask more interesting questions.

That might not last. AI systems have been improving at math so rapidly that it’s unclear what role, if any, human mathematicians will play a decade from now.

The unit distance problem

Paul Erdős was one of the most prolific mathematicians in history. He wrote over 1,500 papers in his lifetime, the most ever. One of his greatest talents was coming up with problems that are simple to state but have deep roots.

In 1946, he introduced the unit distance problem. Imagine you have some points in a 2D plane and you measure the distance between each pair of points:

Credit: Kai Williams / Understanding AI

In this diagram, there are five points and ten pairs of points. Three pairs happen to be exactly 1 unit apart: AD, BE, and CE.

Can we rearrange the points so that more pairs of points are exactly 1 unit apart?

Yes. For instance, we could move points A and D to be closer to the B, C, and E cluster. With a bit more work, we could further rearrange the points so that there are seven pairs exactly one unit apart. But that’s the most we can do.

We could do the same analysis with 6 points, 7 points, and so on. But as the number of points grows, the problem very quickly becomes too complicated to find the exact answer.

The arrangements of 5, 6, 7, 8, and 9 points that have the most pairs of points exactly one unit apart. Figure from the appendix of “The Erdős unit distance problem for small point sets” by Boris Alexeev, Dustin G. Mixon, and Hans Parshall showing the optimal arrangements for 5 through 9 points. Alexeev et al. give the optimal solutions through 21 points; the question is open after that. Credit: Boris Alexeev et al.

So instead of asking exactly how many unit distances are possible for a given number of points, Erdős tried to calculate upper and lower bounds on the number of length-one lines for n points, assuming that n is a large number.

To help calculate a lower bound, Erdős assumed that the points would be laid out in a grid. This is probably not the optimal layout, but if he could demonstrate that points in a grid have a certain number of pairs with unit distance, then the optimal arrangement must have at least that number.

If we make the grid smaller, we can intersect more grid points with the unit circle. This gives more unit distances. Credit: Kai Williams / Understanding AI

The simplest option is to space the grid so that every point is distance 1 from its neighbors directly above, below, left, and right. However, Erdős saw that you could do even better if you took diagonals into account. If you make the grid spacing smaller, you can make each point be distance 1 from a greater number of neighbors. In the diagram above, if the grid spacing is 1, then each individual point is one unit away from four neighbors (the left panel). Instead, if the grid spacing is ⅕ (as shown on the right), then each individual point is one unit away from 12 neighbors:

OpenAI’s write-up of its new result included a confusing diagram showing points in a grid with a bunch of lines connecting them. The diagram becomes easier to understand if we superimpose a circle like this:

A diagram from OpenAI’s announcement of the AI’s disproof of the unit distance conjecture, onto which I superimposed a circle showing the distance-one neighbors for one point. The grid spacing here is 1/√65, which produces unit circles that intersect 16 points on the grid (or would if the grid were larger). Credit: Kai Williams / Understanding AI

This works because of the Pythagorean theorem, which states that if we have a point that is a units to the right and b units above another point, the distance c between those two points satisfies a² + b² = c². The trick is to choose some number c² so that there are a whole bunch of pairs of whole numbers a and b such that a² + b² = c². Then, if we scale the grid down so that each point is 1/c from its neighbors, there will be a bunch of unit distances.

For example, if we choose c² = 25, then the Pythagorean equation can be satisfied by either 0² + 5² = 25 or 3² + 4² = 25. This corresponds to the 12-grid-point circle I showed earlier, with points at (0,5), (3,4), (4,3), (5,0), (-4,3), (-3,4), and so forth. (Technically, these lengths should all be divided by 5 — (⅗, ⅘) for example—but I’m leaving the denominators out for clarity.)

OpenAI’s diagram is based on choosing c² = 65, which can be satisfied by either 1² + 8² = 65 or 4² + 7² = 65. This means that if the grid spacing is 1/√65, each point will be one unit away from 16 other points: (1,8), (4,7), (7,4), (8,1), (-1,8), (-4,7), and so forth. Larger values for c²—if they’re chosen carefully—enable more whole-number diagonals and hence more unit-distance pairs.

However, if c² is too large compared to the number of points in the grid, then many of the potential one-unit-away neighbors will be outside the grid.

In short, we want to choose a c² that’s large enough but not too large. Using insights from number theory, including Jacobi’s two-square theorem, Erdős was able to show that an optimally sized circle will enable the number of unit-distance pairs to grow faster than the number of points, but only barely.

The question became "can you do better?" To find an upper bound, Erdős used an argument from a quite different area of mathematics called graph theory to show that you could only have so many unit distances. But his upper bound grows much, much faster than the best lower bound he was able to construct.

Erdős’s conjecture was that the actual optimum was much closer to the lower bound than the upper one. He predicted, but couldn’t prove, that the maximum number of unit-distance pairs grows just barely faster than the number of points.

To be more precise, Erdős conjectured that the number of unit distances would be n^(1+o(1)). In other words, for a sufficiently large n, the maximum number of unit distances would be less than n^(1+𝜖) for any 𝜖 > 0. That could end up growing a little faster than his lower-bound construction—which was n^(1 + C/(log log n)) for some constant C—but within the same general ballpark.

Proving his guess became known as the unit distance problem. For the next 80 years, it looked like Erdős was right.

Then an OpenAI model proved him wrong.

The AI’s approach

Erdős’s conjecture assumed that, at least for a large number of points, a square grid could yield about as many unit-distance pairs as organizing the points in other ways. OpenAI’s AI proved this wrong by demonstrating that there was another, more complex way to organize n points that allowed more pairs to be exactly one unit apart.

Precisely because the new pattern of points is more complicated, it’s tricky to explain it concisely. But you can think of it as a clever modification of Erdős’s grid.

The AI constructed a grid in a high-dimensional space and then projected this more complex structure into two dimensions. And instead of using a whole-number grid with points like (1,3) or (-3,6), the AI construction used something called algebraic integers to build this more complicated grid. It turns out that this kind of higher-dimensional grid has richer structure, which allows the AI to pack more unit distances into the same number of points.

It’s hard to illustrate this alternative arrangement of points because it only becomes advantageous with a very large number of points. But here’s a simpler arrangement of points that was constructed in a similar way. You can click here if you want to play with the illustration yourself.

It has 1,345 points and only produces 5,916 unit distances, fewer than the 7,632 unit distances that a square 1,296-point grid produces using the Erdős technique. But I think it gives a sense of how a pattern that isn’t a grid could produce more unit distances than a square grid.

A simplified visualization of what the AI model’s arrangement might look like. The 12 red lines emanating from the center are each length one. Click the interactive link to play around with the visualization. Image created with help from ChatGPT, based on an idea by Will Sawin, one of the mathematicians involved in the work. Credit: Kai Williams / Understanding AI

The more complicated patterns pay off. While the OpenAI model’s proof does not explicitly state how many unit-distance pairs are possible for n points, human mathematician Will Sawin was able to show that it grows at least at the rate of n^1.014. This might seem small, but as n gets really big, this number will become much larger than the counts produced by the Erdős approach.

That being said, the AI’s result doesn’t completely resolve the problem. Our best upper bound for the number of unit distances is around n^1.333. More work is needed to close this gap.

How does this result fit into AI for mathematics?

If you’d asked me two weeks ago—before OpenAI's announcement—about the most novel contributions of LLMs to mathematics, I probably would have pointed to the AlphaEvolve system from Google DeepMind.

AlphaEvolve harnesses LLMs to be the engine of an optimization process. If you can turn a math problem into a piece of code to optimize, which you often can, the LLM might find better solutions than humans have for certain types of problems. In November, four mathematicians (including Terence Tao) released a paper that analyzed AlphaEvolve’s performance on 67 optimization problems across the mathematical literature. They found that AlphaEvolve was able to improve on the established literature in some cases.

This was a step up in autonomy from previous LLM contributions, such as literature review, but it still required humans to frame it as an optimization problem and turn the AI’s output into usable mathematics. And only certain types of problems are amenable to this approach. More conceptual questions that don’t include a number to optimize can’t easily be studied with AlphaEvolve.

So AI companies have been working to develop LLM systems that can directly output a correct solution to any math problem. OpenAI’s result is a substantial step in that direction. But it also fits the pattern of previous AI-assisted mathematics.

For one thing, other companies have also worked to solve Erdős problems. Because Erdős posed hundreds of problems over his career—and because mathematician Thomas Bloom has organized an effort to compile all of them at www.erdosproblems.com—AI companies have used them as a testing ground to evaluate AI systems. In January, Cambridge undergraduate Kevin Barreto worked with a friend to ask GPT-5.2 and Harmonic’s Aristotle to produce the first autonomous solution of an Erdős problem. On May 22, two days after OpenAI’s announcement, Google announced that its AI system had solved nine open Erdős problems, including two that had been open for over 50 years.

To be clear, the problem that OpenAI solved is more impressive than any of the other work I just mentioned. But OpenAI’s solution is more in line with past AI efforts than the headline result might suggest.

One reason the unit distance problem was unsolved for 80 years, despite being so well known, is that most people thought Erdős’s conjecture was true. But the mathematical tools we have are nowhere close to being able to prove Erdős’s bound. So mathematicians expected that any proof of the conjecture would involve major new ideas or approaches.

Instead, as we’ve seen, the AI disproved the conjecture by making an extension of Erdős’s initial construction. It was a clever and nonobvious solution, but it also bore some similarity to the kind of optimization work done by a system like AlphaEvolve.

This dynamic is reflected in some of the mathematicians’ responses. Mathematician Tim Gowers wrote that when he first heard about the AI’s result, he thought it had proved the theorem. “I spent the evening adjusting my world view: If the AI could come up with a proof like that, then maybe it would be all over for mathematicians very soon.”

But the next morning, Gowers and other external reviewers received an email about the result, and he realized that the LLM “had disproved the conjecture rather than proving it, which came as a big relief.”

OpenAI’s solution also had two properties that played to the strengths of AI models relative to humans.

First, the eventual solution relied on applying sophisticated techniques from a quite different area of mathematics: algebraic number theory. AI systems have been trained on huge swaths of mathematics—and there’s a lot of math out there—so they have a broader knowledge of previous mathematical work than any human in the world. For a human to solve this, they would have needed to have the relevant algebraic number theory knowledge while also being interested in the unit distance problem, a rare combination.

Second, the reasoning process was such a grind, and seemingly unlikely to succeed, that most humans would not have thought it worth the trouble. Jacob Tsimerman, a University of Toronto professor, remarked in the OpenAI document that he had briefly considered taking a similar approach to disprove the conjecture. But that type of technique “consumes much time and frequently doesn’t work out,” so he abandoned the project.

An AI, on the other hand, can work through many proof strategies that don’t work out before discovering one that does. OpenAI could have run the problem many times before a model found a solution. Indeed, an OpenAI chart revealed that even with the maximum token budget, the internal model solves the problem only half of the time.

To be clear, what the AI system did is still impressive. “It’s always tempting to look at a completed proof and declare it obvious after the fact,” Tsimerman said later in his remark. But as I noted previously, it also played to the strengths of AI systems.

In the short to medium term, this points to a world where AI models complement humans but do not replace them. AI systems will tackle lists of problems curated by human mathematicians or aid humans in finding relevant approaches from seemingly unrelated mathematical fields. But they won’t immediately displace the human role in choosing which questions to ask or developing wholly new techniques.

Even this result was very much a human-AI collaboration. While the AI system found the proof on its own, human mathematicians verified the result. Other humans came up with better-written proofs that extended the AI’s initial ideas, like Will Sawin finding an explicit lower bound as I mentioned above.

It’s unclear how long this complementarity will last, however. Gowers spent the rest of his comment exploring whether the relief he felt on hearing that AI had disproved the conjecture was justified. He more or less concluded that it was, but in a footnote, he wrote that he would guess “that AI will soon reach a high level at other activities such as building theories, formulating definitions and asking interesting questions.”

In the past year, we’ve gone from AI systems that hadn’t yet beaten high school mathematics competitions to ones that can advance mathematics in interesting ways. It seems likely that AI systems will continue to become more autonomous when working on mathematical problems.

At the same time, we haven’t fully explored what current models can achieve in math. Soon after OpenAI’s announcement, University of Michigan postdoc Xiao Ma found that GPT-5.5 was also able to prove Erdős wrong if given a small hint. If a generally available model could disprove this famous conjecture and no one noticed, what other discoveries could happen today that no one has thought to try?

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