Normal Science

A partial digestion of the HRT counterexample

A function of one variable can be translated in space by a spatial shift to obtain a new function and also modulated in frequency by a frequency shift to obtain a new function One can compose these two operations to obtain a time-frequency shift: (This can be viewed of as a portion of the Weyl representation of the Heisenberg group, but we will not adopt a representation-theoretic perspective here.) Some functions obey finite linear relations between their time-frequency shifts. For instance, a ...

The upper limits of biology

This piece is featured in Issue 25 of Works in Progress. Subscribe today to get Issue 25, and an issue every two months after, delivered straight to your door. 9th June 1973: Secretariat leaves the field behind at Belmont Park. Image credit: Getty Images.In June 1973, chestnut colt Secretariat ran the Belmont Stakes so fast that no horse has matched him since. In less than two and a half minutes, Secretariat flew along the track, finally pulling away to win by 31 lengths, nearly three quarters o...

What Will the 21st Century ROAD to Housing Act Do for Housing Supply?

Home under construction, via Wikipedia.A few weeks ago the 21st Century ROAD to Housing Act became law. This is an enormous housing bill that’s been in the works for roughly a year — I first discussed it on Statecraft back in 2025 — and folks are optimistic that it will, per its description, “increase the supply of housing in America.”But actually understanding what, specifically, the law will do is tricky. Like many large pieces of legislation, this is an amalgamation of many smaller laws: the ...

OpenAI's claimed disproof of Connes' Rigidity Conjecture is invalid [pdf]

A Stupid Idea for AI Alignment We Came up with by Looking at the List of Specification Gaming Behaviours

Creatures bred for speed grow really tall and generate high velocities by falling over. An evolved player makes invalid moves far away in the board, causing opponent players to run out of memory and crash. A game-playing agent accrues points by falsely inserting its name as the author of high-value items. These bizarre exploits and dozens more can be found in the list of specification gaming behaviours [sic; British], a document put together by DeepMind Safety Research. “A reinforcement learning...

Corals Spin Tiny Vortices to Get Oxygen, but Not if It’s Too Hot

At first glance, corals present as little more than colorful rocks — piles of lobes, stalagmites, and branches poking out from the seafloor. They are anything but. Corals are complex creatures that form enduring colonies, and just like other animals they need oxygen to live. Across the living surface of coral, a frantic dance of survival takes place, invisible to our eyes and unknown to science… Source

Why the Legendary Erdős Problems Are Falling to AI

On May 20, 2026, OpenAI made an announcement that shook the mathematical world. An internal AI model — one not available to the public — had come up with a counterexample to the “unit distance” problem, a conjecture made in 1946 by Paul Erdős, the prolific, itinerant Hungarian mathematician. Erdős posed thousands of questions, but this one was special: It was both simple to explain and… Source

Indirect lessons from human alignment

(Inspired by a post from Eli Tyre.) Many people make some variant of the following argument: Evolution is an “outer optimizer”. It is trying to make us maximize reproductive fitness. We are “inner optimizers”. We just do what feels good. But what feels good has been set by evolution, which is hoping that it will make us maximize reproductive fitness. But we don’t maximize reproductive fitness. In fact, birth rates are dropping everywhere. Therefore evolution failed. The standard interpretation i...

US–Indian space mission maps extreme subsidence in Mexico City

Why are neural networks and cryptographic ciphers so similar? (2025)

Several neural network design principles are almost identical to cipher design principles.

Fun with polynomials and linear algebra; or, slight abstract nonsense

Longest Path Search About Code Fun Fun with polynomials and linear algebra; or, slight abstract nonsense Posted 2026-04-30 This is mostly a bunch of notes to myself (with some slight expansion) and is a combination/extension/simplification of theorems/ideas/constructions from a bunch of texts, including Wistbauer’s “Foundations of Module and Ring Theory” and Fuhrmann’s “A Polynomial Approach to Linear Algebra”, along with others that at this point I don’t recall. While most of the things here ar

The gauge broke: devs felt 20% faster with AI, measured 19% slower

For two years I argued the feeling of AI speed had come apart from the fact of it, from watching my own teams. This summer it stopped being an anecdote. A controlled trial measured experienced developers feeling about 20% faster while running about 19% slower. The instrument we steer by reads backward.

The Mathematical Dance Inside Plant Cells

The sunlight-collecting organelles known as chloroplasts solve a packing problem: how to optimize photosynthesis without sustaining damage from dangerously intense rays.

Easy Random Trees

Rigorous Nonsense Home Archive Easy Random Trees Posted on February 27, 2026 by Brandon Wilson Tags: math, apl, trees Can you think of a way to efficiently generate a random plane tree? Richard P. Stanley in his book Catalan Numbers has a really nifty combinatorial proof of why Catalan numbers have the formula \[ C_n = {1 \over n+1}{2n \choose n} \] The standard proof uses generating functions applied to an inductive definition of the Catalan numbers, which frankly does little to illumiate their

The extended predicative Mahlo universe in Martin-Löf type theory (2023)

Natural history on canvas: Brueghel knew about bird-eating noctule bats

A 13th-Century Enumeration Algorithm, Ignored for 700 Years

In Or ha-Sekhel, the Kabbalist Abraham Aboulafia prescribes a systematic way to run through every arrangement of the letters of a word. With only two or three simple rules, he describes what we would now call a recursive algorithm for generating permutations.

Deconstructing Datalog

home datafun talon reviews résumé rss Deconstructing Datalog In September 2022, after two rounds of revisions, I submitted the final version of my PhD dissertation, Deconstructing Datalog. Datalog is a logic programming language from the ’80s that augments relational algebra with recursive queries. It has both simple semantics and efficient implementation strategies. Like Lisp and the Velvet Underground, its influence exceeds its popularity; its ideas are still being absorbed into the mainstream

Topological Defect Propagation to Classify Knitted Fabrics

The Riemann Hypothesis – interactive explanation

An interactive journey from 'what is a prime number?' to a genuine understanding of the most important unsolved problem in mathematics.

Principia Artificialis – Mathematical Foundations of Artificial Thought

An open research program exploring the mathematics of artificial thought: information geometry, topology, dynamical systems, and thermodynamics applied to AI inference. - holland202/Principia-Artificialis

Is AI Reasoning Right for the Wrong Reasons?

I’ll just say it: What the hell is going on with AI “reasoning”? Sorry for the air quotes. That punctuational side-eye was more common in 2024, when the specially trained cousins of LLMs now known as “large reasoning models,” or LRMs, were still new. Nowadays it may seem downright churlish, though, given that a “general-purpose reasoning model” from OpenAI solved a famous open mathematical… Source

Why Is Everyone Trying to Build a Solid-State Battery?

A battery technology that’s getting a lot of attention is solid-state batteries, lithium-ion batteries that replace the liquid electrolyte with a solid material. Chinese battery manufacturer CATL alone had more than 1,000 people devoted to solid-state battery research as of 2024, and battery manufacturers like BYD, LG, and Samsung are also working on the technology. US and European startups making solid-state batteries have collectively raised over $4 billion as of 2025.Solid-state batteries hav...

So you want to use plants to reduce CO₂

Humans make carbon dioxide. Carbon dioxide is bad for cognition. But plants turn carbon dioxide back into oxygen. And plants are the one true home decoration strategy. So maybe if you get a lot of plants, you can you can keep carbon dioxide in check and keep your brain working? It’s theoretically possible. It’s probably just barely possible in practice. But it won’t be easy. People produce ~1 kilogram of carbon dioxide per day. That’s around 5.7 × 10²³ molecules or 0.948 moles per hour. (You may...

Can We Lower Construction Costs with Cheaper Labor or Materials?

3D printed structure, via Wikipedia.Last time in our series on the nature of the construction productivity problem, we looked at economies of scale, finding that capturing economies of scale in homebuilding is difficult. Economies of scale primarily operate on the difference between the costs of the material inputs to some process and the cost of the final product. With housing construction, this ratio is already pretty low, not much higher than what we see in high-volume manufacturing industrie...

Does creatine make you smarter?

Is creatine a weird steroid-like hormone or drug? No. Creatine is a nutrient. Most omnivores eat a gram or two per day from meat. Your body also synthesizes a gram or two per day. You need creatine to deliver energy inside of cells. It is normal and non-weird. Does creatine increase testosterone? Unlikely. This concern comes from one study in 2009 on 16 male rugby players.1 But that study is considered extremely suspect. There have been at least twelve other studies that all found no change or p...

A digestion of the Jacobian conjecture counterexample

The notorious Jacobian conjecture can be formulated concretely over the complex numbers as follows. Conjecture 1 (Jacobian Conjecture) Let be a polynomial map in complex variables, whose Jacobian is a non-zero constant. Then is invertible (with polynomial inverse). The condition that the Jacobian is non-zero is equivalent to being locally invertible. (The implication of local invertibility from non-vanishing Jacobian follows from the inverse function theorem; the converse implication can be deri...

Martin Picard’s Mitochondrial Theory of Mind

It was 9 a.m. on a Thursday, and Martin Picard was watching his blood flow from an IV in his arm through a hole in the wall. He was sitting on a twin bed in a claustrophobic chamber less than a shoulder’s width from a stainless steel sink and porcelain toilet. Every hour over 24 hours, including while he slept, a nurse channeled blood from his arm to a research team next door; at each time point… Source

Two more apps: visualizing the zeta process and the motions of the heavens

I believe that the creation of visualization apps to illustrate mathematical or scientific concepts is a particularly favorable use case for modern coding agents, as many of the downside risks attached to other LLM use cases are limited: Not mission-critical. As such apps are not authorative sources of truth and only used for secondary purposes, a small positive error rate in the output can be acceptable. Stand-alone. As the applets are not destined to be incorporated into a larger codebase or l...

How Predictable Are Laws?

An enormous amount of ink has been spilled (some of it by me) on the trials and tribulations of complying with the National Environmental Policy Act, better known as NEPA. NEPA is what requires projects to perform years-long, thousand-plus page environmental impact studies before construction can begin, and suing a project for an insufficiently detailed environmental study is one of the chief ways environmental groups are able to slow down or stop projects they don’t like. And NEPA’s influence g...

Thermodynamic Computers Go With the (Energy) Flow

In the quest to make computers accurate and reliable, noise is the enemy. The thermal jiggling of atoms is a constant threat to the precision needed for detailed calculations. Whether we’re dealing with familiar classical devices like the laptops or supercomputers that we use today, or fancy quantum devices that promise us faster computation tomorrow, we don’t want some haphazard heat fluctuation… Source

Visualizing the Gilbreath expectation sequence

One byproduct of learning how to use coding agents to create visualization apps is that it now becomes straightforward to convert any figure in one’s papers that had already been generated by code (e.g., in Python) into a more interactive, animated applet. I can illustrate this with Figure 1 from my recent paper on the Gilbreath conjecture with Chase and Hunter, reproduced below: This plot displays both exact and numerically simulated values of a certain poorly understood sequence relating to th...

Gilbreath’s conjecture: a Cramér random model and a deterministic analysis

Zachary Chase, Zach Hunter and I have uploaded to the arXiv our preprint Gilbreath’s conjecture: a Cramér random model and a deterministic analysis. This paper is motivated by a notorious conjecture of Gilbreath (also proposed eighty years prior by Proth), which one can state as follows: if one starts with the sequence of primes and repeatedly takes absolute differences of consecutive terms, then the first term of each subsequent row is always : Coming from a PDE background, I like to think of t...

What's new in biology: July 2026

Niko McCarty and Saloni Dattani review important things happening in the world of biotechnology and medicine.The holy grail of cancer therapy. Roughly half of all cancers have a mutation in p53, one of the most important proteins in our body. It protects us against cancer by acting as a checkpoint: when a cell’s DNA is damaged, p53 can pause the cell to help it repair the damage, or trigger the cell to self-destruct before it becomes cancerous. Destroying mutant p53, or restoring the function of...

Researchers Reveal the Power of ‘Quantum Proofs’

More than 30 years ago, researchers discovered that hypothetical computers based on the laws of quantum physics would be able to rapidly solve difficult math problems. Ever since then, they’ve sought to pinpoint cases where quantum computers are more powerful than their ordinary “classical” cousins. For nearly as long, a small band of computer scientists has pursued a related question that gets… Source

The Fall and Rise of Screwworm

Every spring, as sure as the seasons, and for generations unknown, screwworms began their annual march northward from their overwintering sanctuaries in Mexico and South Texas. Pushed by an unknown force as inexorable as gravity, screwworms moved north — ever moving, ever spreading, ever multiplying, ever destroying. No army ever advanced any more surely or methodically. No army was ever more destructive. Attacking, killing, maiming, and destroying, screwworms literally ate their way north. Reac...

A digestion of unit distance constructions

Suppose that one has a set of points in the plane, which we will think of as the complex plane . Let denote the number of unit distances determined by these points, i.e., pairs of points whose displacement obeys the equation (It makes little difference for the asymptotics, but we will count the pair separately from here.) The Erdös unit distance problem asks, for a given large number , what is the largest possible value of amongst all sets of cardinality ? For instance, if one takes to be equall...

Astrophysicists Puzzle Over Webb’s New Universe

When Charlotte Mason ponders cosmic mysteries, she likes to doodle. “I am quite a visual person,” she said. “I usually draw a lot of pictures trying to understand what’s going on.” Mason, an astrophysicist at the Cosmic Dawn Center in Copenhagen, has lately been filling pages with sketches of “little red dots,” perplexing objects discovered by the hundreds in images from the James Webb Space… Source

How p-hacking built the world's most expensive safety regime

Issue 24 of Works in Progress has now arrived with subscribers. Sign up here to receive this issue, plus another every two months – straight to your door.Chernobyl was the world’s worst ever nuclear disaster. The reactor was built to a flawed design. In 1986, its operators disabled automated safety systems and removed control rods to perform a late-night test. The coolant water overheated, speeding up the reaction until it flash-boiled into steam and violently shattered all the nuclear fuel in a...

After 80 Years, Mathematicians Give Famed ‘Erdős Method’ an Upgrade

In 1947, Paul Erdős, the itinerant Hungarian mathematician, introduced what would become one of math’s most powerful tools. He wanted to prove that a certain kind of object existed — in this case, a network made of interconnected nodes. But strangely, his proof didn’t specify how to build it. Instead, he showed that if you consider all networks and select one at random, the chances that you’ll… Source

What's really slowing down the AI buildout

Issue 24 of Works in Progress has now arrived with subscribers. Sign up here to receive this issue, plus another every two months – straight to your door.One of the most expensive projects in history is under construction in Abilene, Texas. This joint venture, Stargate, is the flagship of a bigger project by the same name led by OpenAI and Softbank, and is expected to cost well over $40 billion for a high-performance computing campus that will train new generations of AI models.Stargate is just ...

US Subways Build Too Many Cross Passages

I wrote the following piece for IFP’s Transit Abundance Playbook, a collection of 15 ideas to improve transit delivery in the US. American transit costs and timelines are substantially higher than those of other developed countries: Spain builds tunneled subway for ~$200 million per mile, while New York spends billions; China built an entire network of high-speed rail while California was failing to connect Bakersfield to Merced. This playbook draws on the past decade of research into the causes...

How Physicists Track and Trap the Elusive Neutrino

Seventy years ago, the physicists Clyde Cowan and Frederick Reines took a custom-built 10-ton detector, surrounded it with thick lead walls and wet sandbags, and placed it near a powerful nuclear reactor at the Savannah River Plant in South Carolina. They called the experiment “Project Poltergeist,” designed as it was to catch a ghost. More than a quarter of a century before… Source

Converting Coal Plants to Natural Gas

For the better part of the last several hundred years, coal was the fuel of choice for generating power. Burning coal powered Thomas Newcomen’s steam engine, invented in Britain in the early 18th century, and the first of a line of increasingly efficient converters of coal to usable energy. The Newcomen engine was in fact so inefficient and consumed so much coal that it was almost exclusively used at coal mines, where fuel could be obtained cheaply. The improved steam engines that followed over ...

Why the Human Genome’s Tangled Physicality May Confound AI

Since its molecular structure was deduced in the 1950s, DNA has been hailed by many biologists as the secret of life. They’ve read and studied the information stored in the DNA found in the cells of living organisms, known as their genomes, and claimed that this genetic database must be some kind of blueprint, code script, or computer. But if DNA really does harbor some greater secret about how… Source

Third SAIR competition: inverse Galois challenge

I am happy to announce the third SAIR challenge, which is focused on obtaining numerical data for the infamous inverse Galois problem. This is a collaborative project with the L-functions and modular forms database (LMFDB), and is organized by John Jones, Jen Paulhus, David Roe, Andrew Sutherland, and myself. The challenge is somewhat similar to my own Equational Theories Project, in that one is trying to complete a large mathematical data set in a verified fashion, except that the target data s...

What’s new in biology: June 2026

Niko McCarty and Saloni Dattani review important things happening in the world of biotechnology and medicine.We’ve been writing regular round ups for a little while now, but so much has happened recently that this month’s post feels like it contains a year’s worth of breakthroughs. So pour yourself something cool, get cosy, and enjoy!First, everything new in cancer. In our round up last month, we shared the news of daraxonrasib, the new breakthrough drug that roughly doubles survival in late-sta...

How the Squamish built Senakw

This is the third of three articles from Issue 24 of Works in Progress, which has been arriving with subscribers this week. Subscribe in the next few weeks to receive it, and future issues, straight to your door, workplace, or institution.In a corner of the prestigious Kits Point suburb in Vancouver, eleven great towers are beginning to rise from their foundations. This is Senakw (pronounced sen-AHK), a new development owned, managed and championed by the Squamish Nation, an indigenous group nat...

Biological Evolution and Information Acquisition

A few weeks ago we looked at a simulation of technological evolution by economist Brian Arthur, in which he was able to start with simple building blocks (such as a NAND gate) and evolve surprisingly complex circuits (such as a 12-way AND gate or a 4-bit adder) by randomly combining increasingly useful existing components. We analyzed this as a way of simplifying a search problem: by using existing, working components as modules that can be combined, a few at a time, into more complex modules, a...

How Long Does It Take to Plan a Bridge?

Bear Mountain Bridge in New York, via Wikipedia.Many folks, including me, have observed that it seems to take much longer to build infrastructure in the US than it used to. People point to things like the rapid construction of the Empire State Building (one year) or the Golden Gate Bridge (just over four years) and note that for a modern infrastructure project it can take that long or longer to even get the permits or do the environmental studies.Via Threads. Note that as of 2026 permit timeline...