An Algorithmic Lucidity

You will agree that for a woman who has fought three duels, all of them kills, indulging in such fears is ludicrous. But what's around the corner now is a duel so big that I don't think I have the guts for it; in Faust's words: Verweile doch, du bist so schoen! Keep it as it is. Don't change. [...] Take my life but don't take away the meaning of my life.

—"When It Changed" by Joanna Russ

I've been crashing Prof. Lai's "Advanced Topics" class this semester, on Fourier analysis plus enough functional analysis to get by (since the University can't afford to offer functional analysis as a regular class anymore). It's not obviously a good use of my time, but now that I've started, I have to see it through. I could try to tell a story about impact: I am leveling up my math chops, which will be useful for understanding natural abstractions or singular learning theory or mechanistic explanation, which will be useful for—but maybe it's less embarrassing to not try to sell a story. Call it luxury consumption during garbage time. I've lived a good life, in some ways if not others. I could choose to drown in oceans of regret over all the songs I never finished and stories I never wrote and math I never learned when I was young, in the before-time when there was still enough clock in the timeline for these things to matter—or I could choose to do something with the time I have now, which doesn't.

On Wednesday 16 September, the class had its first examination. (When I had dropped by office hours on Monday, Prof. Lai had said I could take it, but that he might not grade it.) I hadn't prepped enough and did poorly, blanking on two parts. I didn't hand in the paper, but GPT-6 Astra later gave it 58%.

After enjoying lunch and dancing at the mall, I walked back to campus for the math department's occasional seminar. On a whim, I had recently ordered a custom branded department tee (in Hebrew, with the thought of provoking a reaction from the other analysis professor, who is a big free-Palestine guy), which I was sure to wear for the occasion. (Ordering custom T-shirts online was already a thing in the before-time, but my modest GIMP skills would not have yielded as attractive a result in as little effort.) Aida Abiad of the Eindhoven University of Technology was speaking on coding theory applications of spectral graph theory, a topic of nonzero interest to me. (In October 2009, I had ordered a copy of An Introduction to the Theory of Graph Spectra, which sat mostly untouched among my textbook collection for the intervening seventeen years.)

The talk was pretty accessible, as these things go. The independence number \(\alpha\) of a graph is the maximum size of a set of non-adjacent vertices. The exact independence number of a graph is hard to compute, but back in the 1970s, people had come up with some upper bounds in terms of the eigenvalues of the graph's adjacency matrix. Abiad explained her work on extending such techniques to bound the k-independence number \(\alpha_k\) (the maximum size of a set of vertices with graph distance greater than k from each other). Once you have a bound in terms of powers of the eigenvalues, it's natural to ask, what about a polynomial?—and then, which polynomial is the best? Then there are applications: if you're constructing a code and a metric on your set of possible codewords is a graph distance, then the k-independence number of the graph tells you about the largest code you can construct whose codewords are greater than k distance apart.

For the most part, it felt like that talk in that room could have taken place any time in the past seventeen or twenty-seven years. (Much further back than that, and it's harder to believe that SFSU would have had a digital projector rather than still being on transparencies.) Abiad discussed bound improvements published in 2016, 2019, and 2022, but as far as I could tell as a non-specialist, there was no reason the same results couldn't have been reached in 2006 or 2009 or 2012 except that no one had gotten around to it yet. Besides the paper dates, the relative timelessness of the moment was only betrayed by a few odd remarks Abiad made, like, "Now probably the artificial intelligence gives you the graph invariants if you ask for it."

Afterwards, I took the 28 bus back to Daly City BART station. At the station, the bus stopped before pulling all the way up to the curb. Apparently a car was in the way: the bus driver honked a few times, then more frequently, then in one long, continuous honk. Why wasn't the car moving? "It's a Waymo!" one of the passengers said. "Fucking Waymo," said someone else. "Bring back human drivers!" said a man. The bus driver gave up and opened the doors, still short of the curb. Walking to the station among the throng, I saw that the offending Waymo had moved out of the way; it was unclear what had aroused it to motion.

On the train home, inspiration struck. Recently, one Levent Alpöge of Anthropic (and before that, I think someplace called "Harvert") had been in everyone's feed for Tweeting out LLM-assisted solutions to famous open problems. I didn't have access to the frontier magic that Alpöge had inside Anthropic's walled castle, but I wondered if there was more low-hanging fruit accessible to the public models that hadn't been plucked yet just because there were few enough math people monitoring the AI situation that no one had spent the tokens yet. (My impression from hanging out at SFSU had been that everyone knew about LLMs as a way for students to cheat on homework—that's why Prof. Lai couldn't give take-home exams anymore—or for researchers to look up graph invariants, but that few were extrapolating to an AI-dominated future.)

When I got home, I tasked Claude Opus 5 in Claude Code with going through the papers of Prof. Emily Clader (who had let me crash her "Topics" class on algebraic topology this time last year) to look for "open problems that might be interesting to her that might be scoopable with an LLM (so, like, stuff where you can compute a checkable counterexample, which would have been a master's student project)". I said that scooping the result would "probably demand expensive Fable tokens, although Opus is probably fine" for the literature search.

What transpired next happened quickly enough that I can only reconstruct it from the Claude Code session log—with the help of annotations from Claude Opus 5.5 (released 22 September), since the raw JSONL is so unpleasant to read.

Opus 5 downloaded the LaTeX sources of Clader's papers from the ArXiv and grepped for phrases like "open problem", "conjecture" and "future work." "Found a promising one already", it reported after 84 seconds: Clader, Damiolini, Li, and Ramadas 2023 (revised 2025) "Wonderful Compactifications and Rational Curves with Cyclic Action" "conjectures that the Betti numbers of \(\overline{\mathcal{L}}^r_n\) are log-concave."

It continued reading abstracts and grepping for open-problem prompts in a few other papers for another 48 seconds, then decided, "I'll focus on the log-concavity conjecture since it's explicit, computable, and only lightly checked in the original table." Conjecture 4.15 in the paper had been preceded by a table of the aforementioned Betti numbers for eleven values of 2 ≤ r ≤ 4 and 3 ≤ n ≤ 7. "First I'll reproduce their table before searching for any later resolutions." (These words appear in a "type":"thinking" entry in the log, presumably written by a summarizer model based on Opus 5's original chain of thought, which isn't exposed to users.)

Four seconds later, it had written a Python script to reproduce the table, which succeeded. It took another eight seconds to write another Python script to check log-concavity up to r ≤ 15 and n ≤ 40, and "real-rootedness, which implies it" up to r ≤ 15 and n ≤ 25. That program took half a minute to run. "No counterexamples turned up", said the thinking summary, "every case is log-concave and real-rooted, so real-rootedness likely implies log-concavity here. I'll check if this result already exists before attempting a proof."

In parallel with web search and Semantic Scholar citations failing to turn up an existing proof, it did a web search for Athanasiadis binomial Eulerian polynomials colored permutations real-rooted, which turned up a paper on that topic by the named Athanasiadis and another by Haglund and Zhang building off of Athanasiadis. "I see this differs from" the r = 2 case in Clader et al., said the thinking summary on reading the former, "but the same technique applies."

It took another seven seconds to write a Python script to search for a recurrence: given the actual Betti numbers and a guess about what form of recurrence might generate them, finding the coëfficients of the recurrence (if the guess was right) becomes a standard linear algebra computation.

The script crashed. Opus 5 fixed the bug, but the script didn't turn up anything. "Rather than widen the search blindly," reported Opus 5, it would read the Haglund and Zhang paper. "Their methods may apply directly." Its first attempt to locate the theorem statements in the LaTeX of the Haglund and Zhang paper failed, both because it was grepping for "theorem" (and "lemma", "proposition", "conjecture", &c.) whereas the paper used an abbreviation "thm" defined in the preamble, and because the file contained an invalid UTF-8 byte sequence, causing it to be interpreted as a binary file and ignored by Claude Code's version of grep. After converting the file encoding and adjusting the query, Opus 5 was able to read from the paper and write another Python script to check whether the Betti numbers matched anything from the Haglund and Zhang paper: no dice.

Opus 5 thought for 32 seconds. It spent 7 seconds writing another Python script searching for a recurrence, this time using specific numbers for r rather than treating it symbolically. The script took 13 seconds to run and succeeded. "Next I'll derive this recurrence symbolically from the generating function to confirm it rigorously rather than relying on the fit," said the thinking summary. It wrote two more scripts and did some additional web research on whether the conjecture had been resolved elsewhere, then spent 20 seconds writing up a Markdown file of the result and another few seconds on an edit.

"I think we scooped something," it reported, nine minutes and twenty seconds after I had told it to start working. ("We"!)

Back in the human world, I was flabbergasted. "I'll try to read that, but ... that took less than ten minutes of sidereal time!" I wrote. I hadn't even needed to spring for Fable credits.

Basking in the afterglow, I chatted with the model a little more. I asked how long it would have taken me or a typical SFSU master's student to do this. (It said SFSU was unusually strong in this area: a student of Prof. Beck or Prof. Ardila might well have seen real-rootedness.) I asked if it could keep going. (It added a Part II to its writeup, generalizing the result to the setting of a different Clader et al. paper on multimatroids.)

But I knew my work was just beginning. If I wanted to tell Prof. Clader about the solved conjecture without making a fool of myself, I'd need to verify it personally. (There are too many people in the world who blindly trust LLM slop when they shouldn't, so you have to pay a costly signal to prove you're not one of them, even in a case where the output actually is trustworthy with the current models, if not last year's models.)

With some intensive tutoring from other Claude instances, I composed my own writeup of the original Opus 5 proof. Other than the fact that Betti numbers measure how many "d-dimensional holes" a space has (where the hole in a ring is a 1-dimensional hole, but the empty middle of a hollow sphere is a 2-dimensional hole), I didn't know anything about the algebraic geometry that the paper was about, but Conjecture 4.15 turned out to be pretty detachable from the algebraic-geometrical context. \(A^{k}(\overline{\mathcal{L}}_{n}^{r})\) is a free abelian group—basically a product of some number of copies of the integers. The Betti numbers of the conjecture are the number of copies of the integers in \(A^{k}(\overline{\mathcal{L}}_{n}^{r})\) as k varies for a given r and n.

The authors had shown in Theorem 4.13 that \(A^{k}(\overline{\mathcal{L}}_{n}^{r})\) was isomorphic to some other group, which you can use to form a generating function for a given r and n (in fact, a polynomial, because there are only finitely many nonzero Betti numbers), and then an exponential generating function that covers any n. You use the exponential generating function to derive a recurrence (the one that Opus 5 had found in Python), and then an inductive argument on n shows that the polynomials only have real roots, which implies that their coëfficients (which are our Betti numbers) are log-concave, which is quod erat demonstrandum.

It took me five days. And this was very intensive tutoring: Claude told me what to do, while I filled in the algebra between steps to convince myself that I mostly understood what it was saying. (In mathematics, the convention is that authors are sorted alphabetically, but the reason I made Opus 5 the first author on my writeup is not because "Claude" comes before "Davis".)

Despite this—despite everything that's been happening the last half-decade—it was only upon going back to examine the Claude Code session log that it really started to sink in that humans are the inferior race. Paradoxically, it's the mundanity of the reasoning that does it. We're used to brute-force computation performing miracles for us; no one feels intimidated by the magic of a compiler or a chess engine, in which a sufficiently large quantity of "dumb" operations have been arranged to produce an outcome superior to that of a person's intelligence. If the proof of Conjecture 4.15 had been like that, it would be easy to accept as just more magic, like a compiler, and no more threatening.

The difference is that—however much the underlying deep network has the character of large quantities of dumb operations—at the level of abstraction of the Claude Code session log, the operations are person-like, not magic-like. Every action that Claude took is something a human could have done. There are humans who know how to write Python; there are humans who know how to debug file-encoding issues; there are (fewer) humans who know how to prove that a sequence is log-concave using an interlacing-roots induction argument and would have remembered that Christos Athanasiadis wrote a relevant paper on binomial Eulerian polynomials for colored permutations. That's where all these moves came from: human activity leaves behind pretraining data and induction of programs that reproduce the data necessarily reproduce the moves.

But once the moves are inside the computer, they can happen so much faster and be reinforced to choose the next step so much more accurately that the original humans are left far behind, as the machines wield our language and our tools to perform with infallible exactness more than everything that we have performed with a slovenly approximativeness and self-defeating inaccuracy.

Part of the glamor of learning math in the before-time lay in the irreplaceability of expensive human capital investment. Textbooks contained information, but information could only become living knowledge through uncounted hours of laborious struggle, whose pain could only be tolerated on the promise that it kept the sacred flame of the intellect alive. Now it looks like the flame is going to survive, with or without the struggle (and with or without the humans, but that's not exactly what I'm talking about here). Isn't that what we were working for? How could an abundance of the sacred feel this bittersweet?

As a brash but anxious young programmer in the before-time, I affected a standoffish attitude on the topic of games. Why would I learn to play a game as a human, I said, when I could just write a program to do it? When an office chess game got started at my workplace, I wrote a chess engine (a standard minimax search with α–β pruning) rather than learn anything about strategy or tactics. When my insufficiently requited love started hosting a meetup to play Zendo—a deduction game in which one player makes up a rule to positively or negatively classify arrangements of plastic pyramids, while the others submit arrangements to be classified to help them guess the rule—I wrote a program to efficiently propose examples to deduce a true rule from dozens of candidates. (Given a distribution over hypotheses, it generated an arrangement whose positive or negative classification would eliminate half the remaining probability-mass.)

Now, in my old age (and the present end-time in which custom software is no longer such a flex), I no longer feel the need to pretend not to understand the appeal of games. I even learned chess properly back in 'twenty-four.

The one that's captured my interest lately is the arrow-stomping game—known by a couple other brand names in addition to the original, whose inessential differences of implementation accentuate rather than obscure its fundamental nature as the same game. (Except the South Korean one is trash because the timing windows and letter grades are too lenient.)

The arrow-stomping game is notable for being trivial from the perspective of my before-time impulse to write a program for it. When programming (say) chess, the implementation of the rules of the game is quite different from the implementation of the tree search agent to play the game. Moreover, a chess program is an exercise in logic: the program need only represent the abstract structure of the game; robotics to move physical pieces on a physical board would be considered superfluous if they were considered at all.

The arrow-stomping game is just the opposite. If you implemented the game as a sequence of timers (expect Left in 2.0 seconds, Up in 2.25 seconds, Right in 2.5 seconds ...), then the natural implementation of an agent to play the game would be—the same sequence of timers (input Left in 2.0 seconds, &c.). Arrow-stomping is only interesting as a robotics problem: the struggle of an embodied agent to convert visual cues to timing information and a motor plan to deliver a force to the appropriate panel or panels at the requested moment. It is thus particularly suited to the present end-time as an illustration of Moravec's paradox. The games that endure are sports, which appear cognitively trivial only because the form of cognition demanded is more ancient than language, with the effect that we can't talk about it and the pretraining set (for all the vastness of webtext) doesn't cover it.

A related way in which the pre-linguistic nature of arrow-stomping seems thematically suited to the present end-time has to do with how the process of learning it casts doubt on the dream of an intelligence shaped like Reason itself. In the before-time, people thought that the end-times would be brought about by "good old fashioned" engineering turned on itself: the code that writes the code that writes the code that ends the world.

To be clear, that's still plausible as a theoretical possibility for a later end-time, but it turned out to be unnecessary for the beginning of the end-times. For now, we don't know how to write the code that writes the code. We can only write code to train networks that write code. The kind of legible symbolic reasoning embodied in a chess engine is the output of a process that doesn't seem to natively have the shape of legible symbolic reasoning.

Likewise the solution to the robotics problem. Every arrow summons a judgment. In the original-brand implementation, they are Marvelous (±17 ms), Perfect (±34 ms), Great (±84 ms), Good (±124 ms), and Miss. (Apparently historical contingency forces us to suffer "Perfect" not being the best judgment. The American one at least has the decency to call the best judgment "Perfect!!" with two exclamation points, as distinguished from "Perfect".)

The judgments are a dense training signal. Beyond a few words of high-level advice (try to alternate your feet), one does not "study" arrow-stomping. In principle, we could imagine a mind inspecting a sequence of upcoming arrows and searching for a motor plan to fit them, like a chess engine calculating variations. As a human, there's no time for that. One can only let one's policy be shaped by the rewards of the judgments. (The American-brand implementation supplies gradient information by displaying "Early" or "Late" as part of the on-screen judgment; in the original, you have to toggle a config option.)

Arrows often come in patterns (like Left–Right–Left or Right–Up–Right), which can be "chunked" and loaded as one motor program, rather than read as individual notes. The chunking theory correctly predicts that losing the rhythm often results in correlated poor judgments on consecutive notes, in contrast to the independent "Gaussian" error on every step. A Great Fullcombo (completing a song with only Great or better judgments) isn't overwhelmingly harder than a Fullcombo (no Miss judgments): both imply having held the rhythm the whole way, with the Great Fullcombo "only" (only) requiring a tighter Gaussian variance.

The levels of the game provide a smooth progression. As the player masters simpler patterns, higher difficulty levels provide new ones. On level 6 of the original-brand implementation, notes spaced at closer intervals (blue arrows on odd eighths of a measure, contrasted to the red "quarter" notes) become common. Patterns like Left–Down–Right imply either a "crossover" putting one leg in front of the other (Left with left foot, Down with right foot, Right with left foot) or breaking the pattern of alternating feet ("double-stepping"). Novices tend to double-step, but as the arrows become denser at higher levels, crossovers become a practical necessity to keep up with the beat.

And yet I'm reconstructing all this knowledge in words after the fact. No one told me to chunk patterns. At no point did I "practice" crossovers (only realizing I had done them for sure after careful reading and video review). Competences emerge as solutions determined by the training signal. Good (late) tells you to step faster, Great (early), a little slower. A desperate reach to hit a panel in time gets reinforced when it works, until the combined force of thousands of gradient steps inscribe a dance routine which can only in retrospect be described in terms of "crossovers" or "double-stepping."

Ultimately, what makes the arrow-stomping game unusual isn't how impervious its robotics problem is to good old fashioned engineering, but how its rich training signal makes it permeable to anyone with enough time and quarters. I see online chess accounts with thousands of games over years and flat ratings. Apparently, mere experience isn't enough to solve the credit-assignment problem by which the difference between victory and loss might owe to positional weaknesses introduced in the opening. I also tried to learn gymnastics in 2024. It didn't take. Supposedly you can build up to a full handstand by starting against the wall, but even that wasn't enough of a gradient; I wasn't feeling out a clear path through parameter-space from kicking out off the wall for a few seconds to a clean, independent balance. When prodigies do make a better show of learning chess or gymnastics, it's with coaches and puzzles to smooth the way. The alternative to incrementally accreting complex skills from local rewards isn't designing them from clean principles—but not learning them at all.

Apprehension of this strange inversion of reasoning almost makes the present end-time seem like less of a grotesque surprise. The steps we were already taking implied the seeds of the revolution.