# San Francisco State University in Twilight

Author: Zack M. Davis
Originally published: 2026-10-25
Category: mathematics
Tags: artificial intelligence
Canonical URL: https://zackmdavis.net/blog/2026/Oct/san-francisco-state-university-in-twilight/

> 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"](https://www.lexal.net/scifi/scifiction/classics/classics_archive/russ/russ1.html) 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%.

![torso of man wearing purple אוניברסיטת המדינה בסן פרנסיסקו המחלקה למתמטיקה shirt](https://zackmdavis.net/blog/images/sfsu_math_dept_shirt.jpg){: width="300" .alignright }

After enjoying lunch and [dancing](https://zackmdavis.net/blog/2026/Oct/dance-dance/) at the [mall](https://en.wikipedia.org/wiki/Stonestown_Galleria), I walked back to campus for the math department's occasional [seminar](https://sites.google.com/view/sfsuagc/fall-2026). 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](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Independent_set_(graph_theory)). The exact independence number of a graph is [hard to compute](https://en.wikipedia.org/wiki/NP-hardness), but back in the 1970s, people had come up with some upper bounds in terms of the [eigenvalues](https://zackmdavis.net/blog/2012/Dec/eigencritters/) of the graph's [adjacency matrix](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Transparency_(projection)).) 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](https://zackmdavis.net/docs/algebraic_topology-01.pdf)) 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](https://zackmdavis.net/docs/session_2026-09-16_proof-annotated.html) 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"](https://arxiv.org/abs/2208.05463v3) "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](https://arxiv.org/abs/1812.00434) and [another by Haglund and Zhang](https://arxiv.org/abs/1902.01278) 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](https://zackmdavis.net/blog/2026/Aug/so-you-dont-trust-me/), 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](https://zackmdavis.net/docs/betti_numbers_form_a_log-concave_sequence.pdf) of [the original Opus 5 proof](https://zackmdavis.net/docs/log_concavity_Lrn.pdf). 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](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/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](https://zackmdavis.net/blog/2019/May/minimax-search-and-the-structure-of-cognition/), 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](https://www.lesswrong.com/posts/wYpjXRLqbLbnmjbJP/llms-are-still-mostly-powered-by-imitative-learning-not-rl) [from](https://www.lesswrong.com/posts/xvdngZAqFZfek7KGH/pretraining-data-not-verifiability-is-why-llms-are): human activity leaves behind pretraining data and [induction of programs that reproduce the data](https://www.lesswrong.com/posts/Dw8mskAvBX37MxvXo/deep-learning-as-program-synthesis-1) 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](https://www.online-literature.com/george_eliot/theophrastus-such/17/).

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?](https://x.com/jd_pressman/status/1882407537917804759)