Twenty years ago, when the idea of AI taking over the world in our lifetimes still struck most of us as the unconstrained fantasy of those who knew too much science fiction and too little science, many of us would say things like:
Look, the part of the story that’s wildly implausible is that a recursively self-improving superintelligence will just explode from some hacker’s basement and take over the world without warning. If it’s going to happen, we’ll see many warning signs first. We’ll see, I dunno, AI agents breaking out of containment, conspiring with each other to hack websites, in fanatical pursuit of whatever strange goals they have. And then, of course, we’ll see major math problems getting solved by AIs—even the Clay Millennium Problems. That will be the time to panic! Wake me up when that happens!
Twenty years ago, the above was a take that even my most conservative, skeptical colleagues in academic CS would’ve gladly endorsed.
If you want to know my current take, you simply start with the one above, then update on the fact that the wild prophecies have come true. The first rumblings, I’d say, came a decade ago with AlphaGo, they got noticeably louder with LLMs and coding and reasoning agents, and they’ve accelerated this summer and fall into a crescendo of wonders and terrors that one needs to be a particular kind of idiot to deny.
I recoil from the neverending shell game where you say “oh sure, of course AI can now [escape from its sandbox / solve Millennium Problems / whichever dramatic thing it most recently did], no one ever denied that [I did deny it], wake me up when AI does [thing AI hasn’t yet done but is going to do next year], that’s when I’ll reevaluate my whole worldview [no I won’t].” Where no matter how fast the rollercoaster accelerates, even after your whole familiar world has vanished behind you, you’re still inventing reasons why it doesn’t count.
My position on AI is merely the conservative, skeptical position of 2006, updated with intellectual honesty for the reality of late 2026. And that position, if you need me to spell it out, is as follows:
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
It seems to me that the Singularity has already started; it’s just wildly unevenly distributed. Yes, I still unload the dishwasher and clip my toenails. On the other hand, in whatever years I have left, I don’t expect that I’ll ever again prove a theorem because I’m actually needed to prove it. If I do, it will only be for my or others’ enjoyment or edification.
The test is this: if we took the news of these past few weeks and sent it back in time twenty years, would I agree that it looked like the beginning of an AI Singularity? The intellectually honest answer is: yes, absolutely. But then that’s all we need. No backsies.
I feel like it would be healthy for everyone to stop grinding their ideological axes, their sentiments about Dario Amodei or Sam Altman, for long enough simply to acknowledge that the wonders and terrors are here. They couldn’t be here more clearly if the sky had turned reddish-orange like in the Matrix movies.
It’s here clearly enough that, when I put my kids to sleep at night, I now feel it in the pit of my stomach: what sort of future can they possibly have? What could they learn today that could possibly be relevant to that future? (Yesterday, my 13-year-old daughter joked unprompted that, if she wants to become a mathematician, it now looks like she has maybe two more weeks.) Certainly when my grad students want to discuss what sort of careers might await them on graduation, I no longer have any clue what to tell them.
Maybe it will help if I briefly switch topics. Ever since my wife and I moved to Austin, I’ve sometimes gotten some version of the following query: “How can you, as both a Jew and a skeptical scientist, possibly get along well with all those evangelical Christians down there in Texas? Sure, they might seem super friendly to Jews, but don’t you understand that that’s only because of the special role Jews play in their eschatology—when Christ will return in glory, and you’ll either accept Him as Lord or else roast in hell for eternity?” I stare at them and say: “wait, so I get to accept Christ only after He returns? What a great deal! How could I possibly have any objection to that?”
For anyone who says AI doom sounds like an apocalyptic religion, that the rationalists/Singulatarians seem like a Bay Area cult, that Eliezer Yudkowsky gives off the vibes of a messianic prophet: yes, yes, and yes. But crucially, today you’re no longer being asked to believe in arguments and extrapolations, but only in the front-page news. Accepting the reality of the coming machine god after it’s solved Navier-Stokes and dozens of other longstanding open math problems (while dramatically ramping up in capability every month), is sort of like accepting Jesus after he’s returned to earth on the gleaming cloud. It’s the epistemic bare minimum.
Yes, there’s still enormous uncertainty about what the rest of our lives will look like, but as far as I can tell, there’s no longer any real uncertainty that it’ll all mostly revolve around AI, and the extent to which we succeed or fail at directing its power toward human flourishing.
By any accounting that doesn’t stack the deck, Eliezer Yudkowsky was right about what the greatest challenge facing civilization in our lifetimes was going to be, and you and I were wrong about it. Why I was wrong is a question I’ll ask myself every day in whatever time remains. But, you know, at least I updated once the prophesied wonders and terrors actually started arriving! If you haven’t done likewise, why haven’t you?
As you presumably know by now—it was the talk of the nerd internet all week—the Navier-Stokes Millennium Problem appears to be solved, with crucial contributions from both humans and AI, albeit with a tangled dispute about exactly what happened and what ought to have happened. The answer, which an OpenAI model has apparently verified in Lean, is that (as many mathematicians suspected lately) there’s smooth initial data that leads to a singularity in finite time, at least if a smooth external force is applied (the case with no external force is still unresolved). This problem was supposed to carry a $1 million prize, except that OpenAI says they have no interest in collecting the prize and it’s unclear if any human is eligible to collect instead. OpenAI burned at least ~$15 million in compute to produce its 166-page solution, which probably hasn’t yet been read and understood by any human.
See here for the Quanta article, and here for NYU mathematician Tristan Buckmaster’s account of the role played by himself and Levent Alpöge of Anthropic, which substantially differs from OpenAI’s account (you can read a response from OpenAI’s Sebastian Bubeck here). It’s agreed that everything built on an approach pioneered in recent years by the human mathematicians Diego Córdoba and Luis Martínez-Zoroa.
My purpose here is not to adjudicate the dispute. Yes, in swooping in with vastly greater resources once it had gotten wind of progress on Navier-Stokes, OpenAI seems to have acted in a way that some might describe as “unsportsmanlike.” No, I don’t find it plausible that OpenAI’s models meaningfully benefitted from being trained on Buckmaster and Alpöge’s chat logs. But this leaves a crucial question unanswered: what exactly did OpenAI know about Buckmaster and Alpöge‘s work and when did it know it?
Anyway, as Zvi points out, it’s easy to get hung up on the details and lose sight of the high-order bit: namely, that it seems safe to say that human mathematicians are forevermore dethroned as the main theorem-proving entities on planet earth. I feel privileged to have had the traditional kind of career in theoretical computer science in the last decades when that was possible.
If we were just talking about Navier-Stokes, you might accuse me of jumping to conclusions here. But we’re not. In the areas I know best (such as quantum complexity theory), and presumably other areas as well, there’s now a deluge, with longstanding open problems both major and minor falling by the day.
Go to the arXiv or ECCC. Pretty much all the papers that I’d be interested in now include “AI statements” near the acknowledgments (as this is often the central thing I want to know, I wish I didn’t need to scroll to the end of the paper to find it!). These statements can range from “our main result came entirely from GPT-6, but we understood it and take responsibility for it,” to “the results came from an interaction between the human authors and AI” to “we used AI, but only for proofreading and other incidental things” to (mad props!) “the author did not use AI for anything.”
If you talk right now to editors or program committee chairs, it’ll remind you of those ominous scenes from the Lord of the Rings movies where the men of Gondor or Rohan or whatever are grimly fortifying their walled city against the expected onslaught of 50,000 orcs. Reviewing will have to be done partly by AI, because otherwise there’s no way to handle the orc army: the reviewers can’t unilaterally disarm.
Anyway, here’s a small sampling of the significant AI-proved or -assisted results from, like, the last month, besides Navier-Stokes—restricting myself to those that solved longstanding open problems I had previously known or cared about.
- Of course, the counterexample to the Jacobian conjecture, announced by Levent Alpöge in a now-famous tweet: “hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final” (followed by a listing of the counterexample)
- Improved bounds for Grothendieck’s constant (led by friends and colleagues of mine at UT Austin)
- A Lean-verified proof of Fermat’s Last Theorem
- Quantum oracle separation between QMA and QMA(2), and proof of Watrous’s disentangler conjecture, a problem that I and others popularized back in 2007—by a list of authors including my recently graduated PhD student Sabee Grewal
- A proof of perfect completeness for QMA, from (again) Sabee Grewal and Dorian Rudolph, solving a decades-old open problem that I studied back in 2009
- An improved upper bound for shadow tomography of quantum states, from Chen, O’Donnell, Pelecanos, and Wright, improving the dependence on the Hilbert space dimension d from log(d) to √log(d). (When I introduced shadow tomography back in 2017, I raised the question of whether the dependence on d could be eliminated entirely, while preserving polylogarithmic dependence on the number of measurements m.)
Progress on the Aaronson-Ambainis Conjecture (the version that talks directly about quantum algorithms), basically showing that it holds for quantum algorithms that make their queries in a small number of parallel rounds.(Update: Nope, sorry, Jordan Docter points out to me that this one was pre-AI, with AI used only for proofreading and other incidental things!) This was independently achieved by Liu and Mutreja, making more substantial use of AI.- According to rumors that I’ve heard, solutions to some very longstanding open problems in theoretical computer science (no, not P≠NP or other complexity class separations, but think about some of our other biggest problems). I’m told that the AI companies, having been burned by the hostile response to the Navier-Stokes proof, are now sitting on solutions to some very major problems until they figure out a better way to handle things
Feel free to remind me of anything I left out.
Let me try to convey the mood in the mathematical community right now, at least as far as my experience reaches. Nearly every conversation is about the AI tsunami, or eventually circles around to the tsunami even if it’s originally about something else. Often, though, the focus is less on the unknowable future—for how much longer will mathematical research as a human enterprise even exist?—than on immediate questions of how to respond.
What are the new rules for when you get to write a paper with your name on it, and, y’know, get credit for it? That you fully understand the proof, can give talks about the proof, can answer questions about it, take responsibility for its correctness? Do you need to have played any role in finding the proof?
In the cases, likely to become more and more numerous, where all of those conditions are not satisfied, how do you share AI-generated math, if at all? Do you tweet it, like Alpöge hilariously did with Fable’s disproof of the Jacobian Conjecture? Do you post to the arXiv or GitHub? Do you publish a paper that lists “GPT-6 Astra” or “Claude Fable” as the author—but then let the AI profusely thank you in the acknowledgments for suggesting such a wonderful problem to it?
Of course, how one responds to the immediate problems ultimately does depend on one’s broader beliefs about what mathematical research is for and about. Are we just trying to decide whether various conjectures are true or false? Or are we trying to maintain a human community, across the generations, that understands the conjectures and cares about whether they’re true or false and why? If the latter, how do we incentivize people to join that community, to undergo the years of intense training required, if their role will now be reduced to verifiers and explicators (if even that) of gargantuan arguments dumped into their laps by the AI companies?
As many of you will have seen, twenty-five Fields Medalists, including Terence Tao, released an open letter entitled A Severe Misalignment of AI in Mathematics, which articulates some of these concerns in the wake of the Navier-Stokes announcement. As many critics have pointed out, the open letter doesn’t really have a clear ask: mostly, it just eloquently sets out the values of the human mathematical community that the authors consider worth preserving in the age of AI. After reflection, I decided to endorse the statement, because I want to preserve those values as well.
I don’t think any of the signatories are naïve enough to imagine that AI won’t permanently change the way mathematical research is done—indeed, that it isn’t already doing so. There’s surely at most a tiny market for “certified organic theorems.” That isn’t the question. The question is, do we incorporate AI in a way that still puts human understanding, of what either humans or AIs are producing, at the center of the whole enterprise? Maybe someday, it becomes unsustainable to do that. Maybe someday we say: “human math had a great 4,000-year run, but today we close up shop and turn everything over to the machines, continuing to apply our own brains to math, when we do, at most for exercise, recreation, or competition, like chess.”
But, partly because of my worries about AI misalignment, I’m not ready to throw in the towel just yet. I still do want to keep insight and understanding at the center of what mathematicians, computer scientists, and physicists do, for as long as we can keep it there, even as the human race now cedes its supremacy at the task of proving or disproving conjectures.
Speaking of alignment: if you’re any kind of mathematical researcher, and the present age of wonders and terrors has inspired you to want to spend your remaining time confronting the tsunami head-on, rather than pretending it doesn’t exist or is still far away, please join your dozens of colleagues who’ve arrived at the same place!
My friend and colleague Mike Winer was trained as a theoretical physicist, did a postdoc with Juan Maldacena at the Institute for Advanced Study in Princeton, but then got AGI-pilled and decided to switch to full-time work at the Alignment Research Center in Berkeley (founded by Paul Christiano, who moved to AI alignment a decade ago after doing quantum computing theory with me). Mike recently wrote a Substack post entitled From Academia to Alignment, which I enjoyed and which I’d commend to anyone currently considering this transition. In a similar vein, see this from Xiaoyu He. And, one more: a meditation on mathematicians’ possible future as priests or monks, by Stanford math undergrad Logan Graves.
), electrons (
) or gluons and find the best info experiments have provided on properties like their mass and charge. It’s a great place to get an authoritative number for any particle physics property you’re interested in.
means “greater than”; and 
, “greater than or equal to.” Don’t translate 
and 
.
. Evolve it under 
…”).
defines a regime, not a limit. The conditions 
and 
define limits and are inequivalent to 
Then Y.”
then Y.
. Substituting into Eq. (1) yields [equation].
, 
denote the measurement outcome.
denote the Pauli-
operator of qubit 
:
, encoding directionality.
-axis
, which means 
to physicists. Use 
instead, unless you’re writing for engineers (who denote 
(\ell) instead.
, –, 
, or 
. Find the symbol immediately rightward of the initial = sign. Begin the new line directly below that symbol.
in the example under (A). Put the corresponding 
means “approximately equals”; and ~, “scales as.” Approximations convey more information than scaling relations do.
For example, if 
is odd, then 
and 
fit the template.
isn’t a matrix; a matrix represents an operator in terms of a particular basis. Therefore, no equals sign should interrelate an 
of real numbers
and 
︎
case had been proved by Euler over 200 years earlier, I decided that that would be a good place to start: once I had sorted that out, I was optimistic that I would be ready to tackle the general case.
had no solution. Rather, I just worked out some successive differences and stared at them, trying to get some idea of why none of them was a perfect cube. (I should be clear that this story is a reconstruction of what I think probably happened given the few memory traces that remain half a century later rather than a completely reliable account.) At some point, I had the idea of taking the difference sequence of the difference sequence, and discovered that it formed an arithmetic progression. That felt like progress, so I investigated difference sequences a bit more and discovered, purely empirically, the rule that if you start with 
.
case had been proved by Euler over 200 years earlier, I decided that that would be a good place to start: once I had sorted that out, I was optimistic that I would be ready to tackle the general case.
had no solution. Rather, I just worked out some successive differences and stared at them, trying to get some idea of why none of them was a perfect cube. (I should be clear that this story is a reconstruction of what I think probably happened given the few memory traces that remain half a century later rather than a completely reliable account.) At some point, I had the idea of taking the difference sequence of the difference sequence, and discovered that it formed an arithmetic progression. That felt like progress, so I investigated difference sequences a bit more and discovered, purely empirically, the rule that if you start with 
th powers and keep taking successive differences, then eventually you get to the constant sequence 
.
_28_(cropped)).jpg)
Going up to 
gives the room to include one generation of fermions.
Bokor, Boyle and I tried using this to get three generations of fermions. There are things that make this tempting: not just the fact that 
is bigger, but the fact that the Jordan pair you get from it has a kind of three-fold symmetry. But we couldn’t get it to work.
into the Lie algebra 
that fits between them:
of dimension 
acts on 
which is revealed when we put the Standard Model Lie algebra 
for the space of column vectors with two bioctonion entries.![[x,y,z]=\frac{1}{2}(x(y^{\dagger}z)+z(y^{\dagger}x))](https://s0.wp.com/latex.php?latex=%5Bx%2Cy%2Cz%5D%3D%5Cfrac%7B1%7D%7B2%7D%28x%28y%5E%7B%5Cdagger%7Dz%29%2Bz%28y%5E%7B%5Cdagger%7Dx%29%29+&bg=ffffff&fg=333333&s=0&c=20201002)
-graded real Lie algebra
The Lie algebra 
consists of all linear maps from 
to itself that are of this form:![x \mapsto [a,b,x] - [b,a,x]](https://s0.wp.com/latex.php?latex=x+%5Cmapsto+%5Ba%2Cb%2Cx%5D+-+%5Bb%2Ca%2Cx%5D+&bg=ffffff&fg=333333&s=0&c=20201002)
These maps are called real inner derivations. They form a Lie algebra since the commutator of two such maps is another such map. With a bit more work we can define other operations making all of 
into a 
and a Lie subalgebra 
whose Lie algebra is 
whose Lie algebra is 
is a nice kind of manifold called a 
Our original Jordan triple, 
is then the tangent space of that point! So, ![\mathfrak{e}_6 = \big[\mathfrak{so}(10) \textstyle{\oplus} \mathfrak{u}(1)\big] \textstyle{\oplus} \mathbb{O}_\mathbb{C}^2](https://s0.wp.com/latex.php?latex=%5Cmathfrak%7Be%7D_6+%3D+%5Cbig%5B%5Cmathfrak%7Bso%7D%2810%29+%5Ctextstyle%7B%5Coplus%7D+%5Cmathfrak%7Bu%7D%281%29%5Cbig%5D+%5Ctextstyle%7B%5Coplus%7D+%5Cmathbb%7BO%7D_%5Cmathbb%7BC%7D%5E2+&bg=ffffff&fg=333333&s=0&c=20201002)
The even part of our 3-graded Lie algebra, 
generates the stabilizer of a point in the bioctonionic plane. The odd part, our friend 
is the tangent space of that point.
while the odd part itself transforms as the 16-dimensional complex spinor representation of 
Ignoring the extra 
for a moment, this is exactly what we see in a 
grand unified theory: gauge bosons in the adjoint representation, and one generation of fermions in the 16-dimensional spinor representation.
In a Jordan triple 
their role is played by tripotents: elements 
with![[e,e,e] = e](https://s0.wp.com/latex.php?latex=%5Be%2Ce%2Ce%5D+%3D+e+&bg=ffffff&fg=333333&s=0&c=20201002)
![w \mapsto [e,e,w]](https://s0.wp.com/latex.php?latex=w+%5Cmapsto+%5Be%2Ce%2Cw%5D&bg=ffffff&fg=333333&s=0&c=20201002)
has eigenvalues 0, 1/2, and 1, so 
A tripotent is called minimal when its Peirce 1-space is one-dimensional. Two tripotents 
are called colinear when each lies in the other’s Peirce 1/2-space.![[-,-,-]](https://s0.wp.com/latex.php?latex=%5B-%2C-%2C-%5D&bg=ffffff&fg=333333&s=0&c=20201002)
is linear in the first and last slot, but conjugate-linear in the middle slot. So, if you multiply a tripotent by a phase 
you get a new tripotent:![[\alpha e, \alpha e, \alpha e] = \alpha \overline{\alpha} \alpha e = \alpha e](https://s0.wp.com/latex.php?latex=%5B%5Calpha+e%2C+%5Calpha+e%2C+%5Calpha+e%5D+%3D+%5Calpha+%5Coverline%7B%5Calpha%7D+%5Calpha+e+%3D+%5Calpha+e&bg=ffffff&fg=333333&s=0&c=20201002)
(even part in brackets) 
![\mathfrak{e}_6 = [\mathfrak{so}(10) \oplus \mathfrak{u}(1)] \oplus \mathbb{O}_\mathbb{C}^2](https://s0.wp.com/latex.php?latex=%5Cmathfrak%7Be%7D_6+%3D+%5B%5Cmathfrak%7Bso%7D%2810%29+%5Coplus+%5Cmathfrak%7Bu%7D%281%29%5D+%5Coplus+%5Cmathbb%7BO%7D_%5Cmathbb%7BC%7D%5E2&bg=ffffff&fg=333333&s=0&c=20201002)
![\mathfrak{so}(10) = [\mathfrak{su}(5) \oplus \mathfrak{u}(1)] \oplus \mathfrak{a}_5(\mathbb{C})](https://s0.wp.com/latex.php?latex=%5Cmathfrak%7Bso%7D%2810%29+%3D+%5B%5Cmathfrak%7Bsu%7D%285%29+%5Coplus+%5Cmathfrak%7Bu%7D%281%29%5D+%5Coplus+%5Cmathfrak%7Ba%7D_5%28%5Cmathbb%7BC%7D%29&bg=ffffff&fg=333333&s=0&c=20201002)
![\mathfrak{su}(5) = [\mathfrak{g}_{\mathrm{SM}}] \oplus \mathrm{M}_{3,2}(\mathbb{C})](https://s0.wp.com/latex.php?latex=%5Cmathfrak%7Bsu%7D%285%29+%3D+%5B%5Cmathfrak%7Bg%7D_%7B%5Cmathrm%7BSM%7D%7D%5D+%5Coplus+%5Cmathrm%7BM%7D_%7B3%2C2%7D%28%5Cmathbb%7BC%7D%29&bg=ffffff&fg=333333&s=0&c=20201002)
is the Jordan triple of antisymmetric 
complex matrices, 
is the Jordan triple of 
complex matrices, 
and
Descend the table twice:
which has real inner automorphism group 
Its Peirce 
-space is 
with real inner automorphism group 
(colinear with 
so living in 
). Its Peirce 
with real inner automorphism group exactly 
and 
sits inside the original group 
At each we step take the subgroup that acts with determinant 
and preserves the chosen tripotent up to a phase; this gives a chain of subgroups whose members are 
and 
so we get the embeddings
grand unified theory down to the Standard Model. And it’s well known that restricting the 16-dimensional complex spinor representation of 
along this chain gives precisely the Standard Model representation 
on one generation of fermions! So we get one generation of Standard Model fermions this way.
we have projections 
and 
onto its three eigenspaces: its so-called Peirce projectors. Since we get the Standard Model structure using two minimal tripotents 
, there are nine composites of two Peirce projectors we can apply to this triple. This is how we pick out the different kinds of fermions!
Six irreducible pieces, six particle types.
Those six match up one-to-one with the particle types:
1, +1/3)
and 
—all vanish, which is why we land on six pieces and not nine.
the representation 
and even the split of one generation into its six particle multiplets as distinct Peirce components—all comes out of the single object 
you get the same result by choosing three mutually colinear tripotents instead of two—but for that, read our paper!
(where there are 
3’s) grows superexponentially in 
there exists an integer 
such that 
and 
be two 
be a linear map from 
–isomorphism if there exists 
such that 
for every 
. By rescaling we can always take 
to be 1, in which case we have that 
for every 
, then this tells us that 
between 
from 
. (The idea of the proof is as follows: pick inside the unit ball of 
for any two 
and 
. The identity map between these two spaces is an 
Hadamard matrix, then the corresponding linear map is a ![p,q\in[1,\infty]](https://s0.wp.com/latex.php?latex=p%2Cq%5Cin%5B1%2C%5Cinfty%5D&bg=ffffff&fg=333333&s=0&c=20201002)
the Banach-Mazur distance between 
and 
is 
. It is also easy to show that 
, so 
-spaces hardly improve on the easy lower bound, and do not improve on it at all in dimensions 
,
for the set of all 
.) In words, there is a positive constant 
such that for every positive integer 
.
for the set of primes. And yet, Vinogradov’s result is unquestionably a theorem, while Gluskin’s result is unquestionably a counterexample, or at least an example.
, whereas for Gluskin’s theorem the statement to be proved is
and 
,
.
can be converted into an existentially quantifed statement 
. (For this to be an equivalence one needs the axiom of choice, but it is certainly a sufficient condition.) This is not just a piece of logical trickery, but it often reflects quite accurately how we think about some problems. For instance, it is more natural to think of Gluskin’s example as a recipe for constructing (or at least proving the existence of) a pair of suitable normed spaces for any given dimension 
to pairs of normed spaces by giving its value at each 
does not belong to the convex hull of a certain compact set 
and a 
such that 
and 
for every 
. In both these cases it feels natural to regard the result as a theorem that is proved via an existential statement, perhaps because it is the theorem that is ultimately what interests us. But using “what interests us” as a criterion to determine what counts as a counterexample seems a little vague, and is a difficult criterion to use if we want to explain convincingly why AI should be good at finding counterexamples.
, then I may well look for a property 
that follows from 
, it is often the case that the main question in our minds is less, “Why is 
is a counterexample to the universally quantified statement 
. However, we do not describe all existential statements as counterexamples. For example, if I were to say, “The 
are all separable, as is 
, but 
is not separable,” I would not describe the second part of that assertion as a counterexample to the claim that all Banach spaces are separable. Rather, I would present it as probably the most basic example of a non-separable space. The important point seems to be that there was no particular reason to think that all Banach spaces would be separable, and finding an example of a non-separable space is not very difficult.
of them, but that bound can be improved if instead you observe that a complete 5-partite graph can be written as a union of two triangle-free subgraphs, and therefore it is possible to write the complete graph as a union of 
triangle-free graphs. It is then tempting to try to do better, with triangle-free graphs that are less dense but that make up for it with unbounded chromatic number — a necessary condition if one wishes to use a sublogarithmic number of graphs, which is equivalent to showing a superexponential lower bound for 
. All this is to say that when I worked on the problem, my efforts were concentrated on what turned out to be the right direction, so for me OpenAI found an example of what I (weakly) expected, rather than a counterexample.
that is small enough for later arguments to work. But when we search for an example of an object 
), it is often not a good strategy to specify 
. If we are lucky we can describe in a nice way a very general class of objects 
and show that 
satisfies 
of a certain type. The problem is then reduced to finding 
, it can be surprisingly helpful to start by attempting to prove the statement 
. The reason this can be helpful is that using our standard methods of attempting to prove something, we may end up identifying a key lemma that would suffice: that is, we may find an intermediate property 
in a non-trivial way and thus reduce the problem 
. Turning things round again, it may well then be that finding a counterexample to 
not because we think it has a chance of working (if we did, then we would be using the first strategy), but because we hope that if 
that does not have that defect. Again, this strategy can either be iterated or combined with one or more of the other strategies.
, each of which is, individually, quite easy to satisfy. In such situations, we often “build” 
.
︎
is easily checked to be 
. However, the map 
, 
and 
are homogeneous with respect to the grading where 
, 
and 
; their degrees are 
, 
and 
. I’m not sure what to make of this, but it surely matters.
, you get a cubic relation in the remaining variable. Here they are
cubic. Then the discriminants of the three cubics are 
, 
, 
where 
have no common zeroes. Roughly speaking, our map should have special behavior over the loci 
, 
, 
and 
. The fact that $p$, $q$ and $r$ each appear cubed means that the variables 
and 
should have three fold branching over the loci 
and magnetic permeability 
of the vacuum, whose product is 
. This is convenient but not necessary, as explained here:
is zero, since this is typically close to true in a plasma. However Heras does not do this, nor does the original paper:
.
.