Mathe-News! 🚨 Die Jacobi-Vermutung wurde mit einem Tweet widerlegt!

There's more sensational news

from the world of mathematics, because, hey everyone,

by the way, the Jacobi conjecture is

wrong. Levent just

casually writes on X, formerly Twitter. He thanks

his friend who asked around

and his second good friend

Fable, the AI ​​model from

Anthropic, which he started up.

And that provided a counterexample to the

Jacobi hypothesis. Everyone can now

calculate and verify this for themselves

. Indeed, this is a

counterexample and this is

incredible news, because this Jacobi

conjecture has been a part of mathematics for

decades now. It was a

highly renowned, unsolved problem,

and apparently there is a rather

basic counterexample that

no one has noticed yet. What's going on there?

In this video, we'll take a closer look at the mathematics behind it, the history, and what's going on right now.

As early as 1939, Heinrich Keller

suggested that if one

has a polynomial function from the immense space to the immense space and

its codomain matrix

is ​​constantly non-zero in the determinant, then

an inverse mapping also

exists as a polynomial function. So, I'll just

put it simply using

complex vector spaces and complex

numbers. One could

take any body with characteristic zero here.

You can also think about real numbers.

And if we now

map from the nimensional space here, it means that there

are, so to speak, n variables on which

everything depends, and that we are mapping into the

nimensional vector space

. This means we have, so to speak, n

different coordinate components, which are

all individual functions that

depend on the variables.

In the case of n = 2, one could

imagine that there are two variables x and

Y, and that a vector with

two components always results as an image

, which depends on x and Y,

and that the whole thing should be a

polynomial function

, meaning that one is only allowed to

add, subtract, and multiply.

Division by is not even allowed; this function can only be formed using plus,

minus, and multiplication

, then it is a

polynomial function. And with the Jacobi

matrix, one considers, so to speak, the

derivation from this construct. The

problem is, if you

know derivatives from school, then when I

look at the slope of a

function that has one variable,

we now have a problem. We have

several functions and

several variables, and therefore, in a

table, we get a whole

matrix where I go through each individual function in each row

and differentiate in each

column with respect to each individual

variable. So in our

example, we can

calculate all of this using the individual

derivatives. And if you now look at the

determinant, you are essentially looking at

how the volume changes at

this one point where we

made the derivative, if

I evaluate the mapping once at this

point and/or around this

point. And then,

locally, it becomes clear how the

volume will be affected, whether it will be increased or

decreased. And as long as that

is not zero, as long as the volume is

not shrunk to zero,

I can at least reverse it locally at this point

. And

the assumption is that if this

shrinking to zero doesn't

happen everywhere, and even if

this change in

volume is constant everywhere, and

only the directions

change slightly, then I can also

invert it again, even polynomially

. So in our example, if we

calculate it concretely, inverting also means that

we can

rearrange the entire system of equations for X and Y, and we get nice polynomial expressions for U and V again

, only

with plus, minus and multiplication.

However, if we modify the example just a little

and then

look at the Jacobian matrix determinant again, it

now depends on X and is

no longer non-zero everywhere. And

while we might still have

a way to rearrange it for U and V, we'll

no longer have

neat polynomials. And the Jacobi

conjecture is precisely that as long as

we somehow construct a system of equations

where the

corresponding Jacobi matrix is ​​constant and equal to

zero, we will

always find nice polynomials to invert

. This was first written down

as a conjecture in specialist literature in 1939

. And because the

statements here are relatively easy to understand

, there were many

attempts to prove them, although these occasionally

contained errors. And that thing

was still an open, unresolved

problem, and also a very well-known one. When it became

clear in 1998 that a new

century was about to begin, people

remembered how Hilbert had presented around

23 famous problems, or

problems that had since become famous, in 1900

. And then

a Fizz Medal winner also

addressed this question: what are

the problems for the next century?

Several Millennium Problems

and questions that remain unresolved to this day emerged.

Among other things, I believe it was also number 16,

the Jakobi

assumption. So, what I mean to say is that this

problem has actually been

noticed and addressed by the mathematical world.

Many have tried it. There were

also some pieces of evidence that

were presented, but they always

contained errors. And a particularly

tragic case is Yitang Sang,

who addressed this in his doctoral thesis

and was even

able to prove something more general. However, he had used a Lemmer

from his doctoral advisor, which turned out to be

incorrect, and therefore

all his proofs failed. And

then, although he was on a

pretty good path, he simply couldn't

get a position anymore because, to put it bluntly, people said

he was just talking nonsense

in his desert and that's why

he was kicked out of the academic

world, so to speak,

but he had a great comeback

because he was the first to show

that there are infinitely many

prime numbers that have a finite,

predetermined distance between them.

And the Akobi conjecture was also

notorious in that there were some

conjectures where one already knew that

these were still generalized things.

If any of them are true,

then they each imply the

Jacobi hypothesis. So the Jacobi

conjecture was already central in many

places in algebrean geometry,

and it also appeared somewhere else

. In 2005, it even became

clear that the Jakobi conjecture

is equivalent to the Dixm conjecture, where

such a physical connection

could even be established. And

then Levent just comes along, throws out

a tweet like this, all in

lowercase and so casual.

Yes, I had some calculations done here while the World Cup final was on

. The AI

told me something. Here is the result, by the way

. Do whatever you want

with it. Yes, and the crazy thing is, you

can simply do the math now. We

have specified the function here with

three variables X, Y and Z. The

three components are also present. So

now we can set up the entire

Jacobi matrix. I

tried it once. This is going to be

a rather lengthy bill.

Then you would have to

calculate the determinant. However, you can

also have the computer

calculate this perfectly. So Wolfram Alpha

can do that. Levent even provided a

direct link to

Wolfram Alpha in the very next subtweet, where you

can calculate it and see, aha,

the determinant is indeed -2,

not 0, and that's independent of the

values ​​of X, Yy, and Z.

Everything cancels out here, and in the end, you

really do get the constant -2. The

requirements are met, but

three points can still be found.

Two would have been enough; find three points

that all have the same

function value, that are mapped to the

same point, and

therefore it is

no longer possible to invert back from this point. And the

function is not injective, the

function is not invertible,

so in particular it does not have a nice

polynomial inverse. It is simply

not invertible at all and therefore

a crystal-clear, flawless

counterexample to the Jacobi conjecture.

Wow, from here on things really picked up

speed. The news spread

not only via X or Twitter,

but also directly

reached Wikipedia, where someone

had written directly. Incidentally,

a counterexample has apparently been

found here. Then, of course, other

Wikipedia authors said that just because

some guy on Twitter said he had something to say,

we can't just

change the Wikipedia article now.

Everyone said: "Yes, guys, you

just have to do the math, this is a

counterexample that works." He

even

provided links himself where the calculations are done. You

can calculate it yourself. My

other one, no, that doesn't count as a source;

a reputable source has to do that first

. But that's the beauty of

mathematics. Anyone can

calculate it. It works. And

Levent, by the way, isn't just

anyone who casually

ran the thing on the side and

discovered something he

wouldn't understand himself. Firstly, he is currently

employed by Anthropic. That

's perhaps also very important to

say. So, as an employee of Anthropic, he was able to use Fable, the best AI

model from

Anthropic,

and therefore certainly didn't have as many

limitations as some others

who try to get the maximum out of their paid version

. Especially since

Enphropic goes to great lengths to ensure that

Fable cannot be used for any

bad things and

therefore prefers to use weaker models

if you

want to do something with it, even partially during programming. And Levent

also has a top-level mathematical education.

He was a postdog at Harvard.

One also sees Princeton and Cambridge mentioned

in his CV, and his doctorate.

Father Manjul Bagava is himself a

polystyrene carrier.

It doesn't get more prestigious than that in mathematics, and

even his bachelor's supervisor Jacob

Zimmermann is not yet a

medalist, but that could

change this Thursday afternoon, German time. The

International Congress of

Mathematicians begins in Philadelphia this year, and

Jacob Zimmermann will very

likely receive a field medal there

. The Fiels medalists were

apparently already accidentally

awarded a Fiels medal, and therefore it is actually

clear that Jacob Zimmermann will also

receive a Fiels medal, and he was,

as mentioned, the supervisor of

Levent's bachelor's thesis. So he was very well

connected in mathematical circles

and definitely knows something about

mathematics. So perhaps

, in order to truly see new results at the top

, we also need people who ask the right

questions to AI and who know

what is currently possible. And who knows what other

clever hints were hidden in his prompting,

or what other directions he

tried out with Febel and the AI ​​here

. Unfortunately, he only shared the result

and not

the process leading up to it. That's a bit of a

shame, in a way.

But this isn't really a major

academic publication he's

made; he simply wanted to

share this result with the world quickly. Of course, there could also be a

lot of strategic planning and marketing by the company

behind it. We don't know any of that right now

. But even if

the AI ​​gives you such a result

, there's still a bit of a

problem: you might not be able to understand the reasoning behind the AI's

Chain of Thor

and

read its thought process. But the AI

itself also consistently labels its sources as

bad. This is generally a

problem with the latest

developments in mathematics. And these

two problems are precisely the

question now: where does this actually come from? So, firstly,

how did the AI ​​get there in its

thought process, and

secondly, what kind of data

might be contained in the AI's training data

that led it to this point, where

even the AI ​​itself cannot say exactly

where it comes from. Well, at least

for the first time, how might one get there

somehow?

Chat GPT seems to have

provided a fairly convincing answer, where

a result can be achieved without having to

pull a function out of thin air, using three

variables and a few combinations of them

, but rather by

mentioning other abstract objects. I myself

no longer understand what's going on at this

Algebrach level. But

at least here Daniel L said, ah, that

really seems to work. And

that's a plausible argument, which

other Twitter users have also suggested is about on

the level of something that could become

a university exercise

. So, there seem to be some

abstract arguments about how to get there

. And what I

find very interesting is that the example

constructed here seems to be

based on another example, which is somehow contained in

a Russian article that only

cites its own source,

which I found through a translated

Chinese tweet—so some

obscure math paper from

1999. Jakobi is mentioned directly in the title

. I can still

read enough Kürillig to see that Jakobi somehow

appears here. And as example 1 in

this paper, in this five-page

paper on page 4,

a function with two

variables actually appears, and then also

both components, i.e., of R² R²

or c² c², where the

determinant of the Jacobian matrix

is ​​actually -2 constant, independent of x and y. And this example

wasn't explicitly

mentioned again, that there are two points

that map to the same one, but

you can actually find them in this

example here as well. This is at least also

given here as an example of an irreversible mapping. And this

example has, so to speak, only one

problem: dividing

by y is calculated at one point,

and therefore it is not a polynomial. All you were allowed to do was

add, subtract, and

multiply. And if one could, so to speak, find out this

division by here,

one would also have a counterexample to the

Jacobi conjecture. And that's exactly what happens

when you introduce such a clever

third component, add a third

variable; then you

somehow manage to

multiply out this division and

actually get the example that

Fable seems to have

arrived at somehow, perhaps via this

obscure Russian paper—we do

n't know, and that's the problem:

the AI ​​doesn't know anymore whether it

used this specific Russian paper, where perhaps a

scientist, a mathematician, if they

had read it and

come up with the idea based on it

, would have cited it as a source. Well

, as I said, that's a whole topic in

itself, how AI should perhaps also

use sources and reference them again

when it uses ideas from them. Well

, we've added this third

variable, and now we

actually have a counterexample for the case n = 3,

and I could also

simply add a fourth, fifth, or sixth variable here as a dummy variable

. This is, so to speak, a

counterexample, which shows that for all n

greater than 3, the Jacobi conjecture is

false. For n = 1, it is

quite clear, because then this

matrix is ​​just a 1 x 1 matrix containing the

derivative itself. And if

the derivative is constantly non-zero, that

means I have as a function

only a straight line that has a slope

non-zero and

is therefore invertible everywhere as a

second straight line. So the case n = 1 has

always been clear, and the case nö

= 3 has now been disproven.

However, what remains open is the

case n = 2, and in a certain sense,

this is actually the most interesting

case. This is perhaps also called the

planar or

planar Jacobi Conjecture. And it is

historically quite interesting that only

last year, when a

comprehensive research was undertaken, it was

discovered that there is a paper from 1884 that has been

virtually ignored since then, in

which the Jacobi conjecture was actually formulated for the case

n= 2

. So historically,

the

only remaining case n= 2, which

was first mentioned, is

quite an interesting problem; it

should actually be the case. I believe

the 1884 paper even contains a kind of

proof of this, but it

contains an error. So, this pattern of

false evidence has been a recurring theme throughout the history of the Jacobi hypothesis.

However, for the case n = 2,

such a simple polynomial is not possible, since

the degree of Leven's polynomial

is only degree 7 compared to its opposite. This does

not work in the case with two variables. They've already

calculated everything up to over 100 degrees and

seen that it can't work.

So, if there were a counterexample

, it would have to be of at least degree

100. And in a certain sense,

the Jacobi

conjecture has not yet been definitively

defused. The really exciting

problem now is the case with

two-dimensional vector spaces 2, but

the general Jacobi conjecture, which was

often also implied by

other theorems, has now

actually been refuted for the case three and

larger. And that also means

that every theorem—and there is a whole

list of them—that the general Jacobian

conjecture would have implied is also false in

its generality in this form

. And so to speak, for all these

assumptions here there is at least one

dimension, a finite dimension, in which

the original assumption, as

formulated here,

can simply be refuted. And that, based on everything

I gathered yesterday

and have read a bit more about today

, is the current state of affairs in

mathematics. There's this big

bombshell announcement, and now you're fully

informed about what it all means.

At least I hope you enjoyed the fact

that I took a look at all of this

and have

now prepared it for you here. This shows

that sometimes open assumptions

remain open precisely because they

are also false statements, such as

the assumption about the bunk bed. And on

Thursday, not only Levenz's

Bachelor supervisor will probably receive a

Feds medal, but also Hong Wang,

who proved the Kakea suspicion.

I also made a Matte News video about that

. There's a lot going on

in mathematics.