The Irrationality Exponent of Pi is 2 - Over-literal definitions

My First AI-ism.

The FIFTH sentence in the paper reads:

"The expected value for the usual positive circle constant is 2."

I immediately wondered what "the usual positive circle constant" is.  I guessed it was pi, but I also know that I'm attempting to read a paper that I'd usually never be able to read.  So now I'm seeing some strange language, and I can only guess at what it actually means.

I consulted Gemini, and it reassured me that this was an example of the LLM generating 'over-literal definitons'.

Gemini said:

That phrasing is indeed strange, and you aren't the only one who noticed it.

That exact line—"The expected value for the usual positive circle constant is 2"—has been widely highlighted by mathematicians and readers looking through the OpenAI math catalog as a prime example of awkward AI-generated phrasing.

Human mathematicians never casually refer to $\pi$ as "the usual positive circle constant." They just call it $\pi$.

And then it went on explaining:

The circumlocution happens because of how frontier large language models generate academic text:

Over-literal definitions: When an LLM generates text, it sometimes constructs overly descriptive, roundabout names for mathematical objects as if it's introducing them from first principles, rather than using standard shorthand notation.

Training data echoes: The phrase is likely an artifact of how mathematical definitions are textually indexed in massive web-scale corpora (where textbooks might formally introduce $\pi$ as "the positive constant associated with the circumference of a circle").

It is essentially an "AI-ism"—a weird, robotic stylistic tic that slips through when an unedited machine model tries to sound formal and academic, resulting in phrases that sound bizarrely unnatural to human ears.

It'll be interesting to see if I stumble upon any more of these as I go through the paper (proof)

The Irrationality Exponent of Pi is 2 - Tangent #1

The first equation in this paper is in the introduction:

I saw this and pretty much immediately wanted to look at how the parts within the absolute value

|x - p/q|

would change given different values of x, p, and q.

I started writing C code.  Ran a couple of FOR loops over p and q, going from 1 to 1024 (starting at 1 to void a divide by zero, stop at 1024 is mostly arbitrary).  Set x to the value of pi (in the C libraries, that value is M_PI).

Looking at this very simple equation, I see that if x = p/q, then the value is going to be zero.  For numbers like pi, there is no ratio of integers that would be exactly equal.  So for pi, this value would never be zero, but it could probably be close.

So then I can run this code and then plot p vs q vs |x - p/q|, and see what it looks like.

Fascinating is my first reaction, and then confusion.

Here is q (x-axis) versus |pi - p/q| (y-axis) going from 1 to 1024 on the x-axis, and 0 to 5 on the y-axis:

The flat plateau is approximately the value of pi (indicating that the ratio p/q is very very very small).  The two curved lines meet at the bottom at 1024 / pi (about pixel 326).  The curve going up and to the right is asymptotically going towards the value of pi.  The internal structure might be an interference byproduct of the size of the points plotted and their position.

Here is p (x-axis) versus |pi - p/q| (y-axis) going from 1 to 1024 on the x-axis, and 0 to 5 on the y-axis:

This one is confusing.  The far left starts at a value of pi, but I have no idea what that slope is.  There's also other structures in there, plus some of the same likely interference patterns.

I can look at the lowest values versus the value of p.  The red dots indicate the lowest values (very very close to zero, but not quite -- meaning that this ratio is very close to the value of pi)

Here are the corresponding q values (note they end at an x-axis value of about 1024 / pi):

Looking at these two lowest values, the ratios are:

355 / 113 = 3.14159292 (difference of 0.0000002667..., the 7th decimal place)

710 / 226 = 3.14159292

(these are the same thing, an example of what the paper calls 'coprime', I think).  So in this range (letting p and q go from 1 to 1024), there is ONE ratio that is pretty close to pi, and a harmonic.

Here's the same thing, but centered on a difference of pi.  First, for every p:

and now for every q:

I then switched from using pi to using a regular integer number, like 4.

Same plots as above.  For every q:

and every p:

some differences between these and the pi ones.

The Irrationality Exponent of Pi is 2 - Prelude

OpenAI recently published proofs of over 700 problems in mathematics.  A lot of people in this field are very upset about all of this, and no one knows (at this point in time) what's going to happen next.

As soon as they came out, I started looking at the list of titles.  The first one that caught my eye was number 17: "The Irrationality Exponent of Pi is 2".  My response was instant and intense -- what the HELL is this?????

Because of this reaction, I decided to take this paper and thoroughly understand it, with the help of various language models.  I will go through the paper, and when I have a question I'll get an answer that I'll be able to understand.

This will do several things simultaneously:

1. It will test the ability of language models to explain mathematical concepts that I find impossible to understand at first.
2. It will show me areas of mathematics I had no idea about and never otherwise would have known (this is already happening -- see below)
3. It will explain to me every step of the proof process in a way that I can understand, and it can explain to me WHY something is done the way it's done.  I can even jump into history if I want to.  All questions are answered accurately.
4. I'll be able to be convinced that the proof is correct or incorrect, because I'll be able to follow the logic AND the math

Personally, I see this mass publication as a pot of gold that's just been opened.  There's amazing treasure inside!

I will document my journey through this paper on this blog.  It might be a slow process.  Slow because it's going to take a lot of explanation to get through my thick skull.  But slow also because I'll go off on tangents from time to time.  I'll make sure not to get too far off track, but I definitely think that those tangents will help me understand and appreciate the proof more.  I'll copy all of the dialog I have with the language models, although I probably won't publish those besides excerpts or anything relevant to info posted here.

I'm already going off on my first tangent, and I haven't even gotten through the first equation!

After a Heavy Rain

Flowing water art:

Bird Flight

Another long exposure of a couple of birds in flight.  This is about 30 seconds, and there's 5 images per second

There are few other moving objects in this FOV.

Here's the full view at 25 frames per second:

Emergent Patterning 5

Depending on how the 'beginning' and 'next move' are defined, the results are different.  In this example, I use three different objects that can move in these possible ways:

If the index is in a spiral pattern that goes from the center outward, I get this result:

If the index is in a spiral pattern that goes from the outside to the center, I get this different result:

If the index goes horizontally (along the x-axis), I get this:

and finally if the index goes vertically (along the y-axis), I get this:

Where the blank pixels are located is interesting, too.  Here below I'm showing all four of the above images in a mosaic, and highlighting the blank pixels in white.  I'll leave it a task for you to match the right ones together.

Cat Art