negative-pi-predictions

what if π had digits before 3?

this dataset contains 100,000 digits predicted by a neural network at negative positions of π.

yes, this is exactly as stupid as it sounds.

what is this?

normally, we index the fractional digits of π like this:

position:  1 2 3 4 5 6 7 8 9 ...
digit:     1 4 1 5 9 2 6 5 3 ...

so:

π = 3.141592653589793...
      ↑
      position 1

i trained a neural network to predict the digit at a given position using only the position itself.

the model was trained on the first 100,000 fractional digits of π.

then i asked it what happens at:

-1
-2
-3
...
-100000

the model, having absolutely no idea what a negative digit position means, confidently produced 100,000 digits anyway.

those predictions are this dataset.

the "negative π" convention

this is not actually π.

negative positions are outside the mathematical definition used for the training data. they are simply the values produced by the neural network when given negative integers.

for example, the model produced:

position   prediction
-3         9
-2         4
-1         4

while the normal digits begin:

position    digit
1           1
2           4
3           1
4           5
5           9

so, if you squint hard enough, you can imagine:

... 9 4 4 | 3.1415926535...
    ^^^^^
    "negative π"

this interpretation is mathematically meaningless.

that's the point.

model

the predictor is a relatively simple Fourier-feature MLP:

position
   │
   ▼
Fourier Encoding
16 frequencies → 32 dimensions
   │
   ▼
512
   │
   ▼
512
   │
   ▼
512
   │
   ▼
10 logits
   │
   ▼
digit 0–9

the model learns:

f(position) → digit

rather than predicting π autoregressively from previous digits.

why does this work?

inside the training range, the model can approximate the mapping between position and digit reasonably well.

on the training range, it achieved roughly 97% accuracy.

outside the training range, however, the behavior quickly becomes meaningless.

for example, on positions 100001–101000, which were never seen during training, the model achieved:

87 / 1000
8.70% accuracy

which is essentially random guessing over 10 possible digits.

negative positions are therefore not evidence of some hidden structure in π.

they are simply the model extrapolating its learned function into a domain where it has no reason to be correct.

dataset

• 100,000 predictions
• positions: negative integers
• task: neural-network-generated digit predictions
• source: PiPredictor
• format: Hugging Face dataset
• intended use: whatever the fuck you want
• mathematical significance: approximately zero

applications

possible applications include:

• training an AI to distinguish real mathematics from bullshit
• testing models on confidently generated nonsense
• generating cursed numerical datasets
• making your colleagues question your research priorities
• absolutely nothing

license

licensed under WTFPL.

because apparently even the license should acknowledge what is happening here.

do whatever the fuck you want with this dataset.

disclaimer

this dataset does not contain actual digits of π before the decimal point.

there is no such standard sequence.

these are merely predictions generated by a neural network that was asked a question it should never have been asked.

negative π is not real.

unless you believe hard enough.

citation

if you somehow decide to cite this dataset in an actual paper, i am deeply sorry.

akaruineko/negative-pi-predictions

---

TL;DR

i trained a neural network on 100,000 digits of π.

then i asked it:

"what comes before 3?"

it answered.

this dataset is the answer.