Why does this exist?

I have a folder. It is full of half-finished ideas, speculative derivations, and results that are probably interesting but will definitely never make it through formal peer review — at least not at the current level of polish.

So instead of letting them rot, I’m posting them here.

The format is loose. The rigor is variable. The pinky promise is firm:

Peer review is welcome. All criticism will be posted alongside the original entry.

If you find an error, a flawed assumption, or a better framing — open an issue on GitHub. I will read it, respond to it, and append it to the post.

What to expect

Posts will look roughly like this:

  1. An idea — usually the kind that arrives at 11 pm and seems very important.
  2. Some argument — math, code, or prose, depending on what makes sense.
  3. Honest limitations — what would need to be true for this to actually hold up.
  4. Open questions — what I don’t know and am not going to pretend I do.

A taste of the math rendering

Since this blog covers scientific content, equations should work. Here’s a sanity check:

Inline math: the Gaussian integral \( \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi} \) is a classic.

Display math:

$$ \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1} = 1 - \frac{1}{3} + \frac{1}{5} - \cdots = \frac{\pi}{4} $$

And a block with \[...\] delimiters:

\[ \mathcal{F}\{f\}(\xi) = \int_{-\infty}^{\infty} f(x)\, e^{-2\pi i x \xi}\, dx \]

If those rendered correctly, we’re in business.

A taste of code

import numpy as np

def estimate_pi(n_samples: int) -> float:
    """Monte Carlo estimation of pi."""
    x, y = np.random.uniform(-1, 1, (2, n_samples))
    inside = (x**2 + y**2) <= 1.0
    return 4 * inside.mean()

print(f"π ≈ {estimate_pi(10_000_000):.6f}")

See you in the next post.