Meckes is an expert on probability theory. His breakthrough came not here but on The n-Category Café, starting here:

https://golem.ph.utexas.edu/category/2023/01/a_curious_integral.html#c061933

and on until he says "Okay, here we go." He reduces the problem of showing

\[ \displaystyle{\frac{\pi}{8} - \int_0^\infty\cos(2x)\prod_{n=1}^\infty\cos\left(\frac{x}{n} \right) d x } \]

is small to the problem of showing it's improbable that the sum of certain random variables is bigger than 4. Then he uses Hoeffding's inequality!

(2/n)

I haven't carefully sorted out the details, but I believe Meckes has shown

\[ \displaystyle{\frac{\pi}{8} - \int_0^\infty\cos(2x)\prod_{n=1}^\infty\cos\left(\frac{x}{n} \right) d x < 0.00016 } \]

We'd need more sophisticated argument to do better, like show it's less than 10⁻⁴⁰. But this is pretty good! We now have human-comprehensible explanations of why this integral is less than π/8, but not much less.

I think this is a testament to the power of social media. But....

(3/n)