AI Slop or Not? Certifying the Sech⁴ Kernel Against Bochner's and Mercer's Theorems
I posted three equations to social media with one question underneath: does k_H perfectly satisfy Bochner's and Mercer's theorems?
Half the responses assumed it was AI-generated decoration. Plausible-looking symbols, impressive names, no real content underneath.
This post is my answer. I will walk through what the theorems actually demand, show why this kernel meets those demands, and back it up with a numerical certification I ran myself. You can reproduce every result:
Source Code
So Does It Satisfy Both Theorems?
Bochner: The Fourier transform of k_H is non-negative to machine precision across the full frequency grid, and the Gram matrix on 500 random points is strictly positive definite. The kernel is a legitimate positive definite kernel.
Mercer: The integral operator on [0, log N] has non-negative eigenvalues and its eigenfunction expansion reconstructs the kernel to 10⁻¹³ point-wise error with 50 terms. Mercer's theorem applies cleanly.
The honest caveat: this is numerical certification, not a formal analytic proof. The Fourier transform is only checked on a finite grid. A complete proof would derive the sech⁴ Fourier transform analytically and verify non-negativity in closed form — which can be done, but is not shown here.
