Fruit of Preterition

Motivation: Put a budget on the amplitude of a transmitted symbol and the information-optimal input stops being a smooth distribution and becomes a handful of discrete points — a codebook — for no obvious reason. Does gradient descent find it? And does heavy-tailed noise make the codebook sharper or blurrier? The answers are 'no, not really' and 'it depends which question you're asking', both of which are more interesting than the guesses.

Background: Additive noise channels and mutual information. Smith's 1971 theorem on amplitude-constrained capacity. The Blahut-Arimoto algorithm. α-stable laws and the fractional Laplacian as the generator of a Lévy process.

Motivation: The 2D Ising model is everyone's first phase transition. Most expositions show you a picture of magnetisation jumping at Tc and call that the phase diagram. But the deeper story is that you can locate the critical point without ever running a simulation: Kramers-Wannier duality maps low-temperature physics to high-temperature physics, and the only fixed point of that map is the critical temperature. This widget puts the two sides of the duality next to each other so you can watch them collide.

Background: Two-dimensional nearest-neighbour ferromagnetic Ising model. Wolff cluster updates. Two-point correlation function. Onsager's exponent η = 1/4.

Motivation: Spectral algorithms (PCA, spectral clustering, network embeddings) ultimately care about eigenvectors of a noisy estimate of some clean matrix. Why does a big eigengap make those eigenvectors stable? The bound is one line; the geometric reason is one picture.

Background: Symmetric perturbation theory. The Rayleigh quotient as a height function on the unit sphere. Eigenvalue gap as Hessian curvature at a constrained optimum.

Motivation: How much caffeine is still in your body when you go to bed? And how does that change if you swap one big morning coffee for two smaller ones? The shape of the answer is a saturating-then-decaying curve, and the parameters are a few half-lives and one body-weight scaling.

Background: One-compartment pharmacokinetics with first-order absorption. The model is a textbook in two equations; the interesting bits are the timing decisions you make on top of it.