Explainer

The member the shell was extracted from: five stages bind a single neuron's figure, the four steps and the formula — click a step to trace it; the evidence is a training run, the ladder is four architecture eras on log-scale parameter counts.

One query reading four keys — edge weight IS the softmax weight, so the figure shows where the token looks before the steps say why. Evidence: heads sharpen as recall climbs over training; the ladder is context windows across four eras on a log scale.

A ball three steps down a loss bowl: the tangent is the gradient, the horizontal jump is the learning rate's share of it, the dashed circle is where the update lands. Evidence: loss falls under a cosine schedule; the ladder is four optimizers by steps to converge on the same bowl.

A thousand people as a frequency tree: 100 sick, 90 of them test positive — and so do 90 of the 900 well. The two positive boxes are the same size, which is the theorem's whole surprise drawn as area. Evidence: the posterior climbing with each further test; the ladder is base rates on a log scale.

Eight yearly bars, each split into the principal, the interest it earned, and the interest that interest earned — the third band is the only one that curves. Evidence: compound against simple growth over forty years; the ladder is doubling time by rate.

Time on the left — a two-tone signal, the sinusoid it is tested against, their product with its area shaded — and frequency on the right, where that area becomes one bar. Evidence: a square wave's partial sums converging while the Gibbs overshoot refuses to; the ladder is sample rates on a log scale.

A sequence diagram of three messages — SYN, SYN-ACK, ACK — where every arrow's slope is a delay and the numbers on the arrows are the formula, so picture and formula are the same three facts. Evidence: slow start after the handshake; the ladder is one round trip across four distances.

Entropy as area: one rectangle per symbol, width its probability and height its surprise, so every area is a contribution and the total is H — rare symbols tall and thin, common ones short and wide. Evidence: the binary entropy curve against Gini impurity; the ladder is bits per symbol from a coin to a byte.

One skewed parent and, under it on the same axis, the bells that sample means make — wide for n = 4, narrow for n = 25, the means themselves as dots. The population never becomes normal; the averages do, and their spread is σ/√n. Evidence: the standard error falling with √n against the variance falling with n; the ladder is how far the noise drops for each sample size.

A five-page web where every circle's size is its rank, computed by the power iteration the formula describes — links into a page, each linker's rank split across its out-links, a damping term that lets the surfer jump anywhere. Evidence: how many iterations the ranks take to settle at two dampings; the ladder is how far a surfer walks before jumping.

The same four logits three times: as signed bars, as exponentials that make every one positive and stretch the gaps, and as one stacked bar that adds to 100% — the negative class still gets a slice. Evidence: how the top probability and the entropy move with temperature; the ladder is how peaked the same logits become from icy to uniform.

Two bells — the well and the sick — cut by the same line: the sick's tail past it is the sensitivity, the well's is the false-positive rate, and their ratio is one number the test owns whatever the base rate. Evidence: the same ratio moving a low prior far less than a middling one; the ladder is how independent results stack on a log scale.

Three running means of a die, wild for the first rolls and narrowing into the line at 3.5 as n grows, with the ±0.5 band they enter and stay in — the single rolls never settle, their average does. The Central Limit Theorem's pair: that one says what shape the means take, this one says where they go. Evidence: how often a run is still outside the band at each n; the ladder is the rolls it takes to pin the mean to a tolerance.