What the black-box construction costs: the rank has to be chosen before anything is known

At its defaults it draws what the black-box construction costs: the rank has to be chosen before anything is known. The compression route is handed an accuracy and returns whatever rank that costs. This one is handed a rank and returns whatever accuracy that buys, because the random matrix has to be drawn before a single entry of the block has been seen. The curve is a straight line at -0.74 decades a column — from 0.0407 at k = 2 to 1.72·10⁻⁹ at k = 12 — so the guess is a guess about a number of digits, and being two columns short costs about a decade and a half. The lower line is the best representation of the same rank, so the vertical gap is what never seeing the matrix costs: 2.9, 4.5, 4.1, 4.4, 6.6, 8.1 times, widening slowly as the rank grows, because a sample of a block whose spectrum has already fallen off a cliff is sampling noise.

rank-guess is one function in lib/figures/hbuild.js — built from products — a few hundred applications of an operator that is never formed. Everything below came out of it during this build, at arguments taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

The compression route is handed an accuracy and returns whatever rank that costs. This one is handed a rank and returns whatever accuracy that buys, because the random matrix has to be drawn before a single entry of the block has been seen. The curve is a straight line at -0.74 decades a column — from 0.0407 at k = 2 to 1.72·10⁻⁹ at k = 12 — so the guess is a guess about a number of digits, and being two columns short costs about a decade and a half. The lower line is the best representation of the same rank, so the vertical gap is what never seeing the matrix costs: 2.9, 4.5, 4.1, 4.4, 6.6, 8.1 times, widening slowly as the rank grows, because a sample of a block whose spectrum has already fallen off a cliff is sampling noise.

The compression route is handed an accuracy and returns whatever rank that costs. This one is handed a rank and returns whatever accuracy that buys, because the random matrix has to be drawn before a single entry of the block has been seen. The curve is a straight line at -0.74 decades a column — from 0.0407 at k = 2 to 1.72·10⁻⁹ at k = 12 — so the guess is a guess about a number of digits, and being two columns short costs about a decade and a half. The lower line is the best representation of the same rank, so the vertical gap is what never seeing the matrix costs: 2.9, 4.5, 4.1, 4.4, 6.6, 8.1 times, widening slowly as the rank grows, because a sample of a block whose spectrum has already fallen off a cliff is sampling noise.

The compression route is handed an accuracy and returns whatever rank that costs. This one is handed a rank and returns whatever accuracy that buys, because the random matrix has to be drawn before a single entry of the block has been seen. The curve is a straight line at -1.08 decades a column — from 0.0177 at k = 2 to 2.8·10⁻¹³ at k = 12 — so the guess is a guess about a number of digits, and being two columns short costs about a decade and a half. The lower line is the best representation of the same rank, so the vertical gap is what never seeing the matrix costs: 2.2, 2.2, 4.1, 3.5, 2.8, 3.2 times, widening slowly as the rank grows, because a sample of a block whose spectrum has already fallen off a cliff is sampling noise.

Every figure above checked its own claims on the way to being drawn, and a claim that failed would have stopped the picture rather than shipped a wrong one. Those checks used to leave no trace at all: a passing one returned true and the only evidence the figure had checked anything was that nothing crashed. The list below is what they actually said, collected by running this generator with an observer installed — not a description of what it is believed to check.

6 distinct claims across 3 sets of arguments, grouped below by shape — because most of them are one sentence with a different number in it, and how many separate times that sentence was put to the test is the informative part.

It draws a decomposition and prints its residual. It calls rankSweep, and every figure above carries the badge — which residualcheck verifies by looking for it in the emitted SVG rather than by finding the call that builds one. A badge that is constructed and then left out of the body is the failure that check exists for.

Across the library: the rule bites on 217 of 397 generators — 199 print a residual and 18 are exempt with a published reason; 180 factorise nothing. Read from lib/residual-rule.js, which is the same body the gate enforces from, and the gate's last check fails the build if this page and it disagree about any generator.