How many columns a decade of accuracy costs, measured and predicted, at q = 0.500

At its defaults it draws how many columns a decade of accuracy costs, measured and predicted, at q = 0.500. The number of singular values above ε, against the number of digits ε asks for, on a 128 × 128 block between two intervals separated by a gap of 1. The measured curve is a straight line at 0.55 columns a decade: a digit costs the same handful of columns wherever you buy it, which is why the accuracy is a knob and not a cliff. The upper line is what the geometry alone promises — q^(p+1)/(1 − q) below ε, solved for p, using four numbers and no entry of the matrix — at 3.32 columns a decade. Both are straight and they are not the same straight: the bound is right about the shape and loose by 5.3× about the constant, which is the safe direction for a quantity you have to allocate storage from.

rank-accuracy is one function in lib/figures/kernel.js — one block — where a matrix with no zero entries turns out to have five useful columns. 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 number of singular values above ε, against the number of digits ε asks for, on a 128 × 128 block between two intervals separated by a gap of 1. The measured curve is a straight line at 0.55 columns a decade: a digit costs the same handful of columns wherever you buy it, which is why the accuracy is a knob and not a cliff. The upper line is what the geometry alone promises — q^(p+1)/(1 − q) below ε, solved for p, using four numbers and no entry of the matrix — at 3.32 columns a decade. Both are straight and they are not the same straight: the bound is right about the shape and loose by 5.3× about the constant, which is the safe direction for a quantity you have to allocate storage from.

The number of singular values above ε, against the number of digits ε asks for, on a 128 × 128 block between two intervals separated by a gap of 1. The measured curve is a straight line at 0.55 columns a decade: a digit costs the same handful of columns wherever you buy it, which is why the accuracy is a knob and not a cliff. The upper line is what the geometry alone promises — q^(p+1)/(1 − q) below ε, solved for p, using four numbers and no entry of the matrix — at 3.32 columns a decade. Both are straight and they are not the same straight: the bound is right about the shape and loose by 5.3× about the constant, which is the safe direction for a quantity you have to allocate storage from.

The number of singular values above ε, against the number of digits ε asks for, on a 128 × 128 block between two intervals separated by a gap of 8. The measured curve is a straight line at 0.32 columns a decade: a digit costs the same handful of columns wherever you buy it, which is why the accuracy is a knob and not a cliff. The upper line is what the geometry alone promises — q^(p+1)/(1 − q) below ε, solved for p, using four numbers and no entry of the matrix — at 1.00 columns a decade. Both are straight and they are not the same straight: the bound is right about the shape and loose by 3.0× about the constant, which is the safe direction for a quantity you have to allocate storage from.

The number of singular values above ε, against the number of digits ε asks for, on a 128 × 128 block between two intervals separated by a gap of 2. The measured curve is a straight line at 0.50 columns a decade: a digit costs the same handful of columns wherever you buy it, which is why the accuracy is a knob and not a cliff. The upper line is what the geometry alone promises — q^(p+1)/(1 − q) below ε, solved for p, using four numbers and no entry of the matrix — at 2.11 columns a decade. Both are straight and they are not the same straight: the bound is right about the shape and loose by 4.3× about the constant, which is the safe direction for a quantity you have to allocate storage from.

The number of singular values above ε, against the number of digits ε asks for, on a 128 × 128 block between two intervals separated by a gap of 3. The measured curve is a straight line at 0.45 columns a decade: a digit costs the same handful of columns wherever you buy it, which is why the accuracy is a knob and not a cliff. The upper line is what the geometry alone promises — q^(p+1)/(1 − q) below ε, solved for p, using four numbers and no entry of the matrix — at 1.68 columns a decade. Both are straight and they are not the same straight: the bound is right about the shape and loose by 4.0× about the constant, which is the safe direction for a quantity you have to allocate storage from.

The number of singular values above ε, against the number of digits ε asks for, on a 128 × 128 block between two intervals separated by a gap of 4. The measured curve is a straight line at 0.39 columns a decade: a digit costs the same handful of columns wherever you buy it, which is why the accuracy is a knob and not a cliff. The upper line is what the geometry alone promises — q^(p+1)/(1 − q) below ε, solved for p, using four numbers and no entry of the matrix — at 1.45 columns a decade. Both are straight and they are not the same straight: the bound is right about the shape and loose by 3.5× about the constant, which is the safe direction for a quantity you have to allocate storage from.

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.

13 distinct claims across 6 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 rankAgainstAccuracy, 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.