The error that happens before any arithmetic: |g − r| on the target set

At its defaults it draws the error that happens before any arithmetic: |g − r| on the target set. γ√(λ + c) minus its approximant, plotted against the distance from the branch point at −0.4. The target set runs from -0.2 to 8.99, and its left end is 0.2 from the branch point — a choice made before anything is computed. The rational approximant with 8 poles clustered on the cut reaches 8.46·10⁻⁴; the polynomial of degree 9, which linearises to a matrix of the same size, reaches 0.00676. Both errors are committed at this stage. No residual computed later contains either of them, because a solver's residual is taken against whichever of these two functions it was handed.

approximant-error is one function in lib/figures/ratapprox.js — the problem the solver was given — the approximation committed before the arithmetic. 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.

γ√(λ + c) minus its approximant, plotted against the distance from the branch point at −0.4. The target set runs from -0.2 to 8.99, and its left end is 0.2 from the branch point — a choice made before anything is computed. The rational approximant with 8 poles clustered on the cut reaches 8.46·10⁻⁴; the polynomial of degree 9, which linearises to a matrix of the same size, reaches 0.00676. Both errors are committed at this stage. No residual computed later contains either of them, because a solver's residual is taken against whichever of these two functions it was handed.

γ√(λ + c) minus its approximant, plotted against the distance from the branch point at −0.4. The target set runs from -0.2 to 8.99, and its left end is 0.2 from the branch point — a choice made before anything is computed. The rational approximant with 8 poles clustered on the cut reaches 8.46·10⁻⁴; the polynomial of degree 9, which linearises to a matrix of the same size, reaches 0.00676. Both errors are committed at this stage. No residual computed later contains either of them, because a solver's residual is taken against whichever of these two functions it was handed.

γ√(λ + c) minus its approximant, plotted against the distance from the branch point at −0.4. The target set runs from -0.333 to 8.99, and its left end is 0.0668 from the branch point — a choice made before anything is computed. The rational approximant with 8 poles clustered on the cut reaches 0.00331; the polynomial of degree 9, which linearises to a matrix of the same size, reaches 0.0216. Both errors are committed at this stage. No residual computed later contains either of them, because a solver's residual is taken against whichever of these two functions it was handed.

γ√(λ + c) minus its approximant, plotted against the distance from the branch point at −0.4. The target set runs from -0.266 to 8.99, and its left end is 0.134 from the branch point — a choice made before anything is computed. The rational approximant with 8 poles clustered on the cut reaches 0.0015; the polynomial of degree 9, which linearises to a matrix of the same size, reaches 0.0113. Both errors are committed at this stage. No residual computed later contains either of them, because a solver's residual is taken against whichever of these two functions it was handed.

γ√(λ + c) minus its approximant, plotted against the distance from the branch point at −0.4. The target set runs from -0.0661 to 8.99, and its left end is 0.334 from the branch point — a choice made before anything is computed. The rational approximant with 8 poles clustered on the cut reaches 3.54·10⁻⁴; the polynomial of degree 9, which linearises to a matrix of the same size, reaches 0.00295. Both errors are committed at this stage. No residual computed later contains either of them, because a solver's residual is taken against whichever of these two functions it was handed.

γ√(λ + c) minus its approximant, plotted against the distance from the branch point at −0.4. The target set runs from 0.268 to 8.99, and its left end is 0.668 from the branch point — a choice made before anything is computed. The rational approximant with 8 poles clustered on the cut reaches 7.42·10⁻⁵; the polynomial of degree 9, which linearises to a matrix of the same size, reaches 6.09·10⁻⁴. Both errors are committed at this stage. No residual computed later contains either of them, because a solver's residual is taken against whichever of these two functions it was handed.

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 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 fitRational, fitPolynomial, 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.