Asking for more accuracy buys accuracy, and buys no agreement at all

At its defaults it draws asking for more accuracy buys accuracy, and buys no agreement at all. Conjugate gradients on a 200 × 200 matrix with κ = 10⁴, solved at four tolerances, seven partition counts each. The bar at each tolerance spans the smallest and largest forward error the seven runs produced. The bars fall by 1.5·10⁶ across the sweep, which is the tolerance doing exactly what it is for. The ratio between the top and bottom of each bar is 1.34, 1.48, 1.71, 1.17, which does not fall with the tolerance. So the disagreement between machines is not a residue of an insufficiently converged answer that a tighter tolerance would remove; it is proportional to whatever accuracy was reached, and the runs stay a fixed factor apart all the way down.

tolerance-agreement is one function in lib/figures/race.js — what a reduction decides when the decision is discrete — a step count, a rank, and a test's tolerance. 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.

Conjugate gradients on a 200 × 200 matrix with κ = 10⁴, solved at four tolerances, seven partition counts each. The bar at each tolerance spans the smallest and largest forward error the seven runs produced. The bars fall by 1.5·10⁶ across the sweep, which is the tolerance doing exactly what it is for. The ratio between the top and bottom of each bar is 1.34, 1.48, 1.71, 1.17, which does not fall with the tolerance. So the disagreement between machines is not a residue of an insufficiently converged answer that a tighter tolerance would remove; it is proportional to whatever accuracy was reached, and the runs stay a fixed factor apart all the way down.

Conjugate gradients on a 200 × 200 matrix with κ = 10⁴, solved at four tolerances, seven partition counts each. The bar at each tolerance spans the smallest and largest forward error the seven runs produced. The bars fall by 1.5·10⁶ across the sweep, which is the tolerance doing exactly what it is for. The ratio between the top and bottom of each bar is 1.34, 1.48, 1.71, 1.17, which does not fall with the tolerance. So the disagreement between machines is not a residue of an insufficiently converged answer that a tighter tolerance would remove; it is proportional to whatever accuracy was reached, and the runs stay a fixed factor apart all the way down.

Conjugate gradients on a 80 × 80 matrix with κ = 10⁴, solved at four tolerances, seven partition counts each. The bar at each tolerance spans the smallest and largest forward error the seven runs produced. The bars fall by 5.4·10⁶ across the sweep, which is the tolerance doing exactly what it is for. The ratio between the top and bottom of each bar is 1.04, 4.83, 14.83, 1.36, which does not fall with the tolerance. So the disagreement between machines is not a residue of an insufficiently converged answer that a tighter tolerance would remove; it is proportional to whatever accuracy was reached, and the runs stay a fixed factor apart all the way down.

Conjugate gradients on a 120 × 120 matrix with κ = 10⁴, solved at four tolerances, seven partition counts each. The bar at each tolerance spans the smallest and largest forward error the seven runs produced. The bars fall by 5.2·10⁵ across the sweep, which is the tolerance doing exactly what it is for. The ratio between the top and bottom of each bar is 1.06, 3.36, 1.18, 1.19, which does not fall with the tolerance. So the disagreement between machines is not a residue of an insufficiently converged answer that a tighter tolerance would remove; it is proportional to whatever accuracy was reached, and the runs stay a fixed factor apart all the way down.

Conjugate gradients on a 300 × 300 matrix with κ = 10⁴, solved at four tolerances, seven partition counts each. The bar at each tolerance spans the smallest and largest forward error the seven runs produced. The bars fall by 8.7·10⁵ across the sweep, which is the tolerance doing exactly what it is for. The ratio between the top and bottom of each bar is 1.27, 1.36, 1.11, 1.38, which does not fall with the tolerance. So the disagreement between machines is not a residue of an insufficiently converged answer that a tighter tolerance would remove; it is proportional to whatever accuracy was reached, and the runs stay a fixed factor apart all the way down.

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.

7 distinct claims across 5 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 toleranceSweep, 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.