fill-anisotropy

At its defaults it draws what the anisotropy does to the compressibility of the fill, on the operator this site has a field about. The five-point operator's two directions stop being equally weighted, ε running from 1 down to 0.01, and the condition number of what is left on the separator rises from 13.1 to 72.5. Every method in the iterative field gets worse across this sweep: a point smoother stops seeing one direction, a coarse grid stops representing it, and the whole anisotropy anchor exists because of it. The fill goes the other way. Its rank falls 5, 5, 4, 3, 3 — a strongly anisotropic Green's function is *closer* to separable than an isotropic one, because coupling along the weak direction has almost vanished and what is left is nearly a function of one coordinate. The fill stays 100 per cent nonzero throughout, so nothing about the amount of fill is moving; only the amount of information in it.

fill-anisotropy is one function in lib/figures/hfill.js — the fill — dense, and not independent, on the separator nested dissection leaves. 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 five-point operator's two directions stop being equally weighted, ε running from 1 down to 0.01, and the condition number of what is left on the separator rises from 13.1 to 72.5. Every method in the iterative field gets worse across this sweep: a point smoother stops seeing one direction, a coarse grid stops representing it, and the whole anisotropy anchor exists because of it. The fill goes the other way. Its rank falls 5, 5, 4, 3, 3 — a strongly anisotropic Green's function is *closer* to separable than an isotropic one, because coupling along the weak direction has almost vanished and what is left is nearly a function of one coordinate. The fill stays 100 per cent nonzero throughout, so nothing about the amount of fill is moving; only the amount of information in it.

The five-point operator's two directions stop being equally weighted, ε running from 1 down to 0.01, and the condition number of what is left on the separator rises from 13.1 to 72.5. Every method in the iterative field gets worse across this sweep: a point smoother stops seeing one direction, a coarse grid stops representing it, and the whole anisotropy anchor exists because of it. The fill goes the other way. Its rank falls 5, 5, 4, 3, 3 — a strongly anisotropic Green's function is *closer* to separable than an isotropic one, because coupling along the weak direction has almost vanished and what is left is nearly a function of one coordinate. The fill stays 100 per cent nonzero throughout, so nothing about the amount of fill is moving; only the amount of information in it.

The five-point operator's two directions stop being equally weighted, ε running from 1 down to 0.01, and the condition number of what is left on the separator rises from 9.7 to 46.0. Every method in the iterative field gets worse across this sweep: a point smoother stops seeing one direction, a coarse grid stops representing it, and the whole anisotropy anchor exists because of it. The fill goes the other way. Its rank falls 4, 4, 4, 3, 3 — a strongly anisotropic Green's function is *closer* to separable than an isotropic one, because coupling along the weak direction has almost vanished and what is left is nearly a function of one coordinate. The fill stays 100 per cent nonzero throughout, so nothing about the amount of fill is moving; only the amount of information in it.

Every figure above asserted its own claims on the way to being drawn, and a claim that failed would have failed the build rather than drawn a wrong picture. Those assertions 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.

14 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 anisotropySweep, 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 141 of 264 generators — 126 print a residual and 15 are exempt with a published reason; 123 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.