schur-fill

At its defaults it draws the schur complement left on a 15-unknown separator, shaded by the size of each entry. A 15 × 15 grid, its middle column taken as a separator, and both halves eliminated exactly. What is left on the separator is 100 per cent nonzero — the sparsity field's result, unchanged: eliminating a variable couples everything it touched, and by the end everything is coupled to everything. The shading is what that field does not measure. The entries fall away smoothly from the diagonal, because the Schur complement is a discrete Green's function — the operator mapping data on the separator to response on the separator — and away from the diagonal that is an integral operator with a smooth kernel. The outlined block is the 7 × 8 between the separator's two halves: every entry nonzero, and 5 columns describe it to eight digits. The rank structure was not put there by the elimination. It was in the differential operator before anything was discretised.

schur-fill 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.

A 15 × 15 grid, its middle column taken as a separator, and both halves eliminated exactly. What is left on the separator is 100 per cent nonzero — the sparsity field's result, unchanged: eliminating a variable couples everything it touched, and by the end everything is coupled to everything. The shading is what that field does not measure. The entries fall away smoothly from the diagonal, because the Schur complement is a discrete Green's function — the operator mapping data on the separator to response on the separator — and away from the diagonal that is an integral operator with a smooth kernel. The outlined block is the 7 × 8 between the separator's two halves: every entry nonzero, and 5 columns describe it to eight digits. The rank structure was not put there by the elimination. It was in the differential operator before anything was discretised.

A 15 × 15 grid, its middle column taken as a separator, and both halves eliminated exactly. What is left on the separator is 100 per cent nonzero — the sparsity field's result, unchanged: eliminating a variable couples everything it touched, and by the end everything is coupled to everything. The shading is what that field does not measure. The entries fall away smoothly from the diagonal, because the Schur complement is a discrete Green's function — the operator mapping data on the separator to response on the separator — and away from the diagonal that is an integral operator with a smooth kernel. The outlined block is the 7 × 8 between the separator's two halves: every entry nonzero, and 5 columns describe it to eight digits. The rank structure was not put there by the elimination. It was in the differential operator before anything was discretised.

A 11 × 11 grid, its middle column taken as a separator, and both halves eliminated exactly. What is left on the separator is 100 per cent nonzero — the sparsity field's result, unchanged: eliminating a variable couples everything it touched, and by the end everything is coupled to everything. The shading is what that field does not measure. The entries fall away smoothly from the diagonal, because the Schur complement is a discrete Green's function — the operator mapping data on the separator to response on the separator — and away from the diagonal that is an integral operator with a smooth kernel. The outlined block is the 5 × 6 between the separator's two halves: every entry nonzero, and 4 columns describe it to eight digits. The rank structure was not put there by the elimination. It was in the differential operator before anything was discretised.

A 23 × 23 grid, its middle column taken as a separator, and both halves eliminated exactly. What is left on the separator is 100 per cent nonzero — the sparsity field's result, unchanged: eliminating a variable couples everything it touched, and by the end everything is coupled to everything. The shading is what that field does not measure. The entries fall away smoothly from the diagonal, because the Schur complement is a discrete Green's function — the operator mapping data on the separator to response on the separator — and away from the diagonal that is an integral operator with a smooth kernel. The outlined block is the 11 × 12 between the separator's two halves: every entry nonzero, and 6 columns describe it to eight digits. The rank structure was not put there by the elimination. It was in the differential operator before anything was discretised.

A 7 × 7 grid, its middle column taken as a separator, and both halves eliminated exactly. What is left on the separator is 100 per cent nonzero — the sparsity field's result, unchanged: eliminating a variable couples everything it touched, and by the end everything is coupled to everything. The shading is what that field does not measure. The entries fall away smoothly from the diagonal, because the Schur complement is a discrete Green's function — the operator mapping data on the separator to response on the separator — and away from the diagonal that is an integral operator with a smooth kernel. The outlined block is the 3 × 4 between the separator's two halves: every entry nonzero, and 3 columns describe it to eight digits. The rank structure was not put there by the elimination. It was in the differential operator before anything was discretised.

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.

4 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 epsRank, separatorSchur, 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.