fill-spectrum

At its defaults it draws the singular values of the fill between a separator's two halves, and of the same block renumbered. The block is 11 × 12, every entry of it nonzero, and its singular values fall by a factor of 16.2 at the first step and keep falling: 1, 0.062, 0.0039, 1.3·10⁻⁴, 2·10⁻⁶. That is the same cliff a block of a kernel matrix between two intervals has, and it is the same cliff for the same reason — the Schur complement of a discrete Laplacian is a discrete Green's function, so away from the diagonal it is an integral operator with a smooth kernel. The other curve is the same block after one symmetric permutation of the separator's unknowns: 1, 0.84, 0.64, 0.63, 0.57, which is a spectrum with no cliff in it at all. Both matrices have exactly the same entries.

fill-spectrum 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 block is 11 × 12, every entry of it nonzero, and its singular values fall by a factor of 16.2 at the first step and keep falling: 1, 0.062, 0.0039, 1.3·10⁻⁴, 2·10⁻⁶. That is the same cliff a block of a kernel matrix between two intervals has, and it is the same cliff for the same reason — the Schur complement of a discrete Laplacian is a discrete Green's function, so away from the diagonal it is an integral operator with a smooth kernel. The other curve is the same block after one symmetric permutation of the separator's unknowns: 1, 0.84, 0.64, 0.63, 0.57, which is a spectrum with no cliff in it at all. Both matrices have exactly the same entries.

The block is 11 × 12, every entry of it nonzero, and its singular values fall by a factor of 16.2 at the first step and keep falling: 1, 0.062, 0.0039, 1.3·10⁻⁴, 2·10⁻⁶. That is the same cliff a block of a kernel matrix between two intervals has, and it is the same cliff for the same reason — the Schur complement of a discrete Laplacian is a discrete Green's function, so away from the diagonal it is an integral operator with a smooth kernel. The other curve is the same block after one symmetric permutation of the separator's unknowns: 1, 0.84, 0.64, 0.63, 0.57, which is a spectrum with no cliff in it at all. Both matrices have exactly the same entries.

The block is 7 × 8, every entry of it nonzero, and its singular values fall by a factor of 19.1 at the first step and keep falling: 1, 0.052, 0.0019, 2.2·10⁻⁵, 7.9·10⁻⁸. That is the same cliff a block of a kernel matrix between two intervals has, and it is the same cliff for the same reason — the Schur complement of a discrete Laplacian is a discrete Green's function, so away from the diagonal it is an integral operator with a smooth kernel. The other curve is the same block after one symmetric permutation of the separator's unknowns: 1, 0.65, 0.62, 0.53, 0.49, which is a spectrum with no cliff in it at all. Both matrices have exactly the same entries.

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.

3 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 svd, crossSpectrum, 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.