Columns needed by the block between a separator's two halves, against the length of the separator

At its defaults it draws columns needed by the block between a separator's two halves, against the length of the separator. The block is 3×4, 5×6, 7×8, 11×12 across the sweep, so its size more than triples. In the separator's own numbering — the unknowns in the order they sit on the line — it needs 3, 4, 5, 6 columns at 10⁻⁸: about one more per doubling, which is a logarithm. Renumber the same Schur complement by one symmetric permutation and the same block needs 3, 5, 7, 11 — all of them at eight digits and tighter, and a multiple of the geometric count at every tolerance drawn. Nothing about the matrix changed. What changed is that a cluster of a shuffled numbering is scattered along the whole separator, so every block mixes near interactions with far ones and there is no smooth kernel left to compress. The ordering that works here is not chosen: it is the one the separator's geometry already has, which is why nested dissection leaves a compressible fill and a fill-reducing ordering chosen on the graph alone need not.

fill-rank 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 3×4, 5×6, 7×8, 11×12 across the sweep, so its size more than triples. In the separator's own numbering — the unknowns in the order they sit on the line — it needs 3, 4, 5, 6 columns at 10⁻⁸: about one more per doubling, which is a logarithm. Renumber the same Schur complement by one symmetric permutation and the same block needs 3, 5, 7, 11 — all of them at eight digits and tighter, and a multiple of the geometric count at every tolerance drawn. Nothing about the matrix changed. What changed is that a cluster of a shuffled numbering is scattered along the whole separator, so every block mixes near interactions with far ones and there is no smooth kernel left to compress. The ordering that works here is not chosen: it is the one the separator's geometry already has, which is why nested dissection leaves a compressible fill and a fill-reducing ordering chosen on the graph alone need not.

The block is 3×4, 5×6, 7×8, 11×12 across the sweep, so its size more than triples. In the separator's own numbering — the unknowns in the order they sit on the line — it needs 3, 4, 5, 6 columns at 10⁻⁸: about one more per doubling, which is a logarithm. Renumber the same Schur complement by one symmetric permutation and the same block needs 3, 5, 7, 11 — all of them at eight digits and tighter, and a multiple of the geometric count at every tolerance drawn. Nothing about the matrix changed. What changed is that a cluster of a shuffled numbering is scattered along the whole separator, so every block mixes near interactions with far ones and there is no smooth kernel left to compress. The ordering that works here is not chosen: it is the one the separator's geometry already has, which is why nested dissection leaves a compressible fill and a fill-reducing ordering chosen on the graph alone need not.

The block is 3×4, 5×6, 7×8, 11×12 across the sweep, so its size more than triples. In the separator's own numbering — the unknowns in the order they sit on the line — it needs 2, 2, 2, 2 columns at 0.01: about one more per doubling, which is a logarithm. Renumber the same Schur complement by one symmetric permutation and the same block needs 3, 4, 6, 9 — all of them at eight digits and tighter, and a multiple of the geometric count at every tolerance drawn. Nothing about the matrix changed. What changed is that a cluster of a shuffled numbering is scattered along the whole separator, so every block mixes near interactions with far ones and there is no smooth kernel left to compress. The ordering that works here is not chosen: it is the one the separator's geometry already has, which is why nested dissection leaves a compressible fill and a fill-reducing ordering chosen on the graph alone need not.

The block is 3×4, 5×6, 7×8, 11×12 across the sweep, so its size more than triples. In the separator's own numbering — the unknowns in the order they sit on the line — it needs 3, 3, 3, 4 columns at 10⁻⁴: about one more per doubling, which is a logarithm. Renumber the same Schur complement by one symmetric permutation and the same block needs 3, 5, 7, 10 — all of them at eight digits and tighter, and a multiple of the geometric count at every tolerance drawn. Nothing about the matrix changed. What changed is that a cluster of a shuffled numbering is scattered along the whole separator, so every block mixes near interactions with far ones and there is no smooth kernel left to compress. The ordering that works here is not chosen: it is the one the separator's geometry already has, which is why nested dissection leaves a compressible fill and a fill-reducing ordering chosen on the graph alone need not.

The block is 3×4, 5×6, 7×8, 11×12 across the sweep, so its size more than triples. In the separator's own numbering — the unknowns in the order they sit on the line — it needs 3, 4, 4, 5 columns at 10⁻⁶: about one more per doubling, which is a logarithm. Renumber the same Schur complement by one symmetric permutation and the same block needs 3, 5, 7, 10 — all of them at eight digits and tighter, and a multiple of the geometric count at every tolerance drawn. Nothing about the matrix changed. What changed is that a cluster of a shuffled numbering is scattered along the whole separator, so every block mixes near interactions with far ones and there is no smooth kernel left to compress. The ordering that works here is not chosen: it is the one the separator's geometry already has, which is why nested dissection leaves a compressible fill and a fill-reducing ordering chosen on the graph alone need not.

The block is 3×4, 5×6, 7×8, 11×12 across the sweep, so its size more than triples. In the separator's own numbering — the unknowns in the order they sit on the line — it needs 3, 5, 5, 6 columns at 10⁻¹⁰: about one more per doubling, which is a logarithm. Renumber the same Schur complement by one symmetric permutation and the same block needs 3, 5, 7, 11 — all of them at eight digits and tighter, and a multiple of the geometric count at every tolerance drawn. Nothing about the matrix changed. What changed is that a cluster of a shuffled numbering is scattered along the whole separator, so every block mixes near interactions with far ones and there is no smooth kernel left to compress. The ordering that works here is not chosen: it is the one the separator's geometry already has, which is why nested dissection leaves a compressible fill and a fill-reducing ordering chosen on the graph alone need not.

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.

17 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 rankAgainstSeparator, 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.