Same matrix, same arithmetic, 4.13× the fill

At its defaults it draws same matrix, same arithmetic, 4.13× the fill. Gaussian elimination on the Laplacian of the grid 6×6, under five orderings, counting the edges that did not exist before. Eliminating a vertex removes it and joins its neighbours into a clique — the star–mesh transform, which is what the Schur complement of a Laplacian does — so fill is a set of edges rather than a set of nonzeros, and the matrix that remains is still a graph: symmetric, zero row sums, nonpositive off the diagonal, checked at every step and worst at 3·10⁻¹⁵. Minimum degree fills 71 edges against 293 for the worst ordering here, on a graph with 60 to start with. Nothing about the arithmetic differs between these five runs.

elimination-fill is one function in lib/figures/graphres.js — resistance, sparsification and elimination — three routes to one number, and a count that outgrows the format. 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.

Gaussian elimination on the Laplacian of the grid 6×6, under five orderings, counting the edges that did not exist before. Eliminating a vertex removes it and joins its neighbours into a clique — the star–mesh transform, which is what the Schur complement of a Laplacian does — so fill is a set of edges rather than a set of nonzeros, and the matrix that remains is still a graph: symmetric, zero row sums, nonpositive off the diagonal, checked at every step and worst at 3·10⁻¹⁵. Minimum degree fills 71 edges against 293 for the worst ordering here, on a graph with 60 to start with. Nothing about the arithmetic differs between these five runs.

Gaussian elimination on the Laplacian of the grid 6×6, under five orderings, counting the edges that did not exist before. Eliminating a vertex removes it and joins its neighbours into a clique — the star–mesh transform, which is what the Schur complement of a Laplacian does — so fill is a set of edges rather than a set of nonzeros, and the matrix that remains is still a graph: symmetric, zero row sums, nonpositive off the diagonal, checked at every step and worst at 3·10⁻¹⁵. Minimum degree fills 71 edges against 293 for the worst ordering here, on a graph with 60 to start with. Nothing about the arithmetic differs between these five runs.

Gaussian elimination on the Laplacian of the grid 3×3, under five orderings, counting the edges that did not exist before. Eliminating a vertex removes it and joins its neighbours into a clique — the star–mesh transform, which is what the Schur complement of a Laplacian does — so fill is a set of edges rather than a set of nonzeros, and the matrix that remains is still a graph: symmetric, zero row sums, nonpositive off the diagonal, checked at every step and worst at 5.55·10⁻¹⁶. Minimum degree fills 5 edges against 17 for the worst ordering here, on a graph with 12 to start with. Nothing about the arithmetic differs between these five runs.

Gaussian elimination on the Laplacian of the grid 4×4, under five orderings, counting the edges that did not exist before. Eliminating a vertex removes it and joins its neighbours into a clique — the star–mesh transform, which is what the Schur complement of a Laplacian does — so fill is a set of edges rather than a set of nonzeros, and the matrix that remains is still a graph: symmetric, zero row sums, nonpositive off the diagonal, checked at every step and worst at 1.17·10⁻¹⁵. Minimum degree fills 18 edges against 57 for the worst ordering here, on a graph with 24 to start with. Nothing about the arithmetic differs between these five runs.

Gaussian elimination on the Laplacian of the grid 5×5, under five orderings, counting the edges that did not exist before. Eliminating a vertex removes it and joins its neighbours into a clique — the star–mesh transform, which is what the Schur complement of a Laplacian does — so fill is a set of edges rather than a set of nonzeros, and the matrix that remains is still a graph: symmetric, zero row sums, nonpositive off the diagonal, checked at every step and worst at 6.33·10⁻¹⁶. Minimum degree fills 37 edges against 143 for the worst ordering here, on a graph with 40 to start with. Nothing about the arithmetic differs between these five runs.

Gaussian elimination on the Laplacian of the grid 7×7, under five orderings, counting the edges that did not exist before. Eliminating a vertex removes it and joins its neighbours into a clique — the star–mesh transform, which is what the Schur complement of a Laplacian does — so fill is a set of edges rather than a set of nonzeros, and the matrix that remains is still a graph: symmetric, zero row sums, nonpositive off the diagonal, checked at every step and worst at 8.88·10⁻¹⁶. Minimum degree fills 119 edges against 525 for the worst ordering here, on a graph with 84 to start with. Nothing about the arithmetic differs between these five runs.

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 eliminate, 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.