two blocks 40: 40 vertices, 223 edges, and a matrix built from them

At its defaults it draws two blocks 40: 40 vertices, 223 edges, and a matrix built from them. The two blocks 40 laid out at its own second and third Laplacian eigenvectors, so the picture is the same object the measurements are about. The Laplacian L = D − A is built by subtraction of integers, so every row sums to exactly zero — measured at 0, not at the rounding level — and L·1 is the zero vector with no arithmetic error anywhere in it. Its 1 zero eigenvalue counts the connected components, which breadth-first search also puts at 1; the smallest non-zero eigenvalue is 1.458 and the largest computed zero is 4.26·10⁻¹⁶, a gap of 3.42·10¹⁵. The quadratic form xᵀLx is Σ over edges of (xᵢ − xⱼ)², which at the alternating vector is 444.

graph-picture is one function in lib/figures/graphlap.js — the matrix a graph makes — row sums that are exactly zero, a count that is a threshold, and a partition with no vector. 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 two blocks 40 laid out at its own second and third Laplacian eigenvectors, so the picture is the same object the measurements are about. The Laplacian L = D − A is built by subtraction of integers, so every row sums to exactly zero — measured at 0, not at the rounding level — and L·1 is the zero vector with no arithmetic error anywhere in it. Its 1 zero eigenvalue counts the connected components, which breadth-first search also puts at 1; the smallest non-zero eigenvalue is 1.458 and the largest computed zero is 4.26·10⁻¹⁶, a gap of 3.42·10¹⁵. The quadratic form xᵀLx is Σ over edges of (xᵢ − xⱼ)², which at the alternating vector is 444.

The two blocks 40 laid out at its own second and third Laplacian eigenvectors, so the picture is the same object the measurements are about. The Laplacian L = D − A is built by subtraction of integers, so every row sums to exactly zero — measured at 0, not at the rounding level — and L·1 is the zero vector with no arithmetic error anywhere in it. Its 1 zero eigenvalue counts the connected components, which breadth-first search also puts at 1; the smallest non-zero eigenvalue is 1.458 and the largest computed zero is 4.26·10⁻¹⁶, a gap of 3.42·10¹⁵. The quadratic form xᵀLx is Σ over edges of (xᵢ − xⱼ)², which at the alternating vector is 444.

The hypercube 5 laid out at its own second and third Laplacian eigenvectors, so the picture is the same object the measurements are about. The Laplacian L = D − A is built by subtraction of integers, so every row sums to exactly zero — measured at 0, not at the rounding level — and L·1 is the zero vector with no arithmetic error anywhere in it. Its 1 zero eigenvalue counts the connected components, which breadth-first search also puts at 1; the smallest non-zero eigenvalue is 2 and the largest computed zero is 2.7·10⁻¹⁷, a gap of 7.42·10¹⁶. The quadratic form xᵀLx is Σ over edges of (xᵢ − xⱼ)², which at the alternating vector is 64.

The star 30 laid out at its own second and third Laplacian eigenvectors, so the picture is the same object the measurements are about. The Laplacian L = D − A is built by subtraction of integers, so every row sums to exactly zero — measured at 0, not at the rounding level — and L·1 is the zero vector with no arithmetic error anywhere in it. Its 1 zero eigenvalue counts the connected components, which breadth-first search also puts at 1; the smallest non-zero eigenvalue is 1 and the largest computed zero is 1.21·10⁻¹⁶, a gap of 8.26·10¹⁵. The quadratic form xᵀLx is Σ over edges of (xᵢ − xⱼ)², which at the alternating vector is 60.

The preferential 60 laid out at its own second and third Laplacian eigenvectors, so the picture is the same object the measurements are about. The Laplacian L = D − A is built by subtraction of integers, so every row sums to exactly zero — measured at 0, not at the rounding level — and L·1 is the zero vector with no arithmetic error anywhere in it. Its 1 zero eigenvalue counts the connected components, which breadth-first search also puts at 1; the smallest non-zero eigenvalue is 0.5861 and the largest computed zero is 4.82·10⁻¹⁷, a gap of 1.22·10¹⁶. The quadratic form xᵀLx is Σ over edges of (xᵢ − xⱼ)², which at the alternating vector is 248.

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.

10 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 laplacianEig, componentCount, 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.