κ is between 1 and the total stretch, and the bound is 2.28–3.47× loose

At its defaults it draws κ is between 1 and the total stretch, and the bound is 2.28–3.47× loose. Three spanning trees of the grid 5×5 used as preconditioners. For each, the bar spans the smallest and largest eigenvalue of the graph Laplacian preconditioned by the tree's, on the subspace orthogonal to the constant vector, and the tick is the total stretch — the sum over the graph's edges of the resistance of the path the tree makes them take, which is an integer for an unweighted graph and contains no arithmetic at all. The smallest eigenvalue is 1 to 2·10⁻¹⁵ on every tree, which is a theorem rather than a measurement: a tree is a subgraph, so its quadratic form is never the larger. The largest is below the stretch on every tree, by between 2.28 and 3.47. Choosing a preconditioner is therefore choosing a tree, and it can be scored without factorising anything.

tree-stretch 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.

Three spanning trees of the grid 5×5 used as preconditioners. For each, the bar spans the smallest and largest eigenvalue of the graph Laplacian preconditioned by the tree's, on the subspace orthogonal to the constant vector, and the tick is the total stretch — the sum over the graph's edges of the resistance of the path the tree makes them take, which is an integer for an unweighted graph and contains no arithmetic at all. The smallest eigenvalue is 1 to 2·10⁻¹⁵ on every tree, which is a theorem rather than a measurement: a tree is a subgraph, so its quadratic form is never the larger. The largest is below the stretch on every tree, by between 2.28 and 3.47. Choosing a preconditioner is therefore choosing a tree, and it can be scored without factorising anything.

Three spanning trees of the grid 5×5 used as preconditioners. For each, the bar spans the smallest and largest eigenvalue of the graph Laplacian preconditioned by the tree's, on the subspace orthogonal to the constant vector, and the tick is the total stretch — the sum over the graph's edges of the resistance of the path the tree makes them take, which is an integer for an unweighted graph and contains no arithmetic at all. The smallest eigenvalue is 1 to 2·10⁻¹⁵ on every tree, which is a theorem rather than a measurement: a tree is a subgraph, so its quadratic form is never the larger. The largest is below the stretch on every tree, by between 2.28 and 3.47. Choosing a preconditioner is therefore choosing a tree, and it can be scored without factorising anything.

Three spanning trees of the grid 4×4 used as preconditioners. For each, the bar spans the smallest and largest eigenvalue of the graph Laplacian preconditioned by the tree's, on the subspace orthogonal to the constant vector, and the tick is the total stretch — the sum over the graph's edges of the resistance of the path the tree makes them take, which is an integer for an unweighted graph and contains no arithmetic at all. The smallest eigenvalue is 1 to 1.11·10⁻¹⁵ on every tree, which is a theorem rather than a measurement: a tree is a subgraph, so its quadratic form is never the larger. The largest is below the stretch on every tree, by between 2.79 and 3.1. Choosing a preconditioner is therefore choosing a tree, and it can be scored without factorising anything.

Three spanning trees of the grid 6×6 used as preconditioners. For each, the bar spans the smallest and largest eigenvalue of the graph Laplacian preconditioned by the tree's, on the subspace orthogonal to the constant vector, and the tick is the total stretch — the sum over the graph's edges of the resistance of the path the tree makes them take, which is an integer for an unweighted graph and contains no arithmetic at all. The smallest eigenvalue is 1 to 6.66·10⁻¹⁶ on every tree, which is a theorem rather than a measurement: a tree is a subgraph, so its quadratic form is never the larger. The largest is below the stretch on every tree, by between 2.44 and 4.08. Choosing a preconditioner is therefore choosing a tree, and it can be scored without factorising anything.

Three spanning trees of the grid 3×3 used as preconditioners. For each, the bar spans the smallest and largest eigenvalue of the graph Laplacian preconditioned by the tree's, on the subspace orthogonal to the constant vector, and the tick is the total stretch — the sum over the graph's edges of the resistance of the path the tree makes them take, which is an integer for an unweighted graph and contains no arithmetic at all. The smallest eigenvalue is 1 to 2.22·10⁻¹⁵ on every tree, which is a theorem rather than a measurement: a tree is a subgraph, so its quadratic form is never the larger. The largest is below the stretch on every tree, by between 1.59 and 2.55. Choosing a preconditioner is therefore choosing a tree, and it can be scored without factorising anything.

Three spanning trees of the grid 7×7 used as preconditioners. For each, the bar spans the smallest and largest eigenvalue of the graph Laplacian preconditioned by the tree's, on the subspace orthogonal to the constant vector, and the tick is the total stretch — the sum over the graph's edges of the resistance of the path the tree makes them take, which is an integer for an unweighted graph and contains no arithmetic at all. The smallest eigenvalue is 1 to 3.11·10⁻¹⁵ on every tree, which is a theorem rather than a measurement: a tree is a subgraph, so its quadratic form is never the larger. The largest is below the stretch on every tree, by between 4.12 and 4.69. Choosing a preconditioner is therefore choosing a tree, and it can be scored without factorising anything.

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 calls a factoriser without drawing a factorisation (treeSpectrum, stretch), so the rule is written down as not applying, with the reason: the bar is the preconditioned spectrum's own two ends and the tick is the total stretch, which is a combinatorial integer with no arithmetic in it; every quantity a badge could carry is already an endpoint of the bar

The exemption list is the interesting half of the rule rather than an escape hatch — it is where a decision about a figure had to be argued in one line. residualcheck refuses an exemption that is not doing work, and rejected ten of the fifteen written for the expansion's figures on exactly that ground: a figure whose vertical axis is a residual satisfies the rule by construction, and touching a factoriser does not by itself require an entry.

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.