Twenty eigenvalues, one of them zero, and the graph has one piece

At its defaults it draws twenty eigenvalues, one of them zero, and the graph has one piece. The Laplacian spectrum of two ten-vertex paths joined by a single edge of conductance 1, plotted on a logarithmic axis so the zero end is visible. The dashed line is the threshold 10⁻¹⁰‖L‖ = 1.06·10⁻⁹, and the number of eigenvalues below it is the spectral count of connected components: 1. Breadth-first search, which performs no arithmetic on any weight, says 1. The largest computed zero is 5.81·10⁻¹⁷ and the smallest eigenvalue above the threshold is 0.02462. A count of components is therefore a rank decision at a threshold, and the bridge's conductance decides which side of it λ₂ falls on — at nine orders above the rounding level, not at it.

laplacian-spectrum 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 Laplacian spectrum of two ten-vertex paths joined by a single edge of conductance 1, plotted on a logarithmic axis so the zero end is visible. The dashed line is the threshold 10⁻¹⁰‖L‖ = 1.06·10⁻⁹, and the number of eigenvalues below it is the spectral count of connected components: 1. Breadth-first search, which performs no arithmetic on any weight, says 1. The largest computed zero is 5.81·10⁻¹⁷ and the smallest eigenvalue above the threshold is 0.02462. A count of components is therefore a rank decision at a threshold, and the bridge's conductance decides which side of it λ₂ falls on — at nine orders above the rounding level, not at it.

The Laplacian spectrum of two ten-vertex paths joined by a single edge of conductance 1, plotted on a logarithmic axis so the zero end is visible. The dashed line is the threshold 10⁻¹⁰‖L‖ = 1.06·10⁻⁹, and the number of eigenvalues below it is the spectral count of connected components: 1. Breadth-first search, which performs no arithmetic on any weight, says 1. The largest computed zero is 5.81·10⁻¹⁷ and the smallest eigenvalue above the threshold is 0.02462. A count of components is therefore a rank decision at a threshold, and the bridge's conductance decides which side of it λ₂ falls on — at nine orders above the rounding level, not at it.

The Laplacian spectrum of two ten-vertex paths joined by a single edge of conductance 0.01, plotted on a logarithmic axis so the zero end is visible. The dashed line is the threshold 10⁻¹⁰‖L‖ = 1.02·10⁻⁹, and the number of eigenvalues below it is the spectral count of connected components: 1. Breadth-first search, which performs no arithmetic on any weight, says 1. The largest computed zero is 6.99·10⁻¹⁷ and the smallest eigenvalue above the threshold is 0.001891. A count of components is therefore a rank decision at a threshold, and the bridge's conductance decides which side of it λ₂ falls on — at nine orders above the rounding level, not at it.

The Laplacian spectrum of two ten-vertex paths joined by a single edge of conductance 10⁻⁴, plotted on a logarithmic axis so the zero end is visible. The dashed line is the threshold 10⁻¹⁰‖L‖ = 1.02·10⁻⁹, and the number of eigenvalues below it is the spectral count of connected components: 1. Breadth-first search, which performs no arithmetic on any weight, says 1. The largest computed zero is 9.42·10⁻¹⁷ and the smallest eigenvalue above the threshold is 1.999·10⁻⁵. A count of components is therefore a rank decision at a threshold, and the bridge's conductance decides which side of it λ₂ falls on — at nine orders above the rounding level, not at it.

The Laplacian spectrum of two ten-vertex paths joined by a single edge of conductance 10⁻⁶, plotted on a logarithmic axis so the zero end is visible. The dashed line is the threshold 10⁻¹⁰‖L‖ = 1.02·10⁻⁹, and the number of eigenvalues below it is the spectral count of connected components: 1. Breadth-first search, which performs no arithmetic on any weight, says 1. The largest computed zero is 2·10⁻¹⁷ and the smallest eigenvalue above the threshold is 2·10⁻⁷. A count of components is therefore a rank decision at a threshold, and the bridge's conductance decides which side of it λ₂ falls on — at nine orders above the rounding level, not at it.

The Laplacian spectrum of two ten-vertex paths joined by a single edge of conductance 10⁻⁸, plotted on a logarithmic axis so the zero end is visible. The dashed line is the threshold 10⁻¹⁰‖L‖ = 1.02·10⁻⁹, and the number of eigenvalues below it is the spectral count of connected components: 1. Breadth-first search, which performs no arithmetic on any weight, says 1. The largest computed zero is 5.26·10⁻¹⁷ and the smallest eigenvalue above the threshold is 2·10⁻⁹. A count of components is therefore a rank decision at a threshold, and the bridge's conductance decides which side of it λ₂ falls on — at nine orders above the rounding level, not at it.

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.

8 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 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.