The lower bound is attained and the upper one is out by 37.3×

At its defaults it draws the lower bound is attained and the upper one is out by 37.3×. Cheeger's inequality on nine graphs of about 40 vertices each: for the normalised Laplacian's λ₂, λ₂/2 ≤ φ ≤ √(2λ₂), where φ is the conductance the sweep cut actually achieves. Each row shows the two bounds as a bar and the measured conductance as a dot inside it. The lower bound is tight on the complete at a ratio of 1. The upper bound is loosest on the barbell — by a factor of 37.3 — which is the graph in the census with a real bottleneck, and therefore the shape the inequality is always quoted about. The square root is what makes it loose: it is the price of turning a spectral quantity into a combinatorial guarantee, and it is paid where the guarantee is wanted.

cheeger-band 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.

Cheeger's inequality on nine graphs of about 40 vertices each: for the normalised Laplacian's λ₂, λ₂/2 ≤ φ ≤ √(2λ₂), where φ is the conductance the sweep cut actually achieves. Each row shows the two bounds as a bar and the measured conductance as a dot inside it. The lower bound is tight on the complete at a ratio of 1. The upper bound is loosest on the barbell — by a factor of 37.3 — which is the graph in the census with a real bottleneck, and therefore the shape the inequality is always quoted about. The square root is what makes it loose: it is the price of turning a spectral quantity into a combinatorial guarantee, and it is paid where the guarantee is wanted.

Cheeger's inequality on nine graphs of about 40 vertices each: for the normalised Laplacian's λ₂, λ₂/2 ≤ φ ≤ √(2λ₂), where φ is the conductance the sweep cut actually achieves. Each row shows the two bounds as a bar and the measured conductance as a dot inside it. The lower bound is tight on the complete at a ratio of 1. The upper bound is loosest on the barbell — by a factor of 37.3 — which is the graph in the census with a real bottleneck, and therefore the shape the inequality is always quoted about. The square root is what makes it loose: it is the price of turning a spectral quantity into a combinatorial guarantee, and it is paid where the guarantee is wanted.

Cheeger's inequality on nine graphs of about 16 vertices each: for the normalised Laplacian's λ₂, λ₂/2 ≤ φ ≤ √(2λ₂), where φ is the conductance the sweep cut actually achieves. Each row shows the two bounds as a bar and the measured conductance as a dot inside it. The lower bound is tight on the complete at a ratio of 1. The upper bound is loosest on the barbell — by a factor of 13.7 — which is the graph in the census with a real bottleneck, and therefore the shape the inequality is always quoted about. The square root is what makes it loose: it is the price of turning a spectral quantity into a combinatorial guarantee, and it is paid where the guarantee is wanted.

Cheeger's inequality on nine graphs of about 52 vertices each: for the normalised Laplacian's λ₂, λ₂/2 ≤ φ ≤ √(2λ₂), where φ is the conductance the sweep cut actually achieves. Each row shows the two bounds as a bar and the measured conductance as a dot inside it. The lower bound is tight on the complete at a ratio of 1. The upper bound is loosest on the barbell — by a factor of 49.2 — which is the graph in the census with a real bottleneck, and therefore the shape the inequality is always quoted about. The square root is what makes it loose: it is the price of turning a spectral quantity into a combinatorial guarantee, and it is paid where the guarantee is wanted.

Cheeger's inequality on nine graphs of about 32 vertices each: for the normalised Laplacian's λ₂, λ₂/2 ≤ φ ≤ √(2λ₂), where φ is the conductance the sweep cut actually achieves. Each row shows the two bounds as a bar and the measured conductance as a dot inside it. The lower bound is tight on the complete at a ratio of 1. The upper bound is loosest on the barbell — by a factor of 29.4 — which is the graph in the census with a real bottleneck, and therefore the shape the inequality is always quoted about. The square root is what makes it loose: it is the price of turning a spectral quantity into a combinatorial guarantee, and it is paid where the guarantee is wanted.

Cheeger's inequality on nine graphs of about 64 vertices each: for the normalised Laplacian's λ₂, λ₂/2 ≤ φ ≤ √(2λ₂), where φ is the conductance the sweep cut actually achieves. Each row shows the two bounds as a bar and the measured conductance as a dot inside it. The lower bound is tight on the complete at a ratio of 1. The upper bound is loosest on the barbell — by a factor of 61.2 — which is the graph in the census with a real bottleneck, and therefore the shape the inequality is always quoted about. The square root is what makes it loose: it is the price of turning a spectral quantity into a combinatorial guarantee, and it is paid where the guarantee is wanted.

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.

18 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 (cheeger), so the rule is written down as not applying, with the reason: each bar spans λ₂/2 and √(2λ₂) and the dot inside it is the conductance the sweep achieved — the figure is the inequality and its three numbers, and a badge would print the dot a second time

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.