One sweep cut, two conductances: 0.1235 and 0.04

At its defaults it draws one sweep cut, two conductances: 0.1235 and 0.04. Every prefix of the sweep over the second eigenvector of Chung's Laplacian, on two blocks with 1 arc back at 24 vertices, with two quantities plotted for each. The circulation conductance is the probability that a step of the stationary walk crosses the cut, divided by the smaller side's stationary mass; the arc conductance is the weight of arcs crossing it, divided by the smaller side's degree. At the best cut they are 0.1235 and 0.04, a factor of 3.09 apart. The shaded band is Cheeger's, λ₂/2 = 0.03309 to √(2λ₂) = 0.3638 with λ₂ = 0.06619, and it is a statement about the first quantity only. Reading it as a statement about the second is the substitution this figure exists to separate.

two-conductances is one function in lib/figures/digraph.js — arrows — a laplacian that is not symmetric, and the walk that has to be computed before it can be. 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.

Every prefix of the sweep over the second eigenvector of Chung's Laplacian, on two blocks with 1 arc back at 24 vertices, with two quantities plotted for each. The circulation conductance is the probability that a step of the stationary walk crosses the cut, divided by the smaller side's stationary mass; the arc conductance is the weight of arcs crossing it, divided by the smaller side's degree. At the best cut they are 0.1235 and 0.04, a factor of 3.09 apart. The shaded band is Cheeger's, λ₂/2 = 0.03309 to √(2λ₂) = 0.3638 with λ₂ = 0.06619, and it is a statement about the first quantity only. Reading it as a statement about the second is the substitution this figure exists to separate.

Every prefix of the sweep over the second eigenvector of Chung's Laplacian, on two blocks with 1 arc back at 24 vertices, with two quantities plotted for each. The circulation conductance is the probability that a step of the stationary walk crosses the cut, divided by the smaller side's stationary mass; the arc conductance is the weight of arcs crossing it, divided by the smaller side's degree. At the best cut they are 0.1235 and 0.04, a factor of 3.09 apart. The shaded band is Cheeger's, λ₂/2 = 0.03309 to √(2λ₂) = 0.3638 with λ₂ = 0.06619, and it is a statement about the first quantity only. Reading it as a statement about the second is the substitution this figure exists to separate.

Every prefix of the sweep over the second eigenvector of Chung's Laplacian, on two blocks with 2 arcs back at 30 vertices, with two quantities plotted for each. The circulation conductance is the probability that a step of the stationary walk crosses the cut, divided by the smaller side's stationary mass; the arc conductance is the weight of arcs crossing it, divided by the smaller side's degree. At the best cut they are 0.05784 and 0.2, a factor of 3.46 apart. The shaded band is Cheeger's, λ₂/2 = 0.02152 to √(2λ₂) = 0.2934 with λ₂ = 0.04303, and it is a statement about the first quantity only. Reading it as a statement about the second is the substitution this figure exists to separate.

Every prefix of the sweep over the second eigenvector of Chung's Laplacian, on two blocks with 4 arcs back at 20 vertices, with two quantities plotted for each. The circulation conductance is the probability that a step of the stationary walk crosses the cut, divided by the smaller side's stationary mass; the arc conductance is the weight of arcs crossing it, divided by the smaller side's degree. At the best cut they are 0.1436 and 0.1304, a factor of 1.1 apart. The shaded band is Cheeger's, λ₂/2 = 0.06269 to √(2λ₂) = 0.5008 with λ₂ = 0.1254, and it is a statement about the first quantity only. Reading it as a statement about the second is the substitution this figure exists to separate.

Every prefix of the sweep over the second eigenvector of Chung's Laplacian, on two blocks with 1 arc back at 36 vertices, with two quantities plotted for each. The circulation conductance is the probability that a step of the stationary walk crosses the cut, divided by the smaller side's stationary mass; the arc conductance is the weight of arcs crossing it, divided by the smaller side's degree. At the best cut they are 0.06056 and 0.02703, a factor of 2.24 apart. The shaded band is Cheeger's, λ₂/2 = 0.01736 to √(2λ₂) = 0.2635 with λ₂ = 0.03472, and it is a statement about the first quantity only. Reading it as a statement about the second is the substitution this figure exists to separate.

Every prefix of the sweep over the second eigenvector of Chung's Laplacian, on two blocks with 8 arcs back at 44 vertices, with two quantities plotted for each. The circulation conductance is the probability that a step of the stationary walk crosses the cut, divided by the smaller side's stationary mass; the arc conductance is the weight of arcs crossing it, divided by the smaller side's degree. At the best cut they are 0.06916 and 0.125, a factor of 1.81 apart. The shaded band is Cheeger's, λ₂/2 = 0.02056 to √(2λ₂) = 0.2868 with λ₂ = 0.04113, and it is a statement about the first quantity only. Reading it as a statement about the second is the substitution this figure exists to separate.

Every figure above asserted its own claims on the way to being drawn, and a claim that failed would have failed the build rather than drawn a wrong picture. Those assertions 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.

14 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.

The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.

Across the library: the rule bites on 214 of 397 generators — 194 print a residual and 20 are exempt with a published reason; 183 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.