One graph, nudged by 10⁻¹², and 3 different partitions

At its defaults it draws one graph, nudged by 10⁻¹², and 3 different partitions. Three seven-vertex cliques joined in a triangle by three single edges, one of whose conductances is raised by 0. When all three are equal the graph has a three-fold symmetry, λ₂ and λ₃ coincide — measured here at a gap of 1.72·10⁻¹⁵ — and there is no Fiedler vector: there is a plane, and every unit vector in it is an exact eigenvector. Eight runs on this graph with its edge weights nudged by a relative 10⁻¹² — four orders above the rounding, and far below anything anybody would call a difference in the data — return 3 distinct partitions, at conductances 0.0455. All of them are correct answers to the relaxation, and on a symmetric graph they are equally good ones. Which one a library returns is decided by its starting vector, and the site has an essay about a plane surviving what its vectors do not.

partition-degeneracy 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.

Three seven-vertex cliques joined in a triangle by three single edges, one of whose conductances is raised by 0. When all three are equal the graph has a three-fold symmetry, λ₂ and λ₃ coincide — measured here at a gap of 1.72·10⁻¹⁵ — and there is no Fiedler vector: there is a plane, and every unit vector in it is an exact eigenvector. Eight runs on this graph with its edge weights nudged by a relative 10⁻¹² — four orders above the rounding, and far below anything anybody would call a difference in the data — return 3 distinct partitions, at conductances 0.0455. All of them are correct answers to the relaxation, and on a symmetric graph they are equally good ones. Which one a library returns is decided by its starting vector, and the site has an essay about a plane surviving what its vectors do not.

Three seven-vertex cliques joined in a triangle by three single edges, one of whose conductances is raised by 0. When all three are equal the graph has a three-fold symmetry, λ₂ and λ₃ coincide — measured here at a gap of 1.72·10⁻¹⁵ — and there is no Fiedler vector: there is a plane, and every unit vector in it is an exact eigenvector. Eight runs on this graph with its edge weights nudged by a relative 10⁻¹² — four orders above the rounding, and far below anything anybody would call a difference in the data — return 3 distinct partitions, at conductances 0.0455. All of them are correct answers to the relaxation, and on a symmetric graph they are equally good ones. Which one a library returns is decided by its starting vector, and the site has an essay about a plane surviving what its vectors do not.

Three seven-vertex cliques joined in a triangle by three single edges, one of whose conductances is raised by 0.001. When all three are equal the graph has a three-fold symmetry, λ₂ and λ₃ coincide — measured here at a gap of 1.85·10⁻⁴ — and there is no Fiedler vector: there is a plane, and every unit vector in it is an exact eigenvector. Eight runs on this graph with its edge weights nudged by a relative 10⁻¹² — four orders above the rounding, and far below anything anybody would call a difference in the data — return 1 distinct partition, at conductances 0.0455. All of them are correct answers to the relaxation, and on a symmetric graph they are equally good ones. Which one a library returns is decided by its starting vector, and the site has an essay about a plane surviving what its vectors do not.

Three seven-vertex cliques joined in a triangle by three single edges, one of whose conductances is raised by 0.05. When all three are equal the graph has a three-fold symmetry, λ₂ and λ₃ coincide — measured here at a gap of 0.00916 — and there is no Fiedler vector: there is a plane, and every unit vector in it is an exact eigenvector. Eight runs on this graph with its edge weights nudged by a relative 10⁻¹² — four orders above the rounding, and far below anything anybody would call a difference in the data — return 1 distinct partition, at conductances 0.0455. All of them are correct answers to the relaxation, and on a symmetric graph they are equally good ones. Which one a library returns is decided by its starting vector, and the site has an essay about a plane surviving what its vectors do not.

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.

7 distinct claims across 4 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, fiedler, 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.