cospectral-pair

At its defaults it draws two different graphs with the same spectrum, to the last digit. A triangle joined to a square at one vertex, and the complete bipartite graph K₂,₃ with one pendant edge. Six vertices and seven edges each, 12 spanning trees each, and — for the combinatorial Laplacian — the same characteristic polynomial as an integer polynomial, checked in exact arithmetic rather than by comparing computed eigenvalues. Their computed spectra differ by 5.33·10⁻¹⁵, which is the eigensolver's own rounding and not a property of the graphs. Their degree sequences are 4, 2, 2, 2, 2, 2 and 3, 3, 3, 2, 2, 1, and the first contains a triangle while the second is bipartite — so they are not the same graph by two independent readings, and every method in this field that reads only the spectrum answers about the class.

cospectral-pair 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.

A triangle joined to a square at one vertex, and the complete bipartite graph K₂,₃ with one pendant edge. Six vertices and seven edges each, 12 spanning trees each, and — for the combinatorial Laplacian — the same characteristic polynomial as an integer polynomial, checked in exact arithmetic rather than by comparing computed eigenvalues. Their computed spectra differ by 5.33·10⁻¹⁵, which is the eigensolver's own rounding and not a property of the graphs. Their degree sequences are 4, 2, 2, 2, 2, 2 and 3, 3, 3, 2, 2, 1, and the first contains a triangle while the second is bipartite — so they are not the same graph by two independent readings, and every method in this field that reads only the spectrum answers about the class.

A triangle joined to a square at one vertex, and the complete bipartite graph K₂,₃ with one pendant edge. Six vertices and seven edges each, 12 spanning trees each, and — for the combinatorial Laplacian — the same characteristic polynomial as an integer polynomial, checked in exact arithmetic rather than by comparing computed eigenvalues. Their computed spectra differ by 5.33·10⁻¹⁵, which is the eigensolver's own rounding and not a property of the graphs. Their degree sequences are 4, 2, 2, 2, 2, 2 and 3, 3, 3, 2, 2, 1, and the first contains a triangle while the second is bipartite — so they are not the same graph by two independent readings, and every method in this field that reads only the spectrum answers about the class.

A triangle joined to a square at one vertex, and the complete bipartite graph K₂,₃ with one pendant edge. Six vertices and seven edges each, 12 spanning trees each, and — for the combinatorial Laplacian — the same characteristic polynomial as an integer polynomial, checked in exact arithmetic rather than by comparing computed eigenvalues. Drawn here for the NORMALISED Laplacian D^{-1/2}LD^{-1/2}, whose spectra differ by 0.2225: the second graph is bipartite, and a bipartite graph has an eigenvalue at exactly 2, which the first cannot have. A different matrix built from the same object asks a different question. Their degree sequences are 4, 2, 2, 2, 2, 2 and 3, 3, 3, 2, 2, 1, and the first contains a triangle while the second is bipartite — so they are not the same graph by two independent readings, and every method in this field that reads only the spectrum answers about the class.

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.

15 distinct claims across 3 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, spanningTreesExact, 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 214 of 382 generators — 194 print a residual and 20 are exempt with a published reason; 168 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.