spanning-tree-count

At its defaults it draws backward stable at every size, and wrong from n = 16. The number of spanning trees of the complete graph on n vertices, by Kirchhoff's theorem: the determinant of the Laplacian with one row and column deleted, which is nⁿ⁻² by Cayley's formula and is therefore a whole number known in advance. It is computed twice — in BigInt rationals, where the answer is exact, and by the site's own pivoted elimination in binary64. The filled curve is the relative error of the elimination, which never leaves the rounding level: worst 9.38·10⁻¹⁵ across the whole range, so the algorithm is backward stable at every size. The dashed curve is the spacing between consecutive doubles at the answer's own magnitude, relative to the answer. Where they cross, at n = 16, the count has 17 digits and a binary64 has sixteen: rounding to the nearest integer cannot recover it, because the nearest two representable numbers there are 8 apart. The error is committed by the format and not by the elimination, which is a distinction no residual can make.

spanning-tree-count 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.

The number of spanning trees of the complete graph on n vertices, by Kirchhoff's theorem: the determinant of the Laplacian with one row and column deleted, which is nⁿ⁻² by Cayley's formula and is therefore a whole number known in advance. It is computed twice — in BigInt rationals, where the answer is exact, and by the site's own pivoted elimination in binary64. The filled curve is the relative error of the elimination, which never leaves the rounding level: worst 9.38·10⁻¹⁵ across the whole range, so the algorithm is backward stable at every size. The dashed curve is the spacing between consecutive doubles at the answer's own magnitude, relative to the answer. Where they cross, at n = 16, the count has 17 digits and a binary64 has sixteen: rounding to the nearest integer cannot recover it, because the nearest two representable numbers there are 8 apart. The error is committed by the format and not by the elimination, which is a distinction no residual can make.

The number of spanning trees of the complete graph on n vertices, by Kirchhoff's theorem: the determinant of the Laplacian with one row and column deleted, which is nⁿ⁻² by Cayley's formula and is therefore a whole number known in advance. It is computed twice — in BigInt rationals, where the answer is exact, and by the site's own pivoted elimination in binary64. The filled curve is the relative error of the elimination, which never leaves the rounding level: worst 9.38·10⁻¹⁵ across the whole range, so the algorithm is backward stable at every size. The dashed curve is the spacing between consecutive doubles at the answer's own magnitude, relative to the answer. Where they cross, at n = 16, the count has 17 digits and a binary64 has sixteen: rounding to the nearest integer cannot recover it, because the nearest two representable numbers there are 8 apart. The error is committed by the format and not by the elimination, which is a distinction no residual can make.

The number of spanning trees of the complete graph on n vertices, by Kirchhoff's theorem: the determinant of the Laplacian with one row and column deleted, which is nⁿ⁻² by Cayley's formula and is therefore a whole number known in advance. It is computed twice — in BigInt rationals, where the answer is exact, and by the site's own pivoted elimination in binary64. The filled curve is the relative error of the elimination, which never leaves the rounding level: worst 8.88·10⁻¹⁶ across the whole range, so the algorithm is backward stable at every size. The dashed curve is the spacing between consecutive doubles at the answer's own magnitude, relative to the answer. The error is committed by the format and not by the elimination, which is a distinction no residual can make.

The number of spanning trees of the complete graph on n vertices, by Kirchhoff's theorem: the determinant of the Laplacian with one row and column deleted, which is nⁿ⁻² by Cayley's formula and is therefore a whole number known in advance. It is computed twice — in BigInt rationals, where the answer is exact, and by the site's own pivoted elimination in binary64. The filled curve is the relative error of the elimination, which never leaves the rounding level: worst 8.88·10⁻¹⁶ across the whole range, so the algorithm is backward stable at every size. The dashed curve is the spacing between consecutive doubles at the answer's own magnitude, relative to the answer. The error is committed by the format and not by the elimination, which is a distinction no residual can make.

The number of spanning trees of the complete graph on n vertices, by Kirchhoff's theorem: the determinant of the Laplacian with one row and column deleted, which is nⁿ⁻² by Cayley's formula and is therefore a whole number known in advance. It is computed twice — in BigInt rationals, where the answer is exact, and by the site's own pivoted elimination in binary64. The filled curve is the relative error of the elimination, which never leaves the rounding level: worst 3.41·10⁻¹⁵ across the whole range, so the algorithm is backward stable at every size. The dashed curve is the spacing between consecutive doubles at the answer's own magnitude, relative to the answer. The error is committed by the format and not by the elimination, which is a distinction no residual can make.

The number of spanning trees of the complete graph on n vertices, by Kirchhoff's theorem: the determinant of the Laplacian with one row and column deleted, which is nⁿ⁻² by Cayley's formula and is therefore a whole number known in advance. It is computed twice — in BigInt rationals, where the answer is exact, and by the site's own pivoted elimination in binary64. The filled curve is the relative error of the elimination, which never leaves the rounding level: worst 6·10⁻¹⁵ across the whole range, so the algorithm is backward stable at every size. The dashed curve is the spacing between consecutive doubles at the answer's own magnitude, relative to the answer. Where they cross, at n = 16, the count has 17 digits and a binary64 has sixteen: rounding to the nearest integer cannot recover it, because the nearest two representable numbers there are 8 apart. The error is committed by the format and not by the elimination, which is a distinction no residual can make.

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.

32 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 treeCount, 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.