Resistance to 15 vertices, against an answer that is a ratio of integers

At its defaults it draws resistance to 15 vertices, against an answer that is a ratio of integers. Effective resistance from vertex 0 to every other vertex of the grid 4×4, computed twice in floating point — once by grounding a vertex and eliminating, once by summing (uₖᵢ − uₖⱼ)²/λₖ over the whole spectrum — and once in BigInt rationals, where for an unweighted graph the answer is exactly a ratio of two integers and there is no rounding anywhere. The curves are the relative errors of the two floating-point routes against that exact answer: at worst 4.75·10⁻¹⁶ for the elimination and 9.98·10⁻¹⁶ for the spectral sum, which use no arithmetic in common beyond the matrix they are both handed. Foster's theorem says Σ w·R over the edges is exactly n − 1 whatever the graph is; it comes to 15 against 15.

resistance-exact 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.

Effective resistance from vertex 0 to every other vertex of the grid 4×4, computed twice in floating point — once by grounding a vertex and eliminating, once by summing (uₖᵢ − uₖⱼ)²/λₖ over the whole spectrum — and once in BigInt rationals, where for an unweighted graph the answer is exactly a ratio of two integers and there is no rounding anywhere. The curves are the relative errors of the two floating-point routes against that exact answer: at worst 4.75·10⁻¹⁶ for the elimination and 9.98·10⁻¹⁶ for the spectral sum, which use no arithmetic in common beyond the matrix they are both handed. Foster's theorem says Σ w·R over the edges is exactly n − 1 whatever the graph is; it comes to 15 against 15.

Effective resistance from vertex 0 to every other vertex of the grid 4×4, computed twice in floating point — once by grounding a vertex and eliminating, once by summing (uₖᵢ − uₖⱼ)²/λₖ over the whole spectrum — and once in BigInt rationals, where for an unweighted graph the answer is exactly a ratio of two integers and there is no rounding anywhere. The curves are the relative errors of the two floating-point routes against that exact answer: at worst 4.75·10⁻¹⁶ for the elimination and 9.98·10⁻¹⁶ for the spectral sum, which use no arithmetic in common beyond the matrix they are both handed. Foster's theorem says Σ w·R over the edges is exactly n − 1 whatever the graph is; it comes to 15 against 15.

Effective resistance from vertex 0 to every other vertex of the grid 2×3, computed twice in floating point — once by grounding a vertex and eliminating, once by summing (uₖᵢ − uₖⱼ)²/λₖ over the whole spectrum — and once in BigInt rationals, where for an unweighted graph the answer is exactly a ratio of two integers and there is no rounding anywhere. The curves are the relative errors of the two floating-point routes against that exact answer: at worst 4.54·10⁻¹⁶ for the elimination and 6.34·10⁻¹⁶ for the spectral sum, which use no arithmetic in common beyond the matrix they are both handed. Foster's theorem says Σ w·R over the edges is exactly n − 1 whatever the graph is; it comes to 5 against 5.

Effective resistance from vertex 0 to every other vertex of the grid 3×3, computed twice in floating point — once by grounding a vertex and eliminating, once by summing (uₖᵢ − uₖⱼ)²/λₖ over the whole spectrum — and once in BigInt rationals, where for an unweighted graph the answer is exactly a ratio of two integers and there is no rounding anywhere. The curves are the relative errors of the two floating-point routes against that exact answer: at worst 4.7·10⁻¹⁶ for the elimination and 6.27·10⁻¹⁶ for the spectral sum, which use no arithmetic in common beyond the matrix they are both handed. Foster's theorem says Σ w·R over the edges is exactly n − 1 whatever the graph is; it comes to 8 against 8.

Effective resistance from vertex 0 to every other vertex of the grid 3×4, computed twice in floating point — once by grounding a vertex and eliminating, once by summing (uₖᵢ − uₖⱼ)²/λₖ over the whole spectrum — and once in BigInt rationals, where for an unweighted graph the answer is exactly a ratio of two integers and there is no rounding anywhere. The curves are the relative errors of the two floating-point routes against that exact answer: at worst 6.3·10⁻¹⁶ for the elimination and 1.21·10⁻¹⁵ for the spectral sum, which use no arithmetic in common beyond the matrix they are both handed. Foster's theorem says Σ w·R over the edges is exactly n − 1 whatever the graph is; it comes to 11 against 11.

Effective resistance from vertex 0 to every other vertex of the grid 4×5, computed twice in floating point — once by grounding a vertex and eliminating, once by summing (uₖᵢ − uₖⱼ)²/λₖ over the whole spectrum — and once in BigInt rationals, where for an unweighted graph the answer is exactly a ratio of two integers and there is no rounding anywhere. The curves are the relative errors of the two floating-point routes against that exact answer: at worst 7.79·10⁻¹⁶ for the elimination and 1.38·10⁻¹⁵ for the spectral sum, which use no arithmetic in common beyond the matrix they are both handed. Foster's theorem says Σ w·R over the edges is exactly n − 1 whatever the graph is; it comes to 19 against 19.

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.

13 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 resistanceMatrix, resistanceByEig, foster, resistanceExact, 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.