congruence-bars

At its defaults it draws how much two runs of the same fit agree about the factors, from 6 starting points each. Every run here reaches its target to the rounding level — 3·10⁻¹³, 1.16·10⁻¹¹ and 2.39·10⁻¹⁵ at worst — so all three are successful factorisations. The bar is the worst agreement between any two of them about the *factors*, matched over permutations and scalings, which is exactly the freedom the uniqueness theorem allows. Kruskal's condition k_A + k_B + k_C ≥ 2r + 2 holds for the first (9 ≥ 8) and fails for the second (6 < 8), and the bars are 1.0000 and 0.0207. The matrix is the comparison the whole thing rests on: AB = (AM)(M⁻¹B) for every invertible M, so its runs agree to 0.1089 and its factors mean nothing on their own.

congruence-bars is one function in lib/figures/cpals.js — alternating least squares — an iteration that walks out of the set it is searching. 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 run here reaches its target to the rounding level — 3·10⁻¹³, 1.16·10⁻¹¹ and 2.39·10⁻¹⁵ at worst — so all three are successful factorisations. The bar is the worst agreement between any two of them about the *factors*, matched over permutations and scalings, which is exactly the freedom the uniqueness theorem allows. Kruskal's condition k_A + k_B + k_C ≥ 2r + 2 holds for the first (9 ≥ 8) and fails for the second (6 < 8), and the bars are 1.0000 and 0.0207. The matrix is the comparison the whole thing rests on: AB = (AM)(M⁻¹B) for every invertible M, so its runs agree to 0.1089 and its factors mean nothing on their own.

Every run here reaches its target to the rounding level — 3·10⁻¹³, 1.16·10⁻¹¹ and 2.39·10⁻¹⁵ at worst — so all three are successful factorisations. The bar is the worst agreement between any two of them about the *factors*, matched over permutations and scalings, which is exactly the freedom the uniqueness theorem allows. Kruskal's condition k_A + k_B + k_C ≥ 2r + 2 holds for the first (9 ≥ 8) and fails for the second (6 < 8), and the bars are 1.0000 and 0.0207. The matrix is the comparison the whole thing rests on: AB = (AM)(M⁻¹B) for every invertible M, so its runs agree to 0.1089 and its factors mean nothing on their own.

Every run here reaches its target to the rounding level — 3·10⁻¹³, 1.16·10⁻¹¹ and 2.39·10⁻¹⁵ at worst — so all three are successful factorisations. The bar is the worst agreement between any two of them about the *factors*, matched over permutations and scalings, which is exactly the freedom the uniqueness theorem allows. Kruskal's condition k_A + k_B + k_C ≥ 2r + 2 holds for the first (9 ≥ 8) and fails for the second (6 < 8), and the bars are 1.0000 and 0.0207. The matrix is the comparison the whole thing rests on: AB = (AM)(M⁻¹B) for every invertible M, so its runs agree to 0.0432 and its factors mean nothing on their own.

Every run here reaches its target to the rounding level — 1.55·10⁻¹³, 3.59·10⁻¹² and 2.39·10⁻¹⁵ at worst — so all three are successful factorisations. The bar is the worst agreement between any two of them about the *factors*, matched over permutations and scalings, which is exactly the freedom the uniqueness theorem allows. Kruskal's condition k_A + k_B + k_C ≥ 2r + 2 holds for the first (9 ≥ 8) and fails for the second (6 < 8), and the bars are 1.0000 and 0.1450. The matrix is the comparison the whole thing rests on: AB = (AM)(M⁻¹B) for every invertible M, so its runs agree to 0.1089 and its factors mean nothing on their own.

Every run here reaches its target to the rounding level — 3·10⁻¹³, 3.59·10⁻¹² and 2.39·10⁻¹⁵ at worst — so all three are successful factorisations. The bar is the worst agreement between any two of them about the *factors*, matched over permutations and scalings, which is exactly the freedom the uniqueness theorem allows. Kruskal's condition k_A + k_B + k_C ≥ 2r + 2 holds for the first (9 ≥ 8) and fails for the second (6 < 8), and the bars are 1.0000 and 0.1026. The matrix is the comparison the whole thing rests on: AB = (AM)(M⁻¹B) for every invertible M, so its runs agree to 0.1089 and its factors mean nothing on their own.

Every run here reaches its target to the rounding level — 3·10⁻¹³, 1.16·10⁻¹¹ and 2.39·10⁻¹⁵ at worst — so all three are successful factorisations. The bar is the worst agreement between any two of them about the *factors*, matched over permutations and scalings, which is exactly the freedom the uniqueness theorem allows. Kruskal's condition k_A + k_B + k_C ≥ 2r + 2 holds for the first (9 ≥ 8) and fails for the second (6 < 8), and the bars are 1.0000 and 0.0207. The matrix is the comparison the whole thing rests on: AB = (AM)(M⁻¹B) for every invertible M, so its runs agree to 0.0432 and its factors mean nothing on their own.

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.

5 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 146 of 287 generators — 131 print a residual and 15 are exempt with a published reason; 141 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.