ordering-legality

At its defaults it draws 500 random symmetric orderings of a regularised saddle-point matrix, and what each factorisation reproduces. A quasi-definite matrix — [[H + δI, Aᵀ], [A, −δI]] with δ = 10⁻⁶ — has an LDLᵀ factorisation with diagonal D for every symmetric permutation, so the ordering can be chosen for fill with no numerical veto at all. That is the count: 500 of 500 orderings factorise here, against 69.0 per cent for the same matrix with the zero block left where the problem put it. Each bar counts the orderings whose factorisation left a relative residual ‖PKPᵀ − LDLᵀ‖/‖K‖ in that decade, with the tallest holding 136 of the 500. They span 1.56·10⁻¹⁶ to 1.02·10⁻⁷ — 9 orders — and the growth factor across them runs from 1 to 6.43·10⁵ — which is 0.643/δ, an inverse law that holds at every δ this figure is drawn at. Existence is not stability, and the theorem says only the first.

ordering-legality is one function in lib/figures/quasidef.js — quasi-definite — every ordering legal, and the fill a symbolic phase can promise exactly. 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 quasi-definite matrix — [[H + δI, Aᵀ], [A, −δI]] with δ = 10⁻⁶ — has an LDLᵀ factorisation with diagonal D for every symmetric permutation, so the ordering can be chosen for fill with no numerical veto at all. That is the count: 500 of 500 orderings factorise here, against 69.0 per cent for the same matrix with the zero block left where the problem put it. Each bar counts the orderings whose factorisation left a relative residual ‖PKPᵀ − LDLᵀ‖/‖K‖ in that decade, with the tallest holding 136 of the 500. They span 1.56·10⁻¹⁶ to 1.02·10⁻⁷ — 9 orders — and the growth factor across them runs from 1 to 6.43·10⁵ — which is 0.643/δ, an inverse law that holds at every δ this figure is drawn at. Existence is not stability, and the theorem says only the first.

A quasi-definite matrix — [[H + δI, Aᵀ], [A, −δI]] with δ = 10⁻⁸ — has an LDLᵀ factorisation with diagonal D for every symmetric permutation, so the ordering can be chosen for fill with no numerical veto at all. That is the count: 500 of 500 orderings factorise here, against 69.0 per cent for the same matrix with the zero block left where the problem put it. Each bar counts the orderings whose factorisation left a relative residual ‖PKPᵀ − LDLᵀ‖/‖K‖ in that decade, with the tallest holding 130 of the 500. They span 1.65·10⁻¹⁶ to 6.17·10⁻⁴ — 13 orders — and the growth factor across them runs from 1 to 6.43·10⁷ — which is 0.643/δ, an inverse law that holds at every δ this figure is drawn at. Existence is not stability, and the theorem says only the first.

A quasi-definite matrix — [[H + δI, Aᵀ], [A, −δI]] with δ = 10⁻⁶ — has an LDLᵀ factorisation with diagonal D for every symmetric permutation, so the ordering can be chosen for fill with no numerical veto at all. That is the count: 500 of 500 orderings factorise here, against 69.0 per cent for the same matrix with the zero block left where the problem put it. Each bar counts the orderings whose factorisation left a relative residual ‖PKPᵀ − LDLᵀ‖/‖K‖ in that decade, with the tallest holding 136 of the 500. They span 1.56·10⁻¹⁶ to 1.02·10⁻⁷ — 9 orders — and the growth factor across them runs from 1 to 6.43·10⁵ — which is 0.643/δ, an inverse law that holds at every δ this figure is drawn at. Existence is not stability, and the theorem says only the first.

A quasi-definite matrix — [[H + δI, Aᵀ], [A, −δI]] with δ = 10⁻⁴ — has an LDLᵀ factorisation with diagonal D for every symmetric permutation, so the ordering can be chosen for fill with no numerical veto at all. That is the count: 500 of 500 orderings factorise here, against 69.0 per cent for the same matrix with the zero block left where the problem put it. Each bar counts the orderings whose factorisation left a relative residual ‖PKPᵀ − LDLᵀ‖/‖K‖ in that decade, with the tallest holding 151 of the 500. They span 1.34·10⁻¹⁶ to 1.23·10⁻¹¹ — 5 orders — and the growth factor across them runs from 1 to 6428 — which is 0.643/δ, an inverse law that holds at every δ this figure is drawn at. Existence is not stability, and the theorem says only the first.

A quasi-definite matrix — [[H + δI, Aᵀ], [A, −δI]] with δ = 0.01 — has an LDLᵀ factorisation with diagonal D for every symmetric permutation, so the ordering can be chosen for fill with no numerical veto at all. That is the count: 500 of 500 orderings factorise here, against 69.0 per cent for the same matrix with the zero block left where the problem put it. Each bar counts the orderings whose factorisation left a relative residual ‖PKPᵀ − LDLᵀ‖/‖K‖ in that decade, with the tallest holding 301 of the 500. They span 1.17·10⁻¹⁶ to 6.76·10⁻¹⁵ — 2 orders — and the growth factor across them runs from 1 to 63.29 — which is 0.633/δ, an inverse law that holds at every δ this figure is drawn at. Existence is not stability, and the theorem says only the first.

A quasi-definite matrix — [[H + δI, Aᵀ], [A, −δI]] with δ = 10⁻⁶ — has an LDLᵀ factorisation with diagonal D for every symmetric permutation, so the ordering can be chosen for fill with no numerical veto at all. That is the count: 120 of 120 orderings factorise here, against 67.5 per cent for the same matrix with the zero block left where the problem put it. Each bar counts the orderings whose factorisation left a relative residual ‖PKPᵀ − LDLᵀ‖/‖K‖ in that decade, with the tallest holding 28 of the 120. They span 1.71·10⁻¹⁶ to 2.01·10⁻⁸ — 8 orders — and the growth factor across them runs from 1.03 to 5.112·10⁵ — which is 0.511/δ, an inverse law that holds at every δ this figure is drawn at. Existence is not stability, and the theorem says only the first.

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.

10 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 permutationTrial, 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 165 of 306 generators — 150 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.