Three bases for the same null space, at 10 unknowns and 4 constraints

At its defaults it draws three bases for the same null space, at 10 unknowns and 4 constraints. The same constrained problem solved three times, differing only in which basis Z is used for the null space of A. The orthonormal basis, from a QR of Aᵀ, has κ(Z) = 1 exactly and is dense — 100 per cent of its entries are nonzero. The fundamental basis built on the first 4 columns has κ(Z) = 1.994·10⁸, and its reduced Hessian comes out at 3.801·10¹⁶, which is κ(Z)² to within a factor of 0.956 — the square attained rather than bounded. Its answer is wrong by 0.05179, against 1.07·10⁻¹⁵ for the orthonormal one. Choosing the same kind of basis by pivoting instead gives κ(Z) = 2.06, an error of 6.71·10⁻¹⁶, and the same 50 per cent density: all of the sparsity and none of the loss.

null-basis-bars is one function in lib/figures/nullbasis.js — the basis of a constraint — three spans of one null space, and the square between them. 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 same constrained problem solved three times, differing only in which basis Z is used for the null space of A. The orthonormal basis, from a QR of Aᵀ, has κ(Z) = 1 exactly and is dense — 100 per cent of its entries are nonzero. The fundamental basis built on the first 4 columns has κ(Z) = 1.994·10⁸, and its reduced Hessian comes out at 3.801·10¹⁶, which is κ(Z)² to within a factor of 0.956 — the square attained rather than bounded. Its answer is wrong by 0.05179, against 1.07·10⁻¹⁵ for the orthonormal one. Choosing the same kind of basis by pivoting instead gives κ(Z) = 2.06, an error of 6.71·10⁻¹⁶, and the same 50 per cent density: all of the sparsity and none of the loss.

Median over five draws of the resistances. Conditioning after scaling to a unit diagonal; work as the sum over pivots of the squared column count under minimum-degree ordering. least resistance: κ 12.5, work 16913; bands of 0.25 decades: κ 15.8, work 18939; bands of 0.5 decades: κ 18.7, work 19612; bands of 1 decade: κ 42.0, work 17628; bands of 2 decades: κ 504, work 14046; bands of 3 decades: κ 117, work 8782; breadth-first: κ 2.48e+3, work 8547; the node equations: κ 6.96e+3, work 5111.

breadth-first: 49 loops, 613 nonzeros in the loop matrix, factor 376 entries, work 3124; least resistance: 49 loops, 931 nonzeros in the loop matrix, factor 505 entries, work 6161. Blue: entries of the loop matrix; red: fill.

breadth-first: 81 loops, mean 7.31 arcs, longest 12, median 8; least resistance: 81 loops, mean 11.46 arcs, longest 42, median 6.

4 by 4: over breadth-first 1.22 (1.07 to 1.48), over the node equations 0.75 (0.65 to 0.91); 6 by 6: over breadth-first 1.56 (1.27 to 1.93), over the node equations 1.57 (1.29 to 1.95); 8 by 8: over breadth-first 1.44 (1.05 to 1.97), over the node equations 2.00 (1.47 to 2.75); 10 by 10: over breadth-first 1.98 (1.34 to 3.08), over the node equations 3.31 (2.24 to 5.15); 12 by 12: over breadth-first 1.78 (1.50 to 2.27), over the node equations 3.80 (3.21 to 4.86). Median and range over five draws.

Resistances drawn over four decades; tree arcs dark, thicker for lighter resistance, off-tree arcs faint, and the longest fundamental loop in red, 34 arcs. Mean loop length 9.60 arcs; the loop matrix has 1987 nonzeros, its Cholesky factor 1076 entries and 17628 units of work; κ after scaling 45.2.

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.

29 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 basisComparison, 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.