‖A − Q‖ in the Frobenius norm for four orthogonal matrices, on an 8×8 matrix with κ = 10

At its defaults it draws ‖a − q‖ in the frobenius norm for four orthogonal matrices, on an 8×8 matrix with κ = 10. The polar factor is 1.8554 from A. QR with its column signs fixed is 2.1265 — 15 per cent further. QR as Householder returns it, with 7 of 8 columns negated, is 3.8226, which is further than the best of two hundred orthogonal matrices drawn at random. The strip beneath the bars is those two hundred draws, whose best is 2.6928; the polar factor is to the left of all of them, which is the minimisation being checked rather than assumed.

nearest-orthogonal is one function in lib/figures/polar.js — the polar factor — the nearest orthogonal matrix, and two ways to it without an svd. 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 polar factor is 1.8554 from A. QR with its column signs fixed is 2.1265 — 15 per cent further. QR as Householder returns it, with 7 of 8 columns negated, is 3.8226, which is further than the best of two hundred orthogonal matrices drawn at random. The strip beneath the bars is those two hundred draws, whose best is 2.6928; the polar factor is to the left of all of them, which is the minimisation being checked rather than assumed.

Per cent of 600 trials mirrored, against z — the noise over the thickness, divided by the square root of the point count — for alignments in n = 3, 5, 8 and 10 dimensions with k = 1, 2 or 3 directions of thickness 0.01, drawn as points, beside the rate at which the k × k matrix W/m + zG has a negative determinant, W Wishart on m points and G Gaussian, drawn as lines from 20,000 draws each, on a linear axis. At z = 0.7: n = 3, k = 1: 9.7%; n = 5, k = 1: 13%; n = 10, k = 1: 12%; n = 5, k = 2: 23%; n = 10, k = 2: 24%; n = 8, k = 3: 32%; the models give 9.5% for k = 1, 22% for k = 2, 31% for k = 3.

For forty points, the per cent of 20,000 draws in which W/m + zG has a negative determinant, for k × k blocks with k = 1, 2 and 3, against z, and the rate k independent flips at the one-direction rate p would give, (1 − (1 − 2p)ᵏ)/2, dashed, on linear axes. At z = 0.45 one direction mirrors 2.2%; two independent flips would give 4.2% and the determinant gives 7.0%; three would give 6.2% and the determinant gives 14%.

Each trial is a dot: horizontally the smallest singular value of the cross-covariance, signed negative when the polar factor is a reflection; vertically the Frobenius distance between the polar factor and the rotation with its determinant fixed. 198 trials sit at exactly 0 and 102 at exactly 2, with nothing in between: the largest deviation from 2 is 8.9·10⁻¹⁶. In every mirrored trial the rotation's residual exceeds the reflection's by 4σ₃ to a relative 8.3·10⁻¹³.

Median distance of the determinant-fixed rotation from the true one, divided by the noise, over 400 alignments at z = 0.7, split into trials whose polar factor was a rotation and trials where it was a reflection and was fixed. n = 5 with 1 thin direction: 7.5% mirrored, error 0.59σ unmirrored and 0.61σ fixed; n = 5 with 2 thin directions: 22% mirrored, error 10.93σ unmirrored and 15.77σ fixed; n = 8 with 3 thin directions: 30% mirrored, error 24.23σ unmirrored and 29.26σ fixed.

Median distance of the determinant-fixed rotation from the true one, divided by the noise, over 400 alignments at z = 0.45, split into trials whose polar factor was a rotation and trials where it was a reflection and was fixed. n = 5 with 1 thin direction: 1.5% mirrored, error 0.60σ unmirrored and 0.68σ fixed; n = 5 with 2 thin directions: 4.8% mirrored, error 10.98σ unmirrored and 17.93σ fixed; n = 8 with 3 thin directions: 13% mirrored, error 26.08σ unmirrored and 38.99σ fixed.

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.

92 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 qrHouseholder, 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.