The share of random 2 × 2 × 2 tensors with real rank two, against the number drawn, and π/4

At its defaults it draws the share of random 2 × 2 × 2 tensors with real rank two, against the number drawn, and π/4. Each draw is eight independent standard normal entries and its rank is decided exactly, by the sign of the hyperdeterminant, with no iteration involved. Over 19,953 draws 15,705 have rank two — a share of 0.7871 against the exact value π/4 = 0.7854, inside 0.6 standard errors. The band is ±2 of them. A random matrix has one typical rank; this is the picture of a random object having two, each with a probability that is a number rather than an experiment.

rank-share is one function in lib/figures/borderrank.js — border rank — a nearest point that is not there, and a rank that is a sign. 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.

Each draw is eight independent standard normal entries and its rank is decided exactly, by the sign of the hyperdeterminant, with no iteration involved. Over 19,953 draws 15,705 have rank two — a share of 0.7871 against the exact value π/4 = 0.7854, inside 0.6 standard errors. The band is ±2 of them. A random matrix has one typical rank; this is the picture of a random object having two, each with a probability that is a number rather than an experiment.

Eight random 2 × 2 × 2 tensors whose pencils have a complex pair. Horizontally, the distance from each to the zero set of its pencil's discriminant, found by a constrained Newton iteration with no alternating least squares in it; vertically, the relative error alternating least squares reaches after 8000 sweeps. Every point is on or just above the diagonal: the fit's error exceeds the distance by between 0.02 and 1.53 per cent.

The relative error and the size of the largest rank-one term against the sweep, on logarithmic axes, over 8000 sweeps. On the tensor whose pencil has only real eigenvalues the error falls to 1.7e-13 and the terms settle at 2.74. On the tensor whose pencil has a complex pair the error settles at 0.1752 and stays there while the largest term keeps growing, to 67, at about the 0.48 power of the sweep count.

Of 1828 random 2 × 2 × 2 tensors with Gaussian entries, the first 400 whose pencil has a complex pair. Vertically, the relative distance from each to the zero set of its pencil's discriminant, which is the error a rank-two fit settles at; horizontally, the complex pair's imaginary part divided by its distance from the origin of the projective line. The median distance is 0.061, one in ten is below 0.0085 and one in ten above 0.170, and the largest is 0.330. The tilt orders the distances only loosely.

For four random 2 × 2 × 2 tensors with complex pencils, at 30 to 8000 sweeps: the distance between the two eigenvalues of the fitted tensor's pencil — both real at every snapshot — against the size of its largest rank-one term, on logarithmic axes. The product of the two stays within a factor of 2.13 along each run: the eigenvalues close on each other as one over the size of the terms, approaching a double eigenvalue that a sum of two rank-one terms can only reach with terms of infinite size.

Four random 2 × 2 × 2 tensors with complex pencils, at 100 to 8000 sweeps: the fit's relative error minus the distance to the double-eigenvalue surface, against the size of its largest rank-one term, on logarithmic axes. The points fall along a slope of minus two — the excess halves when the terms grow by forty per cent — where on the border-rank tensor, whose distance to the rank-two set is zero, the error falls along a slope of minus one. The two reference lines start from the first point.

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.

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.

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 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.