rank-region

At its defaults it draws a plane of 2 × 2 × 2 tensors, coloured by the sign of the hyperdeterminant that decides their rank. Two of the eight entries are varied over ±2 and the other six are held at the values that make the centre of the picture the rank-three tensor A = e₁⊗e₁⊗e₂ + e₁⊗e₂⊗e₁ + e₂⊗e₁⊗e₁. The light region is Δ > 0, where the tensor has real rank two and two real rank-one terms exist; the dark region is Δ < 0, where the real rank is three and the complex rank is two. 54.2 per cent of this window is rank two. The boundary between them is the curve Δ = 0, on which the rank is three and the distance to the rank-two set is zero — every point of it is a tensor with no nearest rank-two approximation.

rank-region 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.

Two of the eight entries are varied over ±2 and the other six are held at the values that make the centre of the picture the rank-three tensor A = e₁⊗e₁⊗e₂ + e₁⊗e₂⊗e₁ + e₂⊗e₁⊗e₁. The light region is Δ > 0, where the tensor has real rank two and two real rank-one terms exist; the dark region is Δ < 0, where the real rank is three and the complex rank is two. 54.2 per cent of this window is rank two. The boundary between them is the curve Δ = 0, on which the rank is three and the distance to the rank-two set is zero — every point of it is a tensor with no nearest rank-two approximation.

Two of the eight entries are varied over ±2 and the other six are held at the values that make the centre of the picture the rank-three tensor A = e₁⊗e₁⊗e₂ + e₁⊗e₂⊗e₁ + e₂⊗e₁⊗e₁. The light region is Δ > 0, where the tensor has real rank two and two real rank-one terms exist; the dark region is Δ < 0, where the real rank is three and the complex rank is two. 54.2 per cent of this window is rank two. The boundary between them is the curve Δ = 0, on which the rank is three and the distance to the rank-two set is zero — every point of it is a tensor with no nearest rank-two approximation.

Two of the eight entries are varied over ±0.5 and the other six are held at the values that make the centre of the picture the rank-three tensor A = e₁⊗e₁⊗e₂ + e₁⊗e₂⊗e₁ + e₂⊗e₁⊗e₁. The light region is Δ > 0, where the tensor has real rank two and two real rank-one terms exist; the dark region is Δ < 0, where the real rank is three and the complex rank is two. 50.0 per cent of this window is rank two. The boundary between them is the curve Δ = 0, on which the rank is three and the distance to the rank-two set is zero — every point of it is a tensor with no nearest rank-two approximation.

Two of the eight entries are varied over ±6 and the other six are held at the values that make the centre of the picture the rank-three tensor A = e₁⊗e₁⊗e₂ + e₁⊗e₂⊗e₁ + e₂⊗e₁⊗e₁. The light region is Δ > 0, where the tensor has real rank two and two real rank-one terms exist; the dark region is Δ < 0, where the real rank is three and the complex rank is two. 87.2 per cent of this window is rank two. The boundary between them is the curve Δ = 0, on which the rank is three and the distance to the rank-two set is zero — every point of it is a tensor with no nearest rank-two approximation.

Two of the eight entries are varied over ±1 and the other six are held at the values that make the centre of the picture the rank-three tensor A = e₁⊗e₁⊗e₂ + e₁⊗e₂⊗e₁ + e₂⊗e₁⊗e₁. The light region is Δ > 0, where the tensor has real rank two and two real rank-one terms exist; the dark region is Δ < 0, where the real rank is three and the complex rank is two. 50.0 per cent of this window is rank two. The boundary between them is the curve Δ = 0, on which the rank is three and the distance to the rank-two set is zero — every point of it is a tensor with no nearest rank-two approximation.

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.

3 distinct claims across 5 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.