The share of a HOSVD core's energy on its superdiagonal, by family, at rank 6

At its defaults it draws the share of a hosvd core's energy on its superdiagonal, by family, at rank 6. A matrix SVD hands over orthonormal factors and a diagonal middle at the same time. For three indices they come apart, and this is the half that does not survive. Every core here is all-orthogonal — the largest inner product between two slices perpendicular to a mode, relative to the core's own energy, is 3.5·10⁻¹⁶ — and none of them is diagonal. The bars are the fraction of the squared norm carried by the 6 entries on the superdiagonal: smooth 97.7%, hilbert 94.5%, wave 26.6%, noise 0.4%. Keeping only those entries costs 0.152, 0.235, 0.857, 0.998 in relative error against the full core's 7.94·10⁻⁶, 1.77·10⁻⁶, 1.34·10⁻¹⁵, 0.842.

core-energy is one function in lib/figures/hosvd.js — the higher-order svd — d matrix decompositions, and a core that cannot be diagonal. 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 matrix SVD hands over orthonormal factors and a diagonal middle at the same time. For three indices they come apart, and this is the half that does not survive. Every core here is all-orthogonal — the largest inner product between two slices perpendicular to a mode, relative to the core's own energy, is 3.5·10⁻¹⁶ — and none of them is diagonal. The bars are the fraction of the squared norm carried by the 6 entries on the superdiagonal: smooth 97.7%, hilbert 94.5%, wave 26.6%, noise 0.4%. Keeping only those entries costs 0.152, 0.235, 0.857, 0.998 in relative error against the full core's 7.94·10⁻⁶, 1.77·10⁻⁶, 1.34·10⁻¹⁵, 0.842.

A matrix SVD hands over orthonormal factors and a diagonal middle at the same time. For three indices they come apart, and this is the half that does not survive. Every core here is all-orthogonal — the largest inner product between two slices perpendicular to a mode, relative to the core's own energy, is 3.5·10⁻¹⁶ — and none of them is diagonal. The bars are the fraction of the squared norm carried by the 6 entries on the superdiagonal: smooth 97.7%, hilbert 94.5%, wave 26.6%, noise 0.4%. Keeping only those entries costs 0.152, 0.235, 0.857, 0.998 in relative error against the full core's 7.94·10⁻⁶, 1.77·10⁻⁶, 1.34·10⁻¹⁵, 0.842.

A matrix SVD hands over orthonormal factors and a diagonal middle at the same time. For three indices they come apart, and this is the half that does not survive. Every core here is all-orthogonal — the largest inner product between two slices perpendicular to a mode, relative to the core's own energy, is 3.5·10⁻¹⁶ — and none of them is diagonal. The bars are the fraction of the squared norm carried by the 4 entries on the superdiagonal: smooth 97.7%, hilbert 94.5%, wave 26.6%, noise 0.4%. Keeping only those entries costs 0.152, 0.235, 0.857, 0.998 in relative error against the full core's 8.81·10⁻⁴, 4.67·10⁻⁴, 1.33·10⁻¹⁵, 0.925.

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.

7 distinct claims across 3 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.