Why the product route stops: two condition numbers and their product

At its defaults it draws why the product route stops: two condition numbers and their product. The two Gramians of a 22-state model of McMillan degree 10, and what each route to the Hankel singular values is charged. κ(P) = 1.26·10¹⁸ and κ(Q) = 2.16·10¹⁸. The square-root route works with RᵀS, whose condition number is their geometric mean, 1.65·10¹⁸; the route that eigendecomposes PQ works at 2.72·10³⁶, which is past 1/u = 4.5·10¹⁵ — the point at which nothing small survives at all. The Gramians themselves are right: this one agrees with its closed form to 2.15·10⁻¹³. What is lost is lost in the last step, to a product nobody had to form.

balancing-kappa is one function in lib/figures/gramian.js — the bound known before the computation — hankel singular values, and the product that halves 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 two Gramians of a 22-state model of McMillan degree 10, and what each route to the Hankel singular values is charged. κ(P) = 1.26·10¹⁸ and κ(Q) = 2.16·10¹⁸. The square-root route works with RᵀS, whose condition number is their geometric mean, 1.65·10¹⁸; the route that eigendecomposes PQ works at 2.72·10³⁶, which is past 1/u = 4.5·10¹⁵ — the point at which nothing small survives at all. The Gramians themselves are right: this one agrees with its closed form to 2.15·10⁻¹³. What is lost is lost in the last step, to a product nobody had to form.

The two Gramians of a 22-state model of McMillan degree 6, and what each route to the Hankel singular values is charged. κ(P) = 5.66·10¹⁸ and κ(Q) = 5.71·10¹⁸. The square-root route works with RᵀS, whose condition number is their geometric mean, 5.68·10¹⁸; the route that eigendecomposes PQ works at 3.23·10³⁷, which is past 1/u = 4.5·10¹⁵ — the point at which nothing small survives at all. The Gramians themselves are right: this one agrees with its closed form to 2.15·10⁻¹³. What is lost is lost in the last step, to a product nobody had to form.

The two Gramians of a 22-state model of McMillan degree 10, and what each route to the Hankel singular values is charged. κ(P) = 1.26·10¹⁸ and κ(Q) = 2.16·10¹⁸. The square-root route works with RᵀS, whose condition number is their geometric mean, 1.65·10¹⁸; the route that eigendecomposes PQ works at 2.72·10³⁶, which is past 1/u = 4.5·10¹⁵ — the point at which nothing small survives at all. The Gramians themselves are right: this one agrees with its closed form to 2.15·10⁻¹³. What is lost is lost in the last step, to a product nobody had to form.

The two Gramians of a 22-state model of McMillan degree 4, and what each route to the Hankel singular values is charged. κ(P) = 2.69·10¹⁸ and κ(Q) = 4.12·10¹⁸. The square-root route works with RᵀS, whose condition number is their geometric mean, 3.33·10¹⁸; the route that eigendecomposes PQ works at 1.11·10³⁷, which is past 1/u = 4.5·10¹⁵ — the point at which nothing small survives at all. The Gramians themselves are right: this one agrees with its closed form to 2.14·10⁻¹³. What is lost is lost in the last step, to a product nobody had to form.

The two Gramians of a 22-state model of McMillan degree 8, and what each route to the Hankel singular values is charged. κ(P) = 1.05·10¹⁹ and κ(Q) = 7.05·10¹⁸. The square-root route works with RᵀS, whose condition number is their geometric mean, 8.61·10¹⁸; the route that eigendecomposes PQ works at 7.42·10³⁷, which is past 1/u = 4.5·10¹⁵ — the point at which nothing small survives at all. The Gramians themselves are right: this one agrees with its closed form to 2.14·10⁻¹³. What is lost is lost in the last step, to a product nobody had to form.

The two Gramians of a 22-state model of McMillan degree 12, and what each route to the Hankel singular values is charged. κ(P) = 7.9·10¹⁷ and κ(Q) = 7.44·10¹⁷. The square-root route works with RᵀS, whose condition number is their geometric mean, 7.67·10¹⁷; the route that eigendecomposes PQ works at 5.88·10³⁵, which is past 1/u = 4.5·10¹⁵ — the point at which nothing small survives at all. The Gramians themselves are right: this one agrees with its closed form to 2.15·10⁻¹³. What is lost is lost in the last step, to a product nobody had to form.

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.

11 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 calls a factoriser without drawing a factorisation (controllabilityGramian, observabilityGramian, svd), so the rule is written down as not applying, with the reason: the bars are condition numbers of two Gramians; the Gramian's agreement with its closed form is asserted at 10⁻¹⁴ inside the generator and no factor of any matrix is drawn

The exemption list is the interesting half of the rule rather than an escape hatch — it is where a decision about a figure had to be argued in one line. residualcheck refuses an exemption that is not doing work, and rejected ten of the fifteen written for the expansion's figures on exactly that ground: a figure whose vertical axis is a residual satisfies the rule by construction, and touching a factoriser does not by itself require an entry.

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.