Hankel singular values, and the half of them a product cannot see

At its defaults it draws hankel singular values, and the half of them a product cannot see. σₖ divided by σ₁, for a 20-state model of McMillan degree 12. The lower curve is the square-root route — Cholesky-like factors of the two Gramians and one SVD of RᵀS — and it descends to the level at which the Gramians were computed. The upper curve eigendecomposes the product PQ, agrees for the first 6 values and then stops, flattening at 9.67·10⁻¹⁰. The dashed line is σ₁√u = 2.32·10⁻⁹, which is where a route that squares the conditioning must stop: the eigenvalues of PQ are σ², an absolute error of uσ₁² on them is a relative error of √u on σ. κ(PQ) is 1.8·10³⁶ against 1.34·10¹⁸. Every σ below the line is a term in the error bound, so this is not an academic loss.

hankel-values 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.

σₖ divided by σ₁, for a 20-state model of McMillan degree 12. The lower curve is the square-root route — Cholesky-like factors of the two Gramians and one SVD of RᵀS — and it descends to the level at which the Gramians were computed. The upper curve eigendecomposes the product PQ, agrees for the first 6 values and then stops, flattening at 9.67·10⁻¹⁰. The dashed line is σ₁√u = 2.32·10⁻⁹, which is where a route that squares the conditioning must stop: the eigenvalues of PQ are σ², an absolute error of uσ₁² on them is a relative error of √u on σ. κ(PQ) is 1.8·10³⁶ against 1.34·10¹⁸. Every σ below the line is a term in the error bound, so this is not an academic loss.

σₖ divided by σ₁, for a 22-state model of McMillan degree 12. The lower curve is the square-root route — Cholesky-like factors of the two Gramians and one SVD of RᵀS — and it descends to the level at which the Gramians were computed. The upper curve eigendecomposes the product PQ, agrees for the first 6 values and then stops, flattening at 9.44·10⁻¹⁰. The dashed line is σ₁√u = 2.32·10⁻⁹, which is where a route that squares the conditioning must stop: the eigenvalues of PQ are σ², an absolute error of uσ₁² on them is a relative error of √u on σ. κ(PQ) is 5.88·10³⁵ against 7.67·10¹⁷. Every σ below the line is a term in the error bound, so this is not an academic loss.

σₖ divided by σ₁, for a 20-state model of McMillan degree 12. The lower curve is the square-root route — Cholesky-like factors of the two Gramians and one SVD of RᵀS — and it descends to the level at which the Gramians were computed. The upper curve eigendecomposes the product PQ, agrees for the first 6 values and then stops, flattening at 9.67·10⁻¹⁰. The dashed line is σ₁√u = 2.32·10⁻⁹, which is where a route that squares the conditioning must stop: the eigenvalues of PQ are σ², an absolute error of uσ₁² on them is a relative error of √u on σ. κ(PQ) is 1.8·10³⁶ against 1.34·10¹⁸. Every σ below the line is a term in the error bound, so this is not an academic loss.

σₖ divided by σ₁, for a 10-state model of McMillan degree 6. The lower curve is the square-root route — Cholesky-like factors of the two Gramians and one SVD of RᵀS — and it descends to the level at which the Gramians were computed. The upper curve eigendecomposes the product PQ, agrees for the first 5 values and then stops, flattening at 1.45·10⁻⁹. The dashed line is σ₁√u = 2.39·10⁻⁹, which is where a route that squares the conditioning must stop: the eigenvalues of PQ are σ², an absolute error of uσ₁² on them is a relative error of √u on σ. κ(PQ) is 2.01·10³⁵ against 4.49·10¹⁷. Every σ below the line is a term in the error bound, so this is not an academic loss.

σₖ divided by σ₁, for a 14-state model of McMillan degree 10. The lower curve is the square-root route — Cholesky-like factors of the two Gramians and one SVD of RᵀS — and it descends to the level at which the Gramians were computed. The upper curve eigendecomposes the product PQ, agrees for the first 5 values and then stops, flattening at 9.19·10⁻⁹. The dashed line is σ₁√u = 2.23·10⁻⁹, which is where a route that squares the conditioning must stop: the eigenvalues of PQ are σ², an absolute error of uσ₁² on them is a relative error of √u on σ. κ(PQ) is 2.91·10³⁴ against 1.71·10¹⁷. Every σ below the line is a term in the error bound, so this is not an academic loss.

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 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.

It draws a decomposition and prints its residual. It calls hankelSingularValues, 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.