projection-stability

At its defaults it draws a reduced model that interpolates exactly and has a pole in the right half plane. A convection–diffusion system of 30 states at Péclet number 10, whose state matrix is tridiagonal and not symmetric and whose rightmost eigenvalue is -3.49 — stable. It is reduced to order three by two-sided interpolation at {s, 2s, 4s}, and the horizontal axis is s. The curve is the rightmost pole of the reduced model. Above the zero line the model is unstable: it is at 4 of the 19 placements, reaching 24.1. Every one of those unstable models still matches the full system's transfer function at each of its own three interpolation points, to 1.27·10⁻¹⁴ — the construction did exactly what it promised. The dashed line is balanced truncation of the same system at the same order, whose rightmost pole is -0.837: stable, as it is at every order, because stability is a theorem there rather than an outcome.

projection-stability is one function in lib/figures/redstab.js — a model that interpolates exactly and cannot be run — and the method whose stability is a theorem. 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 convection–diffusion system of 30 states at Péclet number 10, whose state matrix is tridiagonal and not symmetric and whose rightmost eigenvalue is -3.49 — stable. It is reduced to order three by two-sided interpolation at {s, 2s, 4s}, and the horizontal axis is s. The curve is the rightmost pole of the reduced model. Above the zero line the model is unstable: it is at 4 of the 19 placements, reaching 24.1. Every one of those unstable models still matches the full system's transfer function at each of its own three interpolation points, to 1.27·10⁻¹⁴ — the construction did exactly what it promised. The dashed line is balanced truncation of the same system at the same order, whose rightmost pole is -0.837: stable, as it is at every order, because stability is a theorem there rather than an outcome.

A convection–diffusion system of 30 states at Péclet number 10, whose state matrix is tridiagonal and not symmetric and whose rightmost eigenvalue is -3.49 — stable. It is reduced to order three by two-sided interpolation at {s, 2s, 4s}, and the horizontal axis is s. The curve is the rightmost pole of the reduced model. Above the zero line the model is unstable: it is at 4 of the 19 placements, reaching 24.1. Every one of those unstable models still matches the full system's transfer function at each of its own three interpolation points, to 1.27·10⁻¹⁴ — the construction did exactly what it promised. The dashed line is balanced truncation of the same system at the same order, whose rightmost pole is -0.837: stable, as it is at every order, because stability is a theorem there rather than an outcome.

A convection–diffusion system of 30 states at Péclet number 25, whose state matrix is tridiagonal and not symmetric and whose rightmost eigenvalue is -6.888 — stable. It is reduced to order three by two-sided interpolation at {s, 2s, 4s}, and the horizontal axis is s. The curve is the rightmost pole of the reduced model. Above the zero line the model is unstable: it is at 4 of the 19 placements, reaching 614. Every one of those unstable models still matches the full system's transfer function at each of its own three interpolation points, to 3.59·10⁻¹³ — the construction did exactly what it promised. The dashed line is balanced truncation of the same system at the same order, whose rightmost pole is -3.73: stable, as it is at every order, because stability is a theorem there rather than an outcome.

A convection–diffusion system of 30 states at Péclet number 4, whose state matrix is tridiagonal and not symmetric and whose rightmost eigenvalue is -3.461 — stable. It is reduced to order three by two-sided interpolation at {s, 2s, 4s}, and the horizontal axis is s. The curve is the rightmost pole of the reduced model. Above the zero line the model is unstable: it is at 3 of the 19 placements, reaching 21.8. Every one of those unstable models still matches the full system's transfer function at each of its own three interpolation points, to 2.94·10⁻¹⁵ — the construction did exactly what it promised. The dashed line is balanced truncation of the same system at the same order, whose rightmost pole is -3.14: stable, as it is at every order, because stability is a theorem there rather than an outcome.

A convection–diffusion system of 30 states at Péclet number 40, whose state matrix is tridiagonal and not symmetric and whose rightmost eigenvalue is -11.53 — stable. It is reduced to order three by two-sided interpolation at {s, 2s, 4s}, and the horizontal axis is s. The curve is the rightmost pole of the reduced model. Above the zero line the model is unstable: it is at 6 of the 19 placements, reaching 106. Every one of those unstable models still matches the full system's transfer function at each of its own three interpolation points, to 2.57·10⁻¹³ — the construction did exactly what it promised. The dashed line is balanced truncation of the same system at the same order, whose rightmost pole is -4.53: stable, as it is at every order, because stability is a theorem there rather than an outcome.

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.

18 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 197 of 363 generators — 179 print a residual and 18 are exempt with a published reason; 166 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.