Six is the degree, and the model reaches it from 24 numbers

At its defaults it draws six is the degree, and the model reaches it from 24 numbers. A reduced model of a forty-state system built out of 24 samples of its transfer function and nothing else — no state matrix, no eigenvalues, no factorisation of anything the model is of. The Loewner pencil of divided differences is formed from the samples, its singular values are read for the degree (6 here, with a gap of 2.01·10⁸ across the cliff), and the pencil is projected onto that many directions. The filled curve is the worst relative error at the samples the model was built from; the open one is the worst error at forty points it never saw. At the true degree of six the first is 6.99·10⁻¹² — interpolation is exact by construction, not by accuracy — and the second is 2.12·10⁻⁹.

loewner-model is one function in lib/figures/graphfit.js — three approximations the data chose the shape of — support points, a restricted perturbation, and a model made of measurements. 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 reduced model of a forty-state system built out of 24 samples of its transfer function and nothing else — no state matrix, no eigenvalues, no factorisation of anything the model is of. The Loewner pencil of divided differences is formed from the samples, its singular values are read for the degree (6 here, with a gap of 2.01·10⁸ across the cliff), and the pencil is projected onto that many directions. The filled curve is the worst relative error at the samples the model was built from; the open one is the worst error at forty points it never saw. At the true degree of six the first is 6.99·10⁻¹² — interpolation is exact by construction, not by accuracy — and the second is 2.12·10⁻⁹.

A reduced model of a forty-state system built out of 24 samples of its transfer function and nothing else — no state matrix, no eigenvalues, no factorisation of anything the model is of. The Loewner pencil of divided differences is formed from the samples, its singular values are read for the degree (6 here, with a gap of 2.01·10⁸ across the cliff), and the pencil is projected onto that many directions. The filled curve is the worst relative error at the samples the model was built from; the open one is the worst error at forty points it never saw. At the true degree of six the first is 6.99·10⁻¹² — interpolation is exact by construction, not by accuracy — and the second is 2.12·10⁻⁹.

A reduced model of a forty-state system built out of 24 samples of its transfer function and nothing else — no state matrix, no eigenvalues, no factorisation of anything the model is of. The Loewner pencil of divided differences is formed from the samples, its singular values are read for the degree (6 here, with a gap of 4.73·10⁶ across the cliff), and the pencil is projected onto that many directions. The filled curve is the worst relative error at the samples the model was built from; the open one is the worst error at forty points it never saw. With relative noise of 10⁻¹² on the samples the singular values stop falling to the rounding level, the cliff becomes a slope, and no order does better than 1.39·10⁻⁹ — the data's own error is a floor, and taking more of the degree past 11 buys nothing.

A reduced model of a forty-state system built out of 24 samples of its transfer function and nothing else — no state matrix, no eigenvalues, no factorisation of anything the model is of. The Loewner pencil of divided differences is formed from the samples, its singular values are read for the degree (9 here, with a gap of 4.41 across the cliff), and the pencil is projected onto that many directions. The filled curve is the worst relative error at the samples the model was built from; the open one is the worst error at forty points it never saw. With relative noise of 10⁻¹⁰ on the samples the singular values stop falling to the rounding level, the cliff becomes a slope, and no order does better than 1.63·10⁻⁷ — the data's own error is a floor, and taking more of the degree past 11 buys nothing.

A reduced model of a forty-state system built out of 24 samples of its transfer function and nothing else — no state matrix, no eigenvalues, no factorisation of anything the model is of. The Loewner pencil of divided differences is formed from the samples, its singular values are read for the degree (10 here, with a gap of 6561 across the cliff), and the pencil is projected onto that many directions. The filled curve is the worst relative error at the samples the model was built from; the open one is the worst error at forty points it never saw. With relative noise of 10⁻⁸ on the samples the singular values stop falling to the rounding level, the cliff becomes a slope, and no order does better than 1.64·10⁻⁵ — the data's own error is a floor, and taking more of the degree past 11 buys nothing.

A reduced model of a forty-state system built out of 24 samples of its transfer function and nothing else — no state matrix, no eigenvalues, no factorisation of anything the model is of. The Loewner pencil of divided differences is formed from the samples, its singular values are read for the degree (12 here, with a gap of ∞ across the cliff), and the pencil is projected onto that many directions. The filled curve is the worst relative error at the samples the model was built from; the open one is the worst error at forty points it never saw. With relative noise of 10⁻⁴ on the samples the singular values stop falling to the rounding level, the cliff becomes a slope, and no order does better than 0.19 — the data's own error is a floor, and taking more of the degree past 11 buys nothing.

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.

17 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 draws a decomposition and prints its residual. It calls orderSweep, orderSweep, 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.