Two rank decisions for one integer: the McMillan degree of a 40-state model

At its defaults it draws two rank decisions for one integer: the mcmillan degree of a 40-state model. Singular values, each divided by the largest of its own set. The Loewner matrix is built from 32 samples of H on the imaginary axis and never touches A: its 6th singular value stands 9.46·10⁴ above the next, so the rank decision is not a close call. The Hankel matrix of Markov parameters CAᵏB is built from A, B and C — strictly more information — and has no cliff, because those parameters grow like ‖A‖₂ᵏ with ‖A‖₂ = 6714.1 — just under 4(n+1)², not the (n+1)² the stencil is scaled by — and its entries span 3·10³⁷. It returns 2 where the answer is 6. The definition is not the algorithm, and the route with less information is the one that works.

mcmillan-degree is one function in lib/figures/lti.js — a model that is a rational function — the degree that is a rank, by two routes that disagree. 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.

Singular values, each divided by the largest of its own set. The Loewner matrix is built from 32 samples of H on the imaginary axis and never touches A: its 6th singular value stands 9.46·10⁴ above the next, so the rank decision is not a close call. The Hankel matrix of Markov parameters CAᵏB is built from A, B and C — strictly more information — and has no cliff, because those parameters grow like ‖A‖₂ᵏ with ‖A‖₂ = 6714.1 — just under 4(n+1)², not the (n+1)² the stencil is scaled by — and its entries span 3·10³⁷. It returns 2 where the answer is 6. The definition is not the algorithm, and the route with less information is the one that works.

Singular values, each divided by the largest of its own set. The Loewner matrix is built from 32 samples of H on the imaginary axis and never touches A: its 6th singular value stands 9.46·10⁴ above the next, so the rank decision is not a close call. The Hankel matrix of Markov parameters CAᵏB is built from A, B and C — strictly more information — and has no cliff, because those parameters grow like ‖A‖₂ᵏ with ‖A‖₂ = 6714.1 — just under 4(n+1)², not the (n+1)² the stencil is scaled by — and its entries span 3·10³⁷. It returns 2 where the answer is 6. The definition is not the algorithm, and the route with less information is the one that works.

Singular values, each divided by the largest of its own set. The Loewner matrix is built from 32 samples of H on the imaginary axis and never touches A: its 6th singular value stands 9.53·10⁵ above the next, so the rank decision is not a close call. The Hankel matrix of Markov parameters CAᵏB is built from A, B and C — strictly more information — and has no cliff, because those parameters grow like ‖A‖₂ᵏ with ‖A‖₂ = 666.18 — just under 4(n+1)², not the (n+1)² the stencil is scaled by — and its entries span 2.7·10³⁶. It returns 2 where the answer is 6. The definition is not the algorithm, and the route with less information is the one that works.

Singular values, each divided by the largest of its own set. The Loewner matrix is built from 32 samples of H on the imaginary axis and never touches A: its 6th singular value stands 3.51·10⁵ above the next, so the rank decision is not a close call. The Hankel matrix of Markov parameters CAᵏB is built from A, B and C — strictly more information — and has no cliff, because those parameters grow like ‖A‖₂ᵏ with ‖A‖₂ = 1754.1 — just under 4(n+1)², not the (n+1)² the stencil is scaled by — and its entries span 1.4·10³⁷. It returns 2 where the answer is 6. The definition is not the algorithm, and the route with less information is the one that works.

Singular values, each divided by the largest of its own set. The Loewner matrix is built from 32 samples of H on the imaginary axis and never touches A: its 6th singular value stands 7.4·10⁴ above the next, so the rank decision is not a close call. The Hankel matrix of Markov parameters CAᵏB is built from A, B and C — strictly more information — and has no cliff, because those parameters grow like ‖A‖₂ᵏ with ‖A‖₂ = 1.4874·10⁴ — just under 4(n+1)², not the (n+1)² the stencil is scaled by — and its entries span 3.5·10³⁷. It returns 2 where the answer is 6. The definition is not the algorithm, and the route with less information is the one that works.

Singular values, each divided by the largest of its own set. The Loewner matrix is built from 32 samples of H on the imaginary axis and never touches A: its 6th singular value stands 2.91·10⁴ above the next, so the rank decision is not a close call. The Hankel matrix of Markov parameters CAᵏB is built from A, B and C — strictly more information — and has no cliff, because those parameters grow like ‖A‖₂ᵏ with ‖A‖₂ = 2.6234·10⁴ — just under 4(n+1)², not the (n+1)² the stencil is scaled by — and its entries span 3.7·10³⁷. It returns 2 where the answer is 6. The definition is not the algorithm, and the route with less information is the one that works.

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.

12 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 loewnerDegree, markovDegree, 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.