36 eigenvalues come back, 6 of them mean something

At its defaults it draws 36 eigenvalues come back, 6 of them mean something. The real eigenvalues of the n(m+1) = 36 that a rational approximant with 5 poles produces, on the real line, with the branch point at −0.4 marked. The 6 discs to the right are the answers, agreeing with the closed form to 7.74·10⁻⁴. The 6 crosses to the left are exact eigenvalues of the approximant lying where γ√(λ + c) is not a real number at all — so the residual that says nothing about accuracy is decisive as a filter: against T̃ all 12 look perfect, and against T the spurious ones do not have a residual. The remaining 24 are complex and are discarded the same way.

spurious-spectrum is one function in lib/figures/ratapprox.js — the problem the solver was given — the approximation committed before the arithmetic. 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 real eigenvalues of the n(m+1) = 36 that a rational approximant with 5 poles produces, on the real line, with the branch point at −0.4 marked. The 6 discs to the right are the answers, agreeing with the closed form to 7.74·10⁻⁴. The 6 crosses to the left are exact eigenvalues of the approximant lying where γ√(λ + c) is not a real number at all — so the residual that says nothing about accuracy is decisive as a filter: against T̃ all 12 look perfect, and against T the spurious ones do not have a residual. The remaining 24 are complex and are discarded the same way.

The real eigenvalues of the n(m+1) = 36 that a rational approximant with 5 poles produces, on the real line, with the branch point at −0.4 marked. The 6 discs to the right are the answers, agreeing with the closed form to 7.74·10⁻⁴. The 6 crosses to the left are exact eigenvalues of the approximant lying where γ√(λ + c) is not a real number at all — so the residual that says nothing about accuracy is decisive as a filter: against T̃ all 12 look perfect, and against T the spurious ones do not have a residual. The remaining 24 are complex and are discarded the same way.

The real eigenvalues of the n(m+1) = 18 that a rational approximant with 2 poles produces, on the real line, with the branch point at −0.4 marked. The 6 discs to the right are the answers, agreeing with the closed form to 0.0424. The 12 crosses to the left are exact eigenvalues of the approximant lying where γ√(λ + c) is not a real number at all — so the residual that says nothing about accuracy is decisive as a filter: against T̃ all 18 look perfect, and against T the spurious ones do not have a residual. The remaining 0 are complex and are discarded the same way.

The real eigenvalues of the n(m+1) = 24 that a rational approximant with 3 poles produces, on the real line, with the branch point at −0.4 marked. The 6 discs to the right are the answers, agreeing with the closed form to 0.0109. The 6 crosses to the left are exact eigenvalues of the approximant lying where γ√(λ + c) is not a real number at all — so the residual that says nothing about accuracy is decisive as a filter: against T̃ all 12 look perfect, and against T the spurious ones do not have a residual. The remaining 12 are complex and are discarded the same way.

The real eigenvalues of the n(m+1) = 30 that a rational approximant with 4 poles produces, on the real line, with the branch point at −0.4 marked. The 6 discs to the right are the answers, agreeing with the closed form to 0.00263. The 0 crosses to the left are exact eigenvalues of the approximant lying where γ√(λ + c) is not a real number at all — so the residual that says nothing about accuracy is decisive as a filter: against T̃ all 6 look perfect, and against T the spurious ones do not have a residual. The remaining 24 are complex and are discarded the same way.

The real eigenvalues of the n(m+1) = 42 that a rational approximant with 6 poles produces, on the real line, with the branch point at −0.4 marked. The 6 discs to the right are the answers, agreeing with the closed form to 1.9·10⁻⁴. The 0 crosses to the left are exact eigenvalues of the approximant lying where γ√(λ + c) is not a real number at all — so the residual that says nothing about accuracy is decisive as a filter: against T̃ all 6 look perfect, and against T the spurious ones do not have a residual. The remaining 36 are complex and are discarded the same way.

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.

8 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 fitRational, rationalEigen, 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.