4 eigenvalues inside a circle of radius 4, and 2 moments to reach them

At its defaults it draws 4 eigenvalues inside a circle of radius 4, and 2 moments to reach them. A delay eigenvalue problem T(λ) = A − λI + exp(−λ)I, which has infinitely many eigenvalues and no linearisation, so the only finite question is what lies inside a given contour. The crosses are the Lambert-W closed form λ = μ + Wₖ(γ·exp(−μ)), one per eigenvalue of A per branch; the discs are what 2 contour moments against a probe block of 2 recover, agreeing to 2.16·10⁻¹². With one moment the method could have returned at most 2 of them and would have said nothing about the rest.

contour-recovery is one function in lib/figures/beyn.js — a ceiling with a knob on it — higher contour moments, and the conditioning that rises with 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.

A delay eigenvalue problem T(λ) = A − λI + exp(−λ)I, which has infinitely many eigenvalues and no linearisation, so the only finite question is what lies inside a given contour. The crosses are the Lambert-W closed form λ = μ + Wₖ(γ·exp(−μ)), one per eigenvalue of A per branch; the discs are what 2 contour moments against a probe block of 2 recover, agreeing to 2.16·10⁻¹². With one moment the method could have returned at most 2 of them and would have said nothing about the rest.

A delay eigenvalue problem T(λ) = A − λI + exp(−λ)I, which has infinitely many eigenvalues and no linearisation, so the only finite question is what lies inside a given contour. The crosses are the Lambert-W closed form λ = μ + Wₖ(γ·exp(−μ)), one per eigenvalue of A per branch; the discs are what 2 contour moments against a probe block of 2 recover, agreeing to 2.16·10⁻¹². With one moment the method could have returned at most 2 of them and would have said nothing about the rest.

A delay eigenvalue problem T(λ) = A − λI + exp(−λ)I, which has infinitely many eigenvalues and no linearisation, so the only finite question is what lies inside a given contour. The crosses are the Lambert-W closed form λ = μ + Wₖ(γ·exp(−μ)), one per eigenvalue of A per branch; the discs are what 3 contour moments against a probe block of 4 recover, agreeing to 6.22·10⁻¹⁵. With one moment the method could have returned at most 4 of them and would have said nothing about the rest.

A delay eigenvalue problem T(λ) = A − λI + exp(−λ)I, which has infinitely many eigenvalues and no linearisation, so the only finite question is what lies inside a given contour. The crosses are the Lambert-W closed form λ = μ + Wₖ(γ·exp(−μ)), one per eigenvalue of A per branch; the discs are what 3 contour moments against a probe block of 4 recover, agreeing to 4.89·10⁻¹⁵. With one moment the method could have returned at most 4 of them and would have said nothing about the rest.

A delay eigenvalue problem T(λ) = A − λI + exp(−λ)I, which has infinitely many eigenvalues and no linearisation, so the only finite question is what lies inside a given contour. The crosses are the Lambert-W closed form λ = μ + Wₖ(γ·exp(−μ)), one per eigenvalue of A per branch; the discs are what 12 contour moments against a probe block of 1 recover, agreeing to 4.14·10⁻¹¹. With one moment the method could have returned at most 1 of them and would have said nothing about the rest.

A delay eigenvalue problem T(λ) = A − λI + exp(−λ)I, which has infinitely many eigenvalues and no linearisation, so the only finite question is what lies inside a given contour. The crosses are the Lambert-W closed form λ = μ + Wₖ(γ·exp(−μ)), one per eigenvalue of A per branch; the discs are what 6 contour moments against a probe block of 2 recover, agreeing to 4.22·10⁻¹³. With one moment the method could have returned at most 2 of them and would have said nothing about the rest.

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 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 contourMoments, momentEigenvalues, 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.