contour-count

At its defaults it draws counting the eigenvalues inside a circle: an integral that is an integer once it has converged. The number of eigenvalues of the delay problem inside |z| = 2, computed as (1/2πi)∮tr(T(z)⁻¹T′(z))dz by the trapezoidal rule. The true count is 2, from the closed form. The error runs 0.00282, 0.0572, 0.0151, 5.25·10⁻⁴, 3.6·10⁻⁷, 1.35·10⁻¹³ at 4 to 128 points — a straight line on this axis is a constant factor a doubling, and this falls faster than that, which is what exponential convergence looks like when the integrand is analytic on the contour. The answer is then ROUNDED, and 0 of the 6 quadratures round to the wrong integer while looking exactly as confident as the ones that do not. The only evidence available is the distance from the nearest integer, which is the quantity plotted.

contour-count is one function in lib/figures/nlevp.js — infinitely many eigenvalues — a contour that counts, and the probe block that is a ceiling. 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 number of eigenvalues of the delay problem inside |z| = 2, computed as (1/2πi)∮tr(T(z)⁻¹T′(z))dz by the trapezoidal rule. The true count is 2, from the closed form. The error runs 0.00282, 0.0572, 0.0151, 5.25·10⁻⁴, 3.6·10⁻⁷, 1.35·10⁻¹³ at 4 to 128 points — a straight line on this axis is a constant factor a doubling, and this falls faster than that, which is what exponential convergence looks like when the integrand is analytic on the contour. The answer is then ROUNDED, and 0 of the 6 quadratures round to the wrong integer while looking exactly as confident as the ones that do not. The only evidence available is the distance from the nearest integer, which is the quantity plotted.

The number of eigenvalues of the delay problem inside |z| = 2, computed as (1/2πi)∮tr(T(z)⁻¹T′(z))dz by the trapezoidal rule. The true count is 2, from the closed form. The error runs 0.00282, 0.0572, 0.0151, 5.25·10⁻⁴, 3.6·10⁻⁷, 1.35·10⁻¹³ at 4 to 128 points — a straight line on this axis is a constant factor a doubling, and this falls faster than that, which is what exponential convergence looks like when the integrand is analytic on the contour. The answer is then ROUNDED, and 0 of the 6 quadratures round to the wrong integer while looking exactly as confident as the ones that do not. The only evidence available is the distance from the nearest integer, which is the quantity plotted.

The number of eigenvalues of the delay problem inside |z| = 1.5, computed as (1/2πi)∮tr(T(z)⁻¹T′(z))dz by the trapezoidal rule. The true count is 1, from the closed form. The error runs 0.473, 0.39, 0.289, 0.142, 0.0268, 7.58·10⁻⁴ at 4 to 128 points — a straight line on this axis is a constant factor a doubling, and this falls faster than that, which is what exponential convergence looks like when the integrand is analytic on the contour. The answer is then ROUNDED, and 0 of the 6 quadratures round to the wrong integer while looking exactly as confident as the ones that do not. The only evidence available is the distance from the nearest integer, which is the quantity plotted.

The number of eigenvalues of the delay problem inside |z| = 4, computed as (1/2πi)∮tr(T(z)⁻¹T′(z))dz by the trapezoidal rule. The true count is 4, from the closed form. The error runs 2.23, 5.64, 1.38, 0.167, 0.00222, 0.00483 at 4 to 128 points — a straight line on this axis is a constant factor a doubling, and this falls faster than that, which is what exponential convergence looks like when the integrand is analytic on the contour. The answer is then ROUNDED, and 3 of the 6 quadratures round to the wrong integer while looking exactly as confident as the ones that do not. The only evidence available is the distance from the nearest integer, which is the quantity plotted.

The number of eigenvalues of the delay problem inside |z| = 5, computed as (1/2πi)∮tr(T(z)⁻¹T′(z))dz by the trapezoidal rule. The true count is 12, from the closed form. The error runs 1.81, 3.37, 0.706, 0.0412, 0.0052, 1.43·10⁻⁵ at 4 to 128 points — a straight line on this axis is a constant factor a doubling, and this falls faster than that, which is what exponential convergence looks like when the integrand is analytic on the contour. The answer is then ROUNDED, and 3 of the 6 quadratures round to the wrong integer while looking exactly as confident as the ones that do not. The only evidence available is the distance from the nearest integer, which is the quantity plotted.

The number of eigenvalues of the delay problem inside |z| = 3, computed as (1/2πi)∮tr(T(z)⁻¹T′(z))dz by the trapezoidal rule. The true count is 3, from the closed form. The error runs 0.151, 0.532, 0.0874, 0.0263, 8.44·10⁻⁴, 7.21·10⁻⁷ at 4 to 128 points — a straight line on this axis is a constant factor a doubling, and this falls faster than that, which is what exponential convergence looks like when the integrand is analytic on the contour. The answer is then ROUNDED, and 1 of the 6 quadratures round to the wrong integer while looking exactly as confident as the ones that do not. The only evidence available is the distance from the nearest integer, which is the quantity plotted.

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.

7 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 countInside, 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 173 of 325 generators — 158 print a residual and 15 are exempt with a published reason; 152 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.