Counting the eigenvalues inside a circle: an integral that is an integer once it has converged

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.

0.822 to 1.587: centred 22 points at c = 1.10, R = 0.88; about the origin 41 at R = 2.064, a geometric mean between two real eigenvalues. 0.822 to 2.686: centred 21 points at c = 1.43, R = 1.67; about the origin 78 at R = 3.129, a geometric mean between two real eigenvalues. 0.822 to 3.644: centred 32 points at c = 2.30, R = 2.85; about the origin 341 at R = 3.908, between a real eigenvalue and a complex pair. 1.587 to 2.686: centred 38 points at c = 2.16, R = 0.88. 1.587 to 3.644: centred 20 points at c = 3.10, R = 1.86. 2.686 to 3.644: centred 26 points at c = 3.25, R = 0.97. Only runs starting at the first eigenvalue can be held by a circle about the origin.

Every pair with an error above 10⁻⁸. The worst relative disagreement is 1.8·10⁻⁸. Circles passing within 0.05 of an eigenvalue are left out.

Radius 0.3234: both eigenvalues sit at ratio 0.551 to the circle, on opposite rays. At even point counts the error falls like 0.192 per point, the next eigenvalue's ratio — 1.9·10⁻⁶ at 8, 6.8·10⁻⁸ at 10, 2.5·10⁻⁹ at 12, 9.1·10⁻¹¹ at 14, 3.4·10⁻¹² at 16. At odd counts it falls like 0.551 — 0.0094 at 9, 0.0029 at 11, 8.7·10⁻⁴ at 13, 2.6·10⁻⁴ at 15, 8·10⁻⁵ at 17.

About the origin: radius 3.908, 341 points for ten digits. About 2.30: radius 2.85, 32 points. The real eigenvalues are 0.822, 1.587, 2.686 and 3.644; the nearest complex pairs have moduli 4.19 to 4.56.

first 2, about 1.10, R 0.88 (22 points): largest ratios 0.554 (1.587, inside), 0.554 (2.686, outside), 0.345 (3.644, outside), 0.317 (0.822, inside), 0.183 (pair 4.19, outside). first 3, about 1.43, R 1.67 (21 points): largest ratios 0.754 (2.686, inside), 0.754 (3.644, outside), 0.361 (0.822, inside), 0.335 (pair 4.19, outside), 0.335 (pair 4.19, outside). first 4, about 2.30, R 2.85 (32 points): largest ratios 0.519 (0.822, inside), 0.508 (pair 4.25, outside), 0.508 (pair 4.25, outside), 0.507 (pair 4.19, outside), 0.507 (pair 4.19, outside). first 4, about the origin, R 3.91 (341 points): largest ratios 0.933 (3.644, inside), 0.933 (pair 4.19, outside), 0.933 (pair 4.19, outside), 0.920 (pair 4.25, outside), 0.920 (pair 4.25, outside). A short vertical line marks two real eigenvalues, one inside and one outside on the same side of the centre, at equal ratio — a pair whose terms cancel.

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.

13 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 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.