The 16 real eigenvalues of an overdamped chain, in two groups of 8, with the certificate between them

At its defaults it draws the 16 real eigenvalues of an overdamped chain, in two groups of 8, with the certificate between them. A chain of 8 masses with C = 8K, which is above the critical damping 5.75877 = 1/sin(π/2(n+1)), so the quadratic is hyperbolic and every one of its 16 eigenvalues is real. The vertical line is the μ at which Q(μ) is negative definite — a Cholesky that completes, and the whole proof that the spectrum is real. It also lies in the gap: exactly 8 eigenvalues above it and 8 below, with a gap of 0.6698 between the two groups. One factorisation answers both questions, which is the second thing a boolean would not have carried.

overdamped-gap is one function in lib/figures/hyperbolic.js — real by class — a cholesky that certifies a spectrum, and a double root that costs half the digits. 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 chain of 8 masses with C = 8K, which is above the critical damping 5.75877 = 1/sin(π/2(n+1)), so the quadratic is hyperbolic and every one of its 16 eigenvalues is real. The vertical line is the μ at which Q(μ) is negative definite — a Cholesky that completes, and the whole proof that the spectrum is real. It also lies in the gap: exactly 8 eigenvalues above it and 8 below, with a gap of 0.6698 between the two groups. One factorisation answers both questions, which is the second thing a boolean would not have carried.

Reference: the closed-form spectrum. The companion route inverts the leading coefficient and takes a real Schur form; the definite route takes a Cholesky of the best positive definite combination of a symmetric linearisation and a Jacobi eigensolver. ε 10⁻²: companion 8.88·10⁻¹⁵, definite 2.35·10⁻¹⁴; ε 10⁻⁴: companion 4.15·10⁻¹⁴, definite 3.95·10⁻¹²; ε 10⁻⁶: companion 5.3·10⁻¹³, definite 1.46·10⁻¹⁰; ε 10⁻⁸: companion 1.13·10⁻¹², definite 5.92·10⁻⁸; ε 10⁻¹⁰: companion 1.07·10⁻¹⁰, definite 9.8·10⁻⁶; ε 10⁻¹²: companion 2.69·10⁻⁹, definite 0.002. The definite route's spectrum is real by construction; the companion route returned no complex value at any of these margins.

ε 10⁻²: Crawford number 0.002288, 0.2288 ε; separation 0.09848, 0.9848 times the square root of ε; ε 10⁻⁴: Crawford number 2.279·10⁻⁵, 0.2279 ε; separation 0.009823, 0.9823 times the square root of ε; ε 10⁻⁶: Crawford number 2.279·10⁻⁷, 0.2279 ε; separation 9.823·10⁻⁴, 0.9823 times the square root of ε; ε 10⁻⁸: Crawford number 2.279·10⁻⁹, 0.2279 ε; separation 9.823·10⁻⁵, 0.9823 times the square root of ε; ε 10⁻¹⁰: Crawford number 2.279·10⁻¹¹, 0.2279 ε; separation 9.823·10⁻⁶, 0.9823 times the square root of ε; ε 10⁻¹²: Crawford number 2.284·10⁻¹³, 0.2284 ε; separation 9.822·10⁻⁷, 0.9822 times the square root of ε.

Where the curve is above zero the combination is positive definite. ε 10⁻²: peak 0.2288 ε at t = 0.337; at t = 0, 0.2158 ε; at t = 1.5, 0.0901 ε; ε 10⁻⁴: peak 0.2279 ε at t = 0.334; at t = 0, 0.2153 ε; at t = 1.5, 0.0901 ε; ε 10⁻⁶: peak 0.2279 ε at t = 0.334; at t = 0, 0.2153 ε; at t = 1.5, 0.0901 ε. Divided by ε, the three profiles coincide.

closed form: -0.347296846423 and -0.347295864245; companion: -0.347296845490 and -0.347295865178; definite pencil: -0.347296355138 and -0.346602323314. The axis is centred on -0.3472963553 and 0.00174 wide.

ε 10⁻⁴: 0 of 20 reading every Schur block exactly, 0 at the default block tolerance; ε 10⁻⁸: 0 of 20 reading every Schur block exactly, 0 at the default block tolerance; ε 10⁻¹²: 0 of 20 reading every Schur block exactly, 0 at the default block tolerance; ε 0: 5 of 20 reading every Schur block exactly, 6 at the default block tolerance, largest imaginary part 5.3·10⁻⁸.

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.

19 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 splitAtCertificate, 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.