overdamped-gap

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.

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.

A chain of 8 masses with C = 7K, 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 0.48 between the two groups. One factorisation answers both questions, which is the second thing a boolean would not have carried.

A chain of 8 masses with C = 20K, 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 2.31 between the two groups. One factorisation answers both questions, which is the second thing a boolean would not have carried.

A chain of 8 masses with C = 10K, 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 0.9861 between the two groups. One factorisation answers both questions, which is the second thing a boolean would not have carried.

A chain of 8 masses with C = 14K, 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 1.539 between the two groups. One factorisation answers both questions, which is the second thing a boolean would not have carried.

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.

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 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 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.