Where a quadratic stops being hyperbolic, located by a Cholesky and by a sine

At its defaults it draws where a quadratic stops being hyperbolic, located by a cholesky and by a sine. The overdamping margin min over modes of (βκ)² − 4κ, for a chain of 8 masses, against β. It reaches zero at β* = 1/sin(π/2(n+1)) = 5.75877048314, which is the closed form. Bisecting on a completely different question — does a Cholesky of −Q(μ) complete for some μ — gives 5.75877048314, agreeing to 13 digits. Neither route computes an eigenvalue. The marks below the axis are the largest imaginary part in the computed spectrum, which is zero to the rounding level above β* and not below it, so a third route agrees with the other two about where the boundary is.

hyperbolic-boundary 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.

The overdamping margin min over modes of (βκ)² − 4κ, for a chain of 8 masses, against β. It reaches zero at β* = 1/sin(π/2(n+1)) = 5.75877048314, which is the closed form. Bisecting on a completely different question — does a Cholesky of −Q(μ) complete for some μ — gives 5.75877048314, agreeing to 13 digits. Neither route computes an eigenvalue. The marks below the axis are the largest imaginary part in the computed spectrum, which is zero to the rounding level above β* and not below it, so a third route agrees with the other two about where the boundary is.

The overdamping margin min over modes of (βκ)² − 4κ, for a chain of 4 masses, against β. It reaches zero at β* = 1/sin(π/2(n+1)) = 3.2360679775, which is the closed form. Bisecting on a completely different question — does a Cholesky of −Q(μ) complete for some μ — gives 3.2360679775, agreeing to 13 digits. Neither route computes an eigenvalue. The marks below the axis are the largest imaginary part in the computed spectrum, which is zero to the rounding level above β* and not below it, so a third route agrees with the other two about where the boundary is.

The overdamping margin min over modes of (βκ)² − 4κ, for a chain of 6 masses, against β. It reaches zero at β* = 1/sin(π/2(n+1)) = 4.49395920743, which is the closed form. Bisecting on a completely different question — does a Cholesky of −Q(μ) complete for some μ — gives 4.49395920743, agreeing to 15 digits. Neither route computes an eigenvalue. The marks below the axis are the largest imaginary part in the computed spectrum, which is zero to the rounding level above β* and not below it, so a third route agrees with the other two about where the boundary is.

The overdamping margin min over modes of (βκ)² − 4κ, for a chain of 10 masses, against β. It reaches zero at β* = 1/sin(π/2(n+1)) = 7.02667418333, which is the closed form. Bisecting on a completely different question — does a Cholesky of −Q(μ) complete for some μ — gives 7.02667418333, agreeing to 14 digits. Neither route computes an eigenvalue. The marks below the axis are the largest imaginary part in the computed spectrum, which is zero to the rounding level above β* and not below it, so a third route agrees with the other two about where the boundary is.

The overdamping margin min over modes of (βκ)² − 4κ, for a chain of 12 masses, against β. It reaches zero at β* = 1/sin(π/2(n+1)) = 8.29622981056, which is the closed form. Bisecting on a completely different question — does a Cholesky of −Q(μ) complete for some μ — gives 8.29622981056, agreeing to 13 digits. Neither route computes an eigenvalue. The marks below the axis are the largest imaginary part in the computed spectrum, which is zero to the rounding level above β* and not below it, so a third route agrees with the other two about where the boundary is.

The overdamping margin min over modes of (βκ)² − 4κ, for a chain of 16 masses, against β. It reaches zero at β* = 1/sin(π/2(n+1)) = 10.8379514475, which is the closed form. Bisecting on a completely different question — does a Cholesky of −Q(μ) complete for some μ — gives 10.8379514475, agreeing to 14 digits. Neither route computes an eigenvalue. The marks below the axis are the largest imaginary part in the computed spectrum, which is zero to the rounding level above β* and not below it, so a third route agrees with the other two about where the boundary is.

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 qepEigen, criticalByCertificate, 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.