A double root, approached: the pair separates like √ε and the accuracy fails like √u

At its defaults it draws a double root, approached: the pair separates like √ε and the accuracy fails like √u. A chain of 8 masses at β = β*(1 + ε), where β* is the critical damping and the smallest mode's two eigenvalues coincide at ε = 0. The separation of that pair is 0.9657√ε at every one of twelve decades — a spread of 1.02 in the constant — and the computed spectrum's worst relative error against the closed form rises as the pair closes, reaching 3.732·10⁻⁸ at the boundary itself. That is √u times a small constant: half the digits, on a problem where κ(K) is 32.16 at every stop and the coefficients are integers. Nothing here is ill conditioned in any sense this site has used before; what is unbounded is the derivative of the map from a coefficient to a double root.

double-root-loss 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 at β = β*(1 + ε), where β* is the critical damping and the smallest mode's two eigenvalues coincide at ε = 0. The separation of that pair is 0.9657√ε at every one of twelve decades — a spread of 1.02 in the constant — and the computed spectrum's worst relative error against the closed form rises as the pair closes, reaching 3.732·10⁻⁸ at the boundary itself. That is √u times a small constant: half the digits, on a problem where κ(K) is 32.16 at every stop and the coefficients are integers. Nothing here is ill conditioned in any sense this site has used before; what is unbounded is the derivative of the map from a coefficient to a double root.

A chain of 4 masses at β = β*(1 + ε), where β* is the critical damping and the smallest mode's two eigenvalues coincide at ε = 0. The separation of that pair is 1.75√ε at every one of twelve decades — a spread of 1.002 in the constant — and the computed spectrum's worst relative error against the closed form rises as the pair closes, reaching 2.855·10⁻⁸ at the boundary itself. That is √u times a small constant: half the digits, on a problem where κ(K) is 9.472 at every stop and the coefficients are integers. Nothing here is ill conditioned in any sense this site has used before; what is unbounded is the derivative of the map from a coefficient to a double root.

A chain of 16 masses at β = β*(1 + ε), where β* is the critical damping and the smallest mode's two eigenvalues coincide at ε = 0. The separation of that pair is 0.5162√ε at every one of twelve decades — a spread of 1.014 in the constant — and the computed spectrum's worst relative error against the closed form rises as the pair closes, reaching 1.085·10⁻⁷ at the boundary itself. That is √u times a small constant: half the digits, on a problem where κ(K) is 116.5 at every stop and the coefficients are integers. Nothing here is ill conditioned in any sense this site has used before; what is unbounded is the derivative of the map from a coefficient to a double root.

A chain of 8 masses at β = β*(1 + ε), where β* is the critical damping and the smallest mode's two eigenvalues coincide at ε = 0. The separation of that pair is 0.9657√ε at every one of twelve decades — a spread of 1.02 in the constant — and the computed spectrum's worst relative error against the closed form rises as the pair closes, reaching 3.732·10⁻⁸ at the boundary itself. That is √u times a small constant: half the digits, on a problem where κ(K) is 32.16 at every stop and the coefficients are integers. Nothing here is ill conditioned in any sense this site has used before; what is unbounded is the derivative of the map from a coefficient to a double root.

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.

12 distinct claims across 4 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.

The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.

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.