The condition number the problem has, and the one the solver's error analysis is written against

At its defaults it draws the condition number the problem has, and the one the solver's error analysis is written against. The same eigenvalue of the same overdamped chain of 8 masses, in seven systems of units. Its condition number as an eigenvalue of the QUADRATIC — Tisseur's, with the three coefficient norms in the numerator and yᵀQ′(λ)x in the denominator — is 4.98 at γ = 1 and 4.98 at γ = 10⁶, a spread of 1 over six decades: it cannot move, because a change of units is not a change of problem. Its condition number as an eigenvalue of the LINEARISED MATRIX runs 49.66 to 7.844·10¹¹, a factor of 1.58·10¹⁰. The forward error follows the second one, and the first one is the honest description of the problem — so the substitution has manufactured an ill conditioning that belongs to the algorithm rather than to the question.

qep-conditioning is one function in lib/figures/qepbe.js — the price of linearising — a residual that stays flat while the answer loses eleven orders. 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 same eigenvalue of the same overdamped chain of 8 masses, in seven systems of units. Its condition number as an eigenvalue of the QUADRATIC — Tisseur's, with the three coefficient norms in the numerator and yᵀQ′(λ)x in the denominator — is 4.98 at γ = 1 and 4.98 at γ = 10⁶, a spread of 1 over six decades: it cannot move, because a change of units is not a change of problem. Its condition number as an eigenvalue of the LINEARISED MATRIX runs 49.66 to 7.844·10¹¹, a factor of 1.58·10¹⁰. The forward error follows the second one, and the first one is the honest description of the problem — so the substitution has manufactured an ill conditioning that belongs to the algorithm rather than to the question.

The same eigenvalue of the same overdamped chain of 8 masses, in seven systems of units. Its condition number as an eigenvalue of the QUADRATIC — Tisseur's, with the three coefficient norms in the numerator and yᵀQ′(λ)x in the denominator — is 4.98 at γ = 1 and 4.98 at γ = 10⁶, a spread of 1 over six decades: it cannot move, because a change of units is not a change of problem. Its condition number as an eigenvalue of the LINEARISED MATRIX runs 49.66 to 7.844·10¹¹, a factor of 1.58·10¹⁰. The forward error follows the second one, and the first one is the honest description of the problem — so the substitution has manufactured an ill conditioning that belongs to the algorithm rather than to the question.

The same eigenvalue of the same overdamped chain of 14 masses, in seven systems of units. Its condition number as an eigenvalue of the QUADRATIC — Tisseur's, with the three coefficient norms in the numerator and yᵀQ′(λ)x in the denominator — is 6.557 at γ = 1 and 6.557 at γ = 10⁶, a spread of 1 over six decades: it cannot move, because a change of units is not a change of problem. Its condition number as an eigenvalue of the LINEARISED MATRIX runs 65.03 to 1.018·10¹², a factor of 1.57·10¹⁰. The forward error follows the second one, and the first one is the honest description of the problem — so the substitution has manufactured an ill conditioning that belongs to the algorithm rather than to the question.

The same eigenvalue of the same overdamped chain of 4 masses, in seven systems of units. Its condition number as an eigenvalue of the QUADRATIC — Tisseur's, with the three coefficient norms in the numerator and yᵀQ′(λ)x in the denominator — is 3.596 at γ = 1 and 3.596 at γ = 10⁶, a spread of 1 over six decades: it cannot move, because a change of units is not a change of problem. Its condition number as an eigenvalue of the LINEARISED MATRIX runs 36.5 to 5.929·10¹¹, a factor of 1.62·10¹⁰. The forward error follows the second one, and the first one is the honest description of the problem — so the substitution has manufactured an ill conditioning that belongs to the algorithm rather than to the question.

The same eigenvalue of the same overdamped chain of 6 masses, in seven systems of units. Its condition number as an eigenvalue of the QUADRATIC — Tisseur's, with the three coefficient norms in the numerator and yᵀQ′(λ)x in the denominator — is 4.338 at γ = 1 and 4.338 at γ = 10⁶, a spread of 1 over six decades: it cannot move, because a change of units is not a change of problem. Its condition number as an eigenvalue of the LINEARISED MATRIX runs 43.48 to 6.925·10¹¹, a factor of 1.59·10¹⁰. The forward error follows the second one, and the first one is the honest description of the problem — so the substitution has manufactured an ill conditioning that belongs to the algorithm rather than to the question.

The same eigenvalue of the same overdamped chain of 10 masses, in seven systems of units. Its condition number as an eigenvalue of the QUADRATIC — Tisseur's, with the three coefficient norms in the numerator and yᵀQ′(λ)x in the denominator — is 5.554 at γ = 1 and 5.554 at γ = 10⁶, a spread of 1 over six decades: it cannot move, because a change of units is not a change of problem. Its condition number as an eigenvalue of the LINEARISED MATRIX runs 55.22 to 8.685·10¹¹, a factor of 1.57·10¹⁰. The forward error follows the second one, and the first one is the honest description of the problem — so the substitution has manufactured an ill conditioning that belongs to the algorithm rather than to the question.

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, 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.