singular-scatter

At its defaults it draws eigenvalues of a singular 6×6 pencil after a perturbation of 10^-8, over 8 seeds. A and B share a null vector, so det(A − λB) is identically zero and every λ is an eigenvalue: the question has no answer. Perturbing by 10^-8 makes the pencil regular, and a solver then returns 6 eigenvalues with residuals no larger than 1.8·10⁻⁹ — small by any standard a caller would apply, and 3.7·10¹⁰ times smaller than the disagreement between seeds. Each row here is one seed. No two rows agree, the values are spread over 66.4, and nothing in the output distinguishes this from a well-posed problem. The only thing that says so is the exact characteristic polynomial, which has no coefficients at all.

singular-scatter is one function in lib/figures/pencil.js — pencils — two matrices, an eigenvalue with no value, and a problem with no answer. 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 and B share a null vector, so det(A − λB) is identically zero and every λ is an eigenvalue: the question has no answer. Perturbing by 10^-8 makes the pencil regular, and a solver then returns 6 eigenvalues with residuals no larger than 1.8·10⁻⁹ — small by any standard a caller would apply, and 3.7·10¹⁰ times smaller than the disagreement between seeds. Each row here is one seed. No two rows agree, the values are spread over 66.4, and nothing in the output distinguishes this from a well-posed problem. The only thing that says so is the exact characteristic polynomial, which has no coefficients at all.

A and B share a null vector, so det(A − λB) is identically zero and every λ is an eigenvalue: the question has no answer. Perturbing by 10^-8 makes the pencil regular, and a solver then returns 6 eigenvalues with residuals no larger than 1.8·10⁻⁹ — small by any standard a caller would apply, and 3.7·10¹⁰ times smaller than the disagreement between seeds. Each row here is one seed. No two rows agree, the values are spread over 66.4, and nothing in the output distinguishes this from a well-posed problem. The only thing that says so is the exact characteristic polynomial, which has no coefficients at all.

A and B share a null vector, so det(A − λB) is identically zero and every λ is an eigenvalue: the question has no answer. Perturbing by 10^-2 makes the pencil regular, and a solver then returns 6 eigenvalues with residuals no larger than 9.63·10⁻⁴ — small by any standard a caller would apply, and 6·10⁴ times smaller than the disagreement between seeds. Each row here is one seed. No two rows agree, the values are spread over 58.1, and nothing in the output distinguishes this from a well-posed problem. The only thing that says so is the exact characteristic polynomial, which has no coefficients at all.

A and B share a null vector, so det(A − λB) is identically zero and every λ is an eigenvalue: the question has no answer. Perturbing by 10^-14 makes the pencil regular, and a solver then returns 6 eigenvalues with residuals no larger than 2.11·10⁻¹⁵ — small by any standard a caller would apply, and 1.9·10²³ times smaller than the disagreement between seeds. Each row here is one seed. No two rows agree, the values are spread over 4.1·10⁸, and nothing in the output distinguishes this from a well-posed problem. The only thing that says so is the exact characteristic polynomial, which has no coefficients at all.

A and B share a null vector, so det(A − λB) is identically zero and every λ is an eigenvalue: the question has no answer. Perturbing by 10^-5 makes the pencil regular, and a solver then returns 6 eigenvalues with residuals no larger than 1.36·10⁻⁶ — small by any standard a caller would apply, and 4.5·10⁷ times smaller than the disagreement between seeds. Each row here is one seed. No two rows agree, the values are spread over 61.5, and nothing in the output distinguishes this from a well-posed problem. The only thing that says so is the exact characteristic polynomial, which has no coefficients at all.

A and B share a null vector, so det(A − λB) is identically zero and every λ is an eigenvalue: the question has no answer. Perturbing by 10^-11 makes the pencil regular, and a solver then returns 6 eigenvalues with residuals no larger than 1.89·10⁻¹² — small by any standard a caller would apply, and 2.2·10¹⁶ times smaller than the disagreement between seeds. Each row here is one seed. No two rows agree, the values are spread over 40800, and nothing in the output distinguishes this from a well-posed problem. The only thing that says so is the exact characteristic polynomial, which has no coefficients at all.

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.

10 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 scatterSweep, 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 113 of 219 generators — 98 print a residual and 15 are exempt with a published reason; 106 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.