pencil-projective

At its defaults it draws the eigenvalues of a 6×6 pencil with 2 algebraic constraints, on the projective line. λ = α/β is a ratio, so an eigenvalue of a pencil is a direction rather than a number and it lives on a line whose two ends are the same point. Drawn as an angle φ = arctan λ, the 4 finite eigenvalues — -0.3769, 0.565, 4.383, 6.428 — sit inside the arc, and the 2 infinite ones sit at its ends, which is one point and not two. Nothing about them is degenerate: each carries a residual ‖βAx − αBx‖ in the same scaling as every other, the largest being 5.53·10⁻¹⁴, and an infinite eigenvalue's residual is ‖Bx‖/‖B‖ — the statement that its eigenvector is a null vector of B. The count is not a rank decision here: it is n minus the degree of det(A − λB), computed in exact rational arithmetic.

pencil-projective 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.

λ = α/β is a ratio, so an eigenvalue of a pencil is a direction rather than a number and it lives on a line whose two ends are the same point. Drawn as an angle φ = arctan λ, the 4 finite eigenvalues — -0.3769, 0.565, 4.383, 6.428 — sit inside the arc, and the 2 infinite ones sit at its ends, which is one point and not two. Nothing about them is degenerate: each carries a residual ‖βAx − αBx‖ in the same scaling as every other, the largest being 5.53·10⁻¹⁴, and an infinite eigenvalue's residual is ‖Bx‖/‖B‖ — the statement that its eigenvector is a null vector of B. The count is not a rank decision here: it is n minus the degree of det(A − λB), computed in exact rational arithmetic.

λ = α/β is a ratio, so an eigenvalue of a pencil is a direction rather than a number and it lives on a line whose two ends are the same point. Drawn as an angle φ = arctan λ, the 4 finite eigenvalues — -0.3769, 0.565, 4.383, 6.428 — sit inside the arc, and the 2 infinite ones sit at its ends, which is one point and not two. Nothing about them is degenerate: each carries a residual ‖βAx − αBx‖ in the same scaling as every other, the largest being 5.53·10⁻¹⁴, and an infinite eigenvalue's residual is ‖Bx‖/‖B‖ — the statement that its eigenvector is a null vector of B. The count is not a rank decision here: it is n minus the degree of det(A − λB), computed in exact rational arithmetic.

λ = α/β is a ratio, so an eigenvalue of a pencil is a direction rather than a number and it lives on a line whose two ends are the same point. Drawn as an angle φ = arctan λ, the 6 finite eigenvalues — -0.7663, -0.01384, 0.9593, 4.611, 5.61, 7.6 — sit inside the arc, and the 0 infinite ones sit at its ends, which is one point and not two. Nothing about them is degenerate: each carries a residual ‖βAx − αBx‖ in the same scaling as every other, the largest being 8.9·10⁻¹⁵, and an infinite eigenvalue's residual is ‖Bx‖/‖B‖ — the statement that its eigenvector is a null vector of B. The count is not a rank decision here: it is n minus the degree of det(A − λB), computed in exact rational arithmetic.

λ = α/β is a ratio, so an eigenvalue of a pencil is a direction rather than a number and it lives on a line whose two ends are the same point. Drawn as an angle φ = arctan λ, the 2 finite eigenvalues — 0.4384, 4.562 — sit inside the arc, and the 4 infinite ones sit at its ends, which is one point and not two. Nothing about them is degenerate: each carries a residual ‖βAx − αBx‖ in the same scaling as every other, the largest being 1.55·10⁻¹⁵, and an infinite eigenvalue's residual is ‖Bx‖/‖B‖ — the statement that its eigenvector is a null vector of B. The count is not a rank decision here: it is n minus the degree of det(A − λB), computed in exact rational arithmetic.

λ = α/β is a ratio, so an eigenvalue of a pencil is a direction rather than a number and it lives on a line whose two ends are the same point. Drawn as an angle φ = arctan λ, the 5 finite eigenvalues — -0.3786, 0.08367, 3.102, 4.705, 6.487 — sit inside the arc, and the 1 infinite ones sit at its ends, which is one point and not two. Nothing about them is degenerate: each carries a residual ‖βAx − αBx‖ in the same scaling as every other, the largest being 1.42·10⁻¹³, and an infinite eigenvalue's residual is ‖Bx‖/‖B‖ — the statement that its eigenvector is a null vector of B. The count is not a rank decision here: it is n minus the degree of det(A − λB), computed in exact rational arithmetic.

λ = α/β is a ratio, so an eigenvalue of a pencil is a direction rather than a number and it lives on a line whose two ends are the same point. Drawn as an angle φ = arctan λ, the 3 finite eigenvalues — -0.04892, 4.357, 4.692 — sit inside the arc, and the 3 infinite ones sit at its ends, which is one point and not two. Nothing about them is degenerate: each carries a residual ‖βAx − αBx‖ in the same scaling as every other, the largest being 1.28·10⁻¹⁴, and an infinite eigenvalue's residual is ‖Bx‖/‖B‖ — the statement that its eigenvector is a null vector of B. The count is not a rank decision here: it is n minus the degree of det(A − λB), computed in exact rational arithmetic.

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.

8 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 pencilEigen, 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.