ghost-ladder
At its defaults it draws copies of each eigenvalue after 80 steps on a 40×40 matrix with a simple spectrum. A bar per eigenvalue that came back more than once, showing how many times. The matrix has 40 distinct eigenvalues by construction; the run returned 25 extra copies of 13 of them, the most-copied arriving 5 times. Every copy is accurate to 1.9·10⁻⁸ relative, which is why nothing but the true spectrum could detect them.
ghost-ladder is one function in lib/figures/lanczos.js —
lanczos — three terms, lost orthogonality, and an eigenvalue that arrives twice. 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.
At its defaults
Drawn even though every essay passes arguments — which on this site is every essay, at 100% of placements since the standard pass. A default nothing exercises is a trap for the next essay to call this with none, and this is the page where a default that has drifted from the figures around it becomes visible.
A bar per eigenvalue that came back more than once, showing how many times. The matrix has 40 distinct eigenvalues by construction; the run returned 25 extra copies of 13 of them, the most-copied arriving 5 times. Every copy is accurate to 1.9·10⁻⁸ relative, which is why nothing but the true spectrum could detect them.
steps: 80
The arguments are the ones An eigenvalue that arrives twice passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
A bar per eigenvalue that came back more than once, showing how many times. The matrix has 40 distinct eigenvalues by construction; the run returned 25 extra copies of 13 of them, the most-copied arriving 5 times. Every copy is accurate to 1.9·10⁻⁸ relative, which is why nothing but the true spectrum could detect them.
steps: 120
The arguments are the ones An eigenvalue that arrives twice passes. A value drawn at the generator's defaults instead would be a picture no essay asked for and no assertion has been run against.
A bar per eigenvalue that came back more than once, showing how many times. The matrix has 40 distinct eigenvalues by construction; the run returned 71 extra copies of 40 of them, the most-copied arriving 8 times. Every copy is accurate to 2.3·10⁻⁸ relative, which is why nothing but the true spectrum could detect them.
What it checked while drawing
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.
9 distinct claims across 3 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.
a run past the dimension of the Krylov space
and every copy is an accurate eigenvalue
and reorthogonalising removes them
and the plain recurrence does not notice
full reorthogonalisation stops when the space is exhausted
Jacobi needs a symmetric matrix
matmul shapes agree
the plain run returns duplicates
there is something to draw
Against the rule
It draws a decomposition and prints its residual. It calls
lanczos,
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 52
of 99 generators —
37 print a residual and
15 are exempt with a published reason;
47 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.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.