threshold-tradeoff
At its defaults it draws fill against growth as the pivot threshold moves, on the 6×6 grid. Two curves against the pivot threshold on a logarithmic horizontal axis. One falls steeply from left to right; the other rises gently. A vertical line marks the value libraries default to.
threshold-tradeoff is one function in lib/figures/sparselu.js —
sparse lu — where the fill argument and the stability argument disagree. 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.
Two curves against the pivot threshold on a logarithmic horizontal axis. One falls steeply from left to right; the other rises gently. A vertical line marks the value libraries default to.
grid: 6
The arguments are the ones A threshold between fill and growth 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.
Two curves against the pivot threshold on a logarithmic horizontal axis. One falls steeply from left to right; the other rises gently. A vertical line marks the value libraries default to.
grid: 5
The arguments are the ones What the symbolic phase can only bound 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.
Two curves against the pivot threshold on a logarithmic horizontal axis. One falls steeply from left to right; the other rises gently. A vertical line marks the value libraries default to.
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.
12 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.
PA = LU at τ = 0.001 — asserted 7 times
a grid between 4×4 and 8×8
a looser threshold gives less fill
and more growth
and the library default sits under both extremes
matmul shapes agree
Against the rule
It draws a decomposition and prints its residual. It calls
sparseLU,
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.
A threshold between fill and growth
One number decides how small a pivot an elimination will accept. At 0.001 the factor holds 172 entries and the matrix grows by 1,330; at 1 it holds 260 and grows by 1.2. The libraries ship 0.1, and the measurement says why.
Sparsity, and what elimination costsStructure and stability stop being separable
The sparsest variable to eliminate on this matrix has a diagonal entry of 10⁻¹². Eliminating it produces the smaller factor, reproduces the matrix to 3.8·10⁻¹⁷ — better than pivoting does — and returns an answer wrong in the fifth digit.
Sparsity, and what elimination costsWhat the symbolic phase can only bound
Without pivoting, the fill can be computed from the graph and the count is exact — 233 predicted, 233 measured. With pivoting it is 233 predicted and 242 measured, and what survives is a bound that is right at every threshold and loose by 1.7 times at the largest grid drawn.