fooling-ladder
At its defaults it draws what hager's estimator returns, as a share of the truth, against the size of the matrix built to defeat it. The estimate over the true 1-norm against n, both axes logarithmic. It falls from 0.376 at n = 4 to 0.0376 at n = 32, along the line 1/t where t is the construction's own multiplier — which is bounded only by the size of the matrix, so the ratio has no floor. A line at one marks a correct estimate.
fooling-ladder is one function in lib/figures/condest.js —
condition estimation — the number a library prints, and the matrix it flatters. 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.
The estimate over the true 1-norm against n, both axes logarithmic. It falls from 0.376 at n = 4 to 0.0376 at n = 32, along the line 1/t where t is the construction's own multiplier — which is bounded only by the size of the matrix, so the ratio has no floor. A line at one marks a correct estimate.
filler: 0.9
The arguments are the ones An estimate that can be fooled 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.
The estimate over the true 1-norm against n, both axes logarithmic. It falls from 0.376 at n = 4 to 0.0376 at n = 32, along the line 1/t where t is the construction's own multiplier — which is bounded only by the size of the matrix, so the ratio has no floor. A line at one marks a correct estimate.
filler: 0.5
The arguments are the ones An estimate that can be fooled 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.
The estimate over the true 1-norm against n, both axes logarithmic. It falls from 0.526 at n = 4 to 0.0658 at n = 32, along the line 1/t where t is the construction's own multiplier — which is bounded only by the size of the matrix, so the ratio has no floor. A line at one marks a correct estimate.
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.
25 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.
and the ratio is exactly one over the construction's multiplier at n = 4 — asserted 7 times
the estimate is the decoy's norm at n = 4 — asserted 7 times
and a larger matrix is fooled harder at n = 6 — asserted 6 times
a filler column below the decoy's own norm
a multiplier the construction supports
an even size, so the alternating column sums to zero
at the same number of products with the matrix at every size
reaching under a tenth inside the sizes drawn
Against the rule
The rule does not apply to it. It factorises nothing, so there is no residual it could be withholding. That is worth stating rather than leaving blank: a site that reported the rule as satisfied by every generator would be counting mostly generators the rule never reached.
Across the library: the rule bites on 90
of 174 generators —
75 print a residual and
15 are exempt with a published reason;
84 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.
An estimate that can be fooled
Nobody computes a condition number, because forming an inverse costs more than the solve did. Every library estimates it instead, from four or five products with a factorisation already in hand. The estimate is exactly right on four random matrices out of five — and there is a matrix, three distinct entries wide, on which it returns a twentieth of the truth.
The arithmetic underneathWhere the box is cut
A branch-and-bound with an interval operator settles a whole square — two roots proved unique, forty-two regions proved empty, nothing left undecided, in 87 evaluations. Move the roots so one lands on the first bisection and it proves nothing at all, at any depth. Cutting at 0.485 instead of 0.5 finds both, in a quarter of the work.