threshold-stops

At its defaults it draws where the arnoldi recurrence stops, against the tolerance it calls a zero, n = 27. The spectrum has 3 coarse clusters of width 10^-4 and 9 fine ones inside them, so this one matrix has three near-invariant subspaces and all three are real. The tolerance decides which the recurrence stops at: it stops at 3 at the loose end, at 9 in the middle, and at 27 at the tight end. The residual it leaves runs from 1.51·10⁻⁵ to 5.48·10⁻¹⁷. Neither end is a mistake and neither is a trade: stopping too early throws away an answer that was available, and stopping too late means dividing by rounding error.

threshold-stops is one function in lib/figures/tolerance.js — tolerances — the zeros nobody meets and everybody writes. 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.

The spectrum has 3 coarse clusters of width 10^-4 and 9 fine ones inside them, so this one matrix has three near-invariant subspaces and all three are real. The tolerance decides which the recurrence stops at: it stops at 3 at the loose end, at 9 in the middle, and at 27 at the tight end. The residual it leaves runs from 1.51·10⁻⁵ to 5.48·10⁻¹⁷. Neither end is a mistake and neither is a trade: stopping too early throws away an answer that was available, and stopping too late means dividing by rounding error.

The spectrum has 3 coarse clusters of width 10^-4 and 9 fine ones inside them, so this one matrix has three near-invariant subspaces and all three are real. The tolerance decides which the recurrence stops at: it stops at 3 at the loose end, at 9 in the middle, and at 27 at the tight end. The residual it leaves runs from 1.51·10⁻⁵ to 5.48·10⁻¹⁷. Neither end is a mistake and neither is a trade: stopping too early throws away an answer that was available, and stopping too late means dividing by rounding error.

The spectrum has 3 coarse clusters of width 10^-8 and 9 fine ones inside them, so this one matrix has three near-invariant subspaces and all three are real. The tolerance decides which the recurrence stops at: it stops at 3 at the loose end, at 9 in the middle, and at 27 at the tight end. The residual it leaves runs from 1.51·10⁻⁹ to 2.92·10⁻¹⁷. Neither end is a mistake and neither is a trade: stopping too early throws away an answer that was available, and stopping too late means dividing by rounding error.

The spectrum has 3 coarse clusters of width 10^-2 and 9 fine ones inside them, so this one matrix has three near-invariant subspaces and all three are real. The tolerance decides which the recurrence stops at: it stops at 3 at the loose end, at 9 in the middle, and at 27 at the tight end. The residual it leaves runs from 0.00152 to 2.52·10⁻¹⁶. Neither end is a mistake and neither is a trade: stopping too early throws away an answer that was available, and stopping too late means dividing by rounding error.

The spectrum has 3 coarse clusters of width 10^-6 and 9 fine ones inside them, so this one matrix has three near-invariant subspaces and all three are real. The tolerance decides which the recurrence stops at: it stops at 3 at the loose end, at 9 in the middle, and at 27 at the tight end. The residual it leaves runs from 1.51·10⁻⁷ to 2.85·10⁻¹⁷. Neither end is a mistake and neither is a trade: stopping too early throws away an answer that was available, and stopping too late means dividing by rounding error.

The spectrum has 3 coarse clusters of width 10^-3 and 9 fine ones inside them, so this one matrix has three near-invariant subspaces and all three are real. The tolerance decides which the recurrence stops at: it stops at 3 at the loose end, at 9 in the middle, and at 27 at the tight end. The residual it leaves runs from 1.51·10⁻⁴ to 5.26·10⁻¹⁷. Neither end is a mistake and neither is a trade: stopping too early throws away an answer that was available, and stopping too late means dividing by rounding error.

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 arnoldiThreshold, 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.