The step at which the Arnoldi recurrence breaks down, against the number of distinct eigenvalues, n = 30

At its defaults it draws the step at which the arnoldi recurrence breaks down, against the number of distinct eigenvalues, n = 30. Every matrix here is 30×30 and every one of them breaks down before step 18. The step is the number of distinct eigenvalues in the spectrum, exactly, at every one of the 8 spectra drawn — the points lie on the diagonal and the horizontal line at n = 30, which is the bound every course states, is off the top of the picture. The relative residual at the breakdown step is at most 5.64·10⁻¹⁶; one step earlier it is at least 3.5·10¹⁰ times larger.

breakdown-onset is one function in lib/figures/breakdown.js — breakdown — the zero that is an answer, and the zero that is nothing. 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.

Every matrix here is 30×30 and every one of them breaks down before step 18. The step is the number of distinct eigenvalues in the spectrum, exactly, at every one of the 8 spectra drawn — the points lie on the diagonal and the horizontal line at n = 30, which is the bound every course states, is off the top of the picture. The relative residual at the breakdown step is at most 5.64·10⁻¹⁶; one step earlier it is at least 3.5·10¹⁰ times larger.

Every matrix here is 30×30 and every one of them breaks down before step 18. The step is the number of distinct eigenvalues in the spectrum, exactly, at every one of the 8 spectra drawn — the points lie on the diagonal and the horizontal line at n = 30, which is the bound every course states, is off the top of the picture. The relative residual at the breakdown step is at most 5.64·10⁻¹⁶; one step earlier it is at least 3.5·10¹⁰ times larger.

Every matrix here is 20×20 and every one of them breaks down before step 18. The step is the number of distinct eigenvalues in the spectrum, exactly, at every one of the 8 spectra drawn — the points lie on the diagonal and the horizontal line at n = 20, which is the bound every course states, is off the top of the picture. The relative residual at the breakdown step is at most 6.38·10⁻¹⁶; one step earlier it is at least 1.5·10¹⁰ times larger.

Every matrix here is 40×40 and every one of them breaks down before step 18. The step is the number of distinct eigenvalues in the spectrum, exactly, at every one of the 8 spectra drawn — the points lie on the diagonal and the horizontal line at n = 40, which is the bound every course states, is off the top of the picture. The relative residual at the breakdown step is at most 8.51·10⁻¹⁶; one step earlier it is at least 8.8·10⁹ times larger.

Every matrix here is 50×50 and every one of them breaks down before step 18. The step is the number of distinct eigenvalues in the spectrum, exactly, at every one of the 8 spectra drawn — the points lie on the diagonal and the horizontal line at n = 50, which is the bound every course states, is off the top of the picture. The relative residual at the breakdown step is at most 9.1·10⁻¹⁶; one step earlier it is at least 3.4·10⁹ times larger.

Every matrix here is 60×60 and every one of them breaks down before step 18. The step is the number of distinct eigenvalues in the spectrum, exactly, at every one of the 8 spectra drawn — the points lie on the diagonal and the horizontal line at n = 60, which is the bound every course states, is off the top of the picture. The relative residual at the breakdown step is at most 8.34·10⁻¹⁶; one step earlier it is at least 1.6·10¹⁰ times larger.

Every figure above checked its own claims on the way to being drawn, and a claim that failed would have stopped the picture rather than shipped a wrong one. Those checks 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.

5 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 onsetSweep, 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 217 of 397 generators — 199 print a residual and 18 are exempt with a published reason; 180 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.