A basis that stops being a basis, and a Ritz value that keeps improving through it

At its defaults it draws a basis that stops being a basis, and a ritz value that keeps improving through it. ‖QᵀQ − I‖ for the two Krylov bases on a chain of 40 masses, against the number of vectors. Both are orthogonalised by the same modified Gram–Schmidt sweep, so what differs is the recurrence: the linearised basis stays at 2.37·10⁻¹³ throughout, and the second-order one runs from 1.29·10⁻¹² at four vectors to 1.347 at twenty — orthogonality gone entirely. The third curve is the distance from the dominant eigenvalue, which falls from 0.03036 to 0.001559 across the same range — the answer improves across the collapse, so a run watching only the answer would report progress at every step.

basis-orthogonality is one function in lib/figures/soar.js — a krylov space with no matrix — half the storage, twice the ritz values, and a basis that collapses. 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.

‖QᵀQ − I‖ for the two Krylov bases on a chain of 40 masses, against the number of vectors. Both are orthogonalised by the same modified Gram–Schmidt sweep, so what differs is the recurrence: the linearised basis stays at 2.37·10⁻¹³ throughout, and the second-order one runs from 1.29·10⁻¹² at four vectors to 1.347 at twenty — orthogonality gone entirely. The third curve is the distance from the dominant eigenvalue, which falls from 0.03036 to 0.001559 across the same range — the answer improves across the collapse, so a run watching only the answer would report progress at every step.

‖QᵀQ − I‖ for the two Krylov bases on a chain of 40 masses, against the number of vectors. Both are orthogonalised by the same modified Gram–Schmidt sweep, so what differs is the recurrence: the linearised basis stays at 2.37·10⁻¹³ throughout, and the second-order one runs from 1.29·10⁻¹² at four vectors to 1.347 at twenty — orthogonality gone entirely. The third curve is the distance from the dominant eigenvalue, which falls from 0.03036 to 0.001559 across the same range — the answer improves across the collapse, so a run watching only the answer would report progress at every step.

‖QᵀQ − I‖ for the two Krylov bases on a chain of 12 masses, against the number of vectors. Both are orthogonalised by the same modified Gram–Schmidt sweep, so what differs is the recurrence: the linearised basis stays at 2.73·10⁻¹³ throughout, and the second-order one runs from 8.86·10⁻¹³ at four vectors to 4 at twenty — orthogonality gone entirely. The third curve is the distance from the dominant eigenvalue, which falls from 0.02521 to 8.833·10⁻¹⁵ across the same range — the answer improves across the collapse, so a run watching only the answer would report progress at every step.

‖QᵀQ − I‖ for the two Krylov bases on a chain of 16 masses, against the number of vectors. Both are orthogonalised by the same modified Gram–Schmidt sweep, so what differs is the recurrence: the linearised basis stays at 6.83·10⁻¹³ throughout, and the second-order one runs from 3.65·10⁻¹² at four vectors to 2.828 at twenty — orthogonality gone entirely. The third curve is the distance from the dominant eigenvalue, which falls from 0.02878 to 0 across the same range — the answer improves across the collapse, so a run watching only the answer would report progress at every step.

‖QᵀQ − I‖ for the two Krylov bases on a chain of 20 masses, against the number of vectors. Both are orthogonalised by the same modified Gram–Schmidt sweep, so what differs is the recurrence: the linearised basis stays at 1.81·10⁻¹³ throughout, and the second-order one runs from 6.59·10⁻¹³ at four vectors to 1.793 at twenty — orthogonality gone entirely. The third curve is the distance from the dominant eigenvalue, which falls from 0.02935 to 9.531·10⁻¹⁶ across the same range — the answer improves across the collapse, so a run watching only the answer would report progress at every step.

‖QᵀQ − I‖ for the two Krylov bases on a chain of 24 masses, against the number of vectors. Both are orthogonalised by the same modified Gram–Schmidt sweep, so what differs is the recurrence: the linearised basis stays at 4.17·10⁻¹³ throughout, and the second-order one runs from 1.81·10⁻¹² at four vectors to 1.558 at twenty — orthogonality gone entirely. The third curve is the distance from the dominant eigenvalue, which falls from 0.02877 to 2.12·10⁻⁴ across the same range — the answer improves across the collapse, so a run watching only the answer would report progress at every step.

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.

11 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.

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 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.