BiCG iterations on one 12×12 system against the distance of the shadow vector from a breakdown

At its defaults it draws bicg iterations on one 12×12 system against the distance of the shadow vector from a breakdown. The matrix, the right-hand side and the answer are the same at every stop. The only thing that moves is r̃₀, the second starting vector, which the method requires and for which every account gives the same non-reason. At η = 0.01 from the surface where the second divisor vanishes, BiCG is the direct method it is advertised as and finishes in 12 steps on 12 unknowns. The steps then run 12, 12, 13, 17, 20, 24, 80, 80, 80, 80 as η falls, and at 10⁻¹¹ the method has not converged after 80. Wherever it does finish it finishes at the same accuracy — the cost is the guarantee, not the answer.

shadow-cost 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.

The matrix, the right-hand side and the answer are the same at every stop. The only thing that moves is r̃₀, the second starting vector, which the method requires and for which every account gives the same non-reason. At η = 0.01 from the surface where the second divisor vanishes, BiCG is the direct method it is advertised as and finishes in 12 steps on 12 unknowns. The steps then run 12, 12, 13, 17, 20, 24, 80, 80, 80, 80 as η falls, and at 10⁻¹¹ the method has not converged after 80. Wherever it does finish it finishes at the same accuracy — the cost is the guarantee, not the answer.

The matrix, the right-hand side and the answer are the same at every stop. The only thing that moves is r̃₀, the second starting vector, which the method requires and for which every account gives the same non-reason. At η = 0.01 from the surface where the second divisor vanishes, BiCG is the direct method it is advertised as and finishes in 12 steps on 12 unknowns. The steps then run 12, 12, 13, 17, 20, 24, 80, 80, 80, 80 as η falls, and at 10⁻¹¹ the method has not converged after 80. Wherever it does finish it finishes at the same accuracy — the cost is the guarantee, not the answer.

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.

6 distinct claims across 2 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 shadowSweep, 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.