The ADI error is the rational function, not merely bounded by it

At its defaults it draws the adi error is the rational function, not merely bounded by it. ‖P − ZZᵀ‖/‖P‖ after k steps of low-rank ADI on a 30-state model, against max|r|² for the same shifts, with P taken from its closed form rather than from another solve. The two curves are parallel: their ratio is 0.756 at k = 2 and 0.746 at k = 10, a spread of 1.0340 while the error itself falls from 0.289 to 3.87·10⁻⁴. A bound that is a constant factor from its quantity over four orders is an identity with a projection in front of it — the constant is how much of P lies where |r| is largest, which depends on the system and not on k.

adi-identity is one function in lib/figures/lowrank.js — why a gramian can be truncated — a rational function's poles, and the decay they explain. 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.

‖P − ZZᵀ‖/‖P‖ after k steps of low-rank ADI on a 30-state model, against max|r|² for the same shifts, with P taken from its closed form rather than from another solve. The two curves are parallel: their ratio is 0.756 at k = 2 and 0.746 at k = 10, a spread of 1.0340 while the error itself falls from 0.289 to 3.87·10⁻⁴. A bound that is a constant factor from its quantity over four orders is an identity with a projection in front of it — the constant is how much of P lies where |r| is largest, which depends on the system and not on k.

‖P − ZZᵀ‖/‖P‖ after k steps of low-rank ADI on a 30-state model, against max|r|² for the same shifts, with P taken from its closed form rather than from another solve. The two curves are parallel: their ratio is 0.756 at k = 2 and 0.746 at k = 10, a spread of 1.0340 while the error itself falls from 0.289 to 3.87·10⁻⁴. A bound that is a constant factor from its quantity over four orders is an identity with a projection in front of it — the constant is how much of P lies where |r| is largest, which depends on the system and not on k.

‖P − ZZᵀ‖/‖P‖ after k steps of low-rank ADI on a 10-state model, against max|r|² for the same shifts, with P taken from its closed form rather than from another solve. The two curves are parallel: their ratio is 0.732 at k = 2 and 0.703 at k = 9, a spread of 1.0415 while the error itself falls from 0.119 to 1.81·10⁻⁵. A bound that is a constant factor from its quantity over four orders is an identity with a projection in front of it — the constant is how much of P lies where |r| is largest, which depends on the system and not on k.

‖P − ZZᵀ‖/‖P‖ after k steps of low-rank ADI on a 40-state model, against max|r|² for the same shifts, with P taken from its closed form rather than from another solve. The two curves are parallel: their ratio is 0.766 at k = 2 and 0.752 at k = 10, a spread of 1.0346 while the error itself falls from 0.336 to 7.87·10⁻⁴. A bound that is a constant factor from its quantity over four orders is an identity with a projection in front of it — the constant is how much of P lies where |r| is largest, which depends on the system and not on k.

‖P − ZZᵀ‖/‖P‖ after k steps of low-rank ADI on a 16-state model, against max|r|² for the same shifts, with P taken from its closed form rather than from another solve. The two curves are parallel: their ratio is 0.759 at k = 2 and 0.748 at k = 10, a spread of 1.0384 while the error itself falls from 0.193 to 5.03·10⁻⁵. A bound that is a constant factor from its quantity over four orders is an identity with a projection in front of it — the constant is how much of P lies where |r| is largest, which depends on the system and not on k.

‖P − ZZᵀ‖/‖P‖ after k steps of low-rank ADI on a 22-state model, against max|r|² for the same shifts, with P taken from its closed form rather than from another solve. The two curves are parallel: their ratio is 0.776 at k = 2 and 0.769 at k = 10, a spread of 1.0296 while the error itself falls from 0.248 to 1.61·10⁻⁴. A bound that is a constant factor from its quantity over four orders is an identity with a projection in front of it — the constant is how much of P lies where |r| is largest, which depends on the system and not on k.

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.

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 adiLowRank, 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.