IRKA's interpolation points walking to the condition that defines them

At its defaults it draws irka's interpolation points walking to the condition that defines them. Each curve is one of 4 interpolation points across 13 iterations of σ ← −λ(Aᵣ). They start logarithmically spread over the frequency range, which is a guess, and settle at 9.861, 40.15, 102.1, 302.8. What is checked at the end is not that they stopped moving but that where they stopped is the first-order condition: the mirrored poles of the model they produced agree with them to 2.81·10⁻¹². A fixed-point iteration that stops moving without satisfying its own condition has converged to nothing, and this one has no convergence proof to lean on — so the condition is checked rather than the movement.

irka-shifts is one function in lib/figures/ratkrylov.js — matching the function where you choose — interpolation points, the basis they need, and the fixed point. 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.

Each curve is one of 4 interpolation points across 13 iterations of σ ← −λ(Aᵣ). They start logarithmically spread over the frequency range, which is a guess, and settle at 9.861, 40.15, 102.1, 302.8. What is checked at the end is not that they stopped moving but that where they stopped is the first-order condition: the mirrored poles of the model they produced agree with them to 2.81·10⁻¹². A fixed-point iteration that stops moving without satisfying its own condition has converged to nothing, and this one has no convergence proof to lean on — so the condition is checked rather than the movement.

Each curve is one of 4 interpolation points across 13 iterations of σ ← −λ(Aᵣ). They start logarithmically spread over the frequency range, which is a guess, and settle at 9.861, 40.15, 102.1, 302.8. What is checked at the end is not that they stopped moving but that where they stopped is the first-order condition: the mirrored poles of the model they produced agree with them to 2.81·10⁻¹². A fixed-point iteration that stops moving without satisfying its own condition has converged to nothing, and this one has no convergence proof to lean on — so the condition is checked rather than the movement.

Each curve is one of 2 interpolation points across 14 iterations of σ ← −λ(Aᵣ). They start logarithmically spread over the frequency range, which is a guess, and settle at 10.31, 67.52. What is checked at the end is not that they stopped moving but that where they stopped is the first-order condition: the mirrored poles of the model they produced agree with them to 1.44·10⁻¹¹. A fixed-point iteration that stops moving without satisfying its own condition has converged to nothing, and this one has no convergence proof to lean on — so the condition is checked rather than the movement.

Each curve is one of 6 interpolation points across 10 iterations of σ ← −λ(Aᵣ). They start logarithmically spread over the frequency range, which is a guess, and settle at 9.857, 39.27, 88.05, 162.2, 297, 524.7. What is checked at the end is not that they stopped moving but that where they stopped is the first-order condition: the mirrored poles of the model they produced agree with them to 1.95·10⁻¹³. A fixed-point iteration that stops moving without satisfying its own condition has converged to nothing, and this one has no convergence proof to lean on — so the condition is checked rather than the movement.

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.

12 distinct claims across 4 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 irka, poles, h2Error, 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.