Nonlinear cost — the series
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The last digit is the cheapest
Every cost curve measured here has the same shape: the first digits are cheap and the last ones are not. One method inverts it. Doubling the work buys twice as many digits as the previous doubling did, so the price of a digit halves every time it is paid.
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The circle between two eigenvalues
A contour count's error is not approximately governed by the nearest eigenvalue; it is exactly one closed-form term per eigenvalue, and summing those terms reproduces the quadrature to a millionth at 1,720 radius and point pairs. Three things follow. The rate is the ratio of the two moduli the circle sits between, not a distance, so two circles 0.2 from their nearest eigenvalue converge three times apart. Ten digits cost about 21 points divided by log₁₀ of that ratio, 71 points with twelve eigenvalues inside and 3,476 with six. And the best circle is not halfway: at the geometric mean of two eigenvalues on one ray their two terms are equal and opposite, and ten digits cost 27 points where the midpoint needs 84.
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A centre that chooses which pair cancels
A contour count's error is one closed-form term per eigenvalue, and two real eigenvalues on one ray cancel at their geometric mean — about the origin. Moved along the real axis, the centre was predicted to buy a further factor of two by putting real eigenvalues on two rays, the far one cancelling with a sign flip. The flip is parity: an eigenvalue behind the centre changes sign only at odd point counts. The factor is not two. Counting the delay problem's first two and three real eigenvalues, the best centre costs 22 and 21 points against the origin's 41 and 78; counting all four, 32 against 341, because about the origin the circle has to pass between a real eigenvalue and a complex pair and about 2.3 it does not. The centre does not add a cancellation. It chooses which pair cancels and pushes everything else away.