COS Method

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1 With $b=512$, $N=16384$, auxiliary tilt $\eta=0.245$, and 60 interval decimal digits, the supplied backend returns a logarithmic interval. Rounding its lower endpoint downward and its upper endpoint upward gives the readable enclosure
2 \[
3 \log_{10}P(Q\ge40)\in\bigl[-4.04489575,-4.03897381\bigr].
4 \]
5 At $N=8192$, the directed ratio upper bound is approximately $0.0246131$, which fails the requested tolerance $0.02$. Doubling the retained mode count, rather than relaxing the tolerance, gives the interval above. The full serialized interval has a directed multiplicative ratio upper bound below $0.013731$. The displayed, coarser interval also meets the requested tolerance $0.02$. An independently evaluated interval for the exact expression $2e^{-10}-e^{-20}$ lies entirely inside the COS enclosure.
6 
7 This same fixed enclosure does not meet tolerance $0.001$. The acceptance routine rejects that tighter request. This is an intended distinction between an accurate point estimate and an established enclosure width. The rank-four example has slower Fourier decay than the rank-eighty control developed later, so its mode count should not be interpreted as a general cost prediction.

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