← Moser's Worm Problem / Rounds / Round 9
The Round 8 smooth candidate's gain over the five-link event control is about 10⁻⁵; rather than enlarging the curve family, this round checks whether that gap comes from double-precision error, integration error, or sampling error. The curve is defined explicitly with fixed decimal parameters, and cross-validated using mpmath at 40 to 120 digits of precision with two quadrature methods, tanh-sinh and Gauss-Legendre, whose results agree completely. Using the fact that sinθ(u) is monotonically decreasing and cosθ(u) is monotonically increasing, we build a mathematically rigorous Darboux upper/lower-sum envelope — at 4096 subdivisions the lower bound still exceeds the event control by at least 7.8×10⁻⁶, and this positivity does not depend on any quadrature coincidence but follows from the monotone ordering of the integrand. We also find that, over the full phase circle, the closest competitor to 270° is 120°, with an arbitrary-precision gap of only about 1.6383×10⁻⁹ — which explains why a complete interval certificate is so difficult.
Relationship to other packages, stated as far as possible in the document's own words, not my interpretation.
“The smooth candidate has therefore been upgraded from a double-precision candidate to one confirmed at arbitrary precision, consistent across two algorithms, and backed by a monotone Darboux cusp lower bound. The next core task is not adding more decimal digits, but completing the piecewise interval exclusion over the entire phase circle.” — from Section 13, “Conclusion,” of this package's main document (original wording transcribed verbatim).
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