← Moser's Worm Problem / Rounds / Round 10
Round 9 verified only the 270° cusp itself; this round splits the entire [0,2π) phase circle into 18 intervals of active support identity — within each interval, the support point is already an extremum of the normal functional, so the envelope theorem lets us compute s'(φ) directly without explicitly differentiating the support-point location. We fully enumerate 12 smooth stationary points and 17 contact switches (4 of which are local minima at cusps), building a minimum-value ledger for each interval. Four phase-sampling densities (32768 to 262144 points) and three derivative-sampling densities (257/1025/4097) all agree, with no new hidden intervals or stationary points emerging as resolution increases. The complete numerical ranking confirms that 270° remains the global minimum, with the closest competitor at 120°, a gap of about 1.6383×10⁻⁹ — leaving this as the only pair that genuinely requires rigorous interval treatment.
Relationship to other packages, stated as far as possible in the document's own words, not my interpretation.
“The Round 8 smooth surpassing result now rests on three layers of support: the arbitrary-precision cusp value; the monotone Darboux positive lower bound; and the complete phase-wise contact-interval numerical exclusion. The key work remaining is turning the third layer from a ‘complete numerical ledger’ into a ‘rigorous per-interval envelope.’” — from Section 14, “Conclusion,” of this package's main document.
Loading…
Loading…
Loading…
sha256 f269d6424bc9027bdfb3d56fb4d317c0f447e7e3d550410b04cf3313dbfbd990