[WynandVStaden]

This updates entry 57c (univalent Bloch constant), whose upper-bound table currently lists only the trivial bound 1.

History. Adds the published upper bounds in chronological order, with references and DOIs: Robinson (1935) 0.658, Goodman (1945) 0.65647, Beller–Hummel (1985) 0.6564155, Carroll–Ortega-Cerdà (2009) 0.6563937 and Ortega-Cerdà–Pridhnani (2010) 0.656319277272. The 2010 value is marked with an asterisk, since its authors state that they did not do a rigorous error analysis.

New bound. Adds $C_{57c} \le 0.65628589$ (asterisked), from a disc with 24 slits, 18 of them curved. The full parameters of the domain are in the comments column. The proof is computer-assisted:

• the inradius is at most $1+10^{-9}$, by an exact covering argument (dyadic cells and boundary witnesses, checked in integer arithmetic);
• the conformal radius is at least 1.523726208329, by a comparison function whose boundary values are bounded in Arb ball arithmetic.

Both programs run in about a minute.

• Preprint: https://www.preprints.org/manuscript/202610.1936/v1 (DOI 10.20944/preprints202610.1936.v1, registration pending)
• Code and certificates: https://github.com/WynandVStaden/bloch-landau-slit-discs

I have run the certificates myself. The README cell becomes "0.6563937 (0.65628589*)", and a line is added under Recent progress.

AI use. The domain, the method, the programs, the proof and this change were produced with the help of Claude (Anthropic) under my direction; this is also noted on the constant page. The result has not been peer reviewed.

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