teorth · Oct 26, 2026 · Oct 14, 2026 · Oct 14, 2026 · Oct 14, 2026 · Oct 20, 2026
diff --git a/README.md b/README.md
@@ -96,7 +96,7 @@ Bounds for which the level of available verification is currently at minimal lev
 | [56](https://teorth.github.io/optimizationproblems/constants/56a.html) | $\mathrm{GL}_2$ Ramanujan conjecture exponent | 0 | $\tfrac{7}{64}=0.109375$ |
 | [57a](https://teorth.github.io/optimizationproblems/constants/57a.html) | Bloch’s constant | $\frac{\sqrt{3}}{4}+2\times 10^{-4}$ | $\dfrac{1}{\sqrt{1+\sqrt{3}}}\,\dfrac{\Gamma(1/3)\Gamma(11/12)}{\Gamma(1/4)}\approx 0.4719$ |
 | [57b](https://teorth.github.io/optimizationproblems/constants/57b.html) | Landau's constant | $\frac{1}{2}+10^{-335}$ | $\dfrac{\Gamma(1/3)\Gamma(5/6)}{\Gamma(1/6)}\approx 0.5433$ |
-| [57c](https://teorth.github.io/optimizationproblems/constants/57c.html) | Univalent Bloch constant | 0.5708858 | 1 |
+| [57c](https://teorth.github.io/optimizationproblems/constants/57c.html) | Univalent Bloch constant | 0.5708858 | 0.6563937 (0.65628589*) |
 | [58](https://teorth.github.io/optimizationproblems/constants/58a.html) | Zaremba’s conjecture constant | 5 | $\infty$ |
 | [59](https://teorth.github.io/optimizationproblems/constants/59a.html) | Bohr radius for the bidisc | 0.3006 | 0.302825279492 |
 | [60](https://teorth.github.io/optimizationproblems/constants/60a.html) | Favard-length decay exponent | $\frac{1}{6}$ | 1 |
@@ -207,6 +207,7 @@ Bounds for which the level of available verification is currently at minimal lev
 - [82](https://teorth.github.io/optimizationproblems/constants/82a.html) **improved upper bound (unverified):** $C_{82} < 0.2534449*$ by [R. Pinho](https://github.com/rpinho/c82-height-certificate), 16 Oct 2026, building on the September 2026 Épi construction (0.2535744749): Doche's limit-point method with $P\_7$ and 25 lattice-found numerator factors, certified by two independently written Arb checkers.
 - [31](https://teorth.github.io/optimizationproblems/constants/31a.html) **improved upper bound (unverified):** $C_{31} \leq 0.824120*$ by [L. Jack](https://github.com/leandrejack-afk/chvatal-sankoff-bound), 17 Oct 2026: Lueker's 2009 automaton method run at $h = 10$ (and $h = 9$, giving $0.825101$).
 - [1a](https://teorth.github.io/optimizationproblems/constants/1a.html), [1b](https://teorth.github.io/optimizationproblems/constants/1b.html) **improved upper bounds (unverified):** $C\_{1a} \leq 1.5027435$* and $C\_{1b} \leq 0.38085856$* by [Atris](https://github.com/atrislabs/records), 13 Oct 2026.
+- [57c](https://teorth.github.io/optimizationproblems/constants/57c.html) **improved upper bound (unverified):** $C\_{57c} \leq 0.65628589^\*$ by [W. van Staden](https://www.preprints.org/manuscript/202610.1936/v1), 20 Oct 2026 (DOI 10.20944/preprints202610.1936.v1); computer-assisted proof (exact covering for the inradius, Arb ball arithmetic for the conformal radius), [code and certificates](https://github.com/WynandVStaden/bloch-landau-slit-discs). The page now also records the earlier upper bounds, including $0.6563937$ (Carroll and Ortega-Cerdà, 2009) and $0.656319277272^\*$ (Ortega-Cerdà and Pridhnani, 2010).
 
 ## Maintainers
 

diff --git a/constants/57c.md b/constants/57c.md
@@ -30,6 +30,12 @@ Also, from $B\le B_l\le L\le B_u$ and $L>\frac{1}{2}+10^{-335}$, one gets the we
 | Bound | Reference | Comments |
 | ----- | --------- | -------- |
 | $1$ | Trivial | The identity function $f(z)=z$ is univalent with $f'(0)=1$ and has $B_f=1$, hence $B_u\le 1$. |
+| $0.658$ | <a href="#Rob1935">[Rob1935]</a> | Disc of radius about $2.956$ minus six radial slits ending at moduli $1$ and $2$. |
+| $0.65647$ | <a href="#Good1945">[Good1945]</a> | Radial slits added in bisecting generations. |
+| $0.6564155$ | <a href="#BH1985">[BH1985]</a> | Radial slits; third generation not bisecting. Numerical. |
+| $0.6563937$ | <a href="#COC2009">[COC2009]</a> | Disc of radius $4.0546358$ minus six radial slits and six harmonically symmetric arcs; numerical evaluation of an explicit theta-function formula, stated correct to 10 decimal places. |
+| $0.656319277272^\*$ | <a href="#OCP2010">[OCP2010]</a> | Disc of radius $5.1195152501$ minus six radial slits and further harmonically symmetric arcs; computed numerically, and the authors state that no rigorous error analysis was done. |
+| $0.65628589^\*$ | <a href="#vS2026">[vS2026]</a> | Disc of radius $R=5.001117628756668$ minus 24 slits $z=\exp w$, $w=\log R+s+i(\theta+b\_1s^2+b\_2s^4)$, $\log(t/R)\le s\le 0$, with $(\theta,t,b\_1,b\_2)$ equal to $(0,1,0,0)$, $(\pi/3,1.9999999974092333,0,0)$, $(0.029930188727091613,2.789342409403121,0.9026921410641155,-0.3301459494660587)$, $(1.0157094972537686,3.473381248030611,-1.2656124700253997,0.9794269477754911)$, $(0.5757776442359243,4.342874184977691,0.03225521024585182,2.9360081049550737)$, and their images under rotation by $2\pi/3$ and complex conjugation. Computer-assisted proof: inradius $\le 1+10^{-9}$ (exact covering), conformal radius $\ge 1.523726208329$ (comparison function, Arb ball arithmetic). Floating-point value of the ratio $0.656285866$. |
 
 ## Known lower bounds
 
@@ -70,10 +76,24 @@ Also, from $B\le B_l\le L\le B_u$ and $L>\frac{1}{2}+10^{-335}$, one gets the we
 	  **loc:** BS2023 PDF p.1270, §1 "Introduction"
 	  **quote:** "In 1995, Yanagihara (see [12]) proved that $B_l > 1/2 + 10^{-335}$. In 2009, Skinner (see [11]) proved that $B_u > 0.5708858$. These bounds are latest bounds and best known so far."
 
+- <a id="BH1985"></a>**[BH1985]** Beller, E.; Hummel, J. A. *On the univalent Bloch constant.* Complex Variables Theory Appl. **4** (1985), no. 3, 243–252. DOI: [10.1080/17476938508814109](https://doi.org/10.1080/17476938508814109).
+
+- <a id="COC2009"></a>**[COC2009]** Carroll, T.; Ortega-Cerdà, J. *The univalent Bloch–Landau constant, harmonic symmetry and conformal glueing.* J. Math. Pures Appl. (9) **92** (2009), no. 4, 396–406. DOI: [10.1016/j.matpur.2009.05.008](https://doi.org/10.1016/j.matpur.2009.05.008). [arXiv:0806.2282](https://arxiv.org/abs/0806.2282).
+
+- <a id="Good1945"></a>**[Good1945]** Goodman, R. E. *On the Bloch–Landau constant for schlicht functions.* Bull. Amer. Math. Soc. **51** (1945), no. 4, 234–239. DOI: [10.1090/S0002-9904-1945-08315-3](https://doi.org/10.1090/S0002-9904-1945-08315-3).
+
+- <a id="OCP2010"></a>**[OCP2010]** Ortega-Cerdà, J.; Pridhnani, B. *The Pólya–Tchebotaröv problem.* In: Harmonic analysis and partial differential equations, Contemp. Math. **505**, Amer. Math. Soc., 2010, 153–170. DOI: [10.1090/conm/505/09921](https://doi.org/10.1090/conm/505/09921). [arXiv:0809.2483](https://arxiv.org/abs/0809.2483).
+
+- <a id="Rob1935"></a>**[Rob1935]** Robinson, R. M. *The Bloch constant 𝔄 for a schlicht function.* Bull. Amer. Math. Soc. **41** (1935), no. 8, 535–540. DOI: [10.1090/S0002-9904-1935-06138-5](https://doi.org/10.1090/S0002-9904-1935-06138-5).
+
 - <a id="Skin2009"></a>**[Skin2009]** Skinner, Brian. *The univalent Bloch constant problem.* Complex Variables and Elliptic Equations **54** (2009), no. 10, 951–955. DOI: [10.1080/17476930903197199](https://doi.org/10.1080/17476930903197199). [Google Scholar](https://scholar.google.com/scholar?q=Skinner+The+univalent+Bloch+constant+problem+2009)
 
 - <a id="Yan1995"></a>**[Yan1995]** Yanagihara, H. *On the locally univalent Bloch constant.* Journal d'Analyse Mathématique **65** (1995), 1–17. DOI: [10.1007/BF02788763](https://doi.org/10.1007/BF02788763). [Google Scholar](https://scholar.google.com/scholar?q=Yanagihara+On+the+locally+univalent+Bloch+constant+1995)
 
+- <a id="vS2026"></a>**[vS2026]** van Staden, W. *An improved upper bound for the univalent Bloch–Landau constant.* Preprints.org, 20 October 2026: [preprints.org/manuscript/202610.1936/v1](https://www.preprints.org/manuscript/202610.1936/v1), DOI 10.20944/preprints202610.1936.v1. Code and certificates: [github.com/WynandVStaden/bloch-landau-slit-discs](https://github.com/WynandVStaden/bloch-landau-slit-discs).
+
 ## Contribution notes
 
 Prepared with assistance from ChatGPT 5.2 Pro.
+
+The history of upper bounds and the bound of [vS2026] were added with the help of Claude (Anthropic). The bound of [vS2026] has not been peer reviewed.