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May 21, 2026
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Improve C_59 upper bound #75

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12 changes: 9 additions & 3 deletions constants/59a.md
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Expand Up @@ -43,17 +43,18 @@ The exact value of $K_d$ is unknown for every $d>1$; in particular, the exact va
The best established range currently is

$$
0.3006\ \le\ K_2\ <\ 0.3177.
0.3006\ \le\ K_2\ <\ 0.3174541.
$$

<a href="#Kne2025-lb-K2-0-3006">[Kne2025-lb-K2-0-3006]</a> <a href="#BPWW2026-ub-K2-0-3177">[BPWW2026-ub-K2-0-3177]</a>
<a href="#Kne2025-lb-K2-0-3006">[Kne2025-lb-K2-0-3006]</a> <a href="#G2026-ub-K2-0-3174541">[G2026-ub-K2-0-3174541]</a>

## Known upper bounds

| Bound | Reference | Comments |
| ----- | --------- | -------- |
| $1/3$ | <a href="#BK1997">[BK1997]</a> | General upper bound $K_n\le 1/3$ (hence $K_2\le 1/3$). <a href="#BK1997-ub-1-3">[BK1997-ub-1-3]</a> |
| $0.3177$ | <a href="#BPWW2026">[BPWW2026]</a> | Explicit construction giving $K_2<0.3177$ (Theorem 6.4). <a href="#BPWW2026-ub-K2-0-3177">[BPWW2026-ub-K2-0-3177]</a> |
| $0.3174541$ | <a href="#G2026">[G2026]</a> | Degree-$(250,250)$ polynomial from a rational-inner Fejer averaging certificate. Exact integer verification gives $B\_{3174541/10000000}(p)>1$. <a href="#G2026-ub-K2-0-3174541">[G2026-ub-K2-0-3174541]</a> |

## Known lower bounds

Expand Down Expand Up @@ -118,6 +119,11 @@ $$
**loc:** arXiv v1 PDF p.18, Theorem 6.4
**quote:** “Theorem 6.4. $K_2<0.3177$.”

- <a id="G2026"></a>**[G2026]** Griego, Sebastian. Rational-inner Fejer averaging certificate for the bidisc Bohr radius bound $K\_2<0.3174541$, [submitted to this repository](https://github.com/teorth/optimizationproblems/pull/75) (2026).
- <a id="G2026-ub-K2-0-3174541"></a>**[G2026-ub-K2-0-3174541]**
**loc:** pull request certificate and exact verifier
**quote:** “The exact integer verifier proves $B\_{3174541/10000000}(p)>1+1/7307638490$ and the rational-inner Fejer averaging certificate proves $\lvert p\rvert\le 1$ on the bidisc.”

## Contribution notes

Prepared with assistance from ChatGPT 5.2 Pro.
Prepared initially with assistance from ChatGPT 5.2 Pro and updated with assistance from ChatGPT 5.5 Pro.