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Put the Aldaz record last on the C₄₇ lower-bound table #179

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Put the Aldaz record last on the C₄₇ lower-bound table #179

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2 changes: 2 additions & 0 deletions README.md
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Expand Up @@ -159,6 +159,8 @@ Bounds for which the level of available verification is currently at minimal lev
- [43](https://teorth.github.io/optimizationproblems/constants/43a.html) **improved lower bound (unverified):** $C_{43} \geq 0.860*$ (exact $43/50$; certificate-layer result conditional on the lemma set of [KHSHGW2026](https://arxiv.org/abs/2601.22365)) by [J. Savva](https://doi.org/10.5281/zenodo.22223485), 1 Sep 2026.
- [88a](https://teorth.github.io/optimizationproblems/constants/88a.html) **improved upper bound:** $C_{88a} \leq 186$ via $\mathrm{DHL}[40,2]$, by [OpenAI](https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/short_gaps.pdf), 30 Aug 2026, with a Lean 4 formalization conditional on three declared axioms.

- [47](https://teorth.github.io/optimizationproblems/constants/47a.html) **presentation:** put Aldaz's $1.6211915$ last in the lower-bound table so the last row is the record.

## Maintainers

This site is maintained by Damek Davis, Paata Ivanisvili and Terence Tao.
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8 changes: 6 additions & 2 deletions constants/47a.md
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Expand Up @@ -39,8 +39,8 @@ the optimal weak-type $(1,1)$ constant of the centered Hardy–Littlewood maxima

| Bound | Reference | Comments |
| ----- | --------- | -------- |
| $\dfrac{3}{4}-\dfrac{\sqrt{2}}{4}+\dfrac{\sqrt{6}}{2}\approx 1.6211915$ | [Ald2000] | Aldaz's Proposition 1.4 gives a lower bound in every dimension $n\ge 2$. Specializing the formula to $n=2$ gives the displayed value. [Ald2000-prop1.4] |
| $\dfrac{11+\sqrt{61}}{12}\approx 1.5675208$ | <a href="#Mel2003">[Mel2003]</a>, <a href="#Ald2011">[Ald2011]</a> | Melas proved $c_1=\dfrac{11+\sqrt{61}}{12}$. Since $c_{d+1}\ge c_d$, we get $c_2\ge c_1$. <a href="#Mel2003-c1-formula">[Mel2003-c1-formula]</a> <a href="#Ald2011-monotone">[Ald2011-monotone]</a> |
| $\dfrac{11+\sqrt{61}}{12}\approx 1.5675208$ | <a href="#Mel2003">[Mel2003]</a>, <a href="#Ald2011">[Ald2011]</a> | Melas proved $c_1=\dfrac{11+\sqrt{61}}{12}$. Since $c_{d+1}\ge c_d$, we get $c_2\ge c_1$. Weaker than the Aldaz bound below; kept for history. <a href="#Mel2003-c1-formula">[Mel2003-c1-formula]</a> <a href="#Ald2011-monotone">[Ald2011-monotone]</a> |
| $\dfrac{3}{4}-\dfrac{\sqrt{2}}{4}+\dfrac{\sqrt{6}}{2}\approx 1.6211915$ | [Ald2000] | Record. Aldaz's Proposition 1.4 in dimension $n=2$. [Ald2000-prop1.4] |

## Additional comments and links

Expand Down Expand Up @@ -105,3 +105,7 @@ the optimal weak-type $(1,1)$ constant of the centered Hardy–Littlewood maxima

Prepared with assistance from ChatGPT 5.2 Pro.
This update was prepared with assistance from GPT-5.5 Pro; citations and mathematical details were reviewed by the human contributor.

## Contribution notes

AI assistance was used to reorder the lower-bound table so the last row is the record.