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Label the C₄₅ certificate as Griego's superseded bound #178

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Label the C₄₅ certificate as Griego's superseded bound #178

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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.

- [45](https://teorth.github.io/optimizationproblems/constants/45a.html) **presentation:** the printed certificate is Griego's $0.490249407811155$, not Yoo's record $0.490180063290061$.

## Maintainers

This site is maintained by Damek Davis, Paata Ivanisvili and Terence Tao.
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5 changes: 4 additions & 1 deletion constants/45a.md
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Expand Up @@ -27,7 +27,9 @@ $C_{45}$ is the asymptotic density (if it exists) of the set of odd integers tha
| $0.107648$ | [CE2018] | |


## Certificate for the $0.490249407811155$ upper-density bound
## Certificate for the superseded $0.490249407811155$ Griego bound

The current record is Yoo's $0.490180063290061$ ([Y2026]); that bound has no printed certificate on this page. The object below recomputes Griego's earlier value.

Let
$$
Expand Down Expand Up @@ -90,3 +92,4 @@ The external verifier independently recomputes the 13-prime seed histogram from

Prepared with assistance from Gemini.
This update was prepared with assistance from GPT-5.5 Pro; the construction, arithmetic, and verification script were reviewed by the human contributor.
AI assistance was used to reword the certificate heading so it is not mistaken for the record bound.