Although it took an unsuccessful attempt at it, the progress is as follows:

"Claude found that combining the results from Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh with the work of Bombieri provides a way to surpass the previous state-of-the-art lower bound proportion of 41.6%, increasing it to 67.2%."

The transcripts, papers, and Claude's explanation are an interesting and a better read than this article, and this is exactly what Anthropic should continue to do and it helps other researchers outside the company as well.

  Claude's paper [0]

  Claude's Formalization [1]

  Anthropic's informal note stating the proof more concisely [2]

  Claude’s explanation of how it arrived at its result; [3]
    
  Detailed transcripts of Claude's process. [4]

[1] https://github.com/anthropics/zeta-23-lean

[2] https://www-cdn.anthropic.com/23455459f8832d06bb175cc0f88d01...

[3] https://www-cdn.anthropic.com/d7f3ecf1d01392d887f8bc974ca187...

[4] https://www-cdn.anthropic.com/8a0d1add3c637b858a9a181e98c40e...