People at Anthropic’s Code W/ Claude event in San Francisco, May 7, 2026. Anthropic sits atop a pyramid of business partners and suppliers, making it a barometer of the broader economy. Its initial public offering will have widespread implications. (Jason Henry/The New York Times) By ERIN GRIFFITH and LAUREN HIRSCH When Anthropic goes public as soon as next month, its initial offering will be more than just a coming-out party for the artificial intelligence company. It will also be a litmus…