SAN FRANCISCO — 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 test for the economy, with trillions of dollars at stake.

Anthropic, which makes the Claude chatbot, is a tent pole of the AI industry. Valued in the private markets at nearly $1 trillion and backed by more than $130 billion in funding, it could raise $100 billion in its offering and promptly become one of the world’s biggest public companies. But the stakes are even higher than that.

Anthropic and OpenAI, its chief rival, sit atop a pyramid of business partners and suppliers — infrastructure providers, chipmakers, energy companies and AI startups — that have all spent heavily over the last four years because they believe demand for AI products will keep growing.

Anthropic’s listing will provide the first in-depth look at the health of a key business underlying the AI boom — and will be a test of its durability.…