The latest investment brings Amazon’s total stake in the San Francisco company to $4 billion. : Amazon said on Wednesday that it had added $2.75 billion to its investment in Anthropic, a start-up that competes with companies like OpenAI and Google in the race to build cutting-edge A.I. systems. Six months ago, Amazon invested $1.25 billion in Anthropic, making the San Francisco start-up Amazon’s most important A.I. partner. Amazon said at the time that it had the option to bring its total investment to $4 billion. It had until the end of March to do so, according to financial filings.