Deep inside Bridgewater Associates LP, the world’s largest hedge-fund firm, software engineers are at work on a secret project that founder Ray Dalio has sometimes called “The Book of the Future.”

The goal is technology that would automate most of the firm’s management. It would represent a culmination of Mr. Dalio’s life work to build Bridgewater into an altar to radical openness—and a place that can endure without him.

At...