While Mr. Murdoch and Fox executives have often couched their defense of “Tucker Carlson Tonight” as a protection of free inquiry and controversial opinions, Mr. Carlson’s on-air provocations have long been something else: part of a painstaking, data-driven experiment that has succeeded wildly in bolstering Fox’s profit machine against the long-term decline in cable news subscriptions.

According to three former Fox employees, Mr. Carlson was among the network’s most avid consumers of what are known as minute-by-minutes — ratings data on an audience’s real-time ebb and flow. “He is going to double down on the white nationalism because the minute-by-minutes show that the audience eats it up,” said a former employee who worked frequently with Mr. Carlson.

Network executives soon began applying the approach to the daytime news shows. They pitched it as “Moneyball” for television: an audience-first approach to deciding what to cover and how to cover it.