That deception, if there was one, occurred on Mr. Tillerson’s watch; he was chairman and chief executive of the company from 2006 until 2017. (After retiring from the company, he served as secretary of state for one tumultuous year.) Attorney General Letitia James’s case relies heavily on public statements from Mr. Tillerson, including his comments at a 2016 shareholder meeting in which he said that, when it comes to the potential costs of issues like climate change regulation to their business, “we have accommodated that uncertainty in the future and everything gets tested against it,” and that the cost “gets put into all our economic models when we make investment decisions as well.”

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The state argues that Exxon did not do tha t and “when it began to appear that maybe Exxon was not following through on what it said about its use of a cost of carbon, the company share price took a hit” to the tune of hundreds of millions of dollars. That loss to investors is the argument that puts the case under the Martin Act, New York’s powerful shareholder protection law.

Exxon counters that the state’s case is fatally flawed and is based on a misunderstanding, perhaps a willful one, of how the company calculated the costs of future projects. The internal number that the state characterizes as part of a subterfuge, the company argues, is in fact an entirely separate financial tool used in a different way. Ted Wells Jr., the lead lawyer representing Exxon in the case, said in his opening statement Tuesday that “The only thing that’s a Potemkin village, an illusion in this case, are the allegations in the complaint.” Mr. Tillerson, he said, “put in place a robust system to manage the risk of increasing climate change regulations.”

During his friendly cross-examination of Mr. Tillerson, Mr. Wells asked Mr. Tillerson a series of questions about whether he and the company had willfully deceived shareholders about the company’s commitment to measuring the effects of climate change, or even themselves, eliciting simple answers.