The criminal tax fraud trial of Donald J. Trump’s family business could come down to three uninteresting words: “on behalf of.” Despite all the talk of posh apartments, free Mercedes-Benzes, and money flowing at Christmas.

The business is accused of giving these off-the-books benefits to a number of executives, who then failed to report them to the IRS. Allen H. Weisselberg, the longtime chief financial officer of the Trump Organization, admitted to being the scheme’s creator and gave evidence in court.

However, the business is not always at fault for his crimes. According to New York law, the Manhattan district attorney’s office must demonstrate that Mr. Weisselberg committed his numerous felonies “on behalf of” the Trump Organization, a cumbersome phrase that the judge presiding over the case has referred to as “a confusing area of the law” in an understatement.

Typically, things don’t look good when a company’s financial whiz becomes a prominent witness against it. But after weeks of embarrassing revelations, the Trump Organization’s only chance of acquittal or a hung jury rests on this phrase, which has sparked a fierce semantic debate. The company’s lawyers have argued that the prosecution must prove that Mr. Weisselberg intended to benefit the corporation when he engineered the scheme — and that “on behalf of” can mean nothing else.

Juan Merchan, the judge, is ultimately responsible for understanding the mysterious words. He also seemed to concur with the defense’s interpretation in court on Tuesday, when the jury was not present.

He stated that “there was some intent to benefit the corporation,” which is what the prosecution will have to prove.

However, he added that the prosecution need not demonstrate that Mr. Weisselberg’s main objective had been to assist the Trump Organization, disproving the defense’s most elaborate claim.

A level-headed, gray-haired judge named Justice Merchan has presided over a variety of cases, including those involving two base jumpers who jumped from One World Trade Center and a man who was fatally stabbed with a fork. This trial, however, has uniquely high-stakes implications: It has enraged the former president and his ranks of fervent supporters, and could permanently stain his family business while reverberating through the 2024 presidential campaign.

And it might all come down to four head-scratching syllables.

Justice Merchan has already ruled that the company’s attorneys are permitted to inform the jury during their closing arguments on Thursday that the prosecution failed to establish that Mr. Weisselberg intended to benefit the business. Additionally, the judge might use the defense’s preferred “intent” interpretation to enlighten the jury once they begin deliberating in the upcoming days if they have any questions about what “on behalf of” means.

According to New York criminal law, prosecutors must also demonstrate that Mr. Weisselberg was acting “on behalf of” the Trump Organization, that he was a “high managerial agent” of the business, and that he committed the crimes “within the scope of his employment.” Both of those requirements are generally accepted.

The “on behalf of” phrase “should limit corporate liability to the conduct engaged in for the corporation’s benefit and not merely personal gain,” the judge claimed after consulting a number of legal treatises.

The discussion intensified last week when the defense and prosecution presented their opposing interpretations and discussed whether or not Mr. Weisselberg’s testimony aided or hindered them.

On the witness stand, Mr. Weisselberg also acknowledged that he had betrayed the business where he had worked for years, and that Mr. Trump had not given him the go-ahead to execute the scheme. Was Mr. Weisselberg’s compensation reduced because he didn’t want to harm the business, Mr. Futerfas questioned. “No, my intention was to save pretax money,” he retorted.

Mr. Futerfas also argued that the New York law’s language was so ambiguous that the case should be dismissed last week when presenting his case to the judge.

The Manhattan District Attorney’s Office’s chief investigations division prosecutor for nearly two decades, Adam S. Kaufmann, said the “on behalf of” issue rarely came up because high-ranking officials’ actions in such cases almost always benefit a company.