If not for the fact that a Texas death row inmate’s life might hang in the balance, Monday’s hour-long argument at the Supreme Court could be dismissed as simply an exercise in semantics.

The federal Criminal Justice Act allows lawyers for indigent defendants facing the death penalty to apply for money for “investigative, expert or other services” that are “reasonably necessary” to assemble the kind of mitigating evidence that might persuade the jury to forgo a recommendation of death.

But a judge denied the funds to attorneys for Carlos Manuel Ayestas, who was convicted of the brutal killing of 67-year-old Santiaga Paneque during an invasion of her Houston home in 1995.

In the U.S. Court of Appeals for the 5th Circuit, which covers Texas, the law has been interpreted to mean a defendant must show that there is a “substantial need” for such services.

As a district court judge and Ayestas’s lawyers pointed out, that creates something of a Catch-22: The defendant would have to demonstrate that there is some relevant evidence he could discover without first having the funding to pursue that evidence.

But Justice Samuel A. Alito Jr. wondered whether the court was devoting itself to debating a distinction without a difference.

“What is the difference between ‘reasonably necessary’ and ‘substantial need’?” Alito asked Ayestas’s lawyer Lee Kovarsky of Baltimore. “I have been racking my brain trying to think of something that it is reasonably necessary for me to obtain but as to which I do not have the substantial need.

“And I can’t think of an example.”

Kovarsky could not satisfy Alito. But Kovarsky said that what a court could not do is speculate on what kind of evidence might be gathered if an investigation was authorized and then base the decision on whether to grant the funds on that kind of speculation.

Liberal justices were clearly more on Ayestas’s side. They signaled they approved of Kovarsky’s test that “services are reasonably necessary when they would be used to identify or develop possible claims by a reasonable attorney representing a paying client of ordinary means.”

Justice Ruth Bader Ginsburg said there could be no other choice in Ayestas’s case.

“This is a horrendous murder; the only chance in the world that this defendant has is if he can put on a . . . mitigation case and convince one juror he shouldn’t get the death penalty,” she said.

Texas Solicitor General Scott A. Keller said reports of schizophrenia and head trauma to Ayestas came after the conviction. But Justice Elena Kagan said that did not matter when the defendant was trying to do what Ayestas had sought to accomplish: prove that his trial lawyer had been negligent in mounting a defense at sentencing.

“A person who has since the incident in question been diagnosed as schizophrenic — you know, some bell goes off that says I think maybe we should do some investigation and try to figure out whether he was suffering from mental-health issues at the time of the incident,” she said.

Justice Sonia Sotomayor pointed out that Ayestas’s trial attorney provided only “two and a half pages of mitigation evidence.” She asked Keller: “How can you stand here and say that this kind of investigation meets any constitutional standard?”

Justice Stephen G. Breyer said there seemed to be a simple way to resolve the case: Tell the 5th Circuit to use the “reasonably necessary” standard instead of its “substantial need” test.

“Follow the statute. And that’s it. Goodbye,” Breyer said. “And all these other arguments are for the lower court.”

The case is Ayestas v. Davis.

Correction: An earlier version of this article incorrectly referred to the Criminal Justice Act as the Capital Justice Act.