The Obama administration, in a memorandum issued shortly after the president took office, provided that the support had to be “substantial” and not “insignificant.” But the memo made clear that the definition applied anywhere in the world and was “not limited to persons captured on the battlefields of Afghanistan” or to those “directly participating in hostilities.”

Congress weighed in again in the National Defense Authorization Act for the 2012 fiscal year, amending the original Authorization for the Use of Military Force to cover detention of those who were “a part of or substantially supported Al Qaeda, the Taliban, or associated forces.” This definition included “any person who has committed a belligerent act or has directly supported such hostilities in aid of such enemy forces.”

These different iterations, each building subtly on what had gone before, have left plenty of room for judicial interpretation. The D.C. Circuit, with exclusive jurisdiction over the Guantánamo habeas cases, has jumped into the gaps. It has endorsed the government’s view that evidence should be viewed holistically, as a composite, even if individual pieces are missing or might have a benign explanation.

The Ali case exemplified this approach. For example, when he was captured, Mr. Ali was staying at a guesthouse with Abu Zubaydah, an Osama bin Laden ally who is now one of the highest of high-value detainees at Guantánamo. Mr. Ali had been at the four-bedroom house for 18 days, and was studying English. The Zubaydah forces were known to teach English to terrorists in training, and others who were later determined to be enemy combatants had been captured at the same or similar houses.

The D.C. Circuit rejected the argument by Mr. Ali’s lawyers that it was applying a standard of “guilt by guesthouse.” The court said that “determining whether an individual is part of Al Qaeda, the Taliban, or an associated force almost always requires drawing inferences from circumstantial evidence, such as that individual’s personal associations.” Mr. Ali, Judge Brett M. Kavanaugh’s opinion concluded, “more likely than not was part of Abu Zubaydah’s force.”

The “more likely than not” burden of proof — formally, a “preponderance of the evidence” — is the other part of the habeas corpus inquiry. This is a low burden of proof —– far lower than the criminal-law standard of “beyond a reasonable doubt” — but it isn’t the lowest. The D.C. Circuit’s judges have debated among themselves whether they shouldn’t be satisfied anytime the government presents some evidence that can be deemed “substantial,” even if it doesn’t measure up to “more likely than not.” But because the Obama administration hasn’t challenged the “preponderance” burden or sought a lower one, the appeals court’s conservative members have grudgingly acquiesced to continuing to use preponderance.

In effect, however, “the preponderance of the evidence and substantial evidence standards have come to be conflated,” one member of the D.C. Circuit, Judge Harry T. Edwards, its former chief judge, complained in an opinion in June. He was concurring only with the result, but not the reasoning, of a decision that rejected the habeas corpus petition of a Yemeni detainee, Abdul al-Qader Ahmed Hussain, held at Guantánamo since early 2002. The other two judges on the panel, Karen LeCraft Henderson and Thomas B. Griffith, said it was appropriate to draw inferences from the facts the government presented about Mr. Hussain’s travels, affiliations and multiple stays in mosques owned by a Qaeda-affiliated Islamic missionary group, Jama’at al-Tablighi, known as J.T. These facts, the two judges said, supported the conclusion that Mr. Hussain, a teenager at the time of his capture, was “a part of Al Qaeda or the Taliban when he was captured.”