Prosecutors also accepted they could not prove beyond reasonable doubt that either Brendan McConville or the murder weapon were transported in co-defendant John Paul Wootton's car.
Instead they argued that the surrounding circumstances confirmed the two men's guilt.
McConville, 42, and Wootton, 22, are both looking to overturn their convictions for the March 2009 murder.
Constable Carroll was ambushed in his car as he responded to a 999 call at Lismore Manor, Craigavon.
He was shot twice in the head by an AK47 rifle recovered during later searches.
He was the first police officer to be murdered following the formation of the PSNI.
McConville, previously of Glenholme Avenue in Lurgan, is serving at least a 25 year sentence for killing the policeman.
Wootton, formerly of Collingdale, Lurgan, received a minimum 14-year term.
The case against them at the non-jury trial involved DNA and other evidence.
Prosecutors contended that Wootton's car was near the scene of the attack and driven off within minutes of the killing.
However, his legal team insist absolutely no evidence exists to link him to any role in the shooting.
Gun residue was found on a coat linked to McConville which was later recovered from the vehicle.
On day three of the two men's joint appeal judges examined whether it could have come from a separate incident.
Some of the residue could have come from another source, the court heard.
Questioned on whether it could be proven that the coat was at the firing point on the night of the killing, prosecution counsel Ciaran Murphy QC said he could prove a connection with the relevant particles.
Lord Chief Justice Sir Declan Morgan then asked: "Are you proving beyond reasonable doubt that the weapon was transported in Mr Wootton's car?"
The barrister replied: "No."
Sir Declan continued: "Are you proving beyond reasonable doubt that Mr McConville was transported in Mr Wootton's car."
Again the response was: "No."
Mr Murphy contended that scientific evidence connects the murder weapon to the jacket.
After it was pointed out to him that some of the discharge could have come from a different incident he stressed the rarity of the particles.
The Crown accepts there's at least a possibility there was some other source which was responsible for some of the gunshot residue.
Lord Chief Justice Sir Declan Morgan
"How then do you get from knowing there's another source as well as the gun that was recovered to concluding that other source was not responsible for the entirety of the gunshot residue?" Sir Declan asked.
According to Mr Murphy the surrounding circumstances provided the answer.
Those established "a sufficient case for the trial judge to reach a conclusion that he was satisfied that at least some of these particles originated from the weapon that night", he argued.
Earlier Mr Murphy defended an account given by a key prosecution witness at the trial.
Defence lawyers attacked his credibility, branding him a compulsive liar and a Walter Mitty-type character whose allegations of what he saw on the night of the shooting were "farcical".
The man, known only as Witness M, has since left Northern Ireland to live under a protection programme.
But Mr Murphy argued that he had reliably recalled seeing McConville near the scene, describing him as someone he knew by nickname and being able to remember exchanging words.
Witness M claimed he was later approached by another man and warned to "keep your mouth shut".
The defence pointed out how he only phoned police to make his claims 11 months later, making the call after drinking into the early hours of the morning.
However, Mr Murphy said: "He was aware in his own mind of the consequences.
"The fact he was drinking at the time has little bearing other than to explain how he may personally have felt and to get up the courage to do what he did because he was nervous and said he had seen someone watching his house on different occasions."
The hearing was adjourned until Friday.