How did Finch's cover identity get blown?

Question

In Person of Interest, in "The Day The World Went Away", Finch's cover identity is exposed after he visited a cafe he brought Grace to. How did Samaritan establish the link between Professor Whistler and Harold Finch? As far as I know, Samaritan knows:

• A person who looks like Finch used to come to the cafe
• A person who looks like Finch used to order the same thing

That, to me, is not enough to conclusively prove that Professor Whistler is indeed Harold Finch. What other information could Samaritan possibly possess to help it expose Finch's cover identity? Was it the fact that Finch was talking to The Machine?

Answer 1

He had visited the cafe as Harold Finch previously and made the same order with Grace. He then repeated the visit in his Professor Whistler identity and was recognised by the waitress. Samaritan could hear what the waitress said to him as well.

Enough was the same for Samaritan to draw a line between the two identities. Bear in mind, in an early episode that Finch mentions how a series of almost totally unrelated facts are linked together by the Machine to determine the perpetrator of a crime. It doesn't take much if you can see and hear everything (and process it quick enough) to spot the connections.

Perhaps it wasn't entirely conclusive, but nobody ever said that Samaritan insists on absolute proof before choosing an action. Once that connection was made, Samaritan could have analysed facial markers, his gait while walking, his voice records etc to achieve whatever level of proof it determined necessary to define Harold as a threat and to send its assets after him.