The term "for any lawful contact" is so overly broad, it's effectively meaningless.
Well then read the bill if you want specificity.
"That means the officer is already engaged in some detention of an individual because he's violated some other law," says Kris Kobach, a University of Missouri Kansas City Law School professor who helped draft the measure. "The most likely context where this law would come into play is a traffic stop."
Read more at the Washington Examiner: http://www.washingtonexaminer.com/opinion/columns/Byron-York/A-carefully-crafted-immigration-law-in-Arizona-92136104.html#ixzz0mQLmXTqj
As an example, police everyday come into "lawful contact" with witnesses and victims of crimes. What the law says, is after lawful contact has been established, the police may ask the "suspect" (for lack of a better word) to evidence of his legal residency or citizenship, if the LEO has a reasonable suspicion that the person they've lawfully contacted is in the country illegally.
So, the police under this law may also ask for the passengers of a car that they've stopped to provide proof of residency or citizenship, even though it's VERY difficult for a passenger to be a suspect in a moving violation.
Kobach was speaking as a man running for office, not rending an accurate characterization of the law as a law professor.
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What you quote is not in the bill, it is an opinion that attempts to clarify. All that matters is that which is stated in the law and, the phrase is overly broad.