Generally these things are set up so that proving someone lied on the form is easy and straightforward. As someone else pointed out, the actual question isn't going to be "are you a terrorist", it's going to be a question about whether you've ever associated with certain people, or organizations which espoused certain views. And that can be very easy to prove.
You see this pattern in more than just immigration cases.
The FBI and other federal agencies infamously "get" people for making untrue statements to their agents, since that's a federal felony right away, and easier to prove than that the person committed some other, more complex crime.
The other day in a reddit thread I got to bring up a case from New Mexico involving someone who literally fabricated evidence of a child in order to claim support payments. Once it was established the child didn't exist, proving all the various elements (like falsifying birth certificate and paternity test) of the crimes involved in getting to that point was difficult, but it was very easy to prove she'd claimed the nonexistent child as a dependent on her income tax returns, which sent her straight to prison (if you want all the details on that one, Google "Barreras v. Trevino").
A few years ago the US Supreme Court (Abramski v. United States) upheld a conviction of someone who purchased a gun, because at the time of the purchase he filled out a form which said he was purchasing the gun for himself, and not on behalf of or as a stand-in for someone else. Except he lied when he checked that box on the form; he was buying it for his uncle. The Supreme Court ruled that it didn't matter whether his uncle was legally eligible to buy the gun (he was); Congress had made it a federal crime to lie on the form, in order to prevent people from skirting background checks by getting someone else to buy a gun on their behalf, and the guy had unquestionably lied on the form.
You see this pattern in more than just immigration cases.
The FBI and other federal agencies infamously "get" people for making untrue statements to their agents, since that's a federal felony right away, and easier to prove than that the person committed some other, more complex crime.
The other day in a reddit thread I got to bring up a case from New Mexico involving someone who literally fabricated evidence of a child in order to claim support payments. Once it was established the child didn't exist, proving all the various elements (like falsifying birth certificate and paternity test) of the crimes involved in getting to that point was difficult, but it was very easy to prove she'd claimed the nonexistent child as a dependent on her income tax returns, which sent her straight to prison (if you want all the details on that one, Google "Barreras v. Trevino").
A few years ago the US Supreme Court (Abramski v. United States) upheld a conviction of someone who purchased a gun, because at the time of the purchase he filled out a form which said he was purchasing the gun for himself, and not on behalf of or as a stand-in for someone else. Except he lied when he checked that box on the form; he was buying it for his uncle. The Supreme Court ruled that it didn't matter whether his uncle was legally eligible to buy the gun (he was); Congress had made it a federal crime to lie on the form, in order to prevent people from skirting background checks by getting someone else to buy a gun on their behalf, and the guy had unquestionably lied on the form.