The Claude C Compiler illustrates the other side: it optimizes for passing tests, not for correctness. It hard-codes values to satisfy the test suite. It will not generalize. Property-based testing would likely catch this particular case, but the general problem remains: for any fixed testing strategy, a sufficiently adversarial system can overfit to it. A proof cannot be gamed. It covers all inputs by construction.
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To get even more radical: I often think unexpected errors should completely crash the program. It’s disruptive in the short term, but I find crashes make software feel more reliable in the long run. You’re more likely to hear about these problems from annoyed users—if not your own testing.