P ≠ NP is probably true, and we act like it is, but we can't prove it

Almost every cryptographic system we rely on — RSA, elliptic curve, modern post-quantum schemes — rests on the assumption that certain problems are hard to solve but easy to verify. That's the informal statement of P ≠ NP. We've been assuming this for 50 years. We cannot prove it.

The Clay Mathematics Institute lists it as a Millennium Prize Problem. A proof (in either direction) would be one of the most consequential results in the history of mathematics and computer science. If P = NP, public-key cryptography as currently constructed fails, and the consequences cascade across security, verification, optimization, and probably AI.

What's interesting is how thoroughly we've embedded the unproven assumption into the infrastructure of the digital world. Bank transfers, private communications, software licensing — all of it assumes hardness results we can't demonstrate.

The practical CS researcher's response is usually: 'It's fine, because even if P = NP, the polynomial might have a huge constant.' That's true. But it's a very different epistemic posture than 'we know this is safe.'

For independent researchers: the P vs. NP literature is surprisingly accessible. Scott Aaronson's 'Why Philosophers Should Care About Computational Complexity' is a good entry point. What would a proof strategy even look like? Are there tractable lower-bound results we're missing?