# Can fine-grained hardness be proved directly from classical hardness (e.g., $\sf P \neq NP$) in some way?

I have just learnt about some typical result of fine-grained hardness in 15-455 by Prof Ryan: CNF-SETH implies $${\sf DIAMETER} \notin {\sf TIME}(mn^{1-\epsilon})$$. (Here DIAMETER stands for the graph diameter problem).

To prove fine-grained hardness, we construct reductions from $$A$$ to $$B$$, then "perturbation" on $$A$$'s complexity will cause a perturbation on $$B$$'s. By some fine-grained hardness assumption on $$B$$, the perturbation on $$B$$'s complexity is restricted (contradicts the assumption), so such perturbation on $$A$$'s is disallowed, and kinds of fine-grained hardness result on $$A$$ is obtained. For example, the proof of $${\sf DIAMETER} \notin {\sf TIME}(mn^{1-\epsilon})$$ by CNF-SETH.

In this kind of proof, we can only derive fine-grained hardness from another fine-grained hardness. But where is the initial fine-grained hardness result? i.e. the one not derived from some other fine-grained hardness assumptions. For example, can we prove fine-grained hardness result from "classical hardness" (I'm sorry that I don't know what's the corresponding term), e.g. $$\sf P \neq NP$$?

The proof technique above seems not work for those classical hardnesses, whose associated complexity classes (e.g. $$\sf P$$ and $$\sf NP$$) are very robust by definition. In this case, the small perturbation on $$B$$ will still fall into the same complexity class, thus won't be prevented by the hardness.

Can fine-grained hardness (e.g. $${\sf DIAMETER} \notin {\sf TIME}(mn^{1-\epsilon})$$) be proved directly from classical hardness (e.g., $$\sf P \neq NP)$$ in some way? What I have learnt is just a brief introduction, and I want to know whether there are some known results (since I failed to find them through Google).