Khot's 2-to-1 Games Conjecture with perfect completeness

Claims Theorem 1.1: for every fixed rational $\delta\in(0,1)$ there are $q=q(\delta)$ and a deterministic polynomial-time reduction from 3-SAT to unweighted 2-to-1 games over $[2q],[q]$ with exact two-element fibers, value 1 on YES and at most $\delta$ on NO instances. Corollaries make the Guruswami-Sinop maximum $k$-colorable subgraph hardness unconditional and recover Hastad's satisfiable Not-Two result. It does NOT prove the Rich 2-to-1 variant of Braverman-Khot-Minzer, the Unique Games Conjecture, or anything about alphabet size as a function of $\delta$ beyond existence.

Claims Theorem 1.1: for every fixed rational $\delta\in(0,1)$ there are $q=q(\delta)$ and a deterministic polynomial-time reduction from 3-SAT to unweighted 2-to-1 games over $[2q],[q]$ with exact two-element fibers, value 1 on YES and at most $\delta$ on NO instances. Corollaries make the Guruswami-Sinop maximum $k$-colorable subgraph hardness unconditional and recover Hastad's satisfiable Not-Two result. It does NOT prove the Rich 2-to-1 variant of Braverman-Khot-Minzer, the Unique Games Conjecture, or anything about alphabet size as a function of $\delta$ beyond existence.