We prove that for all $\varepsilon>0$, there exists a positive integer $k$ such that given a $4$-to-$1$ Game $\Psi$ with alphabet size at most $k$, it is NP-hard to distinguish between the case that val$(\Psi)=1$ and the case that val$(\Psi)\leq \delta$. This confirms the $4$-to-$1$ Games Conjecture from [Khot, CCC 2002]. Previously, the best known result, due to [Dinur, Khot, Kindler, Minzer, Safra], established the almost-perfect completeness version (but applied to the stricter problem of $2$-to-$1$ Games). Using results from the literature, we get the following implications:

1. For all $k\in \mathbb{N}$, given a $3$-colorable graph $G$, it is $\mathbf{NP}$-hard to find a proper $k$-coloring.

2. For all $\delta>0$, given a $2$-colorable $3$-uniform hypergraph $G$, it is $\mathbf{NP}$-hard to find in it an independent set containing at least $\delta$ fraction of the vertices.

Our proof is a three-step construction that builds on the two-step framework of [Dinur, Khot, Kindler, Minzer, Safra]. In the outer-PCP step, we use quadratic equations to gain perfect completeness. We then construct a new middle PCP that performs low-rank tests while preserving a key covering property. Finally, we construct a new inner PCP based on a tensor of the standard Grassmann encoding with its low-rank variant due to [Golowich, FOCS 2023].

The current version of the manuscript is complete mathematically, but it is not in the shape we wished to share in. We have chosen to do so due to rumors that surfaced around Friday, September 11th. We will work on a more complete version of the manuscript in the close future.