Rigid Matrices From Rectangular PCPs or: Hard Claims Have Complex Proofs

We introduce a variant of PCPs, that we refer to as rectangular PCPs, wherein proofs are thought of as square matrices, and the random coins used by the verifier can be partitioned into two disjoint sets, one determining the row of each query and the other determining the *column*. We construct PCPs that are efficient, short, smooth and (almost-)rectangular. As a key application, we show that proofs for hard languages in NTIME$(2^n)$, when viewed as matrices, are rigid infinitely often. This strengthens and considerably simplifies a recent result of Alman and Chen [FOCS, 2019] constructing explicit rigid matrices in FNP. Namely, we prove the following theorem: - There is a constant $\delta \in (0,1)$ such that there is an FNP-machine that, for infinitely many $N$, on input $1^N$ outputs $N \times N$ matrices with entries in $\mathbb{F}_2$ that are $\delta N^2$-far (in Hamming distance) from matrices of rank at most $2^{\log N/\Omega(\log \log N)}$. Our construction of rectangular PCPs starts with an analysis of how randomness yields queries in the Reed--Muller-based outer PCP of Ben-Sasson, Goldreich, Harsha, Sudan and Vadhan [SICOMP, 2006; CCC, 2005]. We then show how to preserve rectangularity under PCP composition and a smoothness-inducing transformation. This warrants refined and stronger notions of rectangularity, which we prove for the outer PCP and its transforms.