The polynomial removal conjecture for ordered binary matrices

Theorem 1.1: there is a fixed binary $66\times66$ matrix $H$ such that for every $h\ge1$, with $n_h=(386h+2)2^h$ and $\epsilon_h=(386h+2)^{-2}$, some $A_h$ is $\epsilon_h$-far from $H$-free yet $N_H(A_h)\le\epsilon_h2^{-h}n_h^{132}$; hence no polynomial removal bound holds for this $H$, and the finite-family question fails for one square pattern. Corollary: canonical row-column sampling testers need $\exp(\Omega(\epsilon^{-1/2}))$ samples along this sequence. Not shown: which patterns do admit polynomial removal, or a counterexample smaller than $66\times66$.

Theorem 1.1: there is a fixed binary $66\times66$ matrix $H$ such that for every $h\ge1$, with $n_h=(386h+2)2^h$ and $\epsilon_h=(386h+2)^{-2}$, some $A_h$ is $\epsilon_h$-far from $H$-free yet $N_H(A_h)\le\epsilon_h2^{-h}n_h^{132}$; hence no polynomial removal bound holds for this $H$, and the finite-family question fails for one square pattern. Corollary: canonical row-column sampling testers need $\exp(\Omega(\epsilon^{-1/2}))$ samples along this sequence. Not shown: which patterns do admit polynomial removal, or a counterexample smaller than $66\times66$.