Obstructions to coloring arithmetic graphs

The arithmetic graph $B_n$ joins distinct $a,b\in\N$ when $\max(a,b)/\gcd(a,b)\le n$. We prove $χ(B_{205})=206$, disproving the conjecture that $χ(B_n)=n$ for every $n$, equivalently the Rainbow Cascades Conjecture. The proof reduces an arbitrary tiling by the arithmetic exponent tile to a periodic