Let \(v\) be the tip of \((a_1, \dots , a_d)\). Then, using that \(\sum _k a_{j, k} = 0\) and \(v_k = a_{j, k}\) for all \(k \ne j\), we see that \(v_j = a_{j, j} + \sum _k v_k\). This means that \((\operatorname{forget}_i(a_1), \dots , \operatorname{forget}_i(a_d)\) is a multidimensional corner by setting \(x = \operatorname{forget}_i(a_i)\) and \(c = \sum _k v_k\).

Assume \((a_1, \dots , a_d)\) is a monochromatic forbidden pattern with tip \(v\), say \(\operatorname{broad}(a_i, t) = b\) for all \(i\). By Lemma 2.10, we also have \(\operatorname{broad}(v, t) = b\). Since \(P\) is a valid NOF protocol for \(\operatorname{eval}\), we get \(\operatorname{eval}(a_i) = \operatorname{eval}(v)\) for all \(i\), meaning that \((a_1, \dots , a_d) = (v, \dots , v)\) is trivial.

Let \(P\) be a protocol valid in time \(t\) for \(\operatorname{eval}\). By Lemma 3.3, \(\operatorname{broad}(\cdot , t)\) is a coloring of \(\{ x \mid \sum _i x_i = 0\} \) in at most \(2^t\) colors (since \(t\) bits were broadcasted) such that all monochromatic forbidden patterns are trivial. By Lemma 3.2, this yields a coloring of \(G^{d - 1}\) in at most \(2^t\) colors such all monochromatic corners are trivial. Hence \(2^t \ge \chi _d(G)\), as wanted.

Putting Theorem 3.4 and Lemma 1.6 together, we get