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![From the ETH proof. Lemma (Eight-square identity [CS03, AM99]). There is a function σ : {0, 1}3 × {0, 1}3 → {−1, 1} such that for all x, y, u, v ∈ {0, 1}3, with ⊕ the bitwise exclusive or, (12) 2 [x = v] [y = u] = [x ⊕ y ⊕ u ⊕ v = 0] (σ(x, y) σ(v, u) + σ(x, u) σ(v, y)). Proof. Let σ(p, q) = (−1)ϕ(p,q) with ϕ(p, q) = ∑ k≤k′ pkqk′ + q1p2p3 + p1q2p3 + p1p2q3. Equa- tion (12) is a statement about 12 bits, and it can be checked at all 4096 settings. (§4 and Appendix F prove it by hand.) ■ The identity is two centuries old: Degen found it in 1818, and Graves found it again in 1843 by showing that the norm of the octonions is multiplicative [Bae02, Section 1].](https://pbs.twimg.com/media/HVvWXQXWIAA-hLM.jpg?name=orig)
ALT From the ETH proof. Lemma (Eight-square identity [CS03, AM99]). There is a function σ : {0, 1}3 × {0, 1}3 → {−1, 1} such that for all x, y, u, v ∈ {0, 1}3, with ⊕ the bitwise exclusive or, (12) 2 [x = v] [y = u] = [x ⊕ y ⊕ u ⊕ v = 0] (σ(x, y) σ(v, u) + σ(x, u) σ(v, y)). Proof. Let σ(p, q) = (−1)ϕ(p,q) with ϕ(p, q) = ∑ k≤k′ pkqk′ + q1p2p3 + p1q2p3 + p1p2q3. Equa- tion (12) is a statement about 12 bits, and it can be checked at all 4096 settings. (§4 and Appendix F prove it by hand.) ■ The identity is two centuries old: Degen found it in 1818, and Graves found it again in 1843 by showing that the norm of the octonions is multiplicative [Bae02, Section 1].
ALT From the ETH proof. Lemma (Eight-square identity [CS03, AM99]). There is a function σ : {0, 1}3 × {0, 1}3 → {−1, 1} such that for all x, y, u, v ∈ {0, 1}3, with ⊕ the bitwise exclusive or, (12) 2 [x = v] [y = u] = [x ⊕ y ⊕ u ⊕ v = 0] (σ(x, y) σ(v, u) + σ(x, u) σ(v, y)). Proof. Let σ(p, q) = (−1)ϕ(p,q) with ϕ(p, q) = ∑ k≤k′ pkqk′ + q1p2p3 + p1q2p3 + p1p2q3. Equa- tion (12) is a statement about 12 bits, and it can be checked at all 4096 settings. (§4 and Appendix F prove it by hand.) ■ The identity is two centuries old: Degen found it in 1818, and Graves found it again in 1843 by showing that the norm of the octonions is multiplicative [Bae02, Section 1].
