Conway’s light on the shadow of Mordell

The paper’s Theorems 3.2 and 3.3 classify real and dual-number solutions of X^2 + Y^2 + Z^2 = 2XYZ + 1 with 1 ≤ x, y ≤ z, giving the parametrization x = cosh u, y = cosh v, z = cosh(u+v), and, over dual numbers, X = cosh u + αε sinh u, Y = cosh v + βε sinh v, Z = cosh(u+v) + γε sinh(u+v) with the constraint γ = α + β, except for the special triple (1,1,1) . The candidate solution reproduces exactly this argument: reduces to real and ε-parts, selects the larger root z = cosh(u+v) under 1 ≤ x, y ≤ z, derives γ = α + β when sinh u, sinh v, sinh(u+v) ≠ 0, and correctly treats the degenerate cases where division by these factors is not allowed. The converse via analytic extension/power series is the same idea used in the paper. No substantive discrepancies found.