Exponent-one blockers and a Mordell-Weil construction of Euler bricks

A body cuboid is a rectangular parallelepiped with integer edges and integer face diagonals; if its space diagonal is also integer, it is a perfect cuboid, whose existence is a long-standing open problem. We make two contributions to the study of body cuboids parametrised by two coprime Pythagorean pairs $(a,b)$ and $(m,n)$ in Euclid form (Master-Hits). The first is a verified exponent-one blocker phenomenon: for every Master-Hit, the space-diagonal norm $f_1:=(W_1{U}_2{)}^2+(U_1{V}_2{)}^2$ admits a prime divisor $\mathrm{\ell }$ of exponent exactly one which is coprime to a fixed list of $29$ canonical expressions in the parameters. This is strictly stronger than the existence of any odd-exponent prime divisor: a prime of exponent $3,5,\dots$ would obstruct $f_1$ from being a square but carry an extra square factor; the observed obstruction is always primitive. The phenomenon is verified on all $\mathrm{151,575}$ Master-Hits whose $f_1$ has been fully factorised. Two natural strengthenings fail: the largest outside-parameter prime need not be a blocker, and the smallest outside-parameter blocker need not have exponent one. The second contribution uses the elliptic fibration of the Master-Hit variety over the $(m,n)$-plane. For coprime $(m,n)$ the Master-Hit equation defines a genus-one quartic $H_{m,n}$; a quartic-to-Weierstrass normalisation gives an elliptic model $E_{m,n}$ with a rational function $\tau$ returning $t^2$. Our generator enumerates bounded Mordell-Weil combinations on $E_{m,n}(\mathbb{Q})$, lifts the points satisfying $\tau (P)\in {\mathbb{Q}}_{>0}^{\mathrm{\square }}$ to admissible Euclid pairs $(a,b)$, and certifies each via exact integer arithmetic. From $\mathrm{61,829}$ classical Master-Hits we generate $\mathrm{1,222,841}$ further ones über $411$ fibres. None of the resulting $\mathrm{1,284,670}$ Master-Hits is a perfect cuboid; all fully factored records satisfy the exponent-one blocker phenomenon.