A torsion-intersection proof of perfect-cuboid nonexistence on 1,072 explicit master-tuple fibers

Abstract

Building on the genus-3 reduction $C_A : w^2 = \lambda^8 + A \lambda^4 + 1$ established in our companion paper (arXiv:2604.09328), we give an unconditional proof of the perfect-cuboid conjecture ("Conjecture B") on $1{,}072$ explicit master-tuple fibers, excluding all rational $(a,b)$-specialisations on each such fiber. Our three main contributions are: (i) a structural classification theorem showing that every primitive Euler-brick arises from the standard $(a,b,m,n)$-parametrisation up to scaling; (ii) a torsion-intersection argument applied to the elliptic quotients $E_A'$ and $E_A''$: whenever the rank-zero hypothesis and the appropriate torsion condition hold for one of them, $|H_{m,n}(\mathbb{Q})| = 8$ is forced, with the eight points all corresponding to degenerate bricks; (iii) two complementary techniques to verify the rank-zero hypothesis algorithmically -- PARI's ellrank (2-descent) and, where this is ambiguous, Sage's exact rational evaluation of $L(E,1)/\Omega_E$ via modular symbols, which combined with the modularity theorem, Kolyvagin's theorem, and Edixhoven's bound on the Manin constant for semistable curves yields an unconditional rank-zero certificate -- together with an explicit lift count refining the naive torsion-intersection bound when the torsion is larger than the leading case. We exhibit $1{,}072$ such fibers with $\max(m,n) \le 100$ on which Conjecture B is thereby established unconditionally.