The Perfect Cuboid: from the Continuous World and Radix Representation to Arithmetic Freedom
G2D.1 and G2D.3 · ranks of the elliptic factors
unsigned in the source
The Local Full-2-Descent Rank Theorem and the Rank-One Theorem for E_−: Exact Local Certification for the Fixed Slice k = 7/32
rank E_f(Q) = 1 (a self-implemented full 2-descent, 65536 → 16 → 8 classes, matched by an explicit non-torsion point); the E_0 torsion is corrected to Z/4; rank E_−(Q) = 1 (2-Selmer dimension 5 reduced by a hand-computed nontrivial Cassels pairing); rank E_0 = 1 by pure descent; hence rank Jac(C_{7/32}) = 2 + 2 + 1 = 5 = g, so classical Chabauty is not applicable.
Two exact rank theorems with no existence or nonexistence claim.
In the pre-publication read-through PARI 2.15.4's ellrank independently gave rank 1 for E_f, E_0 and E_−, with torsion Z/4 × Z/2, Z/4 and Z/4 × Z/2. The shipped verify_g2d1.py is a stub that merely prints a message, so the descent code is not in the package; verify_g2d3.py checks only points and Legendre facts, not the Selmer/Cassels computation. G2D.2 (which proves the rank of E_0 by a parity argument and is superseded by G2D.3) and the early G2D checkpoint (which records the E_0 torsion wrongly as Z/2) are in the download table below.