The Perfect Cuboid: from the Continuous World and Radix Representation to Arithmetic Freedom
G2E.5 and G2E.6 · the bielliptic reduction and the hand-off
unsigned in the source
The Bielliptic Genus-2 Reduction and the Local Exact Verification Hand-off: an Exact Involution, Explicit Elliptic Quotients, a Corrected Dual-Quotient Sieve, Full-2 Descent and the Final Simultaneous Quartic Target
The genus-3 layer maps to a genus-2 curve H_2 with an explicit bielliptic involution τ and quotient E_+ = E_0, so Jac(H_2) ~ E_0 × E_−, and the problem becomes: find points of E_0(Q) with 2 − x a rational square; it also corrects an asymmetric [m]-projection bug in the checker of G2E.4 (clean primes unchanged). The hand-off file gives the fixed slice k = 7/32 to local computation: a full-2 descent of a genus-2 square-value curve, isolation of the covering class (1, 1073, 1073), then the simultaneous quartic system Y² = A4(m), Z² = B4(m) whose only known solutions are the boundary ones (m = 25, 39, 31200/1073); certified and provisional items are listed separately.
Exact structural theorems and an honest hand-off: 'no provisional item should be promoted without an independent exact rerun'; 'if proved, then k = 7/32 is globally excluded as a nondegenerate perfect-cuboid first-channel slice' is conditional.
This page is the end of the G-line: nowhere in the package is the fixed slice claimed to be excluded, let alone the perfect cuboid claimed not to exist; even if k = 7/32 were excluded, k can still vary. Line 482 of G2E.6 says 'this must not be claimed until all other locally soluble classes have been independently eliminated'. The pre-publication read-through re-verified the τ identities, E_+ = E_0 and w = ±1 at m = 25, 39, 31200/1073.