The Perfect Cuboid: from the Continuous World and Radix Representation to Arithmetic Freedom
G2A · Nested-X double square
unsigned in the source
G2A: Nested-X Double-Square Closure — A Two-Parameter Exact Reduction of the Perfect Cuboid Problem
A perfect cuboid exists iff there are rationals k, h ∈ (0, 1) with F_k(h) and F_k(M(h)) both rational squares; for fixed k the simultaneous curve has genus 5, and for k = 7/32 an exact scan of all 121,590,395 reduced h = u/v with v ≤ 20000 finds 4 + 4 single-gate points and no double hit.
A bounded exact computational exclusion, not a proof: the paper says 'this is only a bounded exact computational exclusion' and 'the global problem remains open because k itself is free to vary over Q ∩ (0, 1)'.
The pre-publication read-through redid the gate scan for v ≤ 1500 (684,181 states): first gate {4/7, 4/33, 33/119, 256/817}, second gate {3/11, 29/37, 43/76, 561/1073}, no double hit; the full 121,590,395 states were not re-run.