No Perfect Cuboid with Odd Prime-Power or Odd Squarefree Semiprime Edge: A Divisor-Set Reformulation and Large-Prime Descent

We give a finite divisor-set reformulation of the perfect cuboid problem with a prescribed odd edge. Using this reformulation, we prove that no perfect cuboid has an odd prime-power edge or an odd squarefree semiprime edge. We also prove a large-prime descent theorem, showing that sufficiently large prime-power factors in an odd edge can only occur as common scaling factors. Finally, we show that the valuation-monotonicity obstruction behind the prime-power case is sharp.