Perfect parallelepipeds and the perfect cuboid: two conjectures, four special loci, and a Gaussian criterion

We study the perfect parallelepipeds with two nonparallel rectangular faces -- the nearest known relatives of the conjecturally nonexistent perfect cuboid -- and the cuboid surface itself. First we prove Conjecture 5.2 of Sokolowsky, VanHooft, Volkert and Reiter [Math. Comp. 83 (2014), 2441-2454]: their two-parameter family contains no perfect cuboid. The proof reduces the question to rational points on the rank-zero elliptic curves 20.a3 and 40.a3, and we show that the family coincides, up to similarity, with the lambda = 0 leaning boxes of Wyss and with Rathbun's parametrization. Second, we disprove their Conjecture 4.1 -- the completeness of the family -- by three explicit counterexamples, and behind one of them we construct a new infinite family, fibered in genus-one curves, of perfect parallelepipeds with two nonparallel rectangular faces; the new family, too, is cuboid-free, by Fermat's theorem on z^4 - w^4 = square. Third, we organize all known examples into four special loci F, W, D, L of sharply different arithmetic characters inside the three-dimensional solution variety, prove that all four are cuboid-free, and verify that every one of the 382 primitive examples with common perpendicular edge at most 9,000,000 lies on their union. Finally, returning to the cuboid surface of Stoll and Testa, we make its modular structure explicit in elementary terms: a perfect cuboid exists if and only if there is a Gaussian number t, neither real nor purely imaginary nor of norm one, such that N(t), N(t^2-1) and N(t^2+1) are all rational squares. We compute the modular j-invariants attached to this parameter as explicit degree-24 rational functions and show that the degenerate rational points of the surface do not form a locus of cusps or CM points of its moduli interpretation. (34 pages. Supplementary zip: verification scripts in Python/Sage and the census of all 382 primitive examples up to 9,000,000.)