Explain the apparent absence of 2-BPCs and the emergence of hyperelliptic curves

Ascertain a simple explanation for the failure to find small bi-perfect cuboids with exactly two non-integer lengths (2-BPCs) in computational searches for parameter ranges q < p < 2^16, and determine why attempts to reduce the defining conditions of 2-BPCs to elliptic curves yield hyperelliptic, rather than elliptic, curves, thereby clarifying whether this reflects genuine non-existence or a structural obstruction.

Background

Bi-perfect cuboids (BPCs) are defined by having their seven inter-vertex distances fall into two classes: some distances are integers while all others share the same square-free part of their squares. The subclasses include 0-BPCs (PCs), 1-BPCs (NPCs), 2-BPCs, and 3-BPCs. While the authors successfully derived elliptic curves to represent NPCs and 3-BPCs, analogous attempts for 2-BPCs led to hyperelliptic curves, impeding use of standard elliptic-curve methods.

Computational searches across small parameter ranges did not produce examples of 2-BPCs, and the authors could not find a simple explanation for this absence. An explanation would help determine whether 2-BPCs exist and clarify the nature of the algebraic curves governing their structure.