Equivalence of active and passive discarding representations in causal wiring operads

Prove that representing discarding via a ground-termination symbol and representing discarding by allowing wires to end at an inaccessible "point at infinity" are equivalent formulations within the causal wiring operad W^{⊥}, establishing a formal correspondence between the active and passive perspectives.

Background

In defining causal wiring operads, the authors introduce an active representation of discarding using a ground symbol to terminate wires. They note an alternative passive representation where wires end at a point at infinity, suggesting these may be equivalent perspectives.

While the paper proceeds with the active representation for convenience, it leaves the formal proof of equivalence as future work, indicating an unresolved foundational question about the relationship between these two formulations.