Proposition: Finite-volume mismatch cannot be hidden

On $M$ edges, the total-variation distance between the monitored cost law and the product exponential cost law is at least

At $p=1$, every passage cost vanishes, and the resulting limit is not a norm.

Proof. Use the event that at least one cost is exactly zero. Its probabilities are $1-(1-p{)}^M$ and zero, respectively. For $p=1$, every path has zero cost.