I'm quoting myself here; this is not a formal proof, its a wordy one.
With no competition to mine, the maximum network hashing rate will be impossible to measure, since although transactions may be incoming at a high rate in a mature currency (and assuming a PoW must be submitted with the transaction) , this rate could still be vastly below that of even just one ASIC* such that any adversary looking to double spend would find the task relatively easy.
This also brings into question the entire way in which transaction acceptability can be bounded. In bitcoin the adversary's hashing power relative to the network as a whole is considered, yielding a probability of the best block being reversed, yet with no competition to mine, the maximum hashing rate of the network as a whole cannot be measured since adversaries have nothing to gain by participating in the network's nominal operations, instead they might chose to lie in wait.
edit: the key point is that last one. With no competition to mine, the network hash rate cannot be measured which implies that
the time you need to wait to accept a transaction as confirmed is essentially unbounded. This is not acceptable in any currency.