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Under current circumstances, the second condition has been met.
Does 52% of a 70% turnout count as sufficient? As mandates go, it's marginal at best.
There doesn't appear to be any measure of what Parliament as a whole thinks.
NotThamesWater-
We've accepted less in order to for a government to be formed from a General Election so yes, it is sufficient. We are a democracy without manadatory voting so the remaining 30% potential turnout are viewed as simply failing to have an opinion as to whether we should stay or leave. They have to be considered as not caring either way. 52% of 70% is 36% of the voting population.
Yes, it's a marginal mandate but even a vote difference of 1 still counts in a democracy. And unlike parliamentary representation, where we can objectively say that a governing party lacks a mandate if it doesn't command the support of 50% of the voting population, a referendum is a very different matter. An abstention in a referendum isn't a party political issue and can't be viewed as a rejection of the existing democratic state or system. This is a direct question to the voting population on policy.
The_Seldom_Killer
There is no statutory instrument and Article 50 could be invoked by any single person in the UK.
However, The EU is unlikely to accept notification from anyone other than the Prime Minister or duly appointed successor or sufficient remaining authority. They are also unlikely to accept it if they feel that the notification lacks sufficient backing of the voting population or parliamentary representation.
Under current circumstances, the second condition has been met.