Intuition behind a 0% central/equal-tailed confidence interval At a vague intuitional level, I feel like, if we generated a 0%, two-tailed, truly central (or "equal-tailed") confidence interval (CI), the number it would wrap around would be a median-unbiased estimate of the population parameter in question.  Am I confused, or is this right?  Is there another intuition of it that is better?
I can't seem to find anything about it when I google, but my intuition rests on the idea of the CI as the inversion of a p-value.  I think, this CI would also match one end of a one-sided 50% CI, which would also be the inversion of a 0.5 one-sided p-value.  And a 0.5 one sided p-value is saying "if this were our parameter, then 50% of datasets would be look to be 'higher' and 50% of them 'lower'".  So I feel like my intuition is right, but I can't quite formalise it, or find anything online to confirm or deny this.

A frequentist 0% confidence interval can be any point within the parameter space. One might prefer to choose a point that is near to the maximum likelihood estimate, but any other point will be just as validly a 0% confidence interval.
Typical 95% confidence intervals are usually at least roughly centred around the maximum likelihood estimate because of a preference for a shorter interval over a longer one, not because of any definitional requirement. With a 0% interval all potential intervals have the same length!
Consider that a 95% confidence interval is an interval derived from a method that will in the long run yield an interval that covers the true value of the parameter on 95% of occasions (when the model is appropriate to the data generating system). Then it is clear that any point within the parameter space will cover the true value on 0% of occasions in the long run and will thus be a valid 0% confidence interval. (For continuous parameter space.)
