In other words, does the flat modify C[N] or does it modify the next C up, i.e., C[N+1]?
More like an augmented prime since you have the same letter. Apparently you are mixing up primes and seconds. At any rate any accidental modifies the note behind it.
Strictly speaking C♭6 is in a higher octave than B♯5 even though it's a lower pitch.
Melodic intervals are always measured from the lower note to the higher note, no matter which note comes first.
So yes, if the intervals between Cb and C is just a semitone apart, it’s an augmented prime.
- Cb to Cb: perfect prime
- Cb to C: one semitone bigger —> aug.
If the interval between Cb and C is one semitone greater than an octave, then it’s an augmented octave. :)