Awareness or "Continuous I"
Our Awareness is who has "I" at the moment...although that's an overly simplified way to look at it.
I is a commodity that can be distributed although that's much harder than sharing Front. Several people can blend and share I without discomfort, but splitting it amongst several separate entities is territory heading for a migrane quickly. That's not the same thing as rapid switching, which is playing hot potato with "I" amongst several people who are probably sharing or swapping Front as well.
I and Front need not be synonymous. Our mind can wander, the first-person perspective ("I") can walk out of the Front, and wander around in our head taking awareness with it, or leave our body essentially in a trance-state (at best — someone else *can* take Front over and continue doing things while our mind wanders away — this is tantamount to time loss and doesn't happen to us in any large capacity that we're aware of...but then we might just not be aware of it). If we lay down and meditate, "I" goes with whoever has it and can travel through our body, through shamanic or astral gateways, into other dimensions, visit people elsewhere, perform hypnotic suggestions on our mind, etc. So "I" is not always "Front". Someone takes "I" into dreamscapes when we sleep and has dreams — usually only one of us in the dream. Someone, not always the same person who went to sleep or was dreaming, brings "I" back to Front in the morning when we wake up...
why "Continuous I"
Our entire recorded perspective of time passing is from the point of view of "I". As time passes, "I" continues to record events both internally and externally depending on the focus-of-the-moment. While Front may be essentially unmanned or someone else could theoretically be Front who doesn't have the "I" at the moment, "I" usually does not black out — brown outs can happen (loss of 1 or more senses such as a wash of loss of vision — experienced in some ways similar to vertigo).
Our experience of "time loss" usually occurs between "I" recording something and the data being moved to mid-term (File Cabinet) or long-term (Memory Chamber) storage. So we may remember something for a week (Logbook) then it goes missing. Or we remembered something for weeks (File Cabinet) then about 6 months later it's not to be found anymore. Since "I" is always recording with few exceptions, our lack of memory (whether fully lost, dissociative amnesia, or someone deliberately stealing, obscuring, or hiding things) operates in the background.