EDIT2: This thing uses a DC-DC controller chip! Ignore all of the crossed out section above. You really need to redraw this schematic in a more sensible (less tangled) way to make sense of it, it looks like they drew it compact just to fit it on the minimum of paper.
Yah!
If you look at the AN8086 chip pin 5 is labeled as "out" and it goes directly to transistor Q5.
All the text pointing to that wire is cut off though
but even so, I can still see the end of a "MHz" remark.
L11 apparently has polarized capacitors on each side (C13 and C14) both probably 10V 47uF, so that would make it a passive low pass filter.
Maybe Q3 is used as an active diode that switches synchronous with the SMPS circuit?
It looks like that when Q5 switches to GND the base of Q3 is closed too via C11.
But that does not clarify the voltage drop from 5V1 to 3V4 over Q3.
CN8 is the DC input jack (4V5) and it's built in switch opens up the GND lead of the rechargeable battery, so that definitely is where the power comes from. Charging batteries with only some added series resistance (R19 and R20) does not look very good though...