By 'proper frame' of observer O, I mean any reference frame (coordinate system) in which (Condition A:) The worldline of O is always at the spatial origin for every time coordinate. Clearly such a frame is not unique because spatial rotations do not invalidate (A). What I am interested in is whether it determines a unique foliation of spacetime, so that the time coordinate is unique, modulo a change of time units. My guess is that the answer is No, but that a proper frameof O is 'approximately unique' near O, meaning something like that for any two frames C1 and C2 that are proper relative to O, the time coordinates of a point x under frames C1 and C2 are the same, to first order, for x near O. I can't recall seeing a theorem about this. The closest I can remember is about the existence of Normal Coordinates, but that's a somewhat different issue, as Normal Coordinates are for a locally inertial frame, and the proper frames we are talking about here need not be inertial. Thank you.