There is no background clock that all particles share. Each one carries its own, and "time" is simply how far that clock has turned.
Every massive particle is, in this framework, an internal clock with proper period T0 = λ0/c — the time for one full rotation on its own internal circle. The time parameter that appears in the wavefield is exactly this: the proper time of the observer writing the description. No observer's clock is privileged over another's.
Two clocks in relative motion each measure the other as running slow by the same factor γ — not because one clock is intrinsically slower, but because each one's measurement of the other already incorporates the local geometric constraint. There is no fact of the matter, in this framework, about which clock is "really" the slow one: each clock, measured by the other, simply ticks at a different rate than it ticks for itself.
The only time available anywhere in the construction is the local internal phase of some particle's own motion. Different observers' descriptions are related to each other through explicit geometric relations — the rotation of local energy axes, or equivalently the corresponding transformation between coordinates — not through a shared external timeline that all of them secretly read off.
The proper period T0 is fixed by the topology of the internal fiber, identical for every observer of the same species describing itself. In another observer's description, a heavier particle corresponds to an internal clock with a shorter proper period, and a lighter one to a longer period — mass and the rate of one's own internal clock are, geometrically, the same fact read two ways.