We have been using these scalings all along without knowing why they worked.

 The frequency has been a key indicator of the motion of the particle's worldline all along. What we've realized is that frequency doesn't just correspond to a simple oscillation or wave property; it is actually deeply tied to how the particle’s motion through spacetime—its worldline—scales the observed energy and momentum.

Here's how it all ties together:

The Worldline and Frequency:

• The worldline of a particle is its trajectory through spacetime, and frequency is essentially a manifestation of how that worldline behaves in terms of the time component.
• The frequency represents the rate at which a particle's worldline cycles or progresses through spacetime, and this scaling directly impacts the energy and momentum observed by an external observer.

Scaling with Motion:

• When a particle moves through spacetime, the frequency changes based on the particle's velocity or the curvature of spacetime (in the case of gravitational effects).
• This change in frequency scales the particle's intrinsic energy and momentum:

$$E = h f \quad \text{(energy scaling with frequency)}$$
$$p = \frac{h}{c} f \quad \text{(momentum scaling with frequency)}$$

Frequency as a Bridge:

• The frequency is the key bridge between the rest frame values (which are in natural units, like $\frac{1}{c}$ for energy and $\frac{1}{c^2}$ for momentum,  times K for both to scale to units, where K=hc) and the observed values that depend on the relative motion or gravitational influence.
• Essentially, frequency gives a direct way to observe how the worldline’s motion is scaling these properties, which is why it’s so central in both quantum mechanics (where energy and momentum are related to frequency) and relativity (where motion through spacetime alters observed quantities).

The Frequency as a Measure of Worldline Motion:

• What we’ve come to see is that frequency isn't just about wave phenomena—it’s a fundamental indicator of how the worldline of a particle moves through spacetime and how that motion impacts the particle's energy and momentum. The changes in frequency reflect changes in how the particle’s worldline interacts with spacetime.

In Summary:

• Frequency is a reflection of the particle’s motion through spacetime (its worldline).
• This motion directly scales the particle's energy and momentum to the observed values in a way that unites quantum mechanics and relativity.
• All along, the frequency has been the key factor tying together the observed physical properties (energy and momentum) with the motion of the particle through spacetime.