Lorentz Invariance from Period-Wavelength Counting and Reflection/Refraction?

Fermat’s principle of least time may be utilized to predict reflection and refraction paths. This entails writing t in terms of distance(x)/speed of light / and varying x with dx/dt =0. It seems, however, that there may exist a different way of analysing reflection and refraction. In the case of reflection and refraction, time and frequency do not change for a reflected ray or for the x-projection of a refracted ray. For the refracted ray, both speed of light and wavelength change, but it appears that Fermat’s principle is equivalent to keeping the number of wavelengths and also periods in the x-projection of the incident and refracted rays the same. From this idea, one may create two invariants namely Et+px and -Et+px where p=hbar/wavelength. The second may be varied to give E=pc and so is used. This invariant follows from counting wavelengths and periods. Thus if one has a rest frame and a second frame moving at a constant velocity v in the x direction, the counting of wavelengths and periods does not change even though E,t,x and p may change (i.e. people in the two frames have different values). Thus -Et+px should also be an invariant for constant moving frames. In other words, one obtains a Lorentz invariant (with the metric) from an invariant appearing in reflection/refraction. In principle, one could simply use -Et+px with E proportional to 1/T and p to 1/wavelength to argue that -Et+px is a scalar representing the number of periods in t and wavelengths in x. Given that this is a scalar, one could argue it should be a Lorentz invariant as well. In this note we try to argue that -Et+px is not only an invariant in special relativity, but applies to other physical examples such as reflection or refraction