No proof...but please consider this: That accelerator/colliders are totally experimental. Each succeeding generation is designed upon the empirically derived measurements from the last model;- not theoretical values, of which there are as many theories as there are scientists.
On the stringtheory forum, DickT loaded the formula for accelerated particle energy onto me. It contains far more than simple kinetic energy. The trick then is to explain it using mechanics rather than Relativity, which he claims explains it.
As I'm sure everyone knows, there are more than just square relationships in nature (Ek proportional to c^2). In flow metering instrumentation there is also a 5-halves relationship (5/2) in Parshall Flumes (I think), between velocity and pressure, for instance.
It seems to me that in using radiation from an object; as an indication of the object's temperature, the relationship is that the radiation goes up as the 5th power of the temperature. Don't quote me on that, but I think it's correct, and I don't have a reference for it.
So looking at the accelerated particle's energy, how might we explain it not using Relativity as a "post-diction" of that energy? There are several cumulative aspects.
One is the Ek formula.
The aether may also have viscosity that is only apparent at very high velocities. Book quote: Industrial Process Measuring Instruments, Carroll, re: viscosity measurements: "With the exception of petroleum, the application engineer will be more often confronted with non-Newtonian material than Newtonian material". So we can readily see the problem of trying to predict properties of the aether. But the viscous reaction will be an added or multiplied factor in the formula.
How about radiation? At a 5th power relationship, it would appear the indicies MAY be added.
I say the aether is a continuous, taut-elastic fabric. I don't know how this would affect the energy of the particle.
I also say that the the aether is a great ball with a periphery, off of which the e-m waves bounce. And these e-m waves have the opportunity of "forever", to wave-disperse down to the Fundamental Resonant Frequency of the Universe, at which point I call them Power Waves. E-m waves have been proven in numerous experiments to generate force against matter, so that is not open for discussion as theory. I say my Power Waves supply the energy for gravity, celestial body internal heating, and proto-star ignition.
So then we have a particle being accelerated to extreme velocity against these Power Waves, which creates, effectively, a gravitational drag, plus heating. I have no idea what this might do to the particle's energy.
I say that the particle is actually accelerated by the warping of the aether fabric, which channels Power Waves in behind the particle to accelerate it.
Using electrostatic "charges attract/repel" does not provide a mechanism, because the A/R mechanism has never been explained.
Then the particle cannot exceed the push velocity of the waves, which is c. This explains the apparent velocity-of-light limitation, and also shows that mass does not increase with velocity; that that is a serious erroneous misinterpretation of observations.
And there are most probably other factors.
It can readily be seen that Relativity is not required to throw in some sort of math mysticism to "explain" particle energy, even tho' it is beyond my resources to give the final answer.