Explanation of Concept
Energy is defined as "that which is conserved during transformations." Conserved means totals stay the same when components are rearranged. So the test for the definition of kinetic energy is to rearrange the components and see which total stays the same.
Leibniz used this test in 1686, but he predetermined a false outcome by starting with the claim that force times distance should be a conserved quantity. A correct test compares both alternatives and shows that Leibniz was wrong.
The Test
The definition of energy was argued for 200 years but the wrong alternative was chosen. Now, rocket math can be applied to the original test, and it shows the error. The value of a rocket is that there are no complexities to mask the result.
This model, involving falling objects, was the original basis for the definition of kinetic energy.
Rockets are used to calculate force, which is independent of the definition of energy.
Part 1: Kinetic energy is not ½mv².
A 4kg object dropped 1m (meter) has the same amount of ½mv² as a 1kg object dropped 4m, because force times distance equals ½mv² for an accelerating mass. But a rocket accelerating the masses to those velocities requires twice as much energy as fuel for the large mass as for the small one.
Rocket burn time:
large mass: 1.77177876722800 seconds
small mass: 0.88588938361400 seconds
Therefore, both masses do not have the same energy; the rocket does not transform energy in proportion to ½mv²; ½mv² is not kinetic energy; and a gallon of fuel does not produce a consistent amount of ½mv².
Part 2: Kinetic energy is mv.
A 4kg object dropped for 1s (second) has the same amount of mv (momentum) as a 1kg object dropped for 4s, because force times time equals mv for an accelerating mass. A rocket accelerating the masses to those velocities uses the same amount of energy as fuel for both masses.
Rocket burn time:
large mass: 3.92400000000000 seconds
small mass: 3.92400000000000 seconds
Therefore, both masses have the same amount of energy; the rocket transforms energy in proportion to mv; mv is kinetic energy; and a gallon of fuel produces a consistent amount of mv.
All logic and evidence of energy point to the same conclusion. The logic created the need to derive the mathematical proof.
About ninety percent of physics is corrupted by the error.
Concise Math
Key Point:
With rocket math, the analysis is referenced to the combustion chamber, where equal and opposite forces determine the result. The reference point is where the force acts. Everywhere else, energy is defined relative to the starting point or some similar, external reference frame.
When the reference is the point where the force acts, you get the correct definition of energy, which is momentum. When the reference point is anyplace else, you get the incorrect definition of energy, which is ½mv².
This is because the incorrect definition of energy is equal to force times distance for an accelerating mass. Distance references to something other than the point were the force acts.

Why This is Proof
Physicists mindlessly assume the discrepancies can be accounted for in the exhaust of the rocket. There's no question where the energy is located. It's a question of defining energy in terms of transformation instead of nonsensical math. With ½mv², energy addition (including transformation) varies with reference frames, while addition of mv is independent of reference frames.
The proof shows that, when defining energy as ½mv², the exhaust energy of two rockets can be the same, while the transformation is not the same.
In tests 3 and 4 (Part 2), the rockets have the exact same burn times, while the ½mv² of one mass is 192, and the other is 770. One mass gets four times as much ½mv² as the other, while the rockets do exactly the same thing.
If the rocket and exhaust can do exactly the same thing in both cases and add four times as much supposed energy to the forward mass of one than the other, then the energy is not coming from the fuel or rocket motor; it is simply a mathematical equation with no relationship to laws of nature.
In other words, what I call proof is to show that the incorrect concept of energy has no relationship to fuel use. This should be considered proof in any rational mind.
The correct analysis is that the equal and opposite forces created in the combustion chamber transfer equal and opposite amounts of kinetic energy to the exhaust and forward mass. This is consistent with kinetic energy being proportional to force times time and mv.
What this proof shows is that the erroneous definition of energy separates energy addition from the transformation. Energy addition becomes a mathematical abstraction which changes with reference points and does not maintain a consistent relationship to the transformation which is supposed to be the source of the energy.
The corrected definition of energy maintains a consistent relationship to the transformation, because the transformation produces a consistent and definable amount of force times time. The transformation does not produce a consistent amount of force times distance.
The result of this mathematical proof is totally predictable, because force times time (Ft) does not equal ½mv² for an accelerating mass, and Ft is the only product of a rocket. The rocket produces a constant force, which means unchanging through time, while it has no relation to distance. When a rocket produces a definable amount of force times time, in proportion to fuel use, but does not produce a definable amount of force time distance, which has no relation to fuel use, this shows that energy is transformed in proportion to force times time, not force times distance.
In other words, if a rocket burns for one second, it will do the same thing at one thousand miles per hour as it will at one mile per hour. In fact, the speed is relative to reference frames. But the amount of ½mv² is different in each case.
***** Notice this important conclusion: *****
Physicists have been saying that the rocket equations balance, so there is no problem with the definition of energy. I proved that the definition of energy is incorrect—that means incorrect despite physicists balancing a nonsensical equation.
About ninety percent of physics is based on the faulty premise that if equations balance, it's a law of the universe. String theory is where this standard was highly visible. For several years, physicists discussed equations for string theory; and eventually, they said they had the perfect equations which balance; and therefore, string theory is a law of the universe.
They can no longer claim that balancing equations creates a law of the universe, because the misdefinition of energy shows otherwise. It's possible to balance nonsensical equations. So physicists can throw out about ninety percent of physics which has no other basis that the supposed balancing of equations. There has to be more objective evidence than balancing equations to represent laws of the universe.