[loungitude]

The immeasurable part is acceptable.
The theory part is that no rotational kinetic energy is lost in the ‘everybody jumps up at the same time’ scenario.

[FredZarguna]

Conservation of rotational kinetic energy -- which I agree doesn't change -- requires the angular velocity of the Earth to decrease when the jumpers jump.

The moment of inertia of the system composed of the Earth and jumping people will change. In order to conserve angular momentum -- which must be conserved because there is no external torque -- the angular velocity, ω, must decrease. [In your alternative formulation, there are no external sources of energy, therefore the angular velocity must decrease when I increases in order for the kinetic energy to remain constant: I_iω_i² = I_fω_f²]

The differential dI would (at most, assuming the people ball themselves up into spheres when they jump) would be 2m * r * dr, where r is the radius of the earth, dr is the number of meters they jump, and m is the mass of six billion people. The number is insignificant compared to the moment of inertia of the Earth, which is roughly 2/5 M r², where M is the mass of the Earth. And it is clearly this which dominates.

[Wingy]

The theory part is that no rotational kinetic energy is lost in the ‘everybody jumps up at the same time’ scenario.

Not true! I have it on Congressional integrity (whatever that is..) that Guam will flip over and sink.