What is happening to rotational kinetic energy when moment of inertia is changed?

I know this question is asked here a lot, but I just had to ask this to finalise the concept.

When a system lets say a rod of length $L$ and mass $M$ is rotating with angular speed $omega_1$ its initial angular momentum is $L1 = (1/12)ML^2\omega_1$ and its initial kinetic energy is $KE = (1/24)ML^2{\omega_1}^2$.

Now after some time the rod is folded in half its angular momentum kept conserved i.e. without applying any external force or torque, its new angular velocity becomes $\omega_2 = 4\omega_1$ and its new kinetic energy becomes $KE_2 = (1/6)ML^2{\omega_1}^2$.

This is 4 times the original kinetic energy when no external force works, since the rod is folded, you can even say melted and formed into a smaller and denser rod, it has not undergone compression/expansion of any sort but still there is change in kinetic energy.

The most sense I could make out of this was that all the particles while rotating felt a centripetal force and the particles of half of the rod under this force went in its direction and did some work which appears as the change in kinetic energy. I have written the same and a proof as an answer here.

Now if I am write in my concept where did this energy come from, tension was providing the centripetal force but no work was done against tension as the rod was folded in half not compressed. If I am wrong then where did the energy come from?

Extra: I also tried the analogy of this question in translatory motion, suppose there is body of mass $m$ moving with velocity $v$ suddenly, its mass becomes $m/2$ then its velocity becomes $2v$ and its KE becomes $4KE_1$ there is no need to explain the energy conservation here since mass suddenly does not disappear into thin air, however moment of inertia can be changed and hence the question.

Addendum : Since folding the rod seems to bring about unnecessary questions about ways of folding, you can imagine that if rod was melted and formed into a longer rod all the while the system was rotating and angular momentum conserved, then new length becomes $2L$ new angular velocity becomes ${\omega_1}/4$ and new KE becomes $(1/96)ML^2{\omega_1}^2$. This time the energy becomes ${1/4}^{th}$ of the initial, where did this energy go to ? Certainly movement against centripetal force takes place, but since there is no extension in existing rod, energy can not be stored as spring energy in it, or so I think.