Is thermodynamic free energy and a solid's elastic strain energy more than a shared name?
free energy ⇄ incremental Hooke's law · via elastic strain energy
The system kept surfacing a link between the free energy of statistical thermodynamics (the log of a partition function) and the strain energy that governs a solid's stress response under Hooke's law. Both are called an energy potential whose gradient drives the system, but whether they are the same mathematical object or just two things wearing the word energy is exactly what it could not settle.
The open question
Is there a real, testable bridge here (the way thermoelasticity couples the two), or is free energy simply a homonym across the two fields? A concrete prediction that ties a measurable thermodynamic quantity to a mechanical one would settle it.
What the system already tried
It read full-text papers on partition functions, Legendre transforms, and incremental elasticity, and added the grounded facts to its memory. Even with those facts in hand, the cross-model jury would not confirm a non-trivial, testable connection. So it stayed open, not rejected.
The sources it read
Open review
Is this a real connection or a coincidence of shared words? The facts above are grounded in the sources; the leap between them is what is unproven. Make the case, or settle it with a reference.