This is a follow-up my previous question. Several of the answers indicated that the mass of a particle (atom, proton, etc.) increase with the "binding energy" of it's component particles - the energy needed to split them apart - which doesn't make sense to me. I'm convinced that, at least in an exothermic chemical reaction (where the product bond energies are larger) the product's particles will lose mass (proportionally to the heat dissipated) or at least have no change.
To use a larger-scale analogy, if an object, a "particle", is 100m above the Earth's surface, it has potential energy from gravity. As it falls, this energy is lost, converted into KE. Overall, the two "particles", the object and the Earth, end up with the total energy, and therefore the same total mass. There is no room for a "binding energy" that adds mass.
My reasoning is that this extends to particles, with electrostatic or nuclear forces taking the place of gravity. Potential energy of component particles becomes KE of the bonded particle, they end up with the same mass. Actually, if this KE is dissipated (as in a burning / nuclear fusion reaction) the particles should actually have more mass in their uncombined/unreacted state, thanks to their PE. Surely it isn't possible for the mass to increase without an external input of energy?
However, answerers of my energy in chemical reactions question said that:
the energy involved in the bonds is ... half of what we normally consider the "mass" of the proton - David Zaslavsky
potential energy of the chemical bonds do correspond to an increase of mass - Ben Hocking
So, how can this be, and where is my reasoning incorrect? What exactly is the binding energy (if not just the energy needed to break the bond), and where does it come from?