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But also, because some of the bonds between atoms/molecules were broken, some of the energy was lost. And energy is mass, so you can lose mass even if all the particles are still on either piece.
Two completely different things, on a molecular level that could just be stored potential energy. Like when a mousetrap is set or not, it doesn't change it's weight.
That's how splitting atoms make atomic bombs there's potential energy holding shit together, and we poke one to make it release which cause a giant chain reaction which adds up. Even then the "lost" mass is just blown up. I don't think used my lease fuel gets substantially lighter for example.
Not bonds within atoms, bonds between atoms. Or molecules. That depends on the exact material being cut.
And yeah, on a large scale it gets exploded, but on a scale of cutting paper with scissors it mostly gets lost as heat.