Creating a particle pair requires enough energy to cover the rest-mass cost, so the reaction fails if the supplied energy falls below the threshold. Two photons at 100 J each supply 200 J, exactly meeting the pair-at-rest rest-energy cost 200 J, while a 150 J supply falls short at 150 < 200 for the stated toy energy ledger; real measurements carry beam-energy spread, detector resolution, background, and threshold uncertainty.
A threshold gate checks energy budget before full acceptance. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
Enough energy passes the threshold gate
Two opposite photons can supply 100 J plus 100 J, matching the pair-at-rest rest-energy cost 200 J.
100+100=200≥200
Too little energy fails the threshold gate
If the two-photon setup supplies only 150 J, the energy budget is below required energy 200 J, before momentum closure is even tested.