How Much Unresolved Branching Can a Quantum System Sustain? A Composition-Dependent Mass Threshold for Collapse

We propose that a quantum system can hold its mass in several places at once only up to a fixedlimit, at which the superposition collapses to one branch. The limit applies to the informationeach atom’s position carries about the branch, summed over the system’s atoms and multiplied bythe displaced mass in Planck units. Its value is one nat. The surviving branch is chosen with itsBorn probability, and the average over runs obeys a linear Lindblad equation. Whether the collapseoccurs is fixed by local physics, which no distant operation can change, so it cannot be used tosend a signal. The model has no continuously tunable coupling. For a solid body split into twowell-separated branches, the limit gives a critical mass set by the geometric mean of the atomic andPlanck masses. For silicon it is 3.8 × 10−17 kg, a sphere about 315 nm across. Below this mass themodel is exactly unitary quantum mechanics with decoherence. Its collapse rate there is zero, so theheating and emission bounds on other collapse models do not apply. Within its stated scope, it isindistinguishable from standard quantum mechanics for every quantum system tested so far. Nearthe critical mass it makes five predictions that separate it from continuous-rate collapse models.The most distinctive is a dependence on composition: the critical mass scales as the square root ofthe atomic mass and is 2.7 times larger for gold than for aluminium. Levitated nanospheres, aboutan order of magnitude below the critical mass, are the natural test platform.