A quantum system on a lattice involves two abstractions that must eventually be removed: the lattice spacing (an approximation to continuous space) and Planck's constant (the boundary between quantum and classical behavior). The continuum limit sends the lattice spacing to zero. The semiclassical limit sends Planck's constant to zero. These are different limits, taken for different reasons, and in general they do not commute — taking one first and the other second can give a different answer than the reverse order.
Keller, Pettinari, and van de Ven (2026) show that a single parameter N can govern both limits simultaneously. By setting the semiclassical parameter as a specific power of N — the same N that controls the mesh refinement — the two limits become coupled. The eigenvalues of the discrete quantum operator converge to those of the continuous classical system as N goes to infinity. Both abstractions are removed by the same dial.
The convergence is not trivial. Individually, each limit strips away a layer of structure: the continuum limit removes the lattice, the semiclassical limit removes the quantum coherence. Taken separately, these removals can leave the system in incompatible intermediate states. Coupled through N, the two simplifications proceed in lockstep, and the intermediate states remain consistent at every value of the shared parameter.
The general principle: when a system involves multiple simplifying limits, the order in which you take them can matter — unless the limits are controlled by a single parameter that forces them to proceed together. Finding that parameter transforms a problem of limit ordering into a problem of parameterization. The unity was always there. What was needed was a dial that turned both knobs at once.