The EPRL model is the leading spin foam model for quantum gravity — it assigns transition amplitudes to discrete spacetime geometries built from 4-simplices. The semiclassical limit of these amplitudes was expected to be dominated by flat connections, the discrete analog of flat spacetime. But proving this without invoking the semiclassical approximation has been an open problem: the argument relied on stationary phase analysis, which only applies when the spin labels are large.

The result is exact: the EPRL amplitude is supported on flat connections without any semiclassical assumption. The proof works at finite spin, not just in the large-spin limit. The amplitude vanishes identically on configurations that do not correspond to flat discrete geometries. This is not an approximation or a leading-order statement — it is a property of the full quantum amplitude.

The mechanism is algebraic. The EPRL construction imposes simplicity constraints that reduce the topological BF theory (which knows about all connections) to a theory that knows only about geometry. These constraints, implemented through the Barbero-Immirzi parameter and the embedding of SU(2) representations into SL(2,C), automatically project onto flat connections. The flatness is built into the constraint structure, not selected by the dynamics.

This means the EPRL model at finite spin already knows about flatness — the semiclassical analysis was confirming a property that holds exactly. The model doesn’t approach flat geometry in a limit; it lives on flat geometry at every scale.

The approximation was correct. The surprise is that it wasn’t an approximation.