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The concepts of Lorentz invariance of local (flat space) physics, andunitarity of time evolution and the S-matrix, are famously rigid and robust,admitting no obvious consistent theoretical deformations, and confirmed toincredible accuracy by experiments. But neither of these notions seem to appeardirectly in describing the spatial correlation functions at future infinitycharacterizing the "boundary" observables in cosmology. How then can we seethem emerge as {\it exact} concepts from a possible ab-initio theory for thelate-time wavefunction of the universe? In this letter we examine this questionin a simple but concrete setting, for the perturbative wavefunction in a classof scalar field models where an ab-initio description of the wavefunction hasbeen given by "cosmological polytopes". Singularities of the wavefunction areassociated with facets of the polytope. One of the singularities --corresponding to the "total energy pole" -- is well known to be associated withthe flat-space scattering amplitude. We show how the combinatorics and geometryof this {\it scattering facet} of the cosmological polytope straightforwardlyleads to the emergence of Lorentz invariance and unitarity for the S-matrix.Unitarity follows from the way boundaries of the scattering facet factorizeinto products of lower-dimensional polytopes, while Lorentz invariance followsfrom a contour integral representation of the canonical form, which exists forany polytope, specialized to cosmological polytopes.

On the Emergence of Lorentz Invariance and Unitarity from the Scattering Facet of Cosmological Polytopes