  Our understanding of quantum correlators in cosmological spacetimes,
including those that we can observe in cosmological surveys, has improved
qualitatively in the past few years. Now we know many constraints that these
objects must satisfy as consequences of general physical principles, such as
symmetries, unitarity and locality. Using this new understanding, we derive the
most general scalar four-point correlator, i.e., the trispectrum, to all orders
in derivatives for manifestly local contact interactions. To obtain this result
we use techniques from commutative algebra to write down all possible scalar
four-particle amplitudes without assuming invariance under Lorentz boosts. We
then input these amplitudes into a contact reconstruction formula that
generates a contact cosmological correlator in de Sitter spacetime from a
contact scalar or graviton amplitude. We also show how the same procedure can
be used to derive higher-point contact cosmological correlators. Our results
further extend the reach of the boostless cosmological bootstrap and build a
new connection between flat and curved spacetime physics.
