  We show how to use Exceptional Field Theory to efficiently compute $n$-point
couplings of all Kaluza-Klein modes for vacua that can be uplifted from maximal
gauged supergravities to 10/11 dimensions via a consistent truncation. Via the
AdS/CFT correspondence, these couplings encode the $n$-point functions of
holographic conformal fields theories. Our methods show that these $n$-point
couplings are controlled by the $n$-point invariant of scalar harmonics of the
maximally symmetric point of the truncation, allowing us to show that
infinitely-many $n$-point couplings vanish for any vacua of the truncation,
even though they may be allowed by the remnant symmetry group of the vacua.
This gives new results even for the maximally supersymmetric AdS$_5 \times
S^5$, AdS$_4 \times S^7$ and AdS$_7 \times S^4$ vacua of string and M-theory,
where we prove old conjectures about the vanishing of $n$-point extremal and
near-extremal couplings.
  Focusing in particular on cubic couplings for vacua of 5-dimensional gauged
supergravity, we derive explicit universal formulae encoding these couplings
for any vacuum within a consistent truncation. We use this to compute known and
new couplings involving spin-0, spin-1, spin-2 for the AdS$_5 \times S^5$
vacuum of IIB string theory.
