  We describe a general method for giving $p$-adic interpretations of
$G$-functions arising from degenerating periods of smooth projective algebraic
varieties. Using this, we are able to implement a strategy due to Andr\'e for
bounding heights of moduli points where period functions acquire unusual
algebraic relations. This leads to new results on Galois lower bounds for
special moduli, and new cases of the Zilber-Pink conjecture. In particular, we
establish the first Galois-orbit lower bounds on CM moduli in non-Shimura
settings.
  As a more technical contribution, we introduce a refinement of the
Pila-Zannier strategy capable of handling Zilber-Pink-type atypical
intersection problems in arbitrary dimension and for arbitrary smooth
projective families.
