  We study the Factorization Paradox from the bottom up by adapting methods
from perturbative renormalization. Just as quantum field theories are plagued
with loop divergences that need to be cancelled systematically by introducing
counterterms, gravitational path integrals are plagued by wormhole
contributions that spoil the factorization of the holographic dual. These
wormholes must be cancelled by some stringy effects in a UV complete,
holographic theory of quantum gravity. In a simple model of two-dimensional
topological gravity, we outline a gravitational analog of the recursive BPHZ
procedure in order to systematically introduce ``counter-wormholes" which
parametrize the unknown stringy effects that lead to factorization. Underlying
this procedure is a Hopf algebra of symmetries which is analogous to the
Connes--Kreimer Hopf algebra underlying perturbative renormalization. The group
dual to this Hopf algebra acts to reorganize contributions from spacetimes with
distinct topology, and can be seen as a gauge group relating various equivalent
ways of constructing a factorizing gravitational path integral.
