  Building on the recent derivation of a bare factorization theorem for the
$b$-quark induced contribution to the $h\to\gamma\gamma$ decay amplitude based
on soft-collinear effective theory, we derive the first renormalized
factorization theorem for a process described at subleading power in scale
ratios, where $\lambda=m_b/M_h\ll 1$ in our case. We prove two refactorization
conditions for a matching coefficient and an operator matrix element in the
endpoint region, where they exhibit singularities giving rise to divergent
convolution integrals. The refactorization conditions ensure that the
dependence of the decay amplitude on the rapidity regulator, which regularizes
the endpoint singularities, cancels out to all orders of perturbation theory.
We establish the renormalized form of the factorization formula, proving that
extra contributions arising from the fact that "endpoint regularization" does
not commute with renormalization can be absorbed, to all orders, by a
redefinition of one of the matching coefficients. We derive the
renormalization-group evolution equation satisfied by all quantities in the
factorization formula and use them to predict the large logarithms of order
$\alpha{\hspace{0.3mm}}\alpha_s^2{\hspace{0.3mm}} L^k$ in the three-loop decay
amplitude, where $L=\ln(-M_h^2/m_b^2)$ and $k=6,5,4,3$. We find perfect
agreement with existing numerical results for the amplitude and analytical
results for the three-loop contributions involving a massless quark loop. On
the other hand, we disagree with the results of previous attempts to predict
the series of subleading logarithms
$\sim\alpha{\hspace{0.3mm}}\alpha_s^n{\hspace{0.3mm}} L^{2n+1}$.
