  We use Galois cohomology methods to produce optimal mod $p^d$ level lowering
congruences to a $p$-adic Galois representation that we construct as a well
chosen lift of a given residual mod $p$ representation. Using our explicit
Galois cohomology methods, we construct for a reductive group $G$ and a given
residual representation $\bar{\rho}: \Gamma_F \to G(k)$, ramified at a finite
set of primes $S$, in favorable conditions that we identify, a finite set of
lifts $\rho$, $\{\rho^q\}$ of $\bar{\rho}$ to $G(W(k))$ with the following
properties: $\rho: \Gamma_F \to G(W(k))$ is ramified precisely at $S \cup Q$,
with $Q$ a finite set of primes disjoint from $S$. For $q \in Q$, $\rho^q:G_F
\to G(W(k))$ is unramified outside $S \cup Q \backslash \{q\}$ and $\rho$ and
$\rho^q$ are congruent mod $p^d$ if $\rho$ mod $p^d$ is unramified at $q$.
Furthermore, the Galois representations $\{\rho^q\}$ are "independent".
