Binomial expansion of (u+2v)^{2^n} modulo 2ⁿ⁺²

Let $A$ be a commutative ring, let $n$ be a natural number with $1 \le n$, and let $u, v \in A$. The assertion is that there exists $w \in A$ with

where the exponents $2^n-1$ and $2^n-2$ are truncated natural-number differences (harmless, since $n \ge 1$ gives $2^n \ge 2$) and the integers $2$, $2^{n+1}$, $2^{n+2}$ act through the canonical ring map $\mathbb{Z} \to A$. Equivalently: modulo $2^{n+2}A$ the $2^n$-th power of $u + 2v$ agrees with $u^{2^n}$ plus the two displayed terms of weight $2^{n+1}$, one linear and one quadratic in $v$. No hypothesis beyond commutativity of $A$ and $n \ge 1$ is imposed; in particular $A$ need not be of characteristic $2$, nor $2$-adically complete, nor free of $2$-torsion.

This is the exceptional case $p = 2$, $r = 1$ of the elementary binomial estimate for $p$-th power maps: for odd $p$ all terms past the linear one are divisible by $p^{n+2}$, whereas at $p=2$ the quadratic binomial coefficient $\binom{2^n}{2}(2v)^2 = 2^{n+1}(2^n-1)u^{2^n-2}v^2$ is divisible only by $2^{n+1}$ and survives modulo $2^{n+2}$. It is used in the local deformation-theoretic computations, by Deformation.PLoc.wPartialSum_adicEval_add_sub_sub_algebraMap_mul_add_mem_powSub_two, where the surviving square term forces the normal form of the $w$-series at $p = 2$.