A one-third defect bound at a first coefficient contraction

Let $T:\mathbb N\to\mathbb N$ be the shortcut Collatz map: $T(n)=n/2$ for even $n$ and $T(n)=(3n+1)/2$ for odd $n$. Write $T^k$ for its $k$-fold iterate, with $T^0(n)=n$. Let $q_k(n)$ count the odd inputs among the first $k$ shortcut steps, and let $R_0(n)=0$; recursively $R_{k+1}(n)=R_k(n)$ for an even input $T^k(n)$ and $R_{k+1}(n)=3R_k(n)+2^k$ for an odd input. For $s\in\mathbb N$, define $M(0)=0$, $M(s+1)=3M(s)+2^{\lfloor\log_2(3^s)\rfloor}$, and $e(s)=\lfloor\log_2(3^s)\rfloor+1$. A first coefficient contraction at time $k$ means $k>0$, $3^{q_k(n)}<2^k$, and $2^j\le3^{q_j(n)}$ for every $j<k$. Its existence is a hypothesis.

Let $n,k,d\in\mathbb N$, with $n>0$, and assume a first coefficient contraction at $k$ and $T^k(n)=n+d$. Then

The conclusion applies to an existing first crossing whose endpoint has not fallen below its start. It does not prove existence of a first crossing.