Let $T(x)$ be a partial function $:\mathcal P(\Bbb Z^+)\to\Bbb Z_2$.
I have the rule for a certain $\underline x\in \mathcal P(\Bbb Z^+)$ that at least every fourth digit of $T(\underline x)$ is a $1$ (for the avoidance of doubt, there is no infinitely repeating sequence of zeroes and no sequence of more than three zeroes in a row).
Then for every other $x$ on which the function is defined, $T(x)$ has more zeroes than $\underline x$.
How do I write this condition as a rule or constraint in $\Bbb Z_2$?
I want to write something like $T(x)\leq-\frac1{15}$ for all $x\in\mathcal P(\Bbb Z^+)$. This usage of $\leq$ is derived from the natural ordering of the subset of $\Bbb Q$ that naturally embeds in $\Bbb Z_2$. Have I translated the rule into a bound correctly, and are there alternative or better ways of representing this bound?
I'm happy to accept a bound or rule that's valid only on the subset of $\Bbb Q$ that embeds in $\Bbb Z_2$
Background FWIW: This relates to a lower bound on the path the Collatz graph takes through the positive integers.
The main point is that $y\in \Bbb{Z}_2$ has a 2-adic expansion $$y=\sum_{n=0}^N a_n 2^n, a_n \in \{0,1\},N \in \Bbb{Z}_{\ge 0} \cup \infty$$ with finitely many zeros iff $y \in \Bbb{Z}$.
$\Bbb{Q} \cap \Bbb{Z}_2 = \{ \frac{u}v, u,v\in \Bbb{Z}, 2 \nmid v\}$
Having at least 3 zeros is restrictive only on $\Bbb{Z} \subset \Bbb{Z}_2 $ but it doesn't translate to a $\ge$ condition.
And the condition " no sequence of more than three zeroes in a row" is exactly what doesn't have any easy meaning in $\Bbb{Z}_2$.