Restricted absorption at every 2-adic depth L ≥ 5: explicit 2n+4 batch for (a)·A·(b)

A tree witness is a function $\theta(k,r)\in\{0,1\}$ on the nodes $(k,r)$, $0\le r<2^k$, of the binary tree of residues. For distinct integers $x,y$ with $k=\nu_2(y-x)$, put $x\prec_\theta y$ iff $\mathrm{bit}_k(x)=\theta(k,\,x\bmod 2^k)$. A finite word $Q$ of integers is $\theta$-safe if the sequence "the letters of $Q$ in order, then all remaining integers in $\prec_\theta$-order" contains no monotone $4$-term arithmetic progression. $Q$ is SAFE if it is $\theta$-safe for some tree witness $\theta$. For a word $A=(x_1,\dots,x_n)$ and an affine map $g$, $g(A)=(g(x_1),\dots,g(x_n))$ keeps the original order, and $\cdot$ denotes concatenation.

Theorem (restricted absorption at every depth $L\ge5$). Let $L\ge 5$, let $r$ be odd, and let $A$ be a SAFE word all of whose letters are $\equiv r\pmod{2^L}$. Let $a$ be even, $b\equiv r+2\pmod 4$, and let $y\neq a$ satisfy

Then the word

is SAFE.

Equivalently, put $c=2y-b$. If all letters of $A$ lie in the sibling ball of $c$ at some common $2$-adic depth $\nu=L-1\ge4$, then the prefix $(a)\cdot A\cdot(b)$ absorbs the target $y$ with the explicit batch of $2|A|+4$ letters above. There is no size condition on $y$. For a fixed prefix whose block $A$ is $\equiv r\pmod{2^{L_0}}$, this covers every target with $4\le\nu_2(2y-b-r)\le L_0-1$. The case $L=5$ is Erdos195.restricted_absorption (GPT Astra's mod-32 corollary). This is the general-$L$ form of Astra's corollary. It is a positive absorption result for one infinite class of words in the tree-witness approach to Erdős Problem 195. It does not settle the problem.