Explicit randomized pointer-parity lower bound

For every pair of natural numbers $k,n$ with $k>0$ and $n\geq2$, every real $\varepsilon$ with $0\leq\varepsilon<1/2$, every $v_1\in\{0,\ldots,n-1\}$, every natural number $c$, and every public-randomness Boolean-output classical protocol $P$ on maps $a,b:\{0,\ldots,n-1\}\to\{0,\ldots,n-1\}$, the following holds under the stated communication and correctness assumptions. The protocol consists of finitely many deterministic binary communication trees with nonnegative seed weights summing to $1$; each internal node sends a Boolean function of the speaker's own input, and each leaf outputs a function of either party's own input. For every seed of positive weight, every root-to-leaf path transmits at most $c$ bits, and its number of consecutive same-sender transmission blocks, plus one if its first sender is Alice, is at most $k$, with zero rounds for a path without transmissions. Define an endpoint by taking exactly $k+1$ steps from $L(v_1)$ under $L(v)\mapsto R(a(v))$, $R(v)\mapsto L(b(v))$, and label $L(v)$ by $v$ and $R(v)$ by $n+v$. Assume that, for every $a,b$, the total seed weight of trees correctly reporting whether the sum of the binary digits of this label is odd is at least $1-\varepsilon$. Then

The case $\varepsilon=0$ is included, zero-weight seeds need not obey the communication bounds, and the displayed inequality imposes no positive lower bound on $c$ when its left-hand side is nonpositive.