FALSE: every PP algorithm has bounded error separated from one half by a fixed constant

Let $A:=\{x\in \{0,1{\}}^{\ast }:x\text{ begins with }1\}$. Its deterministic membership test $D(x)$ is computable in linear time. Consider the probabilistic machine that uses one fair bit $b$ and an independent string $r\in \{0,1{\}}^{n+1}$, accepts when $D(x)=1$ and either $b=1$ or $r=0^{n+1}$, and accepts when $D(x)=0$ exactly if $b=1$ and $r\mathrm{\ne }{0}^{n+1}$. Then yes-instances are accepted with probability $1\mathrm{/}2+2^{-(n+2)}$, while no-instances are accepted with probability $1\mathrm{/}2-2^{-(n+2)}$.

Step 1.1 still satisfies the PP threshold condition [L1], but the gap from $1\mathrm{/}2$ shrinks with $n$ and is not bounded below by any fixed positive constant. So the statement is false.