A moving prefix of positive relative length has uniform positive density

Fix a head pattern $P$, real $0<A<C$, and $r_{\min}>0$. There are $c_{\rm prefix}>0$ and $N_0$ such that for every $n\ge N_0$ and integer endpoint $k$ with $\lfloor An\rfloor<k\le\lfloor Cn\rfloor$ and normalized physical prefix fraction at least $r_{\min}$,

Here the prefix fraction is exactly $(k/n-A)/(C-A)$, using the real physical endpoints before rounding. The lower bound is uniform over the moving endpoint.