Black Hole Zeckendorf Games

Let $F_n$ denote the $n$\textsuperscript{th} Fibonacci number (with first few terms 1, 2, 3, 5). The Zeckendorf game starts with $N$ chips on pile $F_1$, and on a turn a player may do one of two moves: if there are chips on piles $F_k$ and $F_{k-1}$ they may remove those and place one on $F_{k+1}$, or if there are two chips on $F_k$ they may remove those and put one on $F_{k+1}$ and one on $F_{k-2}$ (unless $k \le 2$: if there are two chips on $F_2$ we replace that with one each on $F_3$ and $F_1$, while if there are two on $F_1$ we replace with one on $F_2$). Note at any moment in the game if $c_n$ is the number of chips on pile $F_n$ then $\sum c_n F_n = N$. Two players alternate turns until no moves remain. It was quickly proved that if $N > 2$ then Player 2 has a winning strategy and the game ends in the Zeckendorf decomposition of $N$ (i.e., the representation of $N$ as a sum of non-adjacent Fibonacci numbers); unfortunately it's a non-constructive proof. Subsequent papers proved many results about the game and generalizations, but the winning strategy remains a mystery. We thus consider a new variant: the Black Hole Zeckendorf game, where there is a fixed $m$ such that if a chip is placed on pile $F_m$ it becomes inaccessible (we also allow the two players to add chips to set up the initial board, so we do not have to start with $N$ chips on $F_1$). We constructively find winning strategies for some $m$, where the winner depends on $N \bmod 16$.