Is the Binomial Tree Model not self-financing?

Consider a 2-period binomial tree where the derivative price is $f$ and the stock price is $S$. Also, let the bond be deterministic with continuous growth rate $r$ and initial value $B_0$. binomial tree

Recall the replicating strategy is at each time $t_i$ hold $\phi_i = \frac{f_{i+1}^{up} - f_{i+1}^{down}}{S_{i+1}^{up} - S_{i+1}^{down}}$ units of the stock and $\psi_i = B_0^{-1} e^{-r(i+1)\Delta t}(f_{i+1}^{up} - \phi_i S_{i+1}^{up})$ units of the bond. In particular, the value of the portfolio at time $0$ is $V_0 = \phi_0 S_0 + \psi_0 B_0$. When we arrive at time tick 1, lets say our stock price went up to $S_3$. Before rebalancing, our portfolio is worth $V_0|_{end} = \phi_0 S_3 + \psi_0 B_0e^{r \Delta t}$, and after rebalancing it is $V_1 = \phi_1 S_3 + \psi_1 B_0e^{r \Delta t}$. In order for this to be self-financing, we must have $V_1 - V_0|_{end} = 0$. However, \begin{align*} V_1 - V_0|_{end} & = (\phi_1 - \phi_0)S_3 + (\psi_1 - \psi_0)B_0e^{r \Delta t} \\ & = (\phi_1 - \phi_0)S_3 + \left(B_0^{-1} e^{-2r\Delta t}(f_7 - \phi_1 S_{7}) - B_0^{-1} e^{-r\Delta t}(f_{3} - \phi_0 S_{3})\right)B_0e^{r \Delta t} \\ & = (\phi_1 - \phi_0)S_3 + e^{-r\Delta t}(f_7 - \phi_1 S_{7}) - (f_{3} - \phi_0 S_{3}) \\ & = \phi_1 S_3 + e^{-r\Delta t}(f_7 - \phi_1 S_{7}) - f_{3} \\ & = \frac{f_{7} - f_{6}}{S_7 - S_6} S_3 + e^{-r\Delta t}(f_7 - \frac{f_{7} - f_{6}}{S_7 - S_6} S_{7}) - f_{3} \\ & = \frac{1}{S_7 - S_6} \left((f_{7} - f_{6})S_3 + (S_7 - S_6)e^{-r\Delta t}f_7 - e^{-r\Delta t}(f_{7} - f_{6}) S_{7} - (S_7 - S_6)f_{3} \right)\\ & = \frac{1}{S_7 - S_6} \left((f_{7} - f_{6})S_3 - S_6e^{-r\Delta t}f_7 + e^{-r\Delta t}f_{6} S_{7} - (S_7 - S_6)f_{3} \right)\\ & \neq 0. \end{align*}

It seems a lot of effort is put into self-financing strategies, and in fact the binomial representation theorem is used to prove the existence of them in the binomial model. Am I missing something?

I'd add a comment if I could but don't have enough reputation. How do you know your final equation is not equal to zero. The $f_i$ have not yet been calculated in terms of $S$. Certainly the $f_i$ that are in the final set of nodes are known since they are defined in terms of the payoff and the terminal price. The ones in intermediate steps have to be calculated.