only to verify condition (1), by the fact that $$\sum_{r=0}^{K}\#(Sp\phi_{x_0}\cap V_r^i)=\sum_{i=1}^{l}\sum_{r=0}^{K}c_r^il_i \geq\frac{L}{L+1}\omega_i,$$ $${\rm rank}\,P_j\geq k_jt_j+k_j\sum_{i=1}^{l}\alpha_{ij}c_0^i+k_j\sum_{i=1}^{l}\beta_{ij}c_K^i,$$ where $\omega_i=\#(Sp(\phi_{x_0})\cap (0,1)_i)$. Then we have $$\begin{aligned}
        \sum_{j=1}^{p}rankP_j+\sum_{i=1}^{l}\sum_{r=1}^{K-1}rank P_r^i &\geq& \sum_{j=1}^{p}\big(k_jt_j+k_j\sum_{i=1}^{l}\alpha_{ij}c_0^i+k_j\sum_{i=1}^{l}\beta_{ij}c_K^i\big)\\
     & & + \sum_{i=1}^{l}\sum_{r=1}^{K-1}rankP_r^i \\
     &=& \sum_{j=1}^{p}k_jt_j+\sum_{i=1}^{l}c_0^il_i+\sum_{i=1}^{l}c_K^il_i+\sum_{i=1}^{l}\sum_{r=1}^{K-1}c_r^il_i \\
     &\geq& \sum_{j=1}^{p}k_jt_j+\sum_{i=1}^{l}\frac{L}{L+1}\omega_il_i.
   \end{aligned}$$ It is obvious that $$L\cdot{\rm rank}\,\,q<{\rm rank}\,\,(1-q).$$ Moreover, for each projection $e\in A$, we will have $$L\cdot{\rm rank}\,\,(\phi(e)-\psi(e))<{\rm rank}\,\,\psi(e).$$ Hence, (1) holds.

This theorem has some similarity to what was proved in [@EG] and [@EGS], the authors use small spectra variation to find finitely many projections such that (1), (2), (3) hold. But in general, the spectra may not have a regular form and the projections will not belong to $B$, which makes that the small spectra variation may not be sufficient.

If we assume each Elliott-Thomsen building block has the property that there exist an upper bound for all $p$, then we can take $\eta>0$ small enough and choose a suitable decomposition such that for any projection $e\in A$, we have $$\,L\cdot [\phi(e)-\psi(e)]\leq[\psi(e)].$$

Let $A=A(F_1,F_2,\varphi_0,\varphi_1),B=B(F_1',F_2',\varphi_0',\varphi_1')\in \mathcal{C}$, let $G\subset A$ be a finite set, then for any positive integers $J,L$, there exists $\eta>0$ such that if a homomorphism $\phi: A\rightarrow B$ satisfies $\phi(H(\eta))\subset_{\frac{1}{6}}\{ f\in B\, |f\,has\,finite\,spectrum\}$, then there exists a projection $q\in B$ 