User blog comment:Edwin Shade/A Complete Analysis of Taranovsky's Notation/@comment-30118230-20180130182451/@comment-30118230-20180130201408

I'm sorry to be the one to point out a large chunk of your work is false but it's for the better I'm sure. At least it's acurate now!

In general $$C^{k_n}(C(\Omega,0)+\alpha_n,C(.....C^{k_2}(C(\Omega,0)+\alpha_2,C^{k_1}(C(\Omega,0)+\alpha_1,\beta))....))$$ is equal to $$\beta+\varepsilon_0\omega^{\alpha_1}k_1+\varepsilon_0\omega^{\alpha_2}k_2+......+\varepsilon_0\omega^{\alpha_n}k_n$$

While $$C^{k_n}(\Omega+\alpha_n,C(....C^{k_2}(\Omega+\alpha_2,C^{k_1+1}(\Omega+\alpha_1,\beta))....))=\beta+\varepsilon_{\omega^{\alpha_1}k_1+\omega^{\alpha_2}k_2+......+\omega^{\alpha_n}k_n}$$

I hope this will be useful for your analysis for now.