Googology Wiki - 用户贡献 [zh-cn]

FZ Hydra

2025-07-30T13:22:05Z

YukinaKNK：(0)(1,1,1,1,1,1)(1,1,1,1,1)(1,1,1,1)(1,1,1)(1,1)(1)

注：该记号定义程度为7.4，原因是KNK不会写fake系列的规则（

主规则：

<math>\psi_Z(0)=1</math>

<math>\psi_Z[\&](\#+1)=\psi_Z[\&](\#)+\psi_Z[\&](\#)+\psi_Z[\&](\#)+...</math>

兼容规则（只展示KNK会写的，后续会陆续更新出完整的规则）：

<math>\psi_{Z_n}(\psi_{Z_{n+1}}(0))=\psi_{Z_n}[\psi_{Z_{n+1}}(0)](\psi_{Z_{n+1}}[\psi_{Z_{n+2}}(0)](\psi_{Z_{n+2}}[\psi_{Z_{n+3}}(0)](...)))</math>

<math>\psi_{Z_n}[\psi_{Z_{n+1}}(0)](\psi_{Z_{n+1}}(0))=\psi_{Z_n}(\psi_{Z_n}(\psi_{Z_n}(...)))</math>

分析过程：

注：分析使用的OCF如果没有明确标记是B还是M的话，默认的是MOCF

<math>\psi_Z(0)=1</math>

<math>\psi_Z(0)+\psi_Z(0)=2</math>

<math>\psi_Z(0)+\psi_Z(0)+\psi_Z(0)=3</math>

<math>\psi_Z(1)=\psi_Z(0)+\psi_Z(0)+\psi_Z(0)+...=\omega</math>

<math>\psi_Z(1)+\psi_Z(0)=\omega+1</math>

<math>\psi_Z(1)+\psi_Z(0)+\psi_Z(0)=\omega+2</math>

<math>\psi_Z(1)+\psi_Z(0)+\psi_Z(0)+\psi_Z(0)=\omega+3</math>

<math>\psi_Z(1)+\psi_Z(1)=\omega\times2</math>

<math>\psi_Z(1)+\psi_Z(1)+\psi_Z(0)=\omega\times2+1</math>

<math>\psi_Z(1)+\psi_Z(1)+\psi_Z(1)=\omega\times3</math>

<math>\psi_Z(1)+\psi_Z(1)+\psi_Z(1)+\psi_Z(1)=\omega\times4</math>

<math>\psi_Z(2)=\psi_Z(1)+\psi_Z(1)+\psi_Z(1)+...=\omega^2</math>

<math>\psi_Z(2)+\psi_Z(0)=\omega^2+1</math>

<math>\psi_Z(2)+\psi_Z(1)=\omega^2+\omega</math>

<math>\psi_Z(2)+\psi_Z(2)=\omega^2\times2</math>

<math>\psi_Z(2)+\psi_Z(2)+\psi_Z(2)=\omega^2\times3</math>

<math>\psi_Z(3)=\psi_Z(2)+\psi_Z(2)+\psi_Z(2)+...=\omega^3</math>

<math>\psi_Z(4)=\psi_Z(3)+\psi_Z(3)+\psi_Z(3)+...=\omega^4</math>

<math>\psi_Z(\omega)=\psi_Z(\psi_Z(1))=\omega^\omega</math>

<math>\psi_Z(\omega)+\psi_Z(0)=\psi_Z(\psi_Z(1))+\psi_Z(0)=\omega^\omega+1</math>

<math>\psi_Z(\omega)+\psi_Z(1)=\psi_Z(\psi_Z(1))+\psi_Z(1)=\omega^\omega+\omega</math>

<math>\psi_Z(\omega)+\psi_Z(2)=\psi_Z(\psi_Z(1))+\psi_Z(2)=\omega^\omega+\omega^2</math>

<math>\psi_Z(\omega)+\psi_Z(\omega)=\psi_Z(\psi_Z(1))+\psi_Z(\psi_Z(1))=\omega^\omega\times2</math>

<math>\psi_Z(\omega)+\psi_Z(\omega)+\psi_Z(\omega)=\psi_Z(\psi_Z(1))+\psi_Z(\psi_Z(1))+\psi_Z(\psi_Z(1))=\omega^\omega\times3</math>

<math>\psi_Z(\omega+1)=\psi_Z(\psi_Z(1)+\psi_Z(0))=\omega^{\omega+1}</math>

<math>\psi_Z(\omega+2)=\psi_Z(\psi_Z(1)+\psi_Z(0)+\psi_Z(0))=\omega^{\omega+2}</math>

<math>\psi_Z(\omega+3)=\psi_Z(\psi_Z(1)+\psi_Z(0)+\psi_Z(0)+\psi_Z(0))=\omega^{\omega+3}</math>

<math>\psi_Z(\psi_Z(1)+\psi_Z(1))=\omega^{\omega\times2}</math>

<math>\psi_Z(\psi_Z(1)+\psi_Z(1)+\psi_Z(1))=\omega^{\omega\times3}</math>

<math>\psi_Z(\psi_Z(2))=\omega^{\omega^2}</math>

<math>\psi_Z(\psi_Z(3))=\omega^{\omega^3}</math>

<math>\psi_Z(\psi_Z(\psi_Z(1)))=\omega^{\omega^\omega}</math>

<math>\psi_Z(\psi_Z(\psi_Z(\psi_Z(1))))=\omega^{\omega^{\omega^\omega}}</math>

<math>\psi_Z(\psi_Z(\psi_Z(\psi_Z(\psi_Z(1)))))=\omega^{\omega^{\omega^{\omega^\omega}}}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0))=\psi_Z(\psi_Z(\psi_Z(...)))=\varepsilon_0</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0))+\psi_Z(0)=\varepsilon_0+1</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0))+\psi_Z(1)=\varepsilon_0+\omega</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0))+\psi_Z(\psi_Z(1))=\varepsilon_0+\omega^\omega</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0))+\psi_Z(\psi_Z(\psi_Z(1)))=\varepsilon_0+\omega^{\omega^\omega}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0))\times2=\varepsilon_0\times2</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0))\times3=\varepsilon_0\times3</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+1)=\varepsilon_0\times \omega</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+2)=\varepsilon_0\times \omega^2</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+3)=\varepsilon_0\times \omega^3</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+\psi_Z(1))=\varepsilon_0\times\omega^\omega</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+\psi_Z(\psi_Z(1)))=\varepsilon_0\times\omega^{\omega^\omega}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)))=\varepsilon_0^2</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)))+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0))=\varepsilon_0^2+\varepsilon_0</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)))\times2=\varepsilon_0^2\times2</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0))+1)=\varepsilon_0^2\times\omega</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0))+\psi_Z(1))=\varepsilon_0^2\times\omega^\omega</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0))\times2)=\varepsilon_0^3</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0))\times3)=\varepsilon_0^4</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+1))=\varepsilon_0^\omega</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+1)+1)=\varepsilon_0^\omega\times\omega</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+1)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)))=\varepsilon_0^{\omega+1}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+1)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0))\times2)=\varepsilon_0^{\omega+2}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+1)\times2)=\varepsilon_0^{\omega\times2}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+2))=\varepsilon_0^{\omega^2}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+3))=\varepsilon_0^{\omega^3}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+\psi_Z(1)))=\varepsilon_0^{\omega^\omega}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0))))=\varepsilon_0^{\varepsilon_0}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0))))=\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)\times2)=\varepsilon_1</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)\times2+1)=\varepsilon_1\times\omega</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)\times2+\psi_Z(1))=\varepsilon_1\times\omega^\omega</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)\times2+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)))=\varepsilon_1\times\varepsilon_0</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)\times2+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0))))=\varepsilon_1\times\varepsilon_0^{\varepsilon_0}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)\times2+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)\times2))=\varepsilon_1^2</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)\times2+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)\times2+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0))))=\varepsilon_1^{\varepsilon_0}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)\times2+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)\times2+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)\times2)))=\varepsilon_1^{\varepsilon_1}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)\times3)=\varepsilon_2</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(1))=\varepsilon_\omega</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(1)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)))=\varepsilon_\omega\times\varepsilon_0</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(1)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(1)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(1))))=\varepsilon_\omega^{\varepsilon_\omega}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(1)+\psi_{Z_2}(0))=\varepsilon_{\omega+1}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(1)+\psi_{Z_2}(0)\times2)=\varepsilon_{\omega+2}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(1)+\psi_{Z_2}(0)\times3)=\varepsilon_{\omega+3}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(1)\times2)=\varepsilon_{\omega\times2}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(1)\times3)=\varepsilon_{\omega\times3}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(2))=\varepsilon_{\omega^2}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(3))=\varepsilon_{\omega^3}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_Z(1)))=\varepsilon_{\omega^\omega}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0))))=\varepsilon_{\varepsilon_0}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_Z[\psi_{Z_2}(0)](\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)))))=\varepsilon_{\varepsilon_{\varepsilon_0}}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0)))=\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_Z[\psi_{Z_2}(0)](\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(...)))))=\zeta_0</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))+1)=\zeta_0\times\omega</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))))=\zeta_0^2</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0)))))=\zeta_0^{\zeta_0}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))+\psi_{Z_2}(0))=\varepsilon_{\zeta_0+1}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))+\psi_{Z_2}(0)\times2)=\varepsilon_{\zeta_0+2}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))+\psi_{Z_2}(1))=\varepsilon_{\zeta_0+\omega}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))+\psi_{Z_2}(\psi_Z(1)))=\varepsilon_{\zeta_0+\omega^\omega}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))+\psi_{Z_2}(\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0))))=\varepsilon_{\zeta_0+\varepsilon_0}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))+\psi_{Z_2}(\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)\times2)))=\varepsilon_{\zeta_0+\varepsilon_1}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))+\psi_{Z_2}(\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0)))))=\varepsilon_{\zeta_0\times2}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))+\psi_{Z_2}(\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0)))))=\varepsilon_{\zeta_0^{\zeta_0}}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))+\psi_{Z_2}(\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))+\psi_{Z_2}(0))))=\varepsilon_{\varepsilon_{\zeta_0+1}}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))+\psi_{Z_2}(\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))+\psi_{Z_2}(\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))+\psi_{Z_2}(0))))))=\varepsilon_{\varepsilon_{\varepsilon_{\zeta_0+1}}}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))\times2)=\zeta_1</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))\times3)=\zeta_2</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0)+1))=\zeta_\omega</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0))))=\zeta_{\varepsilon_0}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0)))))=\zeta_{\zeta_0}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0)))))))=\zeta_{\zeta_{\zeta_0}}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0)\times2))=\eta_0=\varphi(3,0)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0)\times3))=\theta_0=\varphi(4,0)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0)\times4))=\iota_0=\varphi(5,0)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0)\times5))=\kappa_0=\varphi(6,0)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0)\times6))=\lambda_0=\varphi(7,0)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(1)))=\varphi(\omega,0)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(1))+\psi_{Z_2}(0))=\varphi(1,\varphi(\omega,0)+1)=\varepsilon_{\varphi(\omega,0)+1}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(1))+\psi_{Z_2}(\psi_{Z_2}(0)))=\varphi(2,\varphi(\omega,0)+1)=\zeta_{\varphi(\omega,0)+1}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(1))+\psi_{Z_2}(\psi_{Z_2}(0)\times2))=\varphi(3,\varphi(\omega,0)+1)=\eta_{\varphi(\omega,0)+1}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(1))+\psi_{Z_2}(\psi_{Z_2}(0)\times3))=\varphi(4,\varphi(\omega,0)+1)=\theta_{\varphi(\omega,0)+1}</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(1))\times2)=\varphi(\omega,1)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(1))\times3)=\varphi(\omega,2)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(1)+1))=\varphi(\omega,\omega)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(1)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(1)))))=\varphi(\omega,\varphi(\omega,0))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(1)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(1)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(1)))))))=\varphi(\omega,\varphi(\omega,\varphi(\omega,0))))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(1)+\psi_{Z_2}(0)))=\varphi(\omega+1,0)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(1)+\psi_{Z_2}(0)\times2))=\varphi(\omega+2,0)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(1)\times2))=\varphi(\omega\times2,0)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(1)\times3))=\varphi(\omega\times3,0)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(2)))=\varphi(\omega^2,0)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_Z(1))))=\varphi(\omega^\omega,0)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)))))=\varphi(\varepsilon_0,0)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))))))=\varphi(\zeta_0,0)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(1))))))=\varphi(\varphi(\omega,0),0)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(1)))))))))=\varphi(\varphi(\varphi(\omega,0),0),0)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0))))=\varphi(1,0,0)=\Gamma_0=\psi(\Omega^\Omega)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0)))+\psi_{Z_2}(0))=\varphi(1,\varphi(1,0,0)+1)=\varepsilon_{\Gamma_0+1}=\psi(\Omega^\Omega+1)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0)))+\psi_{Z_2}(\psi_{Z_2}(0)))=\varphi(2,\varphi(1,0,0)+1)=\zeta_{\Gamma_0+1}=\psi(\Omega^\Omega+\Omega)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0)))\times2)=\varphi(1,0,1)=\Gamma_1=\psi(\Omega^\Omega\times2)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0)))\times3)=\varphi(1,0,2)=\Gamma_2=\psi(\Omega^\Omega\times3)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0))+1))=\varphi(1,0,\omega)=\Gamma_\omega=\psi(\Omega^\Omega\times\omega)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0))+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0))))))=\varphi(1,0,\varphi(1,0,0))=\Gamma_{\Gamma_0}=\psi(\Omega^\Omega\times\psi(\Omega^\Omega))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0))+\psi_{Z_2}(0)))=\varphi(1,1,0)=\psi(\Omega^{\Omega+1})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0))+\psi_{Z_2}(0)\times2))=\varphi(1,2,0)=\psi(\Omega^{\Omega+2})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0))+\psi_{Z_2}(1)))=\varphi(1,\omega,0)=\psi(\Omega^{\Omega+\omega})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0))+\psi_{Z_2}(\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)))))=\varphi(1,\varphi(1,0),0)=\psi(\Omega^{\Omega+\psi(0)})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0))+\psi_{Z_2}(\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))))))=\varphi(1,\varphi(2,0),0)=\psi(\Omega^{\Omega+\psi(\Omega)})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0))+\psi_{Z_2}(\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0)))))))=\varphi(1,\varphi(1,0,0),0)=\psi(\Omega^{\Omega+\psi(\Omega^\Omega)})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0))\times2))=\varphi(2,0,0)=\psi(\Omega^{\Omega\times2})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0))\times3))=\varphi(3,0,0)=\psi(\Omega^{\Omega\times3})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0)+1)))=\varphi(\omega,0,0)=\psi(\Omega^{\Omega\times\omega})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0)))))))=\varphi(\varphi(1,0,0),0,0)=\psi(\Omega^{\Omega\times\psi(\Omega^{\Omega})})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0)\times2)))=\varphi(1,0,0,0)=\psi(\Omega^{\Omega^{2}})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0)\times3)))=\varphi(1,0,0,0,0)=\psi(\Omega^{\Omega^{3}})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(1))))=\psi(\Omega^{\Omega^{\omega}})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0))))))=\psi(\Omega^{\Omega^{\psi(0)}})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0)))))))=\psi(\Omega^{\Omega^{\psi(\Omega)}})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0))))))))=\psi(\Omega^{\Omega^{\psi(\Omega^\Omega)}})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0))))))))))))=\psi(\Omega^{\Omega^{\psi(\Omega^{\psi(\Omega^\Omega)})}})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0)))))=\psi(\Omega^{\Omega^{\Omega}})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0))))))=\psi(\Omega^{\Omega^{\Omega^{\Omega}}})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)))=\psi(\psi_1(0))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0))+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(0)))=\psi(\psi_1(0)+1)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0))+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_2}(0))))=\psi(\psi_1(0)+\psi(\Omega))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0))+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0))))=\psi(\psi_1(0)+\psi(\psi_1(0)))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0))+\psi_{Z_2}(0))=\psi(\psi_1(0)+\Omega)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0))+\psi_{Z_2}(\psi_{Z_2}(0)))=\psi(\psi_1(0)+\Omega^\Omega)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0))\times2)=\psi(\psi_1(0)\times2)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0))\times3)=\psi(\psi_1(0)\times3)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)+1))=\psi(\psi_1(0)\times\omega)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)+\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)))))=\psi(\psi_1(0)\times\psi(\psi_1(0)))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)+\psi_{Z_2}(0)))=\psi(\psi_1(0)\times\Omega)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)+\psi_{Z_2}(\psi_{Z_2}(0))))=\psi(\psi_1(0)\times\Omega^\Omega)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)+\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0))))=\psi(\psi_1(0)^2)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)+\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)+1)))=\psi(\psi_1(0)^\omega)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)+\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)+\psi_{Z_2}(0))))=\psi(\psi_1(0)^\Omega)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)+\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)+\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)))))=\psi(\psi_1(0)^{\psi_1(0)})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)+\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)+\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)+\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0))))))=\psi(\psi_1(0)^{\psi_1(0)^{\psi_1(0)}})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)\times2))=\psi(\psi_1(1))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)\times2+\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)\times2)))=\psi(\psi_1(1)^2)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)\times2+\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)\times2+\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)\times2))))=\psi(\psi_1(1)^{\psi_1(1)})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)\times3))=\psi(\psi_1(2))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)\times4))=\psi(\psi_1(3))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(1)))=\psi(\psi_1(\omega))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)))))=\psi(\psi_1(\psi_1(0)))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)))))))=\psi(\psi_1(\psi_1(\psi_1(0))))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0))))=\psi(\Omega_2)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0)))+\psi_{Z_2}(0))=\psi(\Omega_2+\Omega)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0)))+\psi_{Z_2}(\psi_{Z_2}(0)))=\psi(\Omega_2+\Omega^\Omega)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0)))+\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)))=\psi(\Omega_2+\psi_1(0))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0)))+\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)\times2))=\psi(\Omega_2+\psi_1(1))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0)))\times2)=\psi(\Omega_2+\psi_1(\Omega_2))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0)))\times3)=\psi(\Omega_2+\psi_1(\Omega_2)\times2)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0))+1))=\psi(\Omega_2+\psi_1(\Omega_2)\times\omega)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0))+\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0))+\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0))))))=\psi(\Omega_2+\psi_1(\Omega_2)^{\psi_1(\Omega_2)})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0))+\psi_{Z_3}(0)))=\psi(\Omega_2+\psi_1(\Omega_2+1))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0))+\psi_{Z_3}(\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0))+\psi_{Z_3}(0)))))=\psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+1)))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0))\times2))=\psi(\Omega_2\times2)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0)+1)))=\psi(\Omega_2\times\omega)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0)+\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0))))))=\psi(\Omega_2\times\psi_1(\Omega_2))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0)+\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0)+\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0))))))))=\psi(\Omega_2\times\psi_1(\Omega_2\times\psi_1(\Omega_2)))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0)\times2)))=\psi(\Omega_2^2)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(1))))=\psi(\Omega_2^\omega)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0)))))))=\psi(\Omega_2^{\psi_1(\Omega_2)})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(0))))))))))=\psi(\Omega_2^{\psi_1(\Omega_2^{\psi_1(\Omega_2)})})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(\psi_{Z_3}(0)))))=\psi(\Omega_2^{\Omega_2})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(\psi_{Z_3}(\psi_{Z_3}(\psi_{Z_3}(0))))))=\psi(\Omega_2^{\Omega_2^{\Omega_2}})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}[\psi_{Z_4}(0)](\psi_{Z_4}(0))))=\psi(\psi_2(0))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}[\psi_{Z_4}(0)](\psi_{Z_4}(\psi_{Z}[\psi_{Z_2}(0)](\psi_{Z_2}(0))))))=\psi(\psi_2(\psi(0)))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}[\psi_{Z_4}(0)](\psi_{Z_4}(\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0))))))=\psi(\psi_2(\psi_1(0)))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}[\psi_{Z_4}(0)](\psi_{Z_4}(\psi_{Z_3}[\psi_{Z_4}(0)](\psi_{Z_4}(0))))))=\psi(\psi_2(\psi_2(0)))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}[\psi_{Z_4}(0)](\psi_{Z_4}(\psi_{Z_3}[\psi_{Z_4}(0)](\psi_{Z_4}(\psi_{Z_3}[\psi_{Z_4}(0)](\psi_{Z_4}(0))))))))=\psi(\psi_2(\psi_2(\psi_2(0))))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}[\psi_{Z_4}(0)](\psi_{Z_4}(\psi_{Z_4}(0)))))=\psi(\Omega_3)</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}[\psi_{Z_4}(0)](\psi_{Z_4}(\psi_{Z_4}(\psi_{Z_4}(0))))))=\psi(\Omega_3^{\Omega_3})</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}[\psi_{Z_4}(0)](\psi_{Z_4}[\psi_{Z_5}(0)](\psi_{Z_5}(0)))))=\psi(\psi_3(0))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}[\psi_{Z_4}(0)](\psi_{Z_4}[\psi_{Z_5}(0)](\psi_{Z_5}(\psi_{Z_5}[\psi_{Z_6}(0)](\psi_{Z_6}(0)))))))=\psi(\psi_4(0))</math>

<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}[\psi_{Z_4}(0)](\psi_{Z_4}[\psi_{Z_5}(0)](\psi_{Z_5}(\psi_{Z_5}[\psi_{Z_6}(0)](\psi_{Z_6}(\psi_{Z_6}[\psi_{Z_7}(0)](\psi_{Z_7}(0)))))))))=\psi(\psi_5(0))</math>

<math>\psi_Z(\psi_{Z_2}(0))=\psi(\Omega_\omega)=(0)(1,1,1)</math>

<math>\psi_Z(\psi_{Z_2}(0)+1)=\psi(\Omega_\omega)\times\omega=(0)(1,1,1)(1)</math>

<math>\psi_Z(\psi_{Z_2}(0)+\psi(\psi_{Z_2}(0)))=\psi(\Omega_\omega+\psi(\Omega_\omega))=(0)(1,1,1)(1)(2,1,1)</math>

<math>\psi_Z(\psi_{Z_2}(0)+\psi(\psi_{Z_2}(0)+\psi(\psi_{Z_2}(0))))=\psi(\Omega_\omega+\psi(\Omega_\omega+\psi(\Omega_\omega)))=(0)(1,1,1)(1)(2,1,1)(2)(3,1,1)</math>

<math>\psi_Z[\psi_{Z_2}(0)\times2](\psi_{Z_2}(0)\times2)=\psi(\Omega_\omega+\Omega)=(0)(1,1,1)(1,1)</math>

<math>\psi_Z[\psi_{Z_2}(0)\times2](\psi_{Z_2}(0)\times3)=\psi(\Omega_\omega+\Omega\times2)=(0)(1,1,1)(1,1)(1,1)</math>

<math>\psi_Z[\psi_{Z_2}(0)\times2](\psi_{Z_2}(1))=\psi(\Omega_\omega+\Omega\times\omega)=(0)(1,1,1)(1,1)(2)</math>

<math>\psi_Z[\psi_{Z_2}(0)\times2](\psi_{Z_2}(\psi_{Z_2}(0)))=\psi(\Omega_\omega+\Omega^2)=(0)(1,1,1)(1,1)(2,1)</math>

<math>\psi_Z[\psi_{Z_2}(0)\times2](\psi_{Z_2}(\psi_{Z_2}(\psi_{Z_2}(0))))=\psi(\Omega_\omega+\Omega^\Omega)=(0)(1,1,1)(1,1)(2,1)(3,1)</math>

<math>\psi_Z[\psi_{Z_2}(0)\times2](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}(0)))=\psi(\Omega_\omega+\psi_1(0))=(0)(1,1,1)(1,1)(2,2)</math>

<math>\psi_Z[\psi_{Z_2}(0)\times2](\psi_{Z_2}[\psi_{Z_3}(0)](\psi_{Z_3}[\psi_{Z_4}(0)](\psi_{Z_4}(0))))=\psi(\Omega_\omega+\psi_1(\psi_2(0)))=(0)(1,1,1)(1,1)(2,2)(3,3)</math>

<math>\psi_Z[\psi_{Z_2}(0)\times2](\psi_{Z_2}(\psi_{Z_3}(0)))=\psi(\Omega_\omega+\psi_1(\Omega_\omega))=(0)(1,1,1)(1,1)(2,2,1)</math>

<math>\psi_Z[\psi_{Z_2}(0)\times2](\psi_{Z_2}(\psi_{Z_3}(0)+\psi_{Z_2}(\psi_{Z_3}(0))))=\psi(\Omega_\omega+\psi_1(\Omega_\omega+\psi_1(\Omega_\omega)))=(0)(1,1,1)(1,1)(2,2,1)(2,1)(3,2,1)</math>

<math>\psi_Z[\psi_{Z_2}(0)\times2](\psi_{Z_2}[\psi_{Z_3}(0)\times2](\psi_{Z_3}(0)\times2))=\psi(\Omega_\omega+\Omega_2)=(0)(1,1,1)(1,1)(2,2,1)(2,2)</math>

<math>\psi_Z[\psi_{Z_2}(0)\times2](\psi_{Z_2}[\psi_{Z_3}(0)\times2](\psi_{Z_3}[\psi_{Z_4}(0)\times2](\psi_{Z_4}(0)\times2)))=\psi(\Omega_\omega+\Omega_3)=(0)(1,1,1)(1,1)(2,2,1)(2,2)(3,3,1)(3,3)</math>

<math>\psi_Z(\psi_{Z_2}(0)\times2)=\psi(\Omega_\omega\times2)=(0)(1,1,1)(1,1,1)</math>

这里提供一下知乎的错误版本：<math>\psi_Z[\psi_{Z_2}(0)](\psi_{Z_2}(\psi_{Z_3}(0)))=\psi(\Omega_\omega\times2)=(0)(1,1,1)(1,1,1)</math>

<math>\psi_Z(\psi_{Z_2}(0)\times3)=\psi(\Omega_\omega\times3)=(0)(1,1,1)(1,1,1)(1,1,1)</math>

<math>\psi_Z[\psi_{Z_2}(1)](\psi_{Z_2}(1))=\psi(\Omega_\omega\times\omega)=(0)(1,1,1)(2)</math>

<math>\psi_Z(\psi_{Z_2}(1))=\psi(\Omega_\omega\times\Omega)=(0)(1,1,1)(2,1)</math>

<math>\psi_Z(\psi_{Z_2}(1)+\psi_Z(\psi_{Z_2}(1)))=\psi(\Omega_\omega\times\Omega+\psi(\Omega_\omega\times\Omega))=(0)(1,1,1)(2,1)(1)(2,1,1)(3,1)</math>

<math>\psi_Z[\psi_{Z_2}(1)+\psi_{Z_2}(0)](\psi_{Z_2}(1)+\psi_{Z_2}(0))=\psi(\Omega_\omega\times\Omega+\Omega)=(0)(1,1,1)(2,1)(1,1)</math>

<math>\psi_Z(\psi_{Z_2}(1)+\psi_{Z_2}(0))=\psi(\Omega_\omega^2)=(0)(1,1,1)(2,1)(1,1,1)</math>
[[分类:分析]]

BMS分析

2025-07-10T08:34:26Z

YukinaKNK：

目前使用的OCF为M型，后续补充BOCF

==== 1：单行BMS（PrSS） ====
<math>\varnothing=0</math>

<math>(0)=1</math>

<math>(0)(0)=2</math>

<math>(0)(0)(0)=3</math>

<math>(0)(1)=(0)(0)(0)(0)(0)...=\omega</math>

<math>(0)(1)(0)=\omega+1</math>

<math>(0)(1)(0)(0)=\omega+2</math>

<math>(0)(1)(0)(1)=\omega\times2</math>

<math>(0)(1)(0)(1)(0)(1)=\omega\times3</math>

<math>(0)(1)(1)=(0)(1)(0)(1)(0)(1)...=\omega^2</math>

<math>(0)(1)(1)(0)=\omega^2+1</math>

<math>(0)(1)(1)(0)(1)=\omega^2+\omega</math>

<math>(0)(1)(1)(0)(1)(0)(1)=\omega^2+\omega\times2</math>

<math>(0)(1)(1)(0)(1)(1)=\omega^2\times2</math>

<math>(0)(1)(1)(0)(1)(1)(0)(1)=\omega^2\times2+\omega</math>

<math>(0)(1)(1)(0)(1)(1)(0)(1)(1)=\omega^2\times3</math>

<math>(0)(1)(1)(1)=\omega^3</math>

<math>(0)(1)(1)(1)(1)=\omega^4</math>

<math>(0)(1)(2)=(0)(1)(1)(1)(1)...=\omega^\omega</math>

<math>(0)(1)(2)(0)(1)(2)=\omega^\omega\times2</math>

<math>(0)(1)(2)(1)=\omega^{\omega+1}</math>

<math>(0)(1)(2)(1)(1)=\omega^{\omega+2}</math>

<math>(0)(1)(2)(1)(1)(1)=\omega^{\omega+3}</math>

<math>(0)(1)(2)(1)(2)=\omega^{\omega\times2}</math>

<math>(0)(1)(2)(1)(2)(1)(2)=\omega^{\omega\times3}</math>

<math>(0)(1)(2)(2)=\omega^{\omega^2}</math>

<math>(0)(1)(2)(2)(1)=\omega^{\omega^2+1}</math>

<math>(0)(1)(2)(2)(1)(2)=\omega^{\omega^2+\omega}</math>

<math>(0)(1)(2)(2)(1)(2)(2)=\omega^{\omega^2\times2}</math>

<math>(0)(1)(2)(2)(1)(2)(2)(1)(2)(2)=\omega^{\omega^2\times3}</math>

<math>(0)(1)(2)(2)(2)=\omega^{\omega^3}</math>

<math>(0)(1)(2)(2)(2)(2)=\omega^{\omega^4}</math>

<math>(0)(1)(2)(3)=(0)(1)(2)(2)(2)(2)...=\omega^{\omega^\omega}</math>

<math>(0)(1)(2)(3)(4)=\omega^{\omega^{\omega^\omega}}</math>

<math>(0)(1)(2)(3)(4)(5)=\omega^{\omega^{\omega^{\omega^\omega}}}</math>

<math>(0)(1,1)=(0)(1)(2)(3)(4)(5)(6)...=\varepsilon_0</math>

==== 2：双行BMS ====
{| class="wikitable"
|-
! BMS !! Veblen !! MOCF
|-
| <math>(0)(1,1)</math>|| <math>\varepsilon_0</math>
|-
| <math>(0)(1,1)(0,0)</math>|| <math>\varepsilon_0+1</math>
|-
| <math>(0)(1,1)(0,0)(1,0)</math>|| <math>\varepsilon_0+\omega</math>
|-
| <math>(0)(1,1)(0,0)(1,1)</math>|| <math>\varepsilon_0\times2</math>
|-
| <math>(0)(1,1)(0,0)(1,1)(0,0)(1,1)</math>|| <math>\varepsilon_0\times3</math>
|-
| <math>(0)(1,1)(1,0)</math>|| <math>\varepsilon_0\times\omega</math>
|-
| <math>(0)(1,1)(1,0)(1,0)</math>|| <math>\varepsilon_0\times\omega^2</math>
|-
| <math>(0)(1,1)(1,0)(2,0)</math>|| <math>\varepsilon_0\times\omega^\omega</math>
|-
| <math>(0)(1,1)(1,0)(2,0)(3,0)</math>|| <math>\varepsilon_0\times\omega^{\omega^\omega}</math>
|-
| <math>(0)(1,1)(1,0)(2,1)</math>|| <math>\varepsilon_0^2</math>
|-
| <math>(0)(1,1)(1,0)(2,1)(1,0)(2,0)</math>|| <math>\varepsilon_0^2\times\omega</math>
|-
| <math>(0)(1,1)(1,0)(2,1)(1,0)(2,0)(3,0)</math>|| <math>\varepsilon_0^2\times\omega^\omega</math>
|-
| <math>(0)(1,1)(1,0)(2,1)(1,0)(2,1)</math>|| <math>\varepsilon_0^3</math>
|-
| <math>(0)(1,1)(1,0)(2,1)(1,0)(2,1)(1,0)(2,1)</math>|| <math>\varepsilon_0^4</math>
|-
| <math>(0)(1,1)(1,0)(2,1)(2,0)</math>|| <math>\varepsilon_0^\omega</math>
|-
| <math>(0)(1,1)(1,0)(2,1)(2,0)(1,0)(2,1)</math>|| <math>\varepsilon_0^{\omega+1}</math>
|-
| <math>(0)(1,1)(1,0)(2,1)(2,0)(1,0)(2,1)(1,0)(2,1)</math>|| <math>\varepsilon_0^{\omega+2}</math>
|-
| <math>(0)(1,1)(1,0)(2,1)(2,0)(1,0)(2,1)(2,0)</math>|| <math>\varepsilon_0^{\omega\times2}</math>
|-
| <math>(0)(1,1)(1,0)(2,1)(2,0)(1,0)(2,1)(2,0)(1,0)(2,1)(2,0)</math>|| <math>\varepsilon_0^{\omega\times3}</math>
|-
| <math>(0)(1,1)(1,0)(2,1)(2,0)(2,0)</math>|| <math>\varepsilon_0^{\omega^2}</math>
|-
| <math>(0)(1,1)(1,0)(2,1)(2,0)(3,0)</math>|| <math>\varepsilon_0^{\omega^\omega}</math>
|-
| <math>(0)(1,1)(1,0)(2,1)(2,0)(3,0)(4,0)</math>|| <math>\varepsilon_0^{\omega^{\omega^\omega}}</math>
|-
| <math>(0)(1,1)(1,0)(2,1)(2,0)(3,1)</math>|| <math>\varepsilon_0^{\varepsilon_0}</math>
|-
| <math>(0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)</math>|| <math>\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}</math>
|-
| <math>(0)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)(4,0)(5,1)</math>|| <math>\varepsilon_0^{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}</math>
|-
| <math>(0)(1,1)(1,1)</math>|| <math>\varepsilon_1</math>
|-
| <math>(0)(1,1)(1,1)(0,0)(1,1)(1,1)</math>|| <math>\varepsilon_1\times2</math>
|-
| <math>(0)(1,1)(1,1)(1,0)</math>|| <math>\varepsilon_1\times\omega</math>
|-
| <math>(0)(1,1)(1,1)(1,0)(2,1)</math>|| <math>\varepsilon_1\times\varepsilon_0</math>
|-
| <math>(0)(1,1)(1,1)(1,0)(2,1)(2,0)(3,1)</math>|| <math>\varepsilon_1\times\varepsilon_0^2</math>
|-
| <math>(0)(1,1)(1,1)(1,0)(2,1)(2,0)(3,1)(3,0)(4,1)</math>|| <math>\varepsilon_1\times\varepsilon_0^{\varepsilon_0}</math>
|-
| <math>(0)(1,1)(1,1)(1,0)(2,1)(2,1)</math>|| <math>\varepsilon_1^2</math>
|-
| <math>(0)(1,1)(1,1)(1,0)(2,1)(2,1)(2,0)</math>|| <math>\varepsilon_1^\omega</math>
|-
| <math>(0)(1,1)(1,1)(1,0)(2,1)(2,1)(2,0)(3,1)</math>|| <math>\varepsilon_1^{\varepsilon_0}</math>
|-
| <math>(0)(1,1)(1,1)(1,0)(2,1)(2,1)(2,0)(3,1)(3,1)</math>|| <math>\varepsilon_1^{\varepsilon_1}</math>
|-
| <math>(0)(1,1)(1,1)(1,0)(2,1)(2,1)(2,0)(3,1)(3,1)(3,0)(4,1)(4,1)</math>|| <math>\varepsilon_1^{\varepsilon_1^{\varepsilon_1}}</math>
|-
| <math>(0)(1,1)(1,1)(1,1)</math>|| <math>\varepsilon_2</math>
|-
| <math>(0)(1,1)(1,1)(1,1)(1,1)</math>|| <math>\varepsilon_3</math>
|-
| <math>(0)(1,1)(2,0)</math>|| <math>\varepsilon_\omega</math>
|-
| <math>(0)(1,1)(2,0)(1,0)</math>|| <math>\varepsilon_\omega\times\omega</math>
|-
| <math>(0)(1,1)(2,0)(1,0)(2,1)</math>|| <math>\varepsilon_\omega\times\varepsilon_0</math>
|-
| <math>(0)(1,1)(2,0)(1,0)(2,1)(2,1)</math>|| <math>\varepsilon_\omega\times\varepsilon_1</math>
|-
| <math>(0)(1,1)(2,0)(1,0)(2,1)(2,1)(2,1)</math>|| <math>\varepsilon_\omega\times\varepsilon_2</math>
|-
| <math>(0)(1,1)(2,0)(1,0)(2,1)(2,1)(2,1)(2,1)</math>|| <math>\varepsilon_\omega\times\varepsilon_3</math>
|-
| <math>(0)(1,1)(2,0)(1,0)(2,1)(3,0)</math>|| <math>\varepsilon_\omega^2</math>
|-
| <math>(0)(1,1)(2,0)(1,0)(2,1)(3,0)(2,0)(3,1)(4,0)</math>|| <math>\varepsilon_\omega^{\varepsilon_\omega}</math>
|-
| <math>(0)(1,1)(2,0)(1,1)</math>|| <math>\varepsilon_{\omega+1}</math>
|-
| <math>(0)(1,1)(2,0)(1,1)(1,1)</math>|| <math>\varepsilon_{\omega+2}</math>
|-
| <math>(0)(1,1)(2,0)(1,1)(1,1)(1,1)</math>|| <math>\varepsilon_{\omega+3}</math>
|-
| <math>(0)(1,1)(2,0)(1,1)(2,0)</math>|| <math>\varepsilon_{\omega\times2}</math>
|-
| <math>(0)(1,1)(2,0)(1,1)(2,0)(1,1)(2,0)</math>|| <math>\varepsilon_{\omega\times3}</math>
|-
| <math>(0)(1,1)(2,0)(2,0)</math>|| <math>\varepsilon_{\omega^2}</math>
|-
| <math>(0)(1,1)(2,0)(2,0)(2,0)</math>|| <math>\varepsilon_{\omega^3}</math>
|-
| <math>(0)(1,1)(2,0)(3,0)</math>|| <math>\varepsilon_{\omega^\omega}</math>
|-
| <math>(0)(1,1)(2,0)(3,0)(4,0)</math>|| <math>\varepsilon_{\omega^{\omega^\omega}}</math>
|-
| <math>(0)(1,1)(2,0)(3,1)</math>|| <math>\varepsilon_{\varepsilon_0}</math>
|-
| <math>(0)(1,1)(2,0)(3,1)(1,1)</math>|| <math>\varepsilon_{\varepsilon_0+1}</math>
|-
| <math>(0)(1,1)(2,0)(3,1)(3,1)</math>|| <math>\varepsilon_{\varepsilon_1}</math>
|-
| <math>(0)(1,1)(2,0)(3,1)(4,0)</math>|| <math>\varepsilon_{\varepsilon_\omega}</math>
|-
| <math>(0)(1,1)(2,0)(3,1)(4,0)(5,1)</math>|| <math>\varepsilon_{\varepsilon_{\varepsilon_0}}</math>
|-
| <math>(0)(1,1)(2,0)(3,1)(4,0)(5,1)(6,0)(7,1)</math>|| <math>\varepsilon_{\varepsilon_{\varepsilon_{\varepsilon_0}}}</math>
|-
| <math>(0)(1,1)(2,1)</math>|| <math>\zeta_0</math>
|-
| <math>(0)(1,1)(2,1)(1,0)(2,1)(3,1)</math>|| <math>\zeta_0^2</math>
|-
| <math>(0)(1,1)(2,1)(1,0)(2,1)(3,1)(2,0)(3,1)(4,1)</math>|| <math>\zeta_0^{\zeta_0}</math>
|-
| <math>(0)(1,1)(2,1)(1,1)</math>|| <math>\varepsilon_{\zeta_0+1}</math>
|-
| <math>(0)(1,1)(2,1)(1,1)(1,0)(2,1)(3,1)(2,1)</math>|| <math>\varepsilon_{\zeta_0+1}^2</math>
|-
| <math>(0)(1,1)(2,1)(1,1)(1,1)</math>|| <math>\varepsilon_{\zeta_0+2}</math>
|-
| <math>(0)(1,1)(2,1)(1,1)(2,0)</math>|| <math>\varepsilon_{\zeta_0+\omega}</math>
|-
| <math>(0)(1,1)(2,1)(1,1)(2,0)(3,1)</math>|| <math>\varepsilon_{\zeta_0+\varepsilon_0}</math>
|-
| <math>(0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)</math>|| <math>\varepsilon_{\zeta_0\times2}</math>
|-
| <math>(0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)(3,1)</math>|| <math>\varepsilon_{\varepsilon_{\zeta_0+1}}</math>
|-
| <math>(0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)(3,1)(4,0)(5,1)(6,1)(5,1)</math>|| <math>\varepsilon_{\varepsilon_{\varepsilon_{\zeta+1}}}</math>
|-
| <math>(0)(1,1)(2,1)(1,1)(2,1)</math>|| <math>\zeta_1</math>
|-
| <math>(0)(1,1)(2,1)(1,1)(2,1)(1,1)</math>|| <math>\varepsilon_{\zeta_1+1}</math>
|-
| <math>(0)(1,1)(2,1)(1,1)(2,0)</math>|| <math>\varepsilon_{\zeta_1+\omega}</math>
|-
| <math>(0)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1)</math>|| <math>\zeta_2</math>
|-
| <math>(0)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1)</math>|| <math>\zeta_3</math>
|-
| <math>(0)(1,1)(2,1)(2,0)</math>|| <math>\zeta_\omega</math>
|-
| <math>(0)(1,1)(2,1)(2,0)(2,0)</math>|| <math>\zeta_{\omega^2}</math>
|-
| <math>(0)(1,1)(2,1)(2,0)(3,1)</math>|| <math>\zeta_{\varepsilon_0}</math>
|-
| <math>(0)(1,1)(2,1)(2,0)(3,1)(4,1)</math>|| <math>\zeta_{\zeta_0}</math>
|-
| <math>(0)(1,1)(2,1)(2,0)(3,1)(4,1)(4,0)(5,1)(6,1)</math>|| <math>\zeta_{\zeta_{\zeta_0}}</math>
|-
| <math>(0)(1,1)(2,1)(2,1)</math>|| <math>\eta_0</math>
|-
| <math>(0)(1,1)(2,1)(2,1)(1,1)</math>|| <math>\varepsilon_{\eta_0+1}</math>
|-
| <math>(0)(1,1)(2,1)(2,1)(1,1)(2,1)</math>|| <math>\zeta_{\eta_0+1}</math>
|-
| <math>(0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,0)(3,1)(4,1)(4,1)(3,1)(4,1)</math>|| <math>\zeta_{\zeta_{\eta_0+1}}</math>
|-
| <math>(0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)</math>|| <math>\eta_1</math>
|-
| <math>(0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)</math>|| <math>\eta_2</math>
|-
| <math>(0)(1,1)(2,1)(2,1)(2,0)</math>|| <math>\eta_\omega</math>
|-
| <math>(0)(1,1)(2,1)(2,1)(2,1)</math>|| <math>\varphi(4,0)</math>|| <math>\psi(\Omega^3)</math>
|-
| <math>(0)(1,1)(2,1)(3,0)</math>|| <math>\varphi(\omega,0)</math>|| <math>\psi(\Omega^\omega)</math>
|-
| <math>(0)(1,1)(2,1)(3,0)(1,1)</math>|| <math>\varphi(1,\varphi(\omega,0)+1)</math>|| <math>\psi(\Omega^\omega+1)</math>
|-
| <math>(0)(1,1)(2,1)(3,0)(1,1)(2,1)</math>|| <math>\varphi(2,\varphi(\omega,0)+1)</math>|| <math>\psi(\Omega^\omega+\Omega)</math>
|-
| <math>(0)(1,1)(2,1)(3,0)(1,1)(2,1)(2,1)</math>|| <math>\varphi(3,\varphi(\omega,0)+1)</math>|| <math>\psi(\Omega^\omega+\Omega^2)</math>
|-
| <math>(0)(1,1)(2,1)(3,0)(1,1)(2,1)(3,0)</math>|| <math>\varphi(\omega,1)</math>|| <math>\psi(\Omega^\omega\times2)</math>
|-
| <math>(0)(1,1)(2,1)(3,0)(1,1)(2,1)(3,0)(1,1)(2,1)(3,0)</math>|| <math>\varphi(\omega,2)</math>|| <math>\psi(\Omega^\omega\times3)</math>
|-
| <math>(0)(1,1)(2,1)(3,0)(2,0)</math>|| <math>\varphi(\omega,\omega)</math>|| <math>\psi(\Omega^\omega\times\omega)</math>
|-
| <math>(0)(1,1)(2,1)(3,0)(2,1)</math>|| <math>\varphi(\omega+1,0)</math>|| <math>\psi(\Omega^{\omega+1})</math>
|-
| <math>(0)(1,1)(2,1)(3,0)(2,1)(2,1)</math>|| <math>\varphi(\omega+2,0)</math>|| <math>\psi(\Omega^{\omega+2})</math>
|-
| <math>(0)(1,1)(2,1)(3,0)(2,1)(2,1)(2,1)</math>|| <math>\varphi(\omega+3,0)</math>|| <math>\psi(\Omega^{\omega+3})</math>
|-
| <math>(0)(1,1)(2,1)(3,0)(2,1)(3,0)</math>|| <math>\varphi(\omega\times2,0)</math>|| <math>\psi(\Omega^{\omega\times2})</math>
|-
| <math>(0)(1,1)(2,1)(3,0)(2,1)(3,0)(2,1)(3,0)</math>|| <math>\varphi(\omega\times3,0)</math>|| <math>\psi(\Omega^{\omega\times3})</math>
|-
| <math>(0)(1,1)(2,1)(3,0)(3,0)</math>|| <math>\varphi(\omega^2,0)</math>|| <math>\psi(\Omega^{\omega^2})</math>
|-
| <math>(0)(1,1)(2,1)(3,0)(4,1)</math>|| <math>\varphi(\varphi(1,0),0)</math>|| <math>\psi(\Omega^{\psi(0)})</math>
|-
| <math>(0)(1,1)(2,1)(3,0)(4,1)(5,1)</math>|| <math>\varphi(\varphi(2,0),0)</math>|| <math>\psi(\Omega^{\psi(\Omega)})</math>
|-
| <math>(0)(1,1)(2,1)(3,0)(4,1)(5,1)(6,0)</math>|| <math>\varphi(\varphi(\omega,0),0)</math>|| <math>\psi(\Omega^{\psi(\Omega^\omega)})</math>
|-
| <math>(0)(1,1)(2,1)(3,0)(4,1)(5,1)(6,0)(7,1)(8,1)(9,0)</math>|| <math>\varphi(\varphi(\varphi(\omega,0),0),0)</math>|| <math>\psi(\Omega^{\psi(\Omega^{\psi(\Omega^\omega)})})</math>
|-
| <math>(0)(1,1)(2,1)(3,1)</math>|| <math>\varphi(1,0,0)=\Gamma_0</math>|| <math>\psi(\Omega^\Omega)</math>
|-
| <math>(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)</math>|| <math>\varphi(1,0,1)=\Gamma_1</math>|| <math>\psi(\Omega^\Omega\times2)</math>
|-
| <math>(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)</math>|| <math>\varphi(1,0,2)=\Gamma_2</math>|| <math>\psi(\Omega^\Omega\times3)</math>
|-
| <math>(0)(1,1)(2,1)(3,1)(2,0)</math>|| <math>\varphi(1,0,\omega)=\Gamma_\omega</math>|| <math>\psi(\Omega^\Omega\times\omega)</math>
|-
| <math>(0)(1,1)(2,1)(3,1)(2,0)(3,1)(4,1)(5,1)</math>|| <math>\varphi(1,0,\varphi(1,0,0))=\Gamma_{\Gamma_0}</math>|| <math>\psi(\Omega^\Omega\times\psi(\Omega^\Omega))</math>
|-
| <math>(0)(1,1)(2,1)(3,1)(2,0)(3,1)(4,1)(5,1)(4,0)(5,1)(6,1)(7,1)</math>|| <math>\varphi(1,0,\varphi(1,0,\varphi(1,0,0)))=\Gamma_{\Gamma_{\Gamma_0}}</math>|| <math>\psi(\Omega^\Omega\times\psi(\Omega^\Omega\times\psi(\Omega^\Omega)))</math>
|-
| <math>(0)(1,1)(2,1)(3,1)(2,1)</math>|| <math>\varphi(1,1,0)</math>|| <math>\psi(\Omega^{\Omega+1})</math>
|-
| <math>(0)(1,1)(2,1)(3,1)(2,1)(2,1)</math>|| <math>\varphi(1,2,0)</math>|| <math>\psi(\Omega^{\Omega+2})</math>
|-
| <math>(0)(1,1)(2,1)(3,1)(2,1)(3,0)</math>|| <math>\varphi(1,\omega,0)</math>|| <math>\psi(\Omega^{\Omega+\omega})</math>
|-
| <math>(0)(1,1)(2,1)(3,1)(2,1)(3,0)(4,1)(5,1)(6,1)(5,1)(6,0)</math>|| <math>\varphi(1,\varphi(1,\omega,0),0)</math>|| <math>\psi(\Omega^{\Omega+\psi(\Omega^{\Omega+\omega})})</math>
|-
| <math>(0)(1,1)(2,1)(3,1)(2,1)(3,1)</math>|| <math>\psi(2,0,0)</math>|| <math>\psi(\Omega^{\Omega\times2})</math>
|-
| <math>(0)(1,1)(2,1)(3,1)(2,1)(3,1)(2,1)(3,1)</math>|| <math>\psi(3,0,0)</math>|| <math>\psi(\Omega^{\Omega\times3})</math>
|-
| <math>(0)(1,1)(2,1)(3,1)(3,0)</math>|| <math>\varphi(\omega,0,0)</math>|| <math>\psi(\Omega^{\Omega\times\omega})</math>
|-
| <math>(0)(1,1)(2,1)(3,1)(3,1)</math>|| <math>\varphi(1,0,0,0)</math>|| <math>\psi(\Omega^{\Omega^2})</math>
|-
| <math>(0)(1,1)(2,1)(3,1)(3,1)(3,1)</math>|| <math>\varphi(1,0,0,0,0)</math>|| <math>\psi(\Omega^{\Omega^3})</math>
|}

{| class="wikitable"
|-
! BMS !! MOCF
|-
| <math>(0)(1,1)(2,1)(3,1)(4,0)</math>|| <math>\psi(\Omega^{\Omega^\omega})</math>
|-
| <math>(0)(1,1)(2,1)(3,1)(4,0)(5,1)</math>|| <math>\psi(\Omega^{\Omega^{\psi(0)}})</math>
|-
| <math>(0)(1,1)(2,1)(3,1)(4,0)(5,1)(6,1)</math>|| <math>\psi(\Omega^{\Omega^{\psi(\Omega)}})</math>
|-
| <math>(0)(1,1)(2,1)(3,1)(4,0)(5,1)(6,1)(7,1)</math>|| <math>\psi(\Omega^{\Omega^{\psi(\Omega^\Omega)}})</math>
|-
| <math>(0)(1,1)(2,1)(3,1)(4,1)</math>|| <math>\psi(\Omega^{\Omega^\Omega})</math>
|-
| <math>(0)(1,1)(2,1)(3,1)(4,1)(5,1)</math>|| <math>\psi(\Omega^{\Omega^{\Omega^\Omega}})</math>
|-
| <math>(0)(1,1)(2,2)</math>|| <math>\psi(\psi_1(0))</math>
|-
| <math>(0)(1,1)(2,2)(1,1)</math>|| <math>\psi(\psi_1(0)+1)</math>
|-
| <math>(0)(1,1)(2,2)(1,1)(2,1)</math>|| <math>\psi(\psi_1(0)+\Omega)</math>
|-
| <math>(0)(1,1)(2,2)(1,1)(2,1)(3,1)</math>|| <math>\psi(\psi_1(0)+\Omega^\Omega)</math>
|-
| <math>(0)(1,1)(2,2)(1,1)(2,2)</math>|| <math>\psi(\psi_1(0)\times2)</math>
|-
| <math>(0)(1,1)(2,2)(2,0)</math>|| <math>\psi(\psi_1(0)\times\omega)</math>
|-
| <math>(0)(1,1)(2,2)(2,1)</math>|| <math>\psi(\psi_1(0)\times\Omega)</math>
|-
| <math>(0)(1,1)(2,2)(2,1)(3,0)</math>|| <math>\psi(\psi_1(0)\times\Omega^\omega)</math>
|-
| <math>(0)(1,1)(2,2)(2,1)(3,1)</math>|| <math>\psi(\psi_1(0)\times\Omega^\Omega)</math>
|-
| <math>(0)(1,1)(2,2)(2,1)(3,2)</math>|| <math>\psi(\psi_1(0)^2)</math>
|-
| <math>(0)(1,1)(2,2)(2,1)(3,2)(3,0)</math>|| <math>\psi(\psi_1(0)^\omega)</math>
|-
| <math>(0)(1,1)(2,2)(2,1)(3,2)(3,1)(4,2)</math>|| <math>\psi(\psi_1(0)^{\psi_1(0)})</math>
|-
| <math>(0)(1,1)(2,2)(2,2)</math>|| <math>\psi(\psi_1(1))</math>
|-
| <math>(0)(1,1)(2,2)(3,0)</math>|| <math>\psi(\psi_1(\omega))</math>
|-
| <math>(0)(1,1)(2,2)(3,1)(4,2)</math>|| <math>\psi(\psi_1(\psi_1(0)))</math>
|-
| <math>(0)(1,1)(2,2)(3,2)</math>|| <math>\psi(\Omega_2)</math>
|-
| <math>(0)(1,1)(2,2)(3,2)(1,1)</math>|| <math>\psi(\Omega_2+\Omega)</math>
|-
| <math>(0)(1,1)(2,2)(3,2)(1,1)(2,2)</math>|| <math>\psi(\Omega_2+\psi_1(0))</math>
|-
| <math>(0)(1,1)(2,2)(3,2)(1,1)(2,2)(3,1)(4,2)</math>|| <math>\psi(\Omega_2+\psi_1(\psi_1(0)))</math>
|-
| <math>(0)(1,1)(2,2)(3,2)(1,1)(2,2)(3,2)</math>|| <math>\psi(\Omega_2+\psi_1(\Omega_2))</math>
|-
| <math>(0)(1,1)(2,2)(3,2)(2,0)</math>|| <math>\psi(\Omega_2+\psi_1(\Omega_2)\times\omega)</math>
|-
| <math>(0)(1,1)(2,2)(3,2)(2,1)(3,2)(4,2)</math>|| <math>\psi(\Omega_2+\psi_1(\Omega_2)^2)</math>
|-
| <math>(0)(1,1)(2,2)(3,2)(2,1)(3,2)(4,2)(3,0)</math>|| <math>\psi(\Omega_2+\psi_1(\Omega_2)^\omega)</math>
|-
| <math>(0)(1,1)(2,2)(3,2)(2,1)(3,2)(4,2)(3,1)(4,2)(5,2)</math>|| <math>\psi(\Omega_2+\psi_1(\Omega_2)^{\psi_1(\Omega_2)})</math>
|-
| <math>(0)(1,1)(2,2)(3,2)(2,2)</math>|| <math>\psi(\Omega_2+\psi_1(\Omega_2+1))</math>
|-
| <math>(0)(1,1)(2,2)(3,2)(2,2)(3,0)</math>|| <math>\psi(\Omega_2+\psi_1(\Omega_2+\omega))</math>
|-
| <math>(0)(1,1)(2,2)(3,2)(2,2)(3,1)(4,2)(5,2)</math>|| <math>\psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2)))</math>
|-
| <math>(0)(1,1)(2,2)(3,2)(2,2)(3,1)(4,2)(5,2)(3,0)</math>|| <math>\psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2)\times\omega))</math>
|-
| <math>(0)(1,1)(2,2)(3,2)(2,2)(3,1)(4,2)(5,2)(4,0)</math>|| <math>\psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2)^\omega))</math>
|-
| <math>(0)(1,1)(2,2)(3,2)(2,2)(3,1)(4,2)(5,2)(4,1)(5,2)(6,2)</math>|| <math>\psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2)^{\psi_1(\Omega_2)}))</math>
|-
| <math>(0)(1,1)(2,2)(3,2)(2,2)(3,1)(4,2)(5,2)(4,2)</math>|| <math>\psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+1)))</math>
|-
| <math>(0)(1,1)(2,2)(3,2)(2,2)(3,1)(4,2)(5,2)(4,2)(5,0)</math>|| <math>\psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+\omega)))</math>
|-
| <math>(0)(1,1)(2,2)(3,2)(2,2)(3,1)(4,2)(5,2)(4,2)(5,0)(5,1)(6,2)(7,2)</math>|| <math>\psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2))))</math>
|-
| <math>(0)(1,1)(2,2)(3,2)(2,2)(3,2)</math>|| <math>\psi(\Omega_2\times2)</math>
|-
| <math>(0)(1,1)(2,2)(3,2)(3,0)</math>|| <math>\psi(\Omega_2\times\omega)</math>
|-
| <math>(0)(1,1)(2,2)(3,2)(3,1)(4,2)(5,2)</math>|| <math>\psi(\Omega_2\times\psi_1(\Omega_2))</math>
|-
| <math>(0)(1,1)(2,2)(3,2)(3,2)</math>|| <math>\psi(\Omega_2^2)</math>
|-
| <math>(0)(1,1)(2,2)(3,2)(4,0)</math>|| <math>\psi(\Omega_2^\omega)</math>
|-
| <math>(0)(1,1)(2,2)(3,2)(4,2)</math>|| <math>\psi(\Omega_2^{\Omega_2})</math>
|-
| <math>(0)(1,1)(2,2)(3,3)</math>|| <math>\psi(\psi_2(0))</math>
|-
| <math>(0)(1,1)(2,2)(3,3)(4,3)</math>|| <math>\psi(\Omega_3)</math>
|-
| <math>(0)(1,1)(2,2)(3,3)(4,4)</math>|| <math>\psi(\psi_3(0))</math>
|-
| <math>(0)(1,1,1)=(0)(1,1)(2,2)(3,3)\cdots</math>|| <math>\psi(\Omega_\omega)</math>
|}

==== 3：三行BMS (0)(1,1,1)~(0)(1,1,1)(2,1,0) ====
三行之后BMS的行为复杂度急剧上升，因此部分节点的分析可能不会较为详细。

{| class="wikitable"
|-
! BMS !! MOCF
|-
| <math>(0)(1,1,1)</math>|| <math>\psi(\Omega_\omega)</math>
|-
| <math>(0)(1,1,1)(1,1,0)</math>|| <math>\psi(\Omega_\omega+1)</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,0,0)</math>|| <math>\psi(\Omega_\omega+\omega)</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,1,0)</math>|| <math>\psi(\Omega_\omega+\Omega)</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,0)</math>|| <math>\psi(\Omega_\omega+\psi_1(0))</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,0)(2,2,0)</math>|| <math>\psi(\Omega_\omega+\psi_1(1))</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,0)(3,0,0)</math>|| <math>\psi(\Omega_\omega+\psi_1(\omega))</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,0)(3,1,0)(4,2,0)</math>|| <math>\psi(\Omega_\omega+\psi_1(\psi_1(0)))</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,0)(3,2,0)</math>|| <math>\psi(\Omega_\omega+\psi_1(\Omega_2))</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,0)(3,2,0)(2,0,0)</math>|| <math>\psi(\Omega_\omega+\psi_1(\Omega_2)\times\omega)</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,0)(3,2,0)(2,1,0)</math>|| <math>\psi(\Omega_\omega+\psi_1(\Omega_2)^\omega)</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,0)(3,2,0)(2,2,0)</math>|| <math>\psi(\Omega_\omega+\psi_1(\Omega_2+1))</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,0)(3,2,0)(2,2,0)(3,0,0)</math>|| <math>\psi(\Omega_\omega+\psi_1(\Omega_2+\omega))</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,0)(3,2,0)(2,2,0)(3,1,0)(4,2,0)</math>|| <math>\psi(\Omega_\omega+\psi_1(\Omega_2+\psi_1(0)))</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,0)(3,2,0)(2,2,0)(3,2,0)</math>|| <math>\psi(\Omega_\omega+\psi_1(\Omega_2\times2))</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,0)(3,2,0)(3,0,0)</math>|| <math>\psi(\Omega_\omega+\psi_1(\Omega_2\times\omega))</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,0)(3,2,0)(3,2,0)</math>|| <math>\psi(\Omega_\omega+\psi_1(\Omega_2^2))</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,0)(3,2,0)(4,2,0)</math>|| <math>\psi(\Omega_\omega+\psi_1(\Omega_2^{\Omega_2}))</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,0)(3,3,0)</math>|| <math>\psi(\Omega_\omega+\psi_1(\psi_2(0)))</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,0)(3,3,0)(4,4,0)</math>|| <math>\psi(\Omega_\omega+\psi_1(\psi_3(0)))</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,1)</math>|| <math>\psi(\Omega_\omega+\psi_1(\Omega_\omega))</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,1)(2,2,0)</math>|| <math>\psi(\Omega_\omega+\psi_1(\Omega_\omega+1))</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,1)(2,2,0)(3,0,0)</math>|| <math>\psi(\Omega_\omega+\psi_1(\Omega_\omega+\omega))</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,1)(2,2,0)(3,1,0)(4,2,0)</math>|| <math>\psi(\Omega_\omega+\psi_1(\Omega_\omega+\psi_1(0)))</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,1)(2,2,0)(3,1,0)(4,2,0)(5,3,0)</math>|| <math>\psi(\Omega_\omega+\psi_1(\Omega_\omega+\psi_1(\psi_2(0))))</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,1)(2,2,0)(3,1,0)(4,2,1)</math>|| <math>\psi(\Omega_\omega+\psi_1(\Omega_\omega+\psi_1(\Omega_\omega)))</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,1)(2,2,0)(3,2,0)</math>|| <math>\psi(\Omega_\omega+\Omega_2)</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,1)(2,2,0)(3,3,0)</math>|| <math>\psi(\Omega_\omega+\psi_2(0))</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,1)(2,2,0)(3,3,1)(3,3,0)(4,3,0)</math>|| <math>\psi(\Omega_\omega+\Omega_3)</math>
|-
| <math>(0)(1,1,1)(1,1,0)(2,2,1)(2,2,0)(3,3,1)(3,3,0)(4,4,0)</math>|| <math>\psi(\Omega_\omega+\psi_3(0))</math>
|-
| <math>(0)(1,1,1)(1,1,1)</math>|| <math>\psi(\Omega_\omega\times2)</math>
|-
| <math>(0)(1,1,1)(2,0,0)</math>|| <math>\psi(\Omega_\omega\times\omega)</math>
|-
| <math>(0)(1,1,1)(2,1,0)</math>|| <math>\psi(\Omega_\omega\times\Omega)</math>
|}

==== 4： (0)(1,1,1)(2,1,0)~(0)(1,1,1)(2,1,1) ====
这一部分涉及到[[提升效应]]。

{| class="wikitable"
|-
! BMS !! MOCF
|-
| <math>(0)(1,1,1)(2,1,0)</math>|| <math>\psi(\Omega_\omega\times\Omega)</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)</math>|| <math>\psi(\Omega_\omega\times\Omega+\psi_1(\Omega_\omega))</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,0,0)</math>|| <math>\psi(\Omega_\omega\times\Omega+\psi_1(\Omega_\omega\times\omega))</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)</math>|| <math>\psi(\Omega_\omega\times\Omega+\psi_1(\Omega_\omega\times\Omega))</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(2,2,0)</math>|| <math>\psi(\Omega_\omega\times\Omega+\psi_1(\Omega_\omega\times\Omega+1))</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(2,2,0)(3,1,0)(4,2,0)</math>|| <math>\psi(\Omega_\omega\times\Omega+\psi_1(\Omega_\omega\times\Omega+\psi_1(0)))</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(2,2,0)(3,1,0)(4,2,1)(5,1,0)</math>|| <math>\psi(\Omega_\omega\times\Omega+\psi_1(\Omega_\omega\times\Omega+\psi_1(\Omega_\omega\times\Omega)))</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(2,2,0)(3,2,0)</math>|| <math>\psi(\Omega_\omega\times\Omega+\Omega_2)</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(2,2,0)(3,3,0)</math>|| <math>\psi(\Omega_\omega\times\Omega+\psi_2(0))</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(2,2,0)(3,3,1)(4,1,0)(3,3,0)(4,3,0)</math>|| <math>\psi(\Omega_\omega\times\Omega+\Omega_3)</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(2,2,1)</math>|| <math>\psi(\Omega_\omega\times\Omega+\Omega_\omega)</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(2,2,1)(1,1,0)(2,2,1)(3,1,0)</math>|| <math>\psi(\Omega_\omega\times\Omega+\Omega_\omega+\psi_1(\Omega_\omega\times\Omega))</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(2,2,1)(1,1,0)(2,2,1)(3,1,0)(2,2,1)</math>|| <math>\psi(\Omega_\omega\times\Omega+\Omega_\omega+\psi_1(\Omega_\omega\times\Omega+\Omega_\omega))</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(2,2,1)(2,0,0)</math>|| <math>\psi(\Omega_\omega\times\Omega+\Omega_\omega+\psi_1(\Omega_\omega\times\Omega+\Omega_\omega)\times\omega)</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(2,2,1)(2,2,0)</math>|| <math>\psi(\Omega_\omega\times\Omega+\Omega_\omega+\psi_1(\Omega_\omega\times\Omega+\Omega_\omega+1))</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(2,2,1)(2,2,0)(3,2,0)</math>|| <math>\psi(\Omega_\omega\times\Omega+\Omega_\omega+\Omega_2)</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(2,2,1)(2,2,1)</math>|| <math>\psi(\Omega_\omega\times\Omega+\Omega_\omega\times2)</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(2,2,1)(3,0,0)</math>|| <math>\psi(\Omega_\omega\times\Omega+\Omega_\omega\times\omega)</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(2,2,1)(3,1,0)</math>|| <math>\psi(\Omega_\omega\times\Omega\times2)</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(3,0,0)</math>|| <math>\psi(\Omega_\omega\times\Omega\times\omega)</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(4,2,0)</math>|| <math>\psi(\Omega_\omega\times\psi_1(0))</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(4,2,1)</math>|| <math>\psi(\Omega_\omega\times\psi_1(\Omega_\omega))</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(4,2,1)(5,1,0)</math>|| <math>\psi(\Omega_\omega\times\psi_1(\Omega_\omega\times\Omega))</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(4,2,1)(5,1,0)(4,2,1)</math>|| <math>\psi(\Omega_\omega\times\psi_1(\Omega_\omega\times\Omega+\Omega_\omega))</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(4,2,1)(5,1,0)(4,2,1)(5,1,0)</math>|| <math>\psi(\Omega_\omega\times\psi_1(\Omega_\omega\times\Omega\times2)</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(4,2,1)(5,1,0)(6,2,1)</math>|| <math>\psi(\Omega_\omega\times\psi_1(\Omega_\omega\times\psi_1(\Omega_\omega))</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,2,0)</math>|| <math>\psi(\Omega_\omega\times\Omega_2)</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,2,0)(2,2,0)(3,3,1)(4,2,0)(5,3,0)</math>|| <math>\psi(\Omega_\omega\times\psi_2(0))</math>
|-
| <math>(0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,2,0)(2,2,0)(3,3,1)(4,3,0)</math>|| <math>\psi(\Omega_\omega\times\Omega_3)</math>
|-
| <math>\color{red}{(0)(1,1,1)(2,1,0)(1,1,1)}</math>|| <math>\color{red}\psi(\Omega_\omega^2)</math>
|-
|<math>(0)(1,1,1)(2,1,0)(1,1,1)(2,1,0)(1,1,1)</math>
|<math>\psi(\Omega_\omega^2\times2)</math>
|-
|<math>(0)(1,1,1)(2,1,0)(2,0,0)</math>
|<math>\psi(\Omega_\omega^2\times\omega)</math>
|-
|<math>(0)(1,1,1)(2,1,0)(2,1,0)</math>
|<math>\psi(\Omega_\omega^2\times\Omega)</math>
|-
|<math>(0)(1,1,1)(2,1,0)(2,1,0)(1,1,0)(2,2,1)</math>
|<math>\psi(\Omega_\omega^2\times\Omega+\psi_1(\Omega_\omega))</math>
|-
|<math>(0)(1,1,1)(2,1,0)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(3,1,0)(2,2,1)</math>
|<math>\psi(\Omega_\omega^2\times\Omega+\Omega_\omega^2)</math>
|}
<math>(0)(1,1,1)(2,1,0)(2,1,0)(1,1,0)(2,2,1)(3,2,0)(3,2,0)=\psi(\Omega_\omega^2\times\Omega_2)</math>

<math>(0)(1,1,1)(2,1,0)(2,1,0)(1,1,0)(2,2,1)(3,2,0)(3,2,0)(2,2,0)(3,3,1)(4,3,0)(4,3,0)=\psi(\Omega_\omega^2\times\Omega_3)</math>

<math>(0)(1,1,1)(2,1,0)(2,1,0)(1,1,1)=\psi(\Omega_\omega^3)</math>

<math>(0)(1,1,1)(2,1,0)(2,1,0)(2,1,0)(1,1,1)=\psi(\Omega_\omega^4)</math>

<math>(0)(1,1,1)(2,1,0)(3,0,0)=\psi(\Omega_\omega^\omega)</math>

<math>(0)(1,1,1)(2,1,0)(3,1,0)=\psi(\Omega_\omega^\Omega)</math>

<math>(0)(1,1,1)(2,1,0)(3,1,0)(1,1,1)=\psi(\Omega_\omega^{\Omega_\omega})</math>

<math>(0)(1,1,1)(2,1,0)(3,1,0)(4,1,0)(1,1,1)=\psi(\Omega_\omega^{\Omega_\omega^{\Omega_\omega}})</math>

<math>(0)(1,1,1)(2,1,0)(3,2,0)=\psi(\psi_\omega(0))</math>

<math>(0)(1,1,1)(2,1,0)(3,2,0)(1,1,1)=\psi(\psi_\omega(0)+\Omega_\omega)</math>

<math>(0)(1,1,1)(2,1,0)(3,2,0)(1,1,1)(2,1,0)(3,2,0)=\psi(\psi_\omega(0)\times2)</math>

<math>(0)(1,1,1)(2,1,0)(3,2,0)(2,0,0)=\psi(\psi_\omega(0)\times\omega)</math>

<math>(0)(1,1,1)(2,1,0)(3,2,0)(2,1,0)=\psi(\psi_\omega(0)\times\Omega)</math>

<math>(0)(1,1,1)(2,1,0)(3,2,0)(2,1,0)(3,2,0)=\psi(\psi_\omega(0)^2)</math>

<math>(0)(1,1,1)(2,1,0)(3,2,0)(3,0,0)=\psi(\psi_\omega(0)^\omega)</math>

<math>(0)(1,1,1)(2,1,0)(3,2,0)(3,1,0)=\psi(\psi_\omega(0)^\Omega)</math>

<math>(0)(1,1,1)(2,1,0)(3,2,0)(3,1,0)(4,2,0)=\psi(\psi_\omega(0)^{\psi_\omega(0)})</math>

<math>(0)(1,1,1)(2,1,0)(3,2,0)(3,1,0)(4,2,0)(4,1,0)(5,2,0)=\psi(\psi_\omega(0)^{\psi_\omega(0)^{\psi_\omega(0)}})</math>

<math>(0)(1,1,1)(2,1,0)(3,2,0)(3,2,0)=\psi(\psi_\omega(1))</math>

<math>(0)(1,1,1)(2,1,0)(3,2,0)(3,2,0)(3,2,0)=\psi(\psi_\omega(2))</math>

<math>(0)(1,1,1)(2,1,0)(3,2,0)(4,0,0)=\psi(\psi_\omega(\omega))</math>

<math>(0)(1,1,1)(2,1,0)(3,2,0)(4,1,0)=\psi(\psi_\omega(\Omega))</math>

<math>(0)(1,1,1)(2,1,0)(3,2,0)(4,1,0)(5,2,0)=\psi(\psi_\omega(\psi_\omega(0)))</math>

<math>(0)(1,1,1)(2,1,0)(3,2,0)(4,2,0)=\psi(\Omega_{\omega+1})</math>

<math>(0)(1,1,1)(2,1,0)(3,2,0)(4,2,0)(1,1,1)(2,1,1)(3,2,0)(4,2,0)=\psi(\Omega_{\omega+1}+\psi_\omega(\Omega_{\omega+1}))</math>

<math>(0)(1,1,1)(2,1,0)(3,2,0)(4,2,0)(3,2,0)(4,2,0)=\psi(\Omega_{\omega+1}\times2)</math>
[[分类:分析]]

BMS分析

2025-07-05T11:41:01Z

YukinaKNK：创建页面，内容为“目前使用的OCF为M型，后续补充BOCF 注：<math>\eta_0</math>后的<math>\theta,\iota,\kappa,\lambda</math>为个人找规律得出的，没有被公认不要乱用 ==== 1：单行BMS（PrSS） ==== <math>\varnothing=0</math> <math>(0)=1</math> <math>(0)(0)=2</math> <math>(0)(0)(0)=3</math> <math>(0)(1)=(0)(0)(0)(0)(0)...=\omega</math> <math>(0)(1)(0)=\omega+1</math> <math>(0)(1)(0)(0)=\omega+2</math> <math>(0)(1)(0)(1)=\omega\times2</math> <…”

目前使用的OCF为M型，后续补充BOCF

注：<math>\eta_0</math>后的<math>\theta,\iota,\kappa,\lambda</math>为个人找规律得出的，没有被公认不要乱用

==== 1：单行BMS（PrSS） ====
<math>\varnothing=0</math>

<math>(0)=1</math>

<math>(0)(0)=2</math>

<math>(0)(0)(0)=3</math>

<math>(0)(1)=(0)(0)(0)(0)(0)...=\omega</math>

<math>(0)(1)(0)=\omega+1</math>

<math>(0)(1)(0)(0)=\omega+2</math>

<math>(0)(1)(0)(1)=\omega\times2</math>

<math>(0)(1)(0)(1)(0)(1)=\omega\times3</math>

<math>(0)(1)(1)=(0)(1)(0)(1)(0)(1)...=\omega^2</math>

<math>(0)(1)(1)(0)=\omega^2+1</math>

<math>(0)(1)(1)(0)(1)=\omega^2+\omega</math>

<math>(0)(1)(1)(0)(1)(0)(1)=\omega^2+\omega\times2</math>

<math>(0)(1)(1)(0)(1)(1)=\omega^2\times2</math>

<math>(0)(1)(1)(0)(1)(1)(0)(1)=\omega^2\times2+\omega</math>

<math>(0)(1)(1)(0)(1)(1)(0)(1)(1)=\omega^2\times3</math>

<math>(0)(1)(1)(1)=\omega^3</math>

<math>(0)(1)(1)(1)(1)=\omega^4</math>

<math>(0)(1)(2)=(0)(1)(1)(1)(1)...=\omega^\omega</math>

<math>(0)(1)(2)(0)(1)(2)=\omega^\omega\times2</math>

<math>(0)(1)(2)(1)=\omega^{\omega+1}</math>

<math>(0)(1)(2)(1)(1)=\omega^{\omega+2}</math>

<math>(0)(1)(2)(1)(1)(1)=\omega^{\omega+3}</math>

<math>(0)(1)(2)(1)(2)=\omega^{\omega\times2}</math>

<math>(0)(1)(2)(1)(2)(1)(2)=\omega^{\omega\times3}</math>

<math>(0)(1)(2)(2)=\omega^{\omega^2}</math>

<math>(0)(1)(2)(2)(1)=\omega^{\omega^2+1}</math>

<math>(0)(1)(2)(2)(1)(2)=\omega^{\omega^2+\omega}</math>

<math>(0)(1)(2)(2)(1)(2)(2)=\omega^{\omega^2\times2}</math>

<math>(0)(1)(2)(2)(1)(2)(2)(1)(2)(2)=\omega^{\omega^2\times3}</math>

<math>(0)(1)(2)(2)(2)=\omega^{\omega^3}</math>

<math>(0)(1)(2)(2)(2)(2)=\omega^{\omega^4}</math>

<math>(0)(1)(2)(3)=(0)(1)(2)(2)(2)(2)...=\omega^{\omega^\omega}</math>

<math>(0)(1)(2)(3)(4)=\omega^{\omega^{\omega^\omega}}</math>

<math>(0)(1)(2)(3)(4)(5)=\omega^{\omega^{\omega^{\omega^\omega}}}</math>

<math>(0)(1,1)=(0)(1)(2)(3)(4)(5)(6)...=\varepsilon_0</math>

==== 2：双行BMS ====
<math>(0)(1,1)=\varepsilon_0</math>

<math>(0)(1,1)(0)=\varepsilon_0+1</math>

<math>(0)(1,1)(0)(1)=\varepsilon_0+\omega</math>

<math>(0)(1,1)(0)(1,1)=\varepsilon_0\times2</math>

<math>(0)(1,1)(0)(1,1)(0)(1,1)=\varepsilon_0\times3</math>

<math>(0)(1,1)(1)=\varepsilon_0\times\omega</math>

<math>(0)(1,1)(1)(1)=\varepsilon_0\times\omega^2</math>

<math>(0)(1,1)(1)(2)=\varepsilon_0\times\omega^\omega</math>

<math>(0)(1,1)(1)(2)(3)=\varepsilon_0\times\omega^{\omega^\omega}</math>

<math>(0)(1,1)(1)(2,1)=\varepsilon_0^2</math>

<math>(0)(1,1)(1)(2,1)(1)(2)=\varepsilon_0^2\times\omega</math>

<math>(0)(1,1)(1)(2,1)(1)(2)(3)=\varepsilon_0^2\times\omega^\omega</math>

<math>(0)(1,1)(1)(2,1)(1)(2,1)=\varepsilon_0^3</math>

<math>(0)(1,1)(1)(2,1)(1)(2,1)(1)(2,1)=\varepsilon_0^4</math>

<math>(0)(1,1)(1)(2,1)(2)=\varepsilon_0^\omega</math>

<math>(0)(1,1)(1)(2,1)(2)(1)(2,1)=\varepsilon_0^{\omega+1}</math>

<math>(0)(1,1)(1)(2,1)(2)(1)(2,1)(1)(2,1)=\varepsilon_0^{\omega+2}</math>

<math>(0)(1,1)(1)(2,1)(2)(1)(2,1)(2)=\varepsilon_0^{\omega\times2}</math>

<math>(0)(1,1)(1)(2,1)(2)(1)(2,1)(2)(1)(2,1)(2)=\varepsilon_0^{\omega\times3}</math>

<math>(0)(1,1)(1)(2,1)(2)(2)=\varepsilon_0^{\omega^2}</math>

<math>(0)(1,1)(1)(2,1)(2)(3)=\varepsilon_0^{\omega^\omega}</math>

<math>(0)(1,1)(1)(2,1)(2)(3)(4)=\varepsilon_0^{\omega^{\omega^\omega}}</math>

<math>(0)(1,1)(1)(2,1)(2)(3,1)=\varepsilon_0^{\varepsilon_0}</math>

<math>(0)(1,1)(1)(2,1)(2)(3,1)(3)(4,1)=\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}</math>

<math>(0)(1,1)(1)(2,1)(2)(3,1)(3)(4,1)(4)(5,1)=\varepsilon_0^{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}</math>

<math>(0)(1,1)(1,1)=\varepsilon_1</math>

<math>(0)(1,1)(1,1)(0)(1,1)(1,1)=\varepsilon_1\times2</math>

<math>(0)(1,1)(1,1)(1)=\varepsilon_1\times\omega</math>

<math>(0)(1,1)(1,1)(1)(2,1)=\varepsilon_1\times\varepsilon_0</math>

<math>(0)(1,1)(1,1)(1)(2,1)(2)(3,1)=\varepsilon_1\times\varepsilon_0^2</math>

<math>(0)(1,1)(1,1)(1)(2,1)(2)(3,1)(3)(4,1)=\varepsilon_1\times\varepsilon_0^{\varepsilon_0}</math>

<math>(0)(1,1)(1,1)(1)(2,1)(2,1)=\varepsilon_1^2</math>

<math>(0)(1,1)(1,1)(1)(2,1)(2,1)(2)=\varepsilon_1^\omega</math>

<math>(0)(1,1)(1,1)(1)(2,1)(2,1)(2)(3,1)=\varepsilon_1^{\varepsilon_0}</math>

<math>(0)(1,1)(1,1)(1)(2,1)(2,1)(2)(3,1)(3,1)=\varepsilon_1^{\varepsilon_1}</math>

<math>(0)(1,1)(1,1)(1)(2,1)(2,1)(2)(3,1)(3,1)(3)(4,1)(4,1)=\varepsilon_1^{\varepsilon_1^{\varepsilon_1}}</math>

<math>(0)(1,1)(1,1)(1,1)=\varepsilon_2</math>

<math>(0)(1,1)(1,1)(1,1)(1,1)=\varepsilon_3</math>

<math>(0)(1,1)(2)=\varepsilon_\omega</math>

<math>(0)(1,1)(2)(1)=\varepsilon_\omega\times\omega</math>

<math>(0)(1,1)(2)(1)(2,1)=\varepsilon_\omega\times\varepsilon_0</math>

<math>(0)(1,1)(2)(1)(2,1)(2,1)=\varepsilon_\omega\times\varepsilon_1</math>

<math>(0)(1,1)(2)(1)(2,1)(2,1)(2,1)=\varepsilon_\omega\times\varepsilon_2</math>

<math>(0)(1,1)(2)(1)(2,1)(2,1)(2,1)(2,1)=\varepsilon_\omega\times\varepsilon_3</math>

<math>(0)(1,1)(2)(1)(2,1)(3)=\varepsilon_\omega^2</math>

<math>(0)(1,1)(2)(1)(2,1)(3)(2)(3,1)(4)=\varepsilon_\omega^{\varepsilon_\omega}</math>

<math>(0)(1,1)(2)(1,1)=\varepsilon_{\omega+1}</math>

<math>(0)(1,1)(2)(1,1)(1,1)=\varepsilon_{\omega+2}</math>

<math>(0)(1,1)(2)(1,1)(1,1)(1,1)=\varepsilon_{\omega+3}</math>

<math>(0)(1,1)(2)(1,1)(2)=\varepsilon_{\omega\times2}</math>

<math>(0)(1,1)(2)(1,1)(2)(1,1)(2)=\varepsilon_{\omega\times3}</math>

<math>(0)(1,1)(2)(2)=\varepsilon_{\omega^2}</math>

<math>(0)(1,1)(2)(2)(2)=\varepsilon_{\omega^3}</math>

<math>(0)(1,1)(2)(3)=\varepsilon_{\omega^\omega}</math>

<math>(0)(1,1)(2)(3)(4)=\varepsilon_{\omega^{\omega^\omega}}</math>

<math>(0)(1,1)(2)(3,1)=\varepsilon_{\varepsilon_0}</math>

<math>(0)(1,1)(2)(3,1)(1,1)=\varepsilon_{\varepsilon_0+1}</math>

<math>(0)(1,1)(2)(3,1)(3,1)=\varepsilon_{\varepsilon_1}</math>

<math>(0)(1,1)(2)(3,1)(4)=\varepsilon_{\varepsilon_\omega}</math>

<math>(0)(1,1)(2)(3,1)(4)(5,1)=\varepsilon_{\varepsilon_{\varepsilon_0}}</math>

<math>(0)(1,1)(2)(3,1)(4)(5,1)(6)(7,1)=\varepsilon_{\varepsilon_{\varepsilon_{\varepsilon_0}}}</math>

<math>(0)(1,1)(2,1)=\zeta_0</math>

<math>(0)(1,1)(2,1)(1)(2,1)(3,1)=\zeta_0^2</math>

<math>(0)(1,1)(2,1)(1)(2,1)(3,1)(2)(3,1)(4,1)=\zeta_0^{\zeta_0}</math>

<math>(0)(1,1)(2,1)(1,1)=\varepsilon_{\zeta_0+1}</math>

<math>(0)(1,1)(2,1)(1,1)(1)(2,1)(3,1)(2,1)=\varepsilon_{\zeta_0+1}^2</math>

<math>(0)(1,1)(2,1)(1,1)(1,1)=\varepsilon_{\zeta_0+2}</math>

<math>(0)(1,1)(2,1)(1,1)(2)=\varepsilon_{\zeta_0+\omega}</math>

<math>(0)(1,1)(2,1)(1,1)(2)(3,1)=\varepsilon_{\zeta_0+\varepsilon_0}</math>

<math>(0)(1,1)(2,1)(1,1)(2)(3,1)(4,1)=\varepsilon_{\zeta_0\times2}</math>

<math>(0)(1,1)(2,1)(1,1)(2)(3,1)(4,1)(3,1)=\varepsilon_{\varepsilon_{\zeta_0+1}}</math>

<math>(0)(1,1)(2,1)(1,1)(2)(3,1)(4,1)(3,1)(4)(5,1)(6,1)(5,1)=\varepsilon_{\varepsilon_{\varepsilon_{\zeta+1}}}</math>

<math>(0)(1,1)(2,1)(1,1)(2,1)=\zeta_1</math>

<math>(0)(1,1)(2,1)(1,1)(2,1)(1,1)=\varepsilon_{\zeta_1+1}</math>

<math>(0)(1,1)(2,1)(1,1)(2)=\varepsilon_{\zeta_1+\omega}</math>

<math>(0)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1)=\zeta_2</math>

<math>(0)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1)=\zeta_3</math>

<math>(0)(1,1)(2,1)(2)=\zeta_\omega</math>

<math>(0)(1,1)(2,1)(2)(2)=\zeta_{\omega^2}</math>

<math>(0)(1,1)(2,1)(2)(3,1)=\zeta_{\varepsilon_0}</math>

<math>(0)(1,1)(2,1)(2)(3,1)(4,1)=\zeta_{\zeta_0}</math>

<math>(0)(1,1)(2,1)(2)(3,1)(4,1)(4)(5,1)(6,1)=\zeta_{\zeta_{\zeta_0}}</math>

<math>(0)(1,1)(2,1)(2,1)=\eta_0</math>

<math>(0)(1,1)(2,1)(2,1)(1,1)=\varepsilon_{\eta_0+1}</math>

<math>(0)(1,1)(2,1)(2,1)(1,1)(2,1)=\zeta_{\eta_0+1}</math>

<math>(0)(1,1)(2,1)(2,1)(1,1)(2,1)(2)(3,1)(4,1)(4,1)(3,1)(4,1)=\zeta_{\zeta_{\eta_0+1}}</math>

<math>(0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)=\eta_1</math>

<math>(0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)=\eta_2</math>

<math>(0)(1,1)(2,1)(2,1)(2)=\eta_\omega</math>

<math>(0)(1,1)(2,1)(2,1)(2,1)=\theta_0=\varphi(4,0)=\psi(\Omega^3)</math>

<math>(0)(1,1)(2,1)(2,1)(2,1)(2,1)=\iota_0=\varphi(5,0)=\psi(\Omega^4)</math>

<math>(0)(1,1)(2,1)(2,1)(2,1)(2,1)(2,1)=\kappa_0=\varphi(6,0)=\psi(\Omega^5)</math>

<math>(0)(1,1)(2,1)(2,1)(2,1)(2,1)(2,1)(2,1)=\lambda_0=\varphi(7,0)=\psi(\Omega^6)</math>

<math>(0)(1,1)(2,1)(3)=\varphi(\omega,0)=\psi(\Omega^\omega)</math>

<math>(0)(1,1)(2,1)(3)(1,1)=\varphi(1,\varphi(\omega,0)+1)=\psi(\Omega^\omega+1)</math>

<math>(0)(1,1)(2,1)(3)(1,1)(2,1)=\varphi(2,\varphi(\omega,0)+1)=\psi(\Omega^\omega+\Omega)</math>

<math>(0)(1,1)(2,1)(3)(1,1)(2,1)(2,1)=\varphi(3,\varphi(\omega,0)+1)=\psi(\Omega^\omega+\Omega^2)</math>

<math>(0)(1,1)(2,1)(3)(1,1)(2,1)(3)=\varphi(\omega,1)=\psi(\Omega^\omega\times2)</math>

<math>(0)(1,1)(2,1)(3)(1,1)(2,1)(3)(1,1)(2,1)(3)=\varphi(\omega,2)=\psi(\Omega^\omega\times3)</math>

<math>(0)(1,1)(2,1)(3)(2)=\varphi(\omega,\omega)=\psi(\Omega^\omega\times\omega)</math>

<math>(0)(1,1)(2,1)(3)(2,1)=\varphi(\omega+1,0)=\psi(\Omega^{\omega+1})</math>

<math>(0)(1,1)(2,1)(3)(2,1)(2,1)=\varphi(\omega+2,0)=\psi(\Omega^{\omega+2})</math>

<math>(0)(1,1)(2,1)(3)(2,1)(2,1)(2,1)=\varphi(\omega+3,0)=\psi(\Omega^{\omega+3})</math>

<math>(0)(1,1)(2,1)(3)(2,1)(3)=\varphi(\omega\times2,0)=\psi(\Omega^{\omega\times2})</math>

<math>(0)(1,1)(2,1)(3)(2,1)(3)(2,1)(3)=\varphi(\omega\times3,0)=\psi(\Omega^{\omega\times3})</math>

<math>(0)(1,1)(2,1)(3)(3)=\varphi(\omega^2,0)=\psi(\Omega^{\omega^2})</math>

<math>(0)(1,1)(2,1)(3)(4,1)=\varphi(\varphi(1,0),0)=\psi(\Omega^{\psi(0)})</math>

<math>(0)(1,1)(2,1)(3)(4,1)(5,1)=\varphi(\varphi(2,0),0)=\psi(\Omega^{\psi(\Omega)})</math>

<math>(0)(1,1)(2,1)(3)(4,1)(5,1)(6)=\varphi(\varphi(\omega,0),0)=\psi(\Omega^{\psi(\Omega^\omega)})</math>

<math>(0)(1,1)(2,1)(3)(4,1)(5,1)(6)(7,1)(8,1)(9)=\varphi(\varphi(\varphi(\omega,0),0),0)=\psi(\Omega^{\psi(\Omega^{\psi(\Omega^\omega)})})</math>

<math>(0)(1,1)(2,1)(3,1)=\varphi(1,0,0)=\Gamma_0=\psi(\Omega^\Omega)</math>

<math>(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)=\varphi(1,0,1)=\Gamma_1=\psi(\Omega^\Omega\times2)</math>

<math>(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)=\varphi(1,0,2)=\Gamma_2=\psi(\Omega^\Omega\times3)</math>

<math>(0)(1,1)(2,1)(3,1)(2)=\varphi(1,0,\omega)=\Gamma_\omega=\psi(\Omega^\Omega\times\omega)</math>

<math>(0)(1,1)(2,1)(3,1)(2)(3,1)(4,1)(5,1)=\varphi(1,0,\varphi(1,0,0))=\Gamma_{\Gamma_0}=\psi(\Omega^\Omega\times\psi(\Omega^\Omega))</math>

<math>(0)(1,1)(2,1)(3,1)(2)(3,1)(4,1)(5,1)(4)(5,1)(6,1)(7,1)=\varphi(1,0,\varphi(1,0,\varphi(1,0,0)))=\Gamma_{\Gamma_{\Gamma_0}}=\psi(\Omega^\Omega\times\psi(\Omega^\Omega\times\psi(\Omega^\Omega)))</math>

<math>(0)(1,1)(2,1)(3,1)(2,1)=\varphi(1,1,0)=\psi(\Omega^{\Omega+1})</math>

<math>(0)(1,1)(2,1)(3,1)(2,1)(2,1)=\varphi(1,2,0)=\psi(\Omega^{\Omega+2})</math>

<math>(0)(1,1)(2,1)(3,1)(2,1)(3)=\varphi(1,\omega,0)=\psi(\Omega^{\Omega+\omega})</math>

<math>(0)(1,1)(2,1)(3,1)(2,1)(3)(4,1)(5,1)(6,1)(5,1)(6)=\varphi(1,\varphi(1,\omega,0),0)=\psi(\Omega^{\Omega+\psi(\Omega^{\Omega+\omega})})</math>

<math>(0)(1,1)(2,1)(3,1)(2,1)(3,1)=\psi(2,0,0)=\psi(\Omega^{\Omega\times2})</math>

<math>(0)(1,1)(2,1)(3,1)(2,1)(3,1)(2,1)(3,1)=\psi(3,0,0)=\psi(\Omega^{\Omega\times3})</math>

<math>(0)(1,1)(2,1)(3,1)(3)=\varphi(\omega,0,0)=\psi(\Omega^{\Omega\times\omega})</math>

<math>(0)(1,1)(2,1)(3,1)(3,1)=\varphi(1,0,0,0)=\psi(\Omega^{\Omega^2})</math>

<math>(0)(1,1)(2,1)(3,1)(3,1)(3,1)=\varphi(1,0,0,0,0)=\psi(\Omega^{\Omega^3})</math>

<math>(0)(1,1)(2,1)(3,1)(4)=\psi(\Omega^{\Omega^\omega})</math>

<math>(0)(1,1)(2,1)(3,1)(4)(5,1)=\psi(\Omega^{\Omega^{\psi(0)}})</math>

<math>(0)(1,1)(2,1)(3,1)(4)(5,1)(6,1)=\psi(\Omega^{\Omega^{\psi(\Omega)}})</math>

<math>(0)(1,1)(2,1)(3,1)(4)(5,1)(6,1)(7,1)=\psi(\Omega^{\Omega^{\psi(\Omega^\Omega)}})</math>

<math>(0)(1,1)(2,1)(3,1)(4,1)=\psi(\Omega^{\Omega^\Omega})</math>

<math>(0)(1,1)(2,1)(3,1)(4,1)(5,1)=\psi(\Omega^{\Omega^{\Omega^\Omega}})</math>

<math>(0)(1,1)(2,2)=\psi(\psi_1(0))</math>
[[分类:分析]]

FZ Hydra

2025-07-04T05:52:49Z

YukinaKNK：

FZ Hydra

2025-07-04T05:46:57Z

YukinaKNK：

FZ Hydra

2025-07-03T15:54:15Z

YukinaKNK：创建页面，内容为“注：该记号定义程度为7.4，原因是KNK不会写fake系列的规则（主规则： <math>\psi_Z(0)=1</math> <math>\psi_Z[\&](\#+1)=\psi_Z[\&](\#)+\psi_Z[\&](\#)+\psi_Z[\&](\#)+...</math> 兼容规则（只展示KNK会写的，后续会陆续更新出完整的规则）： <math>\psi_{Z_n}(\psi_{Z_{n+1}}(0))=\psi_{Z_n}[\psi_{Z_{n+1}}(0)](\psi_{Z_{n+1}}[\psi_{Z_{n+2}}(0)](\psi_{Z_{n+2}}[\psi_{Z_{n+3}}(0)](...)))</math> <math>\psi_{Z_n}[\psi_{Z_{n+1…”

讨论:首页

2025-07-03T04:18:09Z

YukinaKNK：/* 这里是一个讨论 */ 回复