User:Googleaarex/Extended Up-Arrow Notation/NestedArray

Dimensional/Nested Array Up-Arrow Notation is 4th part of Extended Up-arrow Notation series. {{function
 * name=Dimensional/Nested Array Up-Arrow Notation
 * base=Exponentiation
 * fgh=\psi(\psi_{\chi(\varepsilon_{M+1})}(0))

Rules
7. \(a \square\uparrow_{,_{\square\uparrow}\square\uparrow} b = a \square\uparrow_{\underbrace{,_{\square},_{\square} ... ,_{\square},_{\square}}_{b}\uparrow,_{\square\uparrow}\square} a\)

Analysis
\(\uparrow_{,_{\uparrow\uparrow}\uparrow}\) has level \(\psi(\psi_{I(\omega,0)}(0))\)

\(\uparrow_{,_{\uparrow\uparrow}\uparrow\uparrow}\) has level \(\psi(\psi_{I(\omega,1)}(0))\)

\(\uparrow_{,_{\uparrow\uparrow},\uparrow}\) has level \(\psi(\psi_{I(\omega + 1,0)}(0))\)

\(\uparrow_{,_{\uparrow\uparrow},_{\uparrow\uparrow}\uparrow}\) has level \(\psi(\psi_{I(\omega 2,0)}(0))\)

\(\uparrow_{,_{\uparrow^n}\uparrow}\) has level \(\psi(\psi_{I(\omega^{n-1},0)}(0))\)

\(\uparrow_{,_{,\uparrow},\uparrow}\) has level \(\psi(\psi_{I(1,1,0)}(0))\)

\(\uparrow_{,_{,\uparrow},_{,\uparrow}\uparrow}\) has level \(\psi(\psi_{I(2,0,0)}(0))\)

\(\uparrow_{,_{,\uparrow\uparrow}\uparrow}\) has level \(\psi(\psi_{I(1,0,0,0)}(0))\)

\(\uparrow_{,_{,\uparrow_2}\uparrow}\) has level \(\psi(\psi_{\chi(M^\omega)}(0))\)

\(\uparrow_{,_{,,\uparrow}\uparrow}\) has level \(\psi(\psi_{\chi(M^M)}(0))\)

\(\uparrow_{,_{,_2\uparrow}\uparrow}\) has level \(\psi(\psi_{\chi(M^{M^\omega})}(0))\)

\(\uparrow_{,_{,_{,\uparrow}\uparrow}\uparrow}\) has level \(\psi(\psi_{\chi(M^{M^M})}(0))\)

Limit is level \(\psi(\psi_{\chi(\varepsilon_{M+1})}(0))\)