[cf.\ Lemma 2 of \cite{Cichon1983}]\label{tricklem} For $k\in[2,\omega)$ and $\alpha\in\T\cap\Om_1$ the operations $\Ck$ and $\Pk$ commute, thus \[\Ck\Pk\alpha=\Pk\Ck\alpha,\]
where the latter is equal to $\Ck\alpha-1$.

\begin{theo}\label{maintheo} For $k\in[2,\om)$ the restriction of $\Ck$ to the set $\Tk$ is an order isomorphism with image $\N$.
\end{theo}

\begin{equation}\label{hkHardyeq} \hk(\al)=H_\al(k).\end{equation}

\section{Ordinal algebra}

\subsection{Preliminaries}\label{prelimsec}

Besides denoting the class of limit ordinals as $\Lim$, we denote the class of additive principal numbers as $\Hz$, $\Hz^{>1}:=\Hz\setminus\{1\}$,
and the class of (non-zero) ordinals closed under $\om$-exponentiation, also called $\eps$-numbers, by $\Ez$.

For terms $\al$ we use the following abbreviations. Writing $\al=_\NF\xi+\eta$ means that $\eta\in\Hz$ and $\xi\ge\eta$ is minimal such that $\al=\xi+\eta$,
$\al=_\ANF\xi_1+\ldots+\xi_k$ means that $\xi_1,\ldots,\xi_k\in\Hz$ and $\xi_1\ge\ldots\ge\xi_k$, and to indicate Cantor normal form representation of an ordinal $\al$,
we write $\al=_\CNF\om^{\ze_1}+\ldots+\om^{\ze_k}$ where $\ze_1\ge\ldots\ge\ze_k$.
For the result of $l$-fold addition of the same term $\eta$ to a term $\xi$, where $l<\om$,
we will sometimes use the shorthand $\xi+\eta\cdot l$. We further define $\sumend(0):=0$ and $\sumend(\al):=\xi_k$ if $\al=_\ANF\xi_1+\ldots+\xi_k$ where $k\ge1$.

Setting $\Om_0:=\om$ and $\Omie:=\aleph_{i+1}$, terms in $\T$ are built up from $0$, $1$, ordinal addition, and functions $\thti$ for each $i<\om$, where $\thti(0)=\Omi$
and arguments of $\thti$ are restricted to terms below $\Omiz$. The functions $\thti$ are natural extensions of the fixed point-free Veblen functions and hence injective,
with images contained in the intervals $[\Omi,\Omie)$, respectively for $i<\om$. Compared to the setup in \cite{W26}, defining $\Om_0:=\om$ instead of $\Om_0=1$
and consequently $\thtnod(i):=\om^{i+1}$ instead of $\thtnod(i)=\om^i$ for $i<\om$ is the only modification, which is needed for technical smoothness.

As in subsection 2.1 of \cite{W26}, slightly abusing notation we may consider notation systems
$\Tomie$ to be systems relativized to the initial segment $\Omie$ of ordinals and built up over
$\Omie=\thtie(0), \Om_{i+2}=\tht_{i+2}(0),\ldots$, i.e.\ without renaming the indices of $\tht$-functions. In this terminology $\T=\Tomnod$, and the sequence
$(\Tomi)_{i<\om}$ can be seen as an increasing sequence with larger and larger initial segments of ordinals serving as parameters.

The operation $\cdot^\stari$ searches a $\T$-term for its $\thti$-subterm of largest ordinal value, but under the restriction to treat $\thtj$-subterms
for $j<i$ as atomic. If such largest $\thti$-subterm does not exist, $\cdot^\stari$ returns $0$.  For $\al=\thti(\xi)$ and $\be=\thti(\eta)$ (where $\xi$ and $\eta$ are
any elements of the domain of $\thti$) we have \[\al<\be\Longleftrightarrow(\xi<\eta\mbox{ and }\xi^\stari<\be)\mbox{ or }\al\le\eta^\stari.\]

As in \cite{W26} the arguments of $\thti$-terms $\thti(\xi)$ are often written in a form $\thti(\De+\eta)$, i.e.\ $\xi=\De+\eta$. This by convention always indicates that
$\Omie\mid\De$ (possibly $\De=0$) and $\eta<\Omie$.
Informally, we sometimes call the multiple $\De$ of $\Omie$ the fixed point level of $\thti(\xi)$, and the
predicate $F_i(\De,\eta)$ means that $\eta=\sup^+_{\si<\eta}\thti(\De+\si)$, which according to Proposition 2.6 of \cite{W26} holds if and only if $\eta$ is of a
form $\thti(\Ga+\rho)$ where $\Ga>\De$ and $\eta>\De^\stari$.

Any $\al\in\T\cap\Hz^{>1}$ is of a form $\thti(\De+\eta)$ for unique $i<\om$, $\De\in\T\cap\Omiz$ with $\Omie\mid\De$, and $\eta<\Omie$.
Let $\lv(\al):=i$ denote the \emph{level of $\al$} and for $j\le i$ let $\Par_j(\al)$ denote the set of $\tht$-subterms of level strictly less than $j$
(so that $\Par_0(\al)=\emptyset$).
We will sometimes write $\al\cdot\om$ as a shorthand for the additive principal successor of $\al=\thti(\De+\eta)$ in $\T$,
instead of $\thti(\al)$ in case of $\De>0$ and $\thti(\eta+1)$ if $\De=0$.

For terms $\al$ of a form $\thti(\De+\eta)$ the $\Omi$-localization, cf.\ subsection 2.3 of \cite{W26}, is a sequence $\Omi=\al_0,\ldots,\al_m=\al$ ($m\ge0$),
in which for $\al>\Omi$ the terms $\al_i=\thti(\De_i+\eta_i)$, $i=1,\ldots,m$, provide a strictly increasing sequence of $\thti$-subterms of $\al$ that are not in the scope of any
$\thtj$-function with $j<i$ (i.e.\ that are not collapsed), such that the sequence of fixed point levels $\De_1,\ldots,\De_m$ is strictly \emph{decreasing} and of maximal length,
and where the $\eta_i$ are maximal.
Localization therefore approximates the ordinal $\al$ from below in terms of decreasing fixed point levels.

Restricting ordinal addition occuring in $\T$ to summation in additive normal form, the terms in $\T$ \emph{uniquely} identify ordinal numbers below Takeuti ordinal.
The term algebra developed in \cite{W26} and here therefore can be understood as algebra of ordinals.

We introduce a simple but crucial term measure for inductive definitions along ordinals in $\T$.
\begin{defi}\label{htdefi} We define the height measure $\hti:\T\to\om$ where $i<\om$ as follows.
\[\hti(\al):=\left\{\begin{array}{cl}
   m+1 & {\small \mbox{ if } m=\max\set{k}{\mbox{there is a $\tht_{i+k}$-subterm of $\al$ that is not in the scope of any $\thtj$ with $j<i$}}}\\
   0   & {\small \mbox{ if such $m$ does not exist.}}
   \end{array}\right.\]
\end{defi}

We may extend the definition of $\hti$ also to the set $\Tomi$ which instead of terms $\thtj$ for $j<i$ contains opaque parameters for all ordinals below $\Omi$.

\begin{lem}\label{htlem}
For $\al<\Tomi\cap\Omie$
\[\hti(\al)=\min\set{n}{\al<\thti(\ldots(\tht_{i+n}(0))\ldots)}.\]
$\hti$ is weakly increasing on $\Tomi\cap\Omie$.
\end{lem}

For reader's convenience we recall from \cite{W26} the following crucial definition of a characteristic function regarding uncountable moduli and then
provide the concrete system of fundamental sequences adapted from Definition 3.5 of \cite{W26} to fit with our modified definition of $\thtnod(i)$ for $i<\om$.
\begin{defi}[3.1 and 3.3 of \cite{W26}]\label{chidefi}
We define characteristic functions $\chiomie:\Tomie\to\{0,1\}$ for $i<\om$ by recursion on the build-up of $\Tomie$:
\begin{enumerate}
\item $\chiomie(\al):=\left\{\begin{array}{cl}
                           0&\mbox{ if } \al<\Omie\\
                           1&\mbox{ if } \al=\Omie,
           \end{array}\right.$
\item $\chiomie(\al):=\chiomie(\eta)$ if $\al=_\NF\xi+\eta$,
\item $\chiomie(\al):=\left\{\begin{array}{cl}
          \chiomie(\De) & \mbox{ if } \eta\not\in\Lim\mbox{ or }F_j(\De,\eta)\\[2mm]
          \chiomie(\eta)& \mbox{ otherwise,}
      \end{array}\right.$\\[2mm]
if $\al=\thtj(\De+\eta)>\Omie$ and hence $j\ge i+1$.
\end{enumerate}
For $\al\in\T$ we define $\domf(\al):=i+1$ if $\chiomie(\al)=1$ for some $i<\om$ and $\domf(\al):=0$ otherwise, so that \[\Tcirc=\{\al\in\T\mid\domf(\al)=0\}\]
is the subset of $\T$ containing the terms of ordinals of (at most) countable cofinality.
\end{defi}

Note that we obtain a partitioning of $\T$ in terms of cofinality through the preimages $\domf^{-1}(n)$, $n<\om$, cf.\ Lemma 3.4 of \cite{W26}.
We separately define the \emph{support term} $\ual$ of a $\tht$-term $\al$ first, as follows.

\begin{defi}[cf.\  3.5 of \cite{W26}]\label{supporttermdefi} Let $\al\in\T$ be of the form $\al=\thti(\De+\eta)$ for some $i<\om$, and denote the
$\Omi$-localization of $\al$ by $\Omi=\al_0,\ldots,\alm=\al$. The support term $\ual$ for $\al$ is defined by
\[\ual:=\left\{\begin{array}{cl}
            \almmin & \mbox{ if either } F_i(\De,\eta)\mbox{, or: } \eta=0\mbox{ and }\De[0]^{\star_i}<\almmin=\De^{\star_i}
            \mbox{ where }m>1\\[2mm]
            \thti(\De+\etapr) & \mbox{ if } \eta=\etapr+1\\[2mm]
            1 & \mbox{ if } \al=\om\\[2mm]
            0 & \mbox{ otherwise.}
     \end{array}\right.\]
\end{defi}

\begin{defi}[cf.\ 3.5 of \cite{W26}]\label{bsystemdefi}
Let $\al\in\T$. By recursion on the build-up of $\al$ we define the function \[\al[\cdot]:\aleph_d\to\T^{\Om_d}\] where $d:=\domf(\al)$.
Let $\ze$ range over $\aleph_d$.
\begin{enumerate}
\item $0[\ze]:=0$ and $1[\ze]:=0$.
\item $\al[\ze]:=\xi+\eta[\ze]$ if $\al=_\NF\xi+\eta$.
\item For $\al=\thti(\De+\eta)$ where $i<\om$, noting that $d\le i$, the definition then proceeds as follows.
\begin{enumerate}
\item[3.1.] If $\eta\in\Lim$ and $\neg F_i(\De,\eta)$, that is, $\eta\in\Lim\cap\sup_{\si<\eta}\thti(\De+\si)$, we have $d=d(\eta)$
and define
   \[\al[\ze]:=\left\{\begin{array}{cl}
   \thtnod(\ze)\ & \mbox{ if } \al=\thtnod(\thtnod(0))=\om^\om\\[2mm]
   \thti(\De+\eta[\ze]) & \mbox{ otherwise.}
   \end{array}\right.\]
\item[3.2.] If otherwise $\eta\not\in\Lim$ or $F_i(\De,\eta)$, we distinguish between the following 3 subcases.
\begin{enumerate}
\item[3.2.1.] If $\De=0$, define \[\al[\ze]:=\left\{\begin{array}{cl} \ual\cdot(\ze+1) & \mbox{ if } d=0,\\
            \ze & \mbox{ if } d>0.
       \end{array}\right.\]
\item[3.2.2.] $\chiomie(\De)=1$. This implies that $d=0$, and we define recursively in $n<\om$
\[\al[0]:=\thti(\De[\ual])\quad\mbox{ and }\quad\al[n+1]:=\thti(\De[\al[n]]).\]
\item[3.2.3.] Otherwise. Then $d=d(\De)$ and \[\al[\ze]:=\thti(\De[\ze]+\ual).\]
\end{enumerate}
\end{enumerate}
\end{enumerate}
\end{defi}

The following theorem states that terms in $\T$ satisfy Bachmann property, which is crucial for our approach toward generalizing Goodstein's theorem.

\begin{theo}[Lemma 4.1 and Theorem 4.2 of \cite{W26}]\label{Bachmannprop}
For $\al, \be\in\T$, where $\al$ is not a successor-multiple of a regular cardinal, and for which $\al[\ze]<\be<\al$ holds for some $\ze<\aleph_{\domf(\al)}$, we have \[\al[\ze]\le\be[0].\]
\end{theo}

The following lemma states a basic observation about fundamental sequences with Bachmann property. Note first that whenever $\al[\xi]=\al[\ze]$ where
$\al\in\T$ and $\xi,\ze<\aleph_{\domf(\al)}$, then $\xi=\ze$ since fundamental sequences are strictly increasing. Note further that for any $\al\in\T\cap\Hz$
we have $\al\cdot\om[0]=\al$.

\begin{cor} Let $\al,\be\in\T$ such that $\be<\al$ and $\al$ is not a successor-multiple of any $\Omje$. Let $\ze<\aleph_{\domf(\be)}$, $\xi<\aleph_{\domf(\al)}$,
and suppose that \[\al[\xi]=\be[\ze].\] Then it follows that $\ze=0$.
\end{cor}
{\bf Proof.} This is an immediate consequence of Bachmann property, Theorem \ref{Bachmannprop}.
\qed

By means of the following inversion lemma we will be able to recover $\be$ from terms $\al\in\T$ of a form $\be[n]$ where $n\in[1,\om)$ and $\be\in\Tcirc\cap\Lim$,
as used in Definition \ref{quotientdefi}. As in Definition 3.5 of \cite{W26} we need to formulate the inversion lemma more generally, so as to cover terms of uncountable cofinality.
The inversion is formulated in a way that will cover the cases needed to define the quotients of Definition \ref{quotientdefi}, rather than covering all possible cases.

We first introduce a partial auxiliary function that will smoothen the formulation of the subsequent lemma, cf.\ the support term $\ual$ in Definition \ref{supporttermdefi}.

\begin{defi} For $i<\om$, $\De\in\T$ such that $\Omie\mid\De<\Omiz$, and $\rho\in\T$ we define the partial function $\etafct(i,\De,\rho)$ as follows.
\[\etafct(i,\De,\rho):=\left\{\begin{array}{cl}
           \rho & \mbox{ if } F_i(\De,\rho) \mbox{ holds,} \\[2mm]
           0 & \mbox{ if either } \rho=0,\; \rho=1 \mbox{ and }i=\De=0, \mbox{ or: } \De[0]^\stari<\al_{m-1}=\De^\stari=\rho\\[1mm]
              & \mbox{ where } \al_1,\ldots,\al_m \mbox{ is the $\Omi$-localization of } \thti(\De) \mbox{ and } m>1, \\[2mm]
           \nu+1 & \mbox{ if } \rho=\thti(\De+\nu) \mbox{ for some }\nu\in\T\cap\Omie, \\[2mm]
           \mbox{ undefined } & \mbox{ otherwise.}
      \end{array}\right.\]
\end{defi}

\begin{rmk}
In the cases where $\eta:=\etafct(i,\De,\rho)$ is defined, for $\al:=\thti(\De+\eta)$ we then have $\ual=\rho$, and either $\eta\not\in\Lim$ or $F_i(\De,\eta)$ holds.
\end{rmk}

\begin{lem}[Inversion Lemma]\label{inversionlem} Let $\al\in\T$. $\al$ is of a form $\al=\be[\ze]$ where \begin{enumerate}
\item $\ze\in[1,\om)$ and $\be\in\Tcirc\cap\Lim$, or
\item $\ze\in[\Omi,\Omie)\cap\Hz$ for some $i<\om$ and $\be\in\T$ such that $\chiomie(\be)=1$
\end{enumerate}
if and only if one of the following cases applies:
\begin{enumerate}
\item $\al=\ze\in[\Omi,\Omie)\cap\Hz$ and $\be=\Omie=\thtie(0)$.
\item $\al=\eta\cdot(n+1)$ for some $\eta\in\T\cap\Hz$ and $\ze=n\in[1,\om)$. Then $\be:=\eta\cdot\om$, and we have $\ube=\eta$ and $\al=\be[\ze]$.
\item $\al=\xi_1+\ldots+\xi_k+\rho$ where $\xi_1,\ldots,\xi_k\in\Hz$ is weakly decreasing, $k\ge1$, and $\rho=\eta[\ze]$ according to the lemma's conditions with
$\eta\in(1,\xi_k]\cap\Hz$ so that $\rho<\xi_k$.
Then setting $\be:=\xi_1+\ldots+\xi_k+\eta$ we have $\al=\be[\ze]$.
\item $\al=\thtj(\Ga+\rho)$ and one of the following subcases applies:
\begin{enumerate}
\item $\rho$ is of a form $\rho=\eta[\ze]$ according to the lemma's conditions where $\eta<\Omje$ and $F_j(\Ga,\eta)$ does not hold. Then setting $\De:=\Ga$ and
$\be:=\thtj(\De+\eta)$ we have $\al=\be[\ze]=\thtj(\De+\eta[\ze])$.
\item Otherwise, setting $\xi_1:=\Ga+\rho$, check whether there is a (shortest) sequence $\xi_1,\ldots,\xi_{m+1}$ (where $m\ge1$)
that determines a term $\De$ with $\Omje\mid\De<\Omjz$ and $\chiomje(\De)=1$ in the first step, such that
\begin{enumerate}
\item $\xi_k$ is of a form $\De[\thtj(\xi_{k+1})]$ which is according to the lemma's conditions (here according to condition 2 with i=j)
for $k=1,\dots,m$, and
\item $\xi_{m+1}$ is of a form $\De[\nu]$ where $\eta:=\eta(j,\De,\nu)$ is defined, so that $\be:=\thtj(\De+\eta)$, $\ube=\nu$,
and $\al=\be[m]$.
\end{enumerate}
This case then applies if  $\ze=m\ge1$.
\item $\Ga$ is of a form $\De[\ze]$ according to the lemma's conditions where $\Omje\mid\De<\Omjz$ and $\chiomje(\De)=0$, so that $\al=\thtj(\De[\ze]+\rho)$,
and $\eta:=\eta(j,\De,\rho)$ is defined, so that $\be=\thtj(\De+\eta)$, $\ube=\rho$, and $\al=\be[\ze]$.
\end{enumerate}
\end{enumerate}
\end{lem}
{\bf Proof.} Correctness follows by induction on $\al\in\T$: If one of cases $1$ - $4$ holds with $\be$ and $\ze$, then $\al=\be[\ze]$ matches either condition $1$ or $2$.
The reverse direction, completeness, follows by induction on $\be\in\T$, showing that for any $\be\in\T$ and $\ze$ according to either condition $1$ or $2$, the ordinal
$\al:=\be[\ze]$ satisfies one of cases $1$ - $4$.
\qed

\begin{rmk} Considering the example $\epsn$ as in the introduction, $\epsn[1]=\om^\om$ is the first iterative repetition, while $\epsn[0]=\om=\thtnod(0)$.
\end{rmk}

\subsection{Quotients}
For any $\be\in\T$ it can be syntactically detected whether $\be$ is of a form $\be=\al[n]$ for some $\al\in\Tcirc\cap\Lim$ and $n\in[1,\om)$.
Before we establish this explicitly, we use it in the following
\begin{defi}\label{quotientdefi}
We inductively define the elements of $(\Tquok)_{2\le k<\om}$ as increasing sequence of subsets of $\T$  as follows:
\begin{enumerate}
\item $0, 1\in\Tquok$.
\item If $\xi=_\ANF\xi_1+\ldots+\xi_n$ where $n\ge0$ and $\eta\in\Hz$ such that $\eta<\xi_n$ if $n>0$, then $\xi,\eta\in\Tquok$ implies that $\xi+\eta\cdot l\in\Tquok$
if and only if $l<k$.\item If $\xi=_\ANF\xi_1+\ldots+\xi_m\in\T\cap\Omiz$ where $m\ge0$ and $i<\om$, then $\al:=\thti(\xi)\in\Tquok$, provided that
\begin{enumerate}
\item $\xipr\in\Tquok$ where $\xipr:=1+\xi$ if $i=0$ and $\xipr:=\xi$ otherwise,
\item $\al[0]\in\Tquok$,
\item $\al_1,\ldots,\al_{m-1}\in\Tquok$ where $\al_l:=\thti(\xi_1+\ldots+\xi_l)$ for $l=1,\ldots,m-1$ in case of $m>1$, and
\item $\al$ is not of a form $\be[n]$ where $\be\in\Tcirc\cap\Lim$ and $n\ge k-1$.
\end{enumerate}
\end{enumerate}
The quotients $\Tcircquok\subseteq\Tcirc$ are then defined by $\Tcircquok:=\Tquok\cap\Tcirc$, and we define the term sets relevant for the Goodstein process by
\[\Tk:=\Tcircquokbig\cap\Om_1.\]
\end{defi}

\begin{rmk} $\mbox{ }$
\begin{enumerate}
\item Note the condition $1+j\in\Tquok$ for $\thtnod(j)=\om^{1+j}\in\Tquok$ where $j<\om$, which is why we introduced $\xipr$.
\item Note that any of these quotients $\Tquok$ and $\Tcircquok$ should be closed under the operations $\cdot[l]$ for $l<k-1$. Due to Bachmann property this only needs
to be explicitly stipulated for $l=0$, as we shall see.
\item It is easy to see that we have \[\T=\bigcup_{k\in[2,\om)}\Tquokbig\quad\mbox{ and }\quad\Tcirc=\bigcup_{k\in[2,\om)}\Tcircquokbig.\]
\end{enumerate}
\end{rmk}

We are now going to show that the $<$-order type of the set
$\Tk=\Tcircquok\cap\Om_1$ (where $k\in[2,\om)$) is $\om$ and that $\Ck$ enumerates this set, preserving its $<$-order.
To this end, we first define the analogue $\imc$ (for \emph{iterative maximal coefficient}) of the measure $\mc$ that was used to define quotients $\Equok$ of $\Esc$.
Recall the notions $\lv$, $\Par_i$, and $\hti$ from Section \ref{prelimsec}.

\begin{defi} The iterative maximal coefficient $\imc(\al)$ of terms $\al\in\T$ is defined (in case of $\al\in\Hz^{>1}$) by primary recursion along the $\Par_i(\al)$-subterm
relation where $i:=\lv(\al)$, secondary recursion on $\hti(\al)$, and tertiary $<$-recursion on $\al$ as follows.
\begin{enumerate}
\item $\imc(0):=0$ and $\imc(1):=1$.
\item If $\xi=_\ANF\xi_1+\ldots+\xi_n\in\T$ where $n\ge0$ and $\eta\in\T\cap\Hz$ such that $\eta<\xi_n$ if $n>0$, then for $l\in(0,\om)$
\[\imc(\xi+\eta\cdot l):=\max\{\imc(\xi),\imc(\eta),l\}.\]
\item If $\al\in\T$ is of a form $\thti(\xi)$ where $\xi=_\ANF\xi_1+\ldots+\xi_m$ then, setting
\begin{itemize}
\item $\xipr:=\left\{\begin{array}{cl}
1+\xi & \mbox{  if }i=0\\
\xi     & \mbox{ otherwise,}
\end{array}\right.$ and
\item $\al_l:=\thti(\xi_1+\ldots+\xi_l)$ for $l=1,\ldots,m-1$,
\end{itemize}
we define
\[\imc(\al):=\max\{\imc(\xipr),\imc(\al[0]),\imc(\al_1),\ldots,\imc(\al_{m-1}),n+1\},\]
where either $n\ge1$ is maximal such that $\al$ is of a form $\al=\be[n]$ where $\be\in\Tcirc\cap\Lim$, or otherwise $n=0$.
\end{enumerate}
\end{defi}

\begin{rmk}
Note that we have $\imc(\al)=0$ if and only if $\al=0$. $\imc$ is weakly monotone with respect to the subterm relation, hence weakly increasing along
localization sequences. Since for $\al\in\T\cap\Hz^{>1}$ the definition of $\imc(\al)$ depends on $\imc(\al[0])$, we could not simply define by recursion
along (modified) build-up stages of $\al$. A straightforward induction along the build-up of $\al\in\T\cap\Hz^{>1}$ shows that for $j\le i:=\lv(\al)$ we have
$\Par_j(\al[0])\subseteq\overline{\Par_j(\al)}$ where $\overline{\:\cdot\:}$ denotes the closure under the operation $\cdot[0]$.
\end{rmk}

We now obtain the following characterization of the quotients $\Tquok$ and $\Tcircquok$.
\begin{lem}
For $k\in[2,\om)$ we have \[\Tquok=\{\al\in\T\mid\imc(\al)<k\}\quad\mbox{ and }\quad\Tcircquok=\{\al\in\Tcirc\mid\imc(\al)<k\}.\]
\end{lem}
 {\bf Proof.} This follows immediately by induction along the involved inductive definitions.
\qed

\begin{lem}\label{intervalcollapsinglem} For $\la\in\Tcirc\cap\Lim$ and $\be\in\T\cap[\la[k-1],\la)$ where $k\ge2$ we have $\imc(\be)\ge k$
and hence \[\Tquok\cap[\la[k-1],\la)=\emptyset.\]
\end{lem}
{\bf Proof.} We first observe that by definition of $\imc$, for any $\be\in\T$ \[\imc(\be)<k \Rightarrow \imc(\be[0])<k.\]
Suppose now that for some $\la\in\Tcirc\cap\Lim$, there exists a least $\be\in\T\cap[\la[k-1],\la)$ such that $\imc(\be)<k$.
As $\imc(\la[k-1])\ge k$, this implies that $\la[k-1]<\be<\la$, with $\be\in\Lim$ since clearly $\imc(\ga)\le\imc(\ga+1)$ for any $\ga$.
Bachmann property yields $\la[k-1]\le\be[0]$, which however contradicts the minimality of $\be$ since $\imc(\be[0])<k$.
\qed

The following Theorem \ref{maintheo} is in correspondence with  Lemma 1 and Remarks 1 and 2 of \cite{Cichon1983}. Since the generalized Goodstein principle introduced here induces
much faster growing operations on the natural numbers than just base-$k$ exponentiation, Lemma 1 of \cite{Cichon1983} does not generalize directly. It is therefore instructive
to sketch an alternative proof for the fact that the mapping $\Ck$ restricted to the quotient $\Equok$ is the Mostowski collapse of $\Equok$ onto $\om$.
This proof, which follows from the next lemma, then does generalize to the quotients $\Tk$.
For limit $\al$, by $i_\al$  we count how many times we can iterate the operation $\cdot[k-1]$ while staying within $\Lim$.

\begin{lem}\label{alternativelem}
For $k\in[2,\om)$ and $\al\in\Esc\cap\Lim$ let
\begin{equation}\label{ialeq}
i_\al:=\max\left\{i<\om\mid\forall j\le i\:\left(\al[k-1]^j\in\Lim\right)\right\}
\end{equation}

\begin{equation}\label{slowgrowingeq} \Ck(0):=0,\: \Ck(\al+1):=\Ck(\al)+1, \mbox{ and }\Ck(\la):=\Ck(\la[k-1])\mbox{ for }\la\in\Esc\cap\Lim,\end{equation}

Let $\Esc$ be the familiar notation system for ordinals below $\varepsilon_0$, the proof-theoretic ordinal of Peano arithmetic $\PA$,
built from $0$, $(\xi,\eta)\mapsto\xi+\eta$, and $\xi\mapsto\omega^\xi$ using Cantor normal form, where $\omega$ denotes the least infinite ordinal number.
For $\alpha\in\Esc$ let $\mc(\alpha)$ be the maximum counter of multiples that occur in $\alpha$, that is, the maximum number of times the same summand is consecutively added
in the notation of $\alpha$. For $k\in[2,\omega)$ define the quotient $\Equok\subseteq\Esc$ by \[\Equokbig:=\{\alpha\in\Esc\mid\mc(\alpha)<k\}.\]
Defining the system $(\Esc,\cdot[\cdot])$ of fundamental sequences (limit approximations) for $\Esc$ by
\[0[k]:=0,\;\;1[k]:=0,\;\;(\xi+\eta)[k]:=\xi+\eta[k],\;\;\omega^{\beta+1}[k]:=\omega^\beta\cdot(k+1),\;\mbox{ and }\;\omega^\lambda[k]:=\omega^{\lambda[k]},\]
and the slow-growing hierarchy $\Ck$ for $k>0$ by\footnote{Induction on $\alpha$ shows that our definitions are equivalent to Cichon's in \cite{Cichon1983} for $k>0$, where
Cichon's $\alpha[k]$ is equal to our $\alpha[k-1]$.}
\begin{equation}\label{slowgrowingeq} \Ck(0):=0,\: \Ck(\alpha+1):=\Ck(\alpha)+1, \mbox{ and }\Ck(\lambda):=\Ck(\lambda[k-1])\mbox{ for }\lambda\in\Esc\cap\mathrm{Lim},\end{equation}
where $\mathrm{Lim}$ denotes the class of limit ordinals, as shown in \cite{Cichon1983} the restricted mapping \[\Ck:\Equokbig\to\omega\]
is a bijection via transforming back and forth the bases $k$ and $\omega$, that is, writing a natural number $N$ in hereditary base-$k$ representation using $0$, addition,
and exponentiation to base $k$, replacing the base $k$ by $\omega$ we obtain $\Ckinv(N)$, the unique preimage of $N$ in $\Equok$.
The image of $\Equok$ under $\Ck$ is the Mostowski collapse of $\Equok$ and equal to $\omega$, i.e., the restriction of $\Ck$ to $\Equok$ is an order isomorphism.

Since $(\Equok)_{k\in[2,\omega)}$ is $\subseteq$-increasing, base transformation
from $k$ to $l$ for $2\le k<l<\omega$ is characterized by \[N[k\mapsto l]=\Cl(N[k\mapsto\omega]),\]
where $N<\omega$ is assumed to be given in hereditary base-$k$ representation, so that $N[k\mapsto\omega]\in\Equok\subseteq\Equol$. Thus, ordinals $\alpha\in\Esc$ can be
seen as direct limit representations of natural numbers in hereditary base-$k$ representation, provided that $\mc(\alpha)<k$.

In this article we generalize this mechanism to the notation system $\T$ for ordinals below Takeuti ordinal, that is, to the strength of the fragment $\pioneonecanod$ of second
order number theory, which is the strongest of the ``big five'' theories in Simpson's book on reverse mathematics, using fundamental
sequences and machinery introduced in \cite{W26}. The method is generalizable to stronger notation systems that provide or enable unique terms for ordinals and enjoy Bachmann property.

Adopting the preliminaries of \cite{W26}, subsections 2.1, 2.3, and 2.5, which are summarized here in subsection \ref{prelimsec} for reader's convenience,
let $\T$ be the notation system $\Tt$ for $\tau=1$ introduced there and let $\Tcirc$ be the
subset of terms of countable cofinality. The system $(\Tcirc,\cdot[\cdot])$ of fundamental sequences with Bachmann property given in Definition 3.5 of \cite{W26}
is called a Buchholz system. Bachmann property is a nesting property of fundamental sequences, namely that for terms $\alpha$ and $\beta$ such that $\alpha[\zeta]<\beta<\alpha$
where $\alpha$ is not a successor-multiple of a regular cardinal $\Om_{i+1}$, we have $\alpha[\zeta]\le\beta[0]$, see Theorem \ref{Bachmannprop}.

We will be able to use the approach by Cichon \cite{Cichon1983}, transferring the definition of collapsing functions $\Ck$ as defined in (\ref{slowgrowingeq})
and predecessor functions $\Pk$ for $k>0$, defined equivalently as in \cite{Cichon1983} by
\begin{equation}\label{predeq} \Pk(0):=0,\: \Pk(\alpha+1):=\alpha, \mbox{ and }\Pk(\lambda):=\Pk(\lambda[k-1])\mbox{ for }\lambda\in\mathrm{Lim},\end{equation}
to the Buchholz system $(\Tcirc,\cdot[\cdot])$, more specifically to its countable initial segment.
Note first of all that Cichon's crucial lemma also holds in our context, with the same proof by induction on $\alpha$:

We are going to introduce quotients $\Tquok$ and $\Tcircquok$ of $\T$ and $\Tcirc$, respectively, to base $k$ where $k\in[2,\om)$. The sets $\Tk:=\Tcircquok\cap\Om_1$,
each of which is cofinal in $\T\cap\Om_1$ and which are $\subseteq$-increasing in $k$, will be the canonical analogues of the quotients $\Equok$ mentioned above.
Consider as an instructive basic example the tower $k_k$ of height $k$ of exponentiation to base $k$.
Clearly, $k_k[k\mapsto\om]\in\Equok$, so that the direct limit corresponding to $k_k$ in $\Esc$ is $\om_k$, the tower of height $k$ of exponentiation to base $\om$.
This does not hold in $\Tcircquok$, where the corresponding ideal object is $\epsn=\lim_{n<\om}\om_n$.
The sets $\Tk$ therefore provide more refined hierarchies of
ideal objects uniquely denoting natural numbers, and functions $\Ck$ will act as their enumeration functions, see Theorem \ref{maintheo}.

These preparations allow us to quite canonically generalize Goodstein's theorem \cite{Goodstein} and Cichon's independence proof in \cite{Cichon1983} to obtain independence
of the theory $\pioneonecanod$.
The argumentation, ``\emph{Cichon's trick}'', is as follows: Given a starting base $k\in[2,\om)$ and a natural number $N\in\N$,
by Theorem \ref{maintheo} we obtain a unique $\al\in\Tk$ such that \[N=\Ck(\al)=:N_1.\] The Goodstein operation of incrementing the base of representation and subsequent
subtraction of $1$ is then given by \[N_1[k\mapsto k+1]-1=\Cke(\al)-1=\Pke\Cke\al=\Cke\Pke\al=:N_2,\]
where we used Lemma \ref{tricklem}. Iterating the procedure yields \[N_2[k+1\mapsto k+2]-1=\Ckz(\Pke\al)-1=\Pkz\Ckz(\Pke\al)=\Ckz\Pkz(\Pke\al)=:N_3,\]
so that we obtain the generalized Goodstein sequence \[N_{l+1}=\Ckl(\Pkl\ldots\Pke\al).\]
Termination of the generalized Goodstein sequence is therefore expressed by \[\exists l\:N_{l+1}=0,\]
and noting that $\Ck\al=0$ if and only if $\al=0$, we may reformulate the generalized Goodstein principle as follows:
\begin{equation}\label{reformulation}\forall k\in[2,\om)\:\forall\al\in\Tk\:\exists l\:\:\Pkl\ldots\Pke\al=0.\end{equation}
Thus, transfinite induction up to Takeuti ordinal proves this generalized Goodstein principle.
Independence then follows, once we show that the function $\hk:\T\cap\Om_1\to\N$ defined by
\begin{equation}\label{hkdefieq}\hk(\al):=k + \min\{l\mid\Pkl\ldots\Pke\al=0\}\end{equation}
can easily be expressed in terms of the Hardy function $H_\al$, as in \cite{Cichon1983} for the original Goodstein principle. Defining Hardy hierarchy along Takeuti ordinal
(i.e.\ the initial segment $\T\cap\Om_1$) in the same way as in \cite{Cichon1983} by
\begin{equation}\label{Hardydefi} H_0(x):=x,\:H_{\al+1}(x):=H_\al(x+1), \mbox{ and }H_\la(x):=H_{\la[x]}(x),\end{equation}
we obtain by straightforward $<$-induction on $\al$:
\begin{equation}\label{hkHardyeq} \hk(\al)=H_\al(k).\end{equation}

\begin{defi}[cf.\ 3.5 of \cite{W26}]\label{bsystemdefi}
Let $\al\in\T$. By recursion on the build-up of $\al$ we define the function \[\al[\cdot]:\aleph_d\to\T^{\Om_d}\] where $d:=\domf(\al)$.
Let $\ze$ range over $\aleph_d$.
\begin{enumerate}
\item $0[\ze]:=0$ and $1[\ze]:=0$.
\item $\al[\ze]:=\xi+\eta[\ze]$ if $\al=_\NF\xi+\eta$.
\item For $\al=\thti(\De+\eta)$ where $i<\om$, noting that $d\le i$, the definition then proceeds as follows.
\begin{enumerate}
\item[3.1.] If $\eta\in\Lim$ and $\neg F_i(\De,\eta)$, that is, $\eta\in\Lim\cap\sup_{\si<\eta}\thti(\De+\si)$, we have $d=d(\eta)$
and define
 \[\al[\ze]:=\left\{\begin{array}{cl}
 \thtnod(\ze)\ & \mbox{ if } \al=\thtnod(\thtnod(0))=\om^\om\\[2mm]
 \thti(\De+\eta[\ze]) & \mbox{ otherwise.}
 \end{array}\right.\]
\item[3.2.] If otherwise $\eta\not\in\Lim$ or $F_i(\De,\eta)$, we distinguish between the following 3 subcases.
\begin{enumerate}
\item[3.2.1.] If $\De=0$, define \[\al[\ze]:=\left\{\begin{array}{cl} \ual\cdot(\ze+1) & \mbox{ if } d=0,\\
 \ze & \mbox{ if } d>0.
 \end{array}\right.\]
\item[3.2.2.] $\chiomie(\De)=1$. This implies that $d=0$, and we define recursively in $n<\om$
\[\al[0]:=\thti(\De[\ual])\quad\mbox{ and }\quad\al[n+1]:=\thti(\De[\al[n]]).\]
\item[3.2.3.] Otherwise. Then $d=d(\De)$ and \[\al[\ze]:=\thti(\De[\ze]+\ual).\]
\end{enumerate}
\end{enumerate}
\end{enumerate}
\end{defi}

\begin{lem}[Inversion Lemma]\label{inversionlem} Let $\al\in\T$. $\al$ is of a form $\al=\be[\ze]$ where \begin{enumerate}
\item $\ze\in[1,\om)$ and $\be\in\Tcirc\cap\Lim$, or
\item $\ze\in[\Omi,\Omie)\cap\Hz$ for some $i<\om$ and $\be\in\T$ such that $\chiomie(\be)=1$
\end{enumerate}
if and only if one of the following cases applies:
\begin{enumerate}
\item $\al=\ze\in[\Omi,\Omie)\cap\Hz$ and $\be=\Omie=\thtie(0)$.
\item $\al=\eta\cdot(n+1)$ for some $\eta\in\T\cap\Hz$ and $\ze=n\in[1,\om)$. Then $\be:=\eta\cdot\om$, and we have $\ube=\eta$ and $\al=\be[\ze]$.
\item $\al=\xi_1+\ldots+\xi_k+\rho$ where $\xi_1,\ldots,\xi_k\in\Hz$ is weakly decreasing, $k\ge1$, and $\rho=\eta[\ze]$ according to the lemma's conditions with
$\eta\in(1,\xi_k]\cap\Hz$ so that $\rho<\xi_k$.
Then setting $\be:=\xi_1+\ldots+\xi_k+\eta$ we have $\al=\be[\ze]$.
\item $\al=\thtj(\Ga+\rho)$ and one of the following subcases applies:
\begin{enumerate}
\item $\rho$ is of a form $\rho=\eta[\ze]$ according to the lemma's conditions where $\eta<\Omje$ and $F_j(\Ga,\eta)$ does not hold. Then setting $\De:=\Ga$ and
$\be:=\thtj(\De+\eta)$ we have $\al=\be[\ze]=\thtj(\De+\eta[\ze])$.
\item Otherwise, setting $\xi_1:=\Ga+\rho$, check whether there is a (shortest) sequence $\xi_1,\ldots,\xi_{m+1}$ (where $m\ge1$)
that determines a term $\De$ with $\Omje\mid\De<\Omjz$ and $\chiomje(\De)=1$ in the first step, such that
\begin{enumerate}
\item $\xi_k$ is of a form $\De[\thtj(\xi_{k+1})]$ which is according to the lemma's conditions (here according to condition 2 with i=j)
for $k=1,\dots,m$, and
\item $\xi_{m+1}$ is of a form $\De[\nu]$ where $\eta:=\eta(j,\De,\nu)$ is defined, so that $\be:=\thtj(\De+\eta)$, $\ube=\nu$,
and $\al=\be[m]$.
\end{enumerate}
This case then applies if $\ze=m\ge1$.
\item $\Ga$ is of a form $\De[\ze]$ according to the lemma's conditions where $\Omje\mid\De<\Omjz$ and $\chiomje(\De)=0$, so that $\al=\thtj(\De[\ze]+\rho)$,
and $\eta:=\eta(j,\De,\rho)$ is defined, so that $\be=\thtj(\De+\eta)$, $\ube=\rho$, and $\al=\be[\ze]$.
\end{enumerate}
\end{enumerate}
\end{lem}
{\bf Proof.} Correctness follows by induction on $\al\in\T$: If one of cases $1$ - $4$ holds with $\be$ and $\ze$, then $\al=\be[\ze]$ matches either condition $1$ or $2$.
The reverse direction, completeness, follows by induction on $\be\in\T$, showing that for any $\be\in\T$ and $\ze$ according to either condition $1$ or $2$, the ordinal
$\al:=\be[\ze]$ satisfies one of cases $1$ - $4$.
\qed

\begin{cor}\label{esccollapscor} For $k\ge2$ the mapping $\Ck$ restricted to the quotient $\Equok$ is the Mostowski collapse of $\Equok$ onto $\N$.
\end{cor}
{\bf Proof.} As in Lemma \ref{intervalcollapsinglem} for quotients $\Tquok$, we have a corresponding situation in $\Equok$: For any $\la\in\Esc\cap\Lim$ and $\be\in[\la[k-1],\la)$
we have $\mc(\be)\ge k$, thus \[\Equok\cap[\la[k-1],\la)=\emptyset.\]
By Lemma \ref{alternativelem}, for any $\al\in\Equok\cap\Lim$ the mapping $\Ck$ counts the elements of
\[[\al_0,\al]\cap\Equok=\{\al_0\}\cup\{\al[k-1]^{i_\al}[j]\mid j<k-1\}\cup\{\al\}\] starting from $\Ck(\al_0)$ to $\Ck(\al)$ in increasing order.
 \qed

We close with a characterization of the ordinals collected in the sets $\Tk$ in terms of maximality.
\begin{rmk}[Maximality of elements of $\Tk$]\label{maxrmk}
Any ordinal $\al\in\Tk$ is maximal among all $\be\in\T\cap\Om_1$ such that $\Ck(\al)=\Ck(\be)$.
This is seen as follows: assuming otherwise, let $\be>\al$ be minimal such that $\Ck(\al)=\Ck(\be)$. Then $\be[k-1]<\al<\be$, so that according to Lemma \ref{intervalcollapsinglem} it would
follow that $\imc(\al)\ge k$, which is not the case as $\al\in\Tk$.
\end{rmk}

We are going to introduce quotients $\Tquok$ and $\Tcircquok$ of $\T$ and $\Tcirc$, respectively, to base $k$ where $k\in[2,\omega)$. The sets $\Tk:=\Tcircquok\cap\Om_1$,
each of which is cofinal in $\T\cap\Om_1$ and which are $\subseteq$-increasing in $k$, will be the canonical analogues of the quotients $\Equok$ mentioned above.
Consider as an instructive basic example the tower $k_k$ of height $k$ of exponentiation to base $k$.
Clearly, $k_k[k\mapsto\omega]\in\Equok$, so that the direct limit corresponding to $k_k$ in $\Esc$ is $\om_k$, the tower of height $k$ of exponentiation to base $\omega$.
This does not hold in $\Tcircquok$, where the corresponding ideal object is $\varepsilon_0=\lim_{n<\omega}\om_n$.
The sets $\Tk$ therefore provide more refined hierarchies of
ideal objects uniquely denoting natural numbers, and functions $\Ck$ will act as their enumeration functions, see Theorem \ref{maintheo}.

These preparations allow us to quite canonically generalize Goodstein's theorem \cite{Goodstein} and Cichon's independence proof in \cite{Cichon1983} to obtain independence
of the theory $\pioneonecanod$.
The argumentation, ``\emph{Cichon's trick}'', is as follows: Given a starting base $k\in[2,\omega)$ and a natural number $N\in{\mathbb N}$,
by Theorem \ref{maintheo} we obtain a unique $\alpha\in\Tk$ such that \[N=\Ck(\alpha)=:N_1.\] The Goodstein operation of incrementing the base of representation and subsequent
subtraction of $1$ is then given by \[N_1[k\mapsto k+1]-1=\Cke(\alpha)-1=\Pke\Cke\alpha=\Cke\Pke\alpha=:N_2,\]
where we used Lemma \ref{tricklem}. Iterating the procedure yields \[N_2[k+1\mapsto k+2]-1=\Ckz(\Pke\alpha)-1=\Pkz\Ckz(\Pke\alpha)=\Ckz\Pkz(\Pke\alpha)=:N_3,\]
so that we obtain the generalized Goodstein sequence \[N_{l+1}=\Ckl(\Pkl\ldots\Pke\alpha).\]
Termination of the generalized Goodstein sequence is therefore expressed by \[\exists l\:N_{l+1}=0,\]
and noting that $\Ck\alpha=0$ if and only if $\alpha=0$, we may reformulate the generalized Goodstein principle as follows:
\begin{equation}\label{reformulation}\forall k\in[2,\omega)\:\forall\alpha\in\Tk\:\exists l\:\:\Pkl\ldots\Pke\alpha=0.\end{equation}
Thus, transfinite induction up to Takeuti ordinal proves this generalized Goodstein principle.
Independence then follows, once we show that the function $\hk:\T\cap\Om_1\to{\mathbb N}$ defined by
\begin{equation}\label{hkdefieq}\hk(\alpha):=k + \min\{l\mid\Pkl\ldots\Pke\alpha=0\}\end{equation}
can easily be expressed in terms of the Hardy function $H_\alpha$, as in \cite{Cichon1983} for the original Goodstein principle. Defining Hardy hierarchy along Takeuti ordinal
(i.e.\ the initial segment $\T\cap\Om_1$) in the same way as in \cite{Cichon1983} by
\begin{equation}\label{Hardydefi} H_0(x):=x,\:H_{\alpha+1}(x):=H_\alpha(x+1), \mbox{ and }H_\lambda(x):=H_{\lambda[x]}(x),\end{equation}
we obtain by straightforward $<$-induction on $\alpha$:
\begin{equation}\label{hkHardyeq} \hk(\alpha)=H_\alpha(k).\end{equation}

“Cichon’s trick” is described as follows. Fix a starting base $k\in[2,\omega)$ and $N\in\mathbb N$. By Theorem~\ref{maintheo} choose the unique $\alpha\in\Tk$ with $N=\Ck(\alpha)=:N_1$. The Goodstein step (increment base, then subtract $1$) is computed by
\[
N_1[k\mapsto k+1]-1=\Cke(\alpha)-1=\Pke\Cke\alpha=\Cke\Pke\alpha=:N_2,
\]
using Lemma~\ref{tricklem} (commutation of $\Ck$ and $\Pk$, and “the latter is equal to $\Ck\alpha-1$”). Iterating gives
\[
N_2[k+1\mapsto k+2]-1=\Ckz(\Pke\alpha)-1=\Pkz\Ckz(\Pke\alpha)=\Ckz\Pkz(\Pke\alpha)=:N_3,
\]
and in general the generalized Goodstein sequence
\[
N_{l+1}=\Ckl(\Pkl\ldots\Pke\alpha).
\]
Termination $\exists l\, N_{l+1}=0$ is rewritten (using “$\Ck\alpha=0$ iff $\alpha=0$”) as the generalized Goodstein principle
\[
\forall k\in[2,\omega)\:\forall\alpha\in\Tk\:\exists l\:\:\Pkl\ldots\Pke\alpha=0.\tag{\ref{reformulation}}
\]
It is stated that “transfinite induction up to Takeuti ordinal proves this generalized Goodstein principle.” For independence, define
\[
\hk(\alpha):=k + \min\{l\mid\Pkl\ldots\Pke\alpha=0\},\tag{\ref{hkdefieq}}
\]
introduce the Hardy hierarchy along $\T\cap\Om_1$ by \eqref{Hardydefi}, and “by straightforward $<$-induction on $\alpha$” obtain
\[
\hk(\alpha)=H_\alpha(k).\tag{\ref{hkHardyeq}}
\]
This expression of $\hk$ in terms of Hardy functions is indicated as the key step to conclude independence (as in \cite{Cichon1983}).

“Cichon’s trick” is described as follows. Fix a starting base $k\in[2,\omega)$ and $N\in\mathbb N$. By Theorem~\ref{maintheo} choose the unique $\alpha\in\Tk$ with $N=\Ck(\alpha)=:N_1$. The Goodstein step (increment base, then subtract $1$) is computed by
\[
N_1[k\mapsto k+1]-1=\Cke(\alpha)-1=\Pke\Cke\alpha=\Cke\Pke\alpha=:N_2,
\]
using the following lemma.
\begin{lem}[cf.\ Lemma 2 of \cite{Cichon1983}]\label{tricklem} For $k\in[2,\om)$ and $\al\in\T\cap\Om_1$ the operations $\Ck$ and $\Pk$ commute, thus \[\Ck\Pk\al=\Pk\Ck\al,\]
where the latter is equal to $\Ck\al-1$.
\end{lem}
Iterating gives
\[
N_2[k+1\mapsto k+2]-1=\Ckz(\Pke\alpha)-1=\Pkz\Ckz(\Pke\alpha)=\Ckz\Pkz(\Pke\alpha)=:N_3,
\]
and in general the generalized Goodstein sequence
\[
N_{l+1}=\Ckl(\Pkl\ldots\Pke\alpha).
\]
Termination $\exists l\, N_{l+1}=0$ is rewritten (using “$\Ck\alpha=0$ iff $\alpha=0$”) as the generalized Goodstein principle
\[
\forall k\in[2,\omega)\:\forall\alpha\in\Tk\:\exists l\:\:\Pkl\ldots\Pke\alpha=0.\tag{\ref{reformulation}}
\]
It is stated that “transfinite induction up to Takeuti ordinal proves this generalized Goodstein principle.” For independence, define
\[
\hk(\alpha):=k + \min\{l\mid\Pkl\ldots\Pke\alpha=0\},\tag{\ref{hkdefieq}}
\]
introduce the Hardy hierarchy along $\T\cap\Om_1$ by \eqref{Hardydefi}, and “by straightforward $<$-induction on $\alpha$” obtain
\[
\hk(\alpha)=H_\alpha(k).\tag{\ref{hkHardyeq}}
\]
This expression of $\hk$ in terms of Hardy functions is indicated as the key step to conclude independence (as in \cite{Cichon1983}).

Question:
Let \(\mathcal T\) be the ordinal notation system from the context, let \(\mathcal T\cap \Omega_1\) denote its initial segment below \(\Omega_1\), and let \(\lambda[n]\) be the given fundamental sequence for limit terms. For \(k>0\), define \(\mathcal C_k\) and \(\mathcal P_k\) on ordinals by
\[\mathcal C_k(0)=0,\qquad \mathcal C_k(\beta+1)=\mathcal C_k(\beta)+1,\qquad \mathcal C_k(\lambda)=\mathcal C_k(\lambda[k-1])\]
and
\[\mathcal P_k(0)=0,\qquad \mathcal P_k(\beta+1)=\beta,\qquad \mathcal P_k(\lambda)=\mathcal P_k(\lambda[k-1])\]
for limit \(\lambda\). If \(k\in[2,\omega)\) and \(\alpha\in\mathcal T\cap\Omega_1\), which statement holds?

The operations \(\mathcal C_k\) and \(\mathcal P_k\) commute on \(\alpha\):
\[\mathcal C_k\bigl(\mathcal P_k\alpha\bigr)=\mathcal P_k\bigl(\mathcal C_k\alpha\bigr),\]
and this common value is \(\mathcal C_k\alpha-1\).

Choices:
B.
The operations \(\mathcal C_k\) and \(\mathcal P_k\) commute on \(\alpha\):
\[\mathcal C_k\bigl(\mathcal P_k\alpha\bigr)=\mathcal P_k\bigl(\mathcal C_k\alpha\bigr),\]
and this common value is \(\mathcal C_k\alpha\) for limit \(\alpha\), and \(\mathcal C_k\alpha-1\) for successor \(\alpha\neq 0\).
C.
For every \(k\in[2,\omega)\) and \(\alpha\in\mathcal T\cap\Omega_1\), one has
\[\mathcal C_k\bigl(\mathcal P_k\alpha\bigr)=\mathcal C_k\alpha-1.\]
D.
For every \(k\in[2,\omega)\) and \(\alpha\in\mathcal T\cap\Omega_1\), the predecessor operation is independent of the base in the sense that
\[\mathcal P_k\bigl(\mathcal C_k\alpha\bigr)=\mathcal P_{k+1}\bigl(\mathcal C_{k+1}\alpha\bigr),\]
and hence both sides are equal to \(\mathcal C_k\alpha-1\).
E.
The operations \(\mathcal C_k\) and \(\mathcal P_k\) satisfy the shifted commutation law
\[\mathcal C_k\bigl(\mathcal P_k\alpha\bigr)=\mathcal P_{k+1}\bigl(\mathcal C_k\alpha\bigr),\]
so in particular the common value is \(\mathcal C_k\alpha-1\) after passing from base \(k\) to base \(k+1\).

limit-case recursion for \(\mathcal P_k\) still lowers \(\mathcal C_k\) by one

dropped explicit commutation with \(\mathcal P_k(\mathcal C_k\alpha)\)

same-base commutation replaced by shifted-base Goodstein step


Let $\mathbb{F}_{2^n}$ be the finite field with $2^n$ elements. A function $f$ from $\mathbb{F}_{2^n}$ into itself is called almost perfect nonlinear (APN) if for any non zero $a\in \mathbb{F}_{2^n}$ and any $b\in \mathbb{F}_{2^n}$, the equation
$$
f(x+a)+f(x)=b
$$
admits at most two solutions.

APN functions play a central role in modern cryptography since they provide optimal resistance against differential cryptanalysis (\cite{diffBS}) when used as substitution boxes in block ciphers. Beyond cryptography, they also appear as optimal objects in coding theory, combinatorics, and projective geometry \cite{semibi,DO,hypDE}.

Despite their importance, only a few infinite families of APN functions are currently known, and their classification up to CCZ- or EA-equivalence remains an open problem (see \cite{LK} for a list of known APN families and for the definition of these equivalence relations).

Several of the known families that can be defined in even dimension have been obtained using the so called bivariate construction introduced by Carlet in \cite{carletbiv}. In particular, let $n=2m$, we can decompose $\mathbb{F}_{2^n}$ as $\mathbb{F}_{2^m}\times\mathbb{F}_{2^m}$, and a function from $\mathbb{F}_{2^m}\times\mathbb{F}_{2^m}$ into itself can be represented using bivariate polynomials, that is, $f(x,y)=(f_1(x,y),f_2(x,y))$ with $f_1,f_2\in\mathbb{F}_{2^m}[x,y]$.

In \cite{carletbiv}, the author considered functions $f(x,y)$ where $f_1(x,y)$ was given by the Maiorana-McFarland function $xy$, and provided some necessary and sufficient conditions for the APN property of $f(x,y)$. He also introduced a class of APN function in bivariate form which was later proved (see \cite{APN}) to be equivalent to the hexanomial family constructed in \cite{BChex}.
The bivariate construction was later used for obtaining other classes of APN functions (\cite{APN,tan,ZP}). Recently, in \cite{Golbiproj}, G\"olo\u{g}lu
proposed a generalization of the bivariate construction based on the so-called biprojective polynomials. Bi-projective polynomials has been used for constructing several classes of APN functions lately \cite{CLVbiproj,GKbiproj,LZLQ}.

Within specific families, the APN property is intrinsically connected to the existence of polynomials with well-defined structural properties. Accordingly, a fundamental problem is to determine whether APN functions derived from these constructions exist in infinitely many dimensions or whether they are restricted to finitely many instances \cite{BCPZ,Bhex,BTThex,Golhex}.

In particular, the existence of several classes of bivariate APN families constructed to date relies on the fact that a certain projective polynomial, that is a polynomial of type $x^{2^r+1}+x+a$, admits no roots over $\mathbb{F}_{2^m}$ \cite{CLVbiproj,carletbiv,tan}.

Projective polynomials and their roots have been studied in several works, such as \cite{Bluher,BTThex,HKproj1,HKproj2}.
So, applying Bluher’s results (\cite{Bluher}), one obtains that these constructions yield an APN function in every dimension in which they are defined.

 For the case of the APN class introduced in \cite{APN}, the existence of instances coming from this construction is related to the roots of a certain polynomial, which is not projective.

In particular, the APN class given in \cite{APN} is the following:

Let $\mathbb{F}_{2^n}$ be the finite field with $2^n$ elements. A function $f$ from $\mathbb{F}_{2^n}$ into itself is called almost perfect nonlinear (APN) if for any non zero $a\in \mathbb{F}_{2^n}$ and any $b\in \mathbb{F}_{2^n}$, the equation
$$
f(x+a)+f(x)=b
$$
admits at most two solutions.

Several of the known families that can be defined in even dimension have been obtained using the so called bivariate construction introduced by Carlet in \cite{carletbiv}. In particular, let $n=2m$, we can decompose $\mathbb{F}_{2^n}$ as $\mathbb{F}_{2^m}\times\mathbb{F}_{2^m}$, and a function from $\mathbb{F}_{2^m}\times\mathbb{F}_{2^m}$ into itself can be represented using bivariate polynomials, that is, $f(x,y)=(f_1(x,y),f_2(x,y))$ with $f_1,f_2\in\mathbb{F}_{2^m}[x,y]$.

In particular, the existence of several classes of bivariate APN families constructed to date relies on the fact that a certain projective polynomial, that is a polynomial of type $x^{2^r+1}+x+a$, admits no roots over $\mathbb{F}_{2^m}$ \cite{CLVbiproj,carletbiv,tan}.

Projective polynomials and their roots have been studied in several works, such as \cite{Bluher,BTThex,HKproj1,HKproj2}.
So, applying Bluher’s results (\cite{Bluher}), one obtains that these constructions yield an APN function in every dimension in which they are defined.

For the case of the APN class introduced in \cite{APN}, the existence of instances coming from this construction is related to the roots of a certain polynomial, which is not projective.

In particular, the APN class given in \cite{APN} is the following:

For the case of the APN class introduced in \cite{APN}, the existence of instances coming from this construction is related to the roots of a certain polynomial, which is not projective.

The authors showed that for $n\le 12$ (so $m\le 6$) it was possible to produce new APN functions (up to CCZ-equivalence). However, if such functions exist also for higher dimensions is an open problem.

Denote $q=2^{m/2}$.
First observe that $P_{c,b}(X)$ has a zero in $\mathbb{F}_{q^2}$ if and only if there exists $x\in \mathbb{F}_{q^2}^*$ such that
\begin{align}
\frac{cx^{2^r+1} + bx^{2^r} + 1}{x} \label{eq0}
\end{align}
is a $(q+1)$-root of unity. This is equivalent to ask that
\begin{align*}
\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}=\frac{x^q}{c^qx^{(2^r+1)q} + b^qx^{2^rq} + 1}.
\end{align*}
Let $x=x_0+\xi x_1$, where $\{1,\xi\}$ is an $\mathbb{F}_q$ basis of $\mathbb{F}_{q^2}$ and $x_0,x_1 \in \mathbb{F}_q$. The previous condition (since $x\neq 0$) can be equivalently rewritten as
\begin{align*}
\left(c(x_0+\xi x_1)^{2^r+1} + b(x_0+\xi x_1)^{2^r} + 1\right)\left(c^q(x_0+\xi^q x_1)^{2^r+1} + b^q(x_0+\xi^q x_1)^{2^r} + 1\right)+(x_0+\xi x_1)(x_0+\xi^q x_1)=0.
\end{align*}
In order to prove that for each $b,c \in \mathbb{F}_{q^2}$ there is at least a solution $(\overline{x_0},\overline{x_1}) \in \mathbb{F}_q^2$ to the above equation, we consider the algebraic curve $\mathcal{D}_{b,c,r}$ defined by
\begin{align*}
\left(c(X+\xi Y)^{2^r+1} + b(X+\xi Y)^{2^r} + 1\right)\left(c^q(X+\xi^q Y)^{2^r+1} + b^q(X+\xi^q Y)^{2^r} + 1\right)+(X+\xi Y)(X+\xi^q Y)=0.
\end{align*}
Via the change of variables $(X+\xi Y,X+\xi^q Y)\mapsto (X,Y)$, $\mathcal{D}_{b,c,r}$ is affinely equivalent to the plane curve
$\mathcal{C}_{b,c,r}$ defined by
\begin{align*}
\left(cX^{2^r+1} + bX^{2^r} + 1\right)\left(c^qY^{2^r+1} + b^qY^{2^r} + 1\right)+XY=0.
\end{align*}
Our strategy consists in proving that $\mathcal{C}_{b,c,r}$, $b, c \in \F_{q^2}$, $c \ne 0$, $r \ge 1$, is absolutely irreducible and so is $\mathcal{D}_{b,c,r}$. Hence, by the Hasse-Weil bound we obtain the existence of at least one point $(\overline{x_0},\overline{x_1}) \in \mathbb{F}_q^2$ in $\mathcal{D}_{b,c,r}$. The case $c = 0$ is treated separataly. Therefore, by Theorem \ref{Th:APN} the function $f_{b,c,r}(x,y)$ is not APN.

\begin{lemma}\label{lm:H90}
Let $\alpha\in\mathbb{F}_{2^m}$ and let $j$ be such that $\gcd(j,m)=1$. Then, $Tr_{2}^{2^m}(\alpha)=0$ if and only if there exists $\beta\in\mathbb{F}_{2^m}$ such that $\alpha=\beta^{2^j}-\beta$.
Here $Tr_{2}^{2^m}$ is the trace map from $\mathbb{F}_{2^m}$ onto $\mathbb{F}_2$.
\end{lemma}

\begin{lemma}
Let $b\in \mathbb{F}_{q^2}^*$, and $r$ be such that $\gcd(r,m)=1$. Then, the polynomial
 $\big(bX^{2^r} + 1\big)^{q+1} + X^{q+1}$ has a zero in $\mathbb{F}_{q^2}$.
\end{lemma}
\begin{proof}
 We note that $\big(bX^{2^r} + 1\big)^{q+1} + X^{q+1}$ has a zero in $\mathbb{F}_{q^2}$ if and only if there exist $x\in\mathbb{F}_{q^2}$ and $u\in\mu_{q+1}$ such that
 \begin{equation}\label{eq:c0}
 bx^{2^r} + ux+1=0.
 \end{equation}
 Now, $b=u't$ for some $t\in\mathbb{F}_{q}^*$ and $u' \in \mu_{q+1}$. Therefore, performing the substitution $x\mapsto b^{-2^{-r}}x$ and considering $u=u'^{2^{-r}}\in\mu_{q+1}$, Equation \eqref{eq:c0} becomes
 \begin{equation}\label{eq:c02}
 x^{2^r}+t'x+1=0,
 \end{equation}
 where $t'=t^{-2^{-r}}$.

As a consequences we get the following:
\begin{theorem}\label{th:b0}
Let $n = 2m$ with $m$ even, and let $r < m/2$ be such that $\gcd(r,m) = 1$. Then, for any $b\in \mathbb{F}_{2^{m}}^*$, the function $f_{b,0,r}(x,y)$ defined as in Theorem \ref{Th:APN} is not APN.
\end{theorem}

In \cite{APN}, the authors show that for $m\le 6$, we have instance of APN functions coming from Theorem \ref{Th:APN} for $r=1$. To check that $f_{b,c,1}$ cannot be APN for $8 \le m\le 16$ we need the following proposition which allows us to reduce the number of pairs $(b,c)$.
\begin{prop}\label{prop:red}
 Let $k\ge 0$ be an integer, and $u\in\mu_{q+1}$. Then, for any $b,c\in\mathbb{F}_{q^2}$ the equation
 \begin{equation}\label{eq:sol1}
 \big(cx^{2^r+1} + bx^{2^r} + 1\big)^{q+1} + x^{q+1}=0
 \end{equation}

By \eqref{eq0}, the existence of a root of the polynomial
\[
P_{c,b}(X) = \bigl(cX^{2^r+1} + bX^{2^r} + 1\bigr)^{q+1} + X^{q+1}
\]
is equivalent to the existence of an element $u \in \mu_{q+1}$ such that the equation
\[
cx^{2^r+1} + bx^{2^r} + ux + 1 = 0
\]
admits a root in $\mathbb{F}_{q^2}$. This equation can be transformed into
\begin{equation}\label{eq:projpol}
x^{2^r+1} + x + A = 0,
\end{equation}
where
\[
 A = \frac{(ub+c)c^{2^r-1}}{\bigl(uc^{2^r-1} + b^{2^r}\bigr)^{2^{-r}+1}},
\]
under the assumption that $uc^{2^r-1}+b^{2^r} \neq 0$, see for instance \cite{Bluher}. In \cite[Theorem 2.1]{BTThex} it has been proved that equation \eqref{eq:projpol} admits no solution over $\mathbb{F}_{q^2}$ if and only if
\begin{equation}\label{eq:condition}
A = \frac{a(a+1)^{2^r+2^{-r}}}{(a+a^{2^{-r}})^{2^r+1}},
\end{equation}
for some non-cube $a$.
For the case $r=1$, the previous request is equivalent to ask that
\begin{align*}
A=a+\frac{1}{a},
\end{align*}
for some non-cube $a$.
So, for $r=1$, using MAGMA \cite{Magma} it is possible to check that one can always find some $u \in \mu_{q+1}$ such that {$uc+b^{2} \neq 0$ and} the associated value of $A$ does \emph{not} belong to the set
\[
\Biggl\{ a+\frac{1}{a} : a \text{ not a cube} \Biggr\},
\]
for any choice of $b,c\in\mathbb{F}_{2^m}$ and $8\le m\le 16$. Therefore, the function $f_{b,c,1}$ cannot be APN. So, we get the following result.
\begin{cor}
 Let $m\ge 8$ be an even integer. Then, for any choice $b,c \in \mathbb{F}_{2^m}$ the function $f_{b,c,1}$ as in Theorem \ref{Th:APN} is not APN.
\end{cor}

\begin{theorem}\label{Th:APN}\cite[Theorem 6.2]{APN}
Let $n = 2m$ with $m$ even, and let $r < m/2$ be such that $\gcd(r,m) = 1$.  Consider $b,c \in \mathbb{F}_{2^{m}}$
Then
\[
f_{b,c,r}(x,y) = \big(xy,\, x^{2^r+1} + x^{2^{r+m/2}} y^{2^{m/2}} + bxy^{2^r} + cy^{2^r+1}\big)
\]
is APN if and only if
\[
P_{c,b}(X)=\big(cX^{2^r+1} + bX^{2^r} + 1\big)^{2^{m/2}+1} + X^{2^{m/2}+1}
\]
has no zero in $\mathbb{F}_{2^m}$.
\end{theorem}

The authors showed that for $n\le 12$ (so $m\le 6$) it was possible to produce new APN functions (up to CCZ-equivalence). However, if such functions exist also for higher dimensions is an open problem.

The aim of this work is to investigate such an open question. In particular, we prove that for each $r <m/8-1$ there are no instances of $b,c \in \mathbb{F}_{2^{m}}$ for which $f_{b,c,r}(x,y)$ is APN.

The main tool is given by application of techniques from algebraic varieties over finite fields.

Denote $q=2^{m/2}$.
First observe that $P_{c,b}(X)$ has a zero in $\mathbb{F}_{q^2}$ if and only if there exists $x\in \mathbb{F}_{q^2}^*$ such that
\begin{align}
\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}  \label{eq0}
\end{align}
is a $(q+1)$-root of unity. This is equivalent to ask that
\begin{align*}
\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}=\frac{x^q}{c^qx^{(2^r+1)q} + b^qx^{2^rq} + 1}.
\end{align*}
Let $x=x_0+\xi x_1$, where $\{1,\xi\}$ is an $\mathbb{F}_q$ basis of $\mathbb{F}_{q^2}$ and $x_0,x_1 \in \mathbb{F}_q$. The previous condition (since $x\neq 0$) can be equivalently rewritten as
\begin{align*}
\left(c(x_0+\xi x_1)^{2^r+1} + b(x_0+\xi x_1)^{2^r} + 1\right)\left(c^q(x_0+\xi^q x_1)^{2^r+1} + b^q(x_0+\xi^q x_1)^{2^r} + 1\right)+(x_0+\xi x_1)(x_0+\xi^q x_1)=0.
\end{align*}
In order to prove that for each $b,c \in \mathbb{F}_{q^2}$ there is at least a solution $(\overline{x_0},\overline{x_1}) \in \mathbb{F}_q^2$ to the above equation, we consider the algebraic curve $\mathcal{D}_{b,c,r}$ defined by
\begin{align*}
\left(c(X+\xi Y)^{2^r+1} + b(X+\xi Y)^{2^r} + 1\right)\left(c^q(X+\xi^q Y)^{2^r+1} + b^q(X+\xi^q Y)^{2^r} + 1\right)+(X+\xi Y)(X+\xi^q Y)=0.
\end{align*}
Via the change of variables $(X+\xi Y,X+\xi^q Y)\mapsto (X,Y)$, $\mathcal{D}_{b,c,r}$ is affinely equivalent to the plane curve
$\mathcal{C}_{b,c,r}$ defined by
\begin{align*}
\left(cX^{2^r+1} + bX^{2^r} + 1\right)\left(c^qY^{2^r+1} + b^qY^{2^r} + 1\right)+XY=0.
\end{align*}
Our strategy consists in proving that $\mathcal{C}_{b,c,r}$, $b, c \in \F_{q^2}$, $c \ne 0$, $r \ge 1$, is absolutely irreducible and so is $\mathcal{D}_{b,c,r}$. Hence, by the Hasse-Weil bound we obtain the existence of at least one point $(\overline{x_0},\overline{x_1}) \in \mathbb{F}_q^2$ in $\mathcal{D}_{b,c,r}$. The case $c = 0$ is treated separataly. Therefore, by Theorem \ref{Th:APN} the function $f_{b,c,r}(x,y)$ is not APN.

Let $q=2^{m/2}$. Using Theorem~\ref{Th:APN}, it suffices to show that $P_{c,b}(X)$ has a zero in $\mathbb{F}_{q^2}$ (hence in $\mathbb{F}_{2^m}$) for the relevant parameters, so that $f_{b,c,r}(x,y)$ is not APN.

First, observe that “$P_{c,b}(X)$ has a zero in $\mathbb{F}_{q^2}$ if and only if there exists $x\in \mathbb{F}_{q^2}^*$ such that
\[
\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}
\]
is a $(q+1)$-root of unity,” which is equivalently rewritten as
\[
\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}=\frac{x^q}{c^qx^{(2^r+1)q} + b^qx^{2^rq} + 1}.
\]
Write $x=x_0+\xi x_1$ with $\{1,\xi\}$ an $\mathbb{F}_q$-basis of $\mathbb{F}_{q^2}$ and $x_0,x_1\in\mathbb{F}_q$. Then (since $x\neq 0$) the condition becomes a plane equation in $(x_0,x_1)$.

To prove that “for each $b,c\in\mathbb{F}_{q^2}$ there is at least a solution $(\overline{x_0},\overline{x_1})\in\mathbb{F}_q^2$,” introduce the algebraic curve $\mathcal{D}_{b,c,r}$ over $\mathbb{F}_q$ defined by that equation, and via the change of variables $(X+\xi Y, X+\xi^q Y)\mapsto (X,Y)$ pass to an affinely equivalent plane curve $\mathcal{C}_{b,c,r}$:
\[
\left(cX^{2^r+1} + bX^{2^r} + 1\right)\left(c^qY^{2^r+1} + b^qY^{2^r} + 1\right)+XY=0.
\]
The stated strategy is to prove $\mathcal{C}_{b,c,r}$ (hence $\mathcal{D}_{b,c,r}$) is “absolutely irreducible” for $b,c\in\F_{q^2}$, $c\neq 0$, $r\ge 1$ (with “the case $c=0$ treated separately”). Then “by the Hasse-Weil bound we obtain the existence of at least one point $(\overline{x_0},\overline{x_1})\in\mathbb{F}_q^2$ in $\mathcal{D}_{b,c,r}$,” yielding a zero of $P_{c,b}$ in $\mathbb{F}_{q^2}$. “Therefore, by Theorem~\ref{Th:APN} the function $f_{b,c,r}(x,y)$ is not APN.”

Let $q=2^{m/2}$. In establishing the main theorem, it suffices to show that $P_{c,b}(X)$ has a zero in $\mathbb{F}_{q^2}$ (hence in $\mathbb{F}_{2^m}$) for the relevant parameters, so that $f_{b,c,r}(x,y)$ is not APN.

First, observe that “$P_{c,b}(X)$ has a zero in $\mathbb{F}_{q^2}$ if and only if there exists $x\in \mathbb{F}_{q^2}^*$ such that
\[
\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}
\]
is a $(q+1)$-root of unity,” which is equivalently rewritten as
\[
\frac{cx^{2^r+1} + bx^{2^r} + 1}{x}=\frac{x^q}{c^qx^{(2^r+1)q} + b^qx^{2^rq} + 1}.
\]
Write $x=x_0+\xi x_1$ with $\{1,\xi\}$ an $\mathbb{F}_q$-basis of $\mathbb{F}_{q^2}$ and $x_0,x_1\in\mathbb{F}_q$. Then (since $x\neq 0$) the condition becomes a plane equation in $(x_0,x_1)$.

To prove that “for each $b,c\in\mathbb{F}_{q^2}$ there is at least a solution $(\overline{x_0},\overline{x_1})\in\mathbb{F}_q^2$,” introduce the algebraic curve $\mathcal{D}_{b,c,r}$ over $\mathbb{F}_q$ defined by that equation, and via the change of variables $(X+\xi Y, X+\xi^q Y)\mapsto (X,Y)$ pass to an affinely equivalent plane curve $\mathcal{C}_{b,c,r}$:
\[
\left(cX^{2^r+1} + bX^{2^r} + 1\right)\left(c^qY^{2^r+1} + b^qY^{2^r} + 1\right)+XY=0.
\]
The stated strategy is to prove $\mathcal{C}_{b,c,r}$ (hence $\mathcal{D}_{b,c,r}$) is “absolutely irreducible” for $b,c\in\F_{q^2}$, $c\neq 0$, $r\ge 1$ (with “the case $c=0$ treated separately”). Then “by the Hasse-Weil bound we obtain the existence of at least one point $(\overline{x_0},\overline{x_1})\in\mathbb{F}_q^2$ in $\mathcal{D}_{b,c,r}$,” yielding a zero of $P_{c,b}$ in $\mathbb{F}_{q^2}$. This completes the proof of the main theorem.

\label{Th:APN}To prove the main theorem, we use Theorem 6.2 of \cite[Theorem 6.2]{APN}.
Let $n = 2m$ with $m$ even, and let $r < m/2$ be such that $\gcd(r,m) = 1$. Consider $b,c \in \mathbb{F}_{2^{m}}$.
Then
\[
f_{b,c,r}(x,y) = \big(xy,\, x^{2^r+1} + x^{2^{r+m/2}} y^{2^{m/2}} + bxy^{2^r} + cy^{2^r+1}\big)
\]
is APN if and only if
\[
P_{c,b}(X)=\big(cX^{2^r+1} + bX^{2^r} + 1\big)^{2^{m/2}+1} + X^{2^{m/2}+1}
\]
has no zero in $\mathbb{F}_{2^m}$.

Biconditional or Equivalence
Nonexistence

Question:
Let $n=2m$ with $m$ even, and let $r<m/2$ satisfy $\gcd(r,m)=1$. For $b,c\in\mathbb F_{2^m}$, define a map $f_{b,c,r}:\mathbb F_{2^m}\times\mathbb F_{2^m}\to \mathbb F_{2^m}\times\mathbb F_{2^m}$ by
\[
f_{b,c,r}(x,y)=\big(xy,\,x^{2^r+1}+x^{2^{r+m/2}}y^{2^{m/2}}+bxy^{2^r}+cy^{2^r+1}\big).
\]
Recall that a function $f$ on a vector space over $\mathbb F_2$ is almost perfect nonlinear (APN) if for every nonzero $a$ and every $d$, the equation $f(z+a)+f(z)=d$ has at most two solutions. Which statement holds for $f_{b,c,r}$ under these assumptions?

$f_{b,c,r}$ is APN if and only if the polynomial
\[
P_{c,b}(X)=\big(cX^{2^r+1}+bX^{2^r}+1\big)^{2^{m/2}+1}+X^{2^{m/2}+1}
\]
has no zero in $\mathbb F_{2^m}$.

Choices:
B.
$f_{b,c,r}$ is APN if and only if the polynomial
\[
P_{c,b}(X)=\big(cX^{2^r+1}+bX^{2^r}+1\big)^{2^{m/2}+1}+X^{2^{m/2}+1}
\]
has no zero in $\mathbb F_{2^{m/2}}$.
C.
If the polynomial
\[
P_{c,b}(X)=\big(cX^{2^r+1}+bX^{2^r}+1\big)^{2^{m/2}+1}+X^{2^{m/2}+1}
\]
has no zero in $\mathbb F_{2^m}$, then $f_{b,c,r}$ is APN.
D.
$f_{b,c,r}$ is APN if and only if there exist $b,c\in\mathbb F_{2^m}$ such that the polynomial
\[
P_{c,b}(X)=\big(cX^{2^r+1}+bX^{2^r}+1\big)^{2^{m/2}+1}+X^{2^{m/2}+1}
\]
has no zero in $\mathbb F_{2^m}$.
E.
$f_{b,c,r}$ is APN whenever the polynomial
\[
P_{c,b}(X)=\big(cX^{2^r+1}+bX^{2^r}+1\big)^{2^{m/2}+1}+X^{2^{m/2}+1}
\]
has at most one zero in $\mathbb F_{2^m}$.


\begin{thm}\label{XiaFock}
		We have
		\begin{equation*}
			C^{*} (\mathcal{WL})=\mathcal{T}^{(1)}.
		\end{equation*}
	\end{thm}

For some \(\alpha>0\), \(p>0\) and \(dV\) the standard volume measure on \(\mathbb{C}^n\), let \(L^{p}_{\alpha}(\mathbb{C}^n,dV)\) be the Lebesgue space of measurable functions \(f\) on \(\mathbb{C}^n\) such that
	\begin{equation}\label{Fnorm1}
		\left\lVert f \right\rVert^{p}_{\alpha} := \left(\frac{p\alpha}{2\pi}\right)^{n} \int_{\mathbb{C}^n} |f(z)e^{-\frac{\alpha}{2}|z|^2}|^{p} dV(z) < \infty~.
	\end{equation}
	Similarly, for \(\alpha>0\) and \(p=\infty\), we denoted by \(L^{\infty}_{\alpha}\) the space of Lebesgue measurable function \(f\) on \(\mathbb{C}^n\) such that
	\begin{equation*}
		\left\lVert f \right\rVert_{\infty,\alpha} := \text{esssup}\{|f(z)|e^{-\frac{\alpha|z|^2}{2}} : ~z \in \mathbb{C}^n\}< \infty ~.
	\end{equation*}
	The classical Fock space \(F^{p}_{\alpha}\) is the space of entire functions on \(\mathbb{C}^n\) which belong to \( L^{p}_{\alpha}(\mathbb{C}^n, dV)\). Similarly, the Fock space \(F^{\infty}_{\alpha}\) is the space of entire functions on \(\mathbb{C}^n\) which belong to \(L^{\infty}_{\alpha}\).

	Let \(d\mu\) be the Gaussian measure on \(\mathbb{C}^n\), \(n\ge 1\). In terms of the standard volume measure \(dV\) on \(\mathbb{C}^n\), it is given by
	\begin{equation*}
		d\mu(z) = \pi^{-n} e^{-|z|^2} dV(z)~.
	\end{equation*}

	 The Fock space \(H^{2}(\mathbb{C}^n,d\mu)\) is defined to be the subspace of the (Hilbert-) Lebesgue space \(L^{2}(\mathbb{C}^n,d\mu)\) consisting of entire functions. Notice that \(H^{2}(\mathbb{C}^n,d\mu) = F^{2}_{1} \). The symbol \(K_z\) denotes the reproducing kernel and the symbol \(k_z\) denotes the normalized reproducing kernel for \(H^{2}(\mathbb{C}^n,d\mu)\). That is,
	\begin{equation*}
		K_z(\zeta) = e^{\langle \zeta, z \rangle}, \quad k_{z}(\zeta) = e^{\langle \zeta, z\rangle}e^{-\frac{|z|^2}{2}}~, \quad z,\zeta \in \mathbb{C}^n.
	\end{equation*}

	In \cite{xia2015localization}, J. Xia showed in the case of the Bergman space on the unit ball of \(\mathbb{C}^n\) that the norm closure of \(\{T_{f}: f \in L^{\infty}(B,dv) \}\) coincides with the \(C^{*}\)-algebra of weakly localized operators. Also, he stated in \cite[Section 4 ]{xia2015localization}  that  the analogue of \cite[Theorem 1.5]{xia2015localization} on the Fock space \(H^{2}(\mathbb{C}^n, d\mu)\) was true. In this paper, we define the notion of weakly localized operators, state Xia's theorem for the Fock space \(H^{2}(\mathbb{C}^n, d\mu)\) and provide details of its proof. Further, we present a consequence of this theorem on the compactness of operators on \(H^{2}(\mathbb{C}^n,d\mu)\).  We begin with the following definitions and we state the main theorem, the proof of which will retain our attention in the following sections.

	\begin{defn}
		For \(f \in L^{\infty}(\mathbb{C}^n,dV) \), the \textbf{Toeplitz operator} \(T_{f}\) is defined by the formula
		\begin{equation*}
			T_{f}h = P(fh)~,\quad h \in H^{2}(\mathbb{C}^n,d\mu)~,
		\end{equation*}
		where \(P \colon L^{2}(\mathbb{C}^n,d\mu) \rightarrow H^{2}(\mathbb{C}^n,d\mu)\) is the orthogonal projection.
	\end{defn}

	The \textbf{standard lattice} in \(\mathbb{C}^n\) is denoted by
	\begin{equation*}
		\mathbb{Z}^{2n} = \{(m_{1}+il_{1}, \dots, m_{n}+il_{n}): m_{1}, l_{1}, \dots, m_{n},l_{n} \in \mathbb{Z}\}~.
	\end{equation*}
	We fix an orthonormal set \(\{e_{u}: u \in \mathbb{Z}^{2n}\}\) in \(H^{2}(\mathbb{C}^n,d\mu)\). We let \(S\) denote the \textbf{fundamental unit cube in }\(\mathbb{C}^{n}\). That is,
	\begin{equation*}
		S = \{(x_{1}+iy_{1}, \dots, x_{n}+iy_{n}) : x_{1},y_{1}, \dots , x_{n},y_{n} \in [0,1)\}.
	\end{equation*}
	With \(\mathbb{Z}^{2n}\) and \(S\), we have
	\begin{equation*}
		\cup_{u \in \mathbb{Z}^{2n}} \{S+u\} = \mathbb{C}^{n} = \cup_{u \in \mathbb{Z}^{2n}} \{u-S\} ~,
	\end{equation*}
	which is a tiling of the space, meaning that there is no overlap between \(S+u\) and \(S+v\) for \(u \neq v\) in \(\mathbb{Z}^{2n}\) (resp. between \(u-S\) and \(v-S\) for \(u \neq v \in \mathbb{Z}^{2n}\) ).

	\begin{defn}
		Let \(\mathcal{T}^{(1)}\) denote the norm closure of \(\{T_{f}: f\in L^{\infty}(\mathbb{C}^n,dV)\}\) in \(\mathcal{B}(H^2(\mathbb{C}^n,d\mu))\) with respect to the operator norm. That is
		\begin{equation*}
			\mathcal{T}^{(1)} = \{B : \lim_{k \to \infty } \norm{B-T_{b_{k}}}=0 , b_{k} \in L^{\infty}(\mathbb{C}^{n},dV)\}.
		\end{equation*}
	\end{defn}

	\begin{defn}
		We denote by \(\mathcal{S}\) the linear span of the normalized reproducing kernels \(k_z\). A linear operator \(B : \mathcal{S} \rightarrow H^{2}(\mathbb{C}^n,d\mu) \) is said to be \textbf{admissible} on \( H^{2}(\mathbb{C}^n,d\mu) \) if there exists a linear operator \( B^{*} : \mathcal{S} \rightarrow H^{2}(\mathbb{C}^n,d\mu) \) such that the duality relation
		\begin{equation}\label{R1}
			\langle Bk_{z}, k_{w} \rangle = \langle k_{z}, B^{*}k_w \rangle
		\end{equation}
		holds for all \(z,w \in \mathbb{C}^n\).
	\end{defn}
	The inner product here is with respect to \(d\mu\).

	We define below sufficiently localized operators following J. Xia and D. Zheng (XZ) in \cite{xia2013localization}.
	\begin{defn}
		A bounded linear operator \(B\) on \(H^{2}(\mathbb{C}^n,d\mu)\) is said to be XZ-\textbf{sufficiently localized} if there exist constants \(2n<\beta<\infty\) and \(0<C<\infty\) such that
		\begin{equation}\label{SL1}
			|\langle Bk_{z},k_{w}\rangle| \le \frac{C}{(1+|z-w|)^{\beta}}
		\end{equation}
		for all \(z,w\in \mathbb{C}^n\).
	\end{defn}

		\begin{defn}
		An admissible operator \(B\) on \(H^{2}(\mathbb{C}^n,d\mu)\) is said to be \textbf{weakly localized} if it satisfies the following four conditions
		\begin{equation*}
			\sup_{z\in \mathbb{C}^n}\int_{\mathbb{C}^n} |\langle Bk_{z}, k_{w} \rangle| dV(w) <\infty, \quad \sup_{z\in \mathbb{C}^n}\int_{\mathbb{C}^n} |\langle B^{*} k_{z}, k_{w} \rangle| dV(w) <\infty
		\end{equation*}

		and
		\begin{equation*}
			\lim_{r\to\infty}\sup_{z\in \mathbb{C}^n}\int_{|z-w|\ge r} |\langle Bk_{z}, k_{w} \rangle| dV(w) =0, \quad \lim_{r\to\infty}\sup_{z\in \mathbb{C}^n}\int_{|z-w|\ge r} |\langle B^{*} k_{z}, k_{w} \rangle| dV(w) =0~.
		\end{equation*}
	\end{defn}

	\begin{exa}\label{Ex1}
		If \(f\) is a \textbf{bounded measurable function} on \(\mathbb{C}^n\), then there is a positive constant \(0<C=C(f)<\infty\) such that
		\begin{equation*}
			|\langle T_{f}k_{z}, k_{w}\rangle| \le C e^{-(1/8) |z-w|^2}
		\end{equation*}
		for all \(z,w \in \mathbb{C}^n\). That is, \(T_{f} \in \text{XZ-}\mathcal{SL}\). Also, \(T_{f} \in \mathcal{L}_{p}\) for \(2<p<\frac{8}{3}\).
	\end{exa}

	\begin{proof}
		For each \(z, w, \xi \in \mathbb{C}^n\), we have
		\begin{equation*}
			|k_{z}(\xi)k_{w}(\xi)|e^{-|\xi|^2} = |e^{\langle \xi , z \rangle - \frac{|z|^2}{2} } e^{\langle \xi, w \rangle - \frac{|w|^2}{2}} | e^{-|\xi|^2} = e^{-\frac{1}{2}\left(|z-\xi|^2 + |w-\xi|^2\right)} ~.
		\end{equation*}
		By the triangular inequality we have
		\begin{equation*}
			|z-w|^2 \le |z-\xi + \xi -w|^2 \le (|z-\xi| + |w-\xi|)^2 \le 2 (|z-\xi|^2 + |w-\xi|^2)~.
		\end{equation*}
		It follows that
		\begin{eqnarray}\label{kzw}
			|k_{z}(\xi)k_{w}(\xi)|e^{-|\xi|^2} = e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)}e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)} \le e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)} e^{-\frac{1}{8}|z-w|^2}~.
		\end{eqnarray}
		Let \(f\) be a bounded measurable function on \(\mathbb{C}^n\), then for all \(g,h\in H^{2}(\mathbb{C}^n,d\mu)\), it holds
		\begin{equation}\label{Toepfgh}
			\langle T_{f}g, h \rangle = \int_{\mathbb{C}^{n}} f(\xi)g(\xi)\overline{h(\xi)} d\mu(\xi)~.
		\end{equation}
		In fact,
		\begin{eqnarray*}
			\langle T_{f}g, h \rangle &=&  \int_{\mathbb{C}^{n}} T_{f}g(w)\overline{h(w)} d\mu(w) \\
			&=& \int_{\mathbb{C}^{n}}\int_{\mathbb{C}^{n}} K(w,\xi) f(\xi)g(\xi) d\mu(\xi) \overline{h(w)} d\mu(w) \\
			&=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \int_{\mathbb{C}^{n}} K_{\xi}(w) \overline{h(w)} d\mu(w) d\mu(\xi) \\
			&=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \langle K_{\xi} , h \rangle d\mu(\xi) = \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \overline{\langle  h, K_{\xi}  \rangle} d\mu(\xi) \\
			&=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi)\overline{h(\xi)} d\mu(\xi)~.
		\end{eqnarray*}
		Therefore,
		\begin{eqnarray*}
			|\langle T_{f}k_{z}, k_{w} \rangle| &=& \left|\int_{\mathbb{C}^n}
			f(\xi)k_{z}(\xi) \overline{k_{w}(\xi)} d\mu(\xi)\right|  \\
			&\le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n} \int_{\mathbb{C}^n} |k_{z}(\xi) k_{w}(\xi)| e^{-|\xi|^2} dV(\xi) \\
			& \le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n}  e^{-\frac{1}{8}|z-w|^2} \int_{\mathbb{C}^n} e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)}  dV(\xi) \\
			& \le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n}  e^{-\frac{1}{8}|z-w|^2} \left(\int_{\mathbb{C}^n} e^{-\frac{1}{2}|z-\xi|^2} dV(\xi) \right)^{\frac{1}{2}} \left(\int_{\mathbb{C}^n} e^{-\frac{1}{2}|w-\xi|^2}dV(\xi) \right)^{\frac{1}{2}} \\
			&=& \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n}  e^{-\frac{1}{8}|z-w|^2} \int_{\mathbb{C}^n} e^{-\frac{1}{2}|\zeta|^2} dV(\zeta) \\
			&=& (\sqrt{2})^n \left\lVert f \right\rVert_{\infty} e^{-\frac{1}{8}|z-w|^2}~.
		\end{eqnarray*}
		This completes the proof.
	\end{proof}

	 \begin{pro}
	 	Any XZ-sufficiently localized operator is weakly localized.
	 \end{pro}
	\begin{proof}
	Let \(B\) be a XZ-sufficiently localized operator, then  for all \(z\in \mathbb{C}^n\), we have
	\begin{eqnarray*}
		\int_{\mathbb{C}^n} |\langle Bk_z, k_w \rangle| dV(w) &\le& \frac{C}{\pi^n} \int_{\mathbb{C}^n} \frac{1}{(1+|\zeta|)^{\beta}} dV(\zeta) \\
		&=& \frac{Cc_n}{\pi^n} \int_{0}^{\infty} \frac{r^{2n-1}}{(1+r)^{\beta}} dr \\
		&=& C_n \left[ \int_{0}^{1} \frac{r^{2n-1}}{(1+r)^{\beta}} dr  + \int_{1}^{\infty } \frac{r^{2n-1}}{(1+r)^{\beta}} dr\right] \\
		&\le & C_n \left[ \int_{0}^{1} \frac{1}{(1+r)^{\beta}} dr +  \int_{1}^{\infty } \frac{r^{2n-1}}{r^{\beta}} dr\right] \\
		&=& C_n \left\{\frac{1}{\beta-1}\left(1-\frac{1}{2^{\beta}-1}\right) + \frac{1}{\beta-2n}\right\} \\
		&:=& C(n,\beta)~,
	\end{eqnarray*}
	which does not depend on \(z\). Therefore,
	\begin{equation*}
		\sup_{z\in \mathbb{C}^n}\int_{\mathbb{C}^n} |\langle Bk_z, k_w \rangle| dV(w) \le C(n,\beta)  <\infty~.
	\end{equation*}
	And in the same way, we also have
	\begin{equation*}
		\sup_{z\in \mathbb{C}^n}\int_{\mathbb{C}^n} |\langle B^*k_z, k_w \rangle| dV(w) \le C(n,\beta) <\infty~.
	\end{equation*}
	We now show that \(B\) satisfies the third condition of weakly localized operator.

	Let \(z\in \mathbb{C}^n\), using the change of variables \(\zeta=z-w\) and the spherical coordinates, it holds
	\begin{eqnarray*}
		\int_{|z-w|\ge r} |\langle Bk_z, k_w \rangle dV(w) &\le& C \int_{|z-w|\ge r} \frac{1}{(1+|z-w|)^{\beta}} dV(w) \\
		&=& C \int_{|\zeta|\ge r} \frac{1}{(1+|\zeta|)^{\beta}} dV(\zeta) \\
		&=& C c_n \int_{\rho\ge r} \frac{\rho^{2n-1}}{(1+\rho)^{\beta}} d \rho \\
		&\le& C c_n \int_{\rho\ge r} \frac{\rho^{2n-1}}{\rho^{\beta}} d \rho =  C c_n \int_{\rho\ge r} \frac{1}{\rho^{\beta-2n+1}} d \rho \\
		&=& \frac{Cc_n}{\beta-2n}\frac{1}{r^{\beta-2n}}~.
	\end{eqnarray*}
	It follows that,
	\begin{equation*}
		\lim_{r\to\infty} \sup_{z\in \mathbb{C}^n} \int_{|z-w|\ge r} |\langle Bk_z, k_w \rangle dV(w) \le \frac{Cc_n}{\beta-2n} \lim_{r\to\infty} \frac{1}{r^{\beta-2n}} =0~.
	\end{equation*}
	Using the duality relation (\ref{R1}) of an admissible operator and similarly like with \(B\), we also have:
	\begin{equation*}
		\lim_{r\to\infty} \sup_{z\in \mathbb{C}^n} \int_{|z-w|\ge r} |\langle B^*k_z, k_w \rangle dV(w) \le \frac{Cc_n}{\beta-2n} \lim_{r\to\infty} \frac{1}{r^{\beta-2n}} =0~.
	\end{equation*}
\end{proof}

	We denote by \(\mathcal{WL}\) the collection of weakly localized operators on \(H^{2}(\mathbb{C}^n,d\mu)\). We recall the following definition from \cite{zhu2007}.

		\begin{defn}
		A \textbf{Banach algebra} is a complex algebra together with a complete norm satisfying the condition \(\left\lVert xy \right\rVert \le \left\lVert x \right\rVert\left\lVert y \right\rVert\).
		A \(C^{*}\)\textbf{-algebra} is a Banach algebra \(\mathcal{A}\) with an involution \(x \mapsto x^{*}\) on it satisfying the following conditions:
		\begin{enumerate}[label=$\blacktriangleright$]
			\item \(x^{**}=x\) for all vectors \(x \in \mathcal{A} \).
			\item \((ax + by)^{*} = \bar{a}x^{*} + \bar{b}y^{*} \) for all vectors \(x,y \in \mathcal{A}\) and \(a,b \in \mathbb{C}\).
			\item \((xy)^{*}=y^{*}x^{*}\) for all vectors \(x,y \in \mathcal{A}\).
			\item \(\norm{xx^{*}} = \left\lVert x \right\rVert^{2}\) for all vector \(x \in \mathcal{A}\).
		\end{enumerate}
	\end{defn}

	\begin{defn}
		A Banach algebra \(\mathcal{A}\) is called a \textbf{\(\star\)-algebra} if for every \(A\in \mathcal{A}\), we have \(A^{*} \in \mathcal{A} \).
	\end{defn}

	\begin{defn}
		We denote by \(C^{*}(\mathcal{WL})\) the \(C^{*}\)-algebra generated by weakly localized operators on \(H^{2}(\mathbb{C}^n, d\mu)\). Also, \(C^{*}(\mathcal{WL})\) is actually the norm closure of \(\mathcal{WL}\) since \(\mathcal{WL} \) is a \(\star\)-algebra.
	\end{defn}

	We will prove the following main result.

For some \(\alpha>0\), \(p>0\) and \(dV\) the standard volume measure on \(\mathbb{C}^n\), let \(L^{p}_{\alpha}(\mathbb{C}^n,dV)\) be the Lebesgue space of measurable functions \(f\) on \(\mathbb{C}^n\) such that
 \begin{equation}\label{Fnorm1}
 \left\lVert f \right\rVert^{p}_{\alpha} := \left(\frac{p\alpha}{2\pi}\right)^{n} \int_{\mathbb{C}^n} |f(z)e^{-\frac{\alpha}{2}|z|^2}|^{p} dV(z) < \infty~.
 \end{equation}
 Similarly, for \(\alpha>0\) and \(p=\infty\), we denoted by \(L^{\infty}_{\alpha}\) the space of Lebesgue measurable function \(f\) on \(\mathbb{C}^n\) such that
 \begin{equation*}
 \left\lVert f \right\rVert_{\infty,\alpha} := \text{esssup}\{|f(z)|e^{-\frac{\alpha|z|^2}{2}} : ~z \in \mathbb{C}^n\}< \infty ~.
 \end{equation*}
 The classical Fock space \(F^{p}_{\alpha}\) is the space of entire functions on \(\mathbb{C}^n\) which belong to \( L^{p}_{\alpha}(\mathbb{C}^n, dV)\). Similarly, the Fock space \(F^{\infty}_{\alpha}\) is the space of entire functions on \(\mathbb{C}^n\) which belong to \(L^{\infty}_{\alpha}\).

\begin{defn}
 We denote by \(\mathcal{S}\) the linear span of the normalized reproducing kernels \(k_z\). A linear operator \(B : \mathcal{S} \rightarrow H^{2}(\mathbb{C}^n,d\mu) \) is said to be \textbf{admissible} on \( H^{2}(\mathbb{C}^n,d\mu) \) if there exists a linear operator \( B^{*} : \mathcal{S} \rightarrow H^{2}(\mathbb{C}^n,d\mu) \) such that the duality relation
 \begin{equation}\label{R1}
 \langle Bk_{z}, k_{w} \rangle = \langle k_{z}, B^{*}k_w \rangle
 \end{equation}
 holds for all \(z,w \in \mathbb{C}^n\).
 \end{defn}
 The inner product here is with respect to \(d\mu\).

\begin{exa}\label{Ex1}
 If \(f\) is a \textbf{bounded measurable function} on \(\mathbb{C}^n\), then there is a positive constant \(0<C=C(f)<\infty\) such that
 \begin{equation*}
 |\langle T_{f}k_{z}, k_{w}\rangle| \le C e^{-(1/8) |z-w|^2}
 \end{equation*}
 for all \(z,w \in \mathbb{C}^n\). That is, \(T_{f} \in \text{XZ-}\mathcal{SL}\). Also, \(T_{f} \in \mathcal{L}_{p}\) for \(2<p<\frac{8}{3}\).
 \end{exa}

\begin{proof}
 For each \(z, w, \xi \in \mathbb{C}^n\), we have
 \begin{equation*}
 |k_{z}(\xi)k_{w}(\xi)|e^{-|\xi|^2} = |e^{\langle \xi , z \rangle - \frac{|z|^2}{2} } e^{\langle \xi, w \rangle - \frac{|w|^2}{2}} | e^{-|\xi|^2} = e^{-\frac{1}{2}\left(|z-\xi|^2 + |w-\xi|^2\right)} ~.
 \end{equation*}
 By the triangular inequality we have
 \begin{equation*}
 |z-w|^2 \le |z-\xi + \xi -w|^2 \le (|z-\xi| + |w-\xi|)^2 \le 2 (|z-\xi|^2 + |w-\xi|^2)~.
 \end{equation*}
 It follows that
 \begin{eqnarray}\label{kzw}
 |k_{z}(\xi)k_{w}(\xi)|e^{-|\xi|^2} = e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)}e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)} \le e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)} e^{-\frac{1}{8}|z-w|^2}~.
 \end{eqnarray}
 Let \(f\) be a bounded measurable function on \(\mathbb{C}^n\), then for all \(g,h\in H^{2}(\mathbb{C}^n,d\mu)\), it holds
 \begin{equation}\label{Toepfgh}
 \langle T_{f}g, h \rangle = \int_{\mathbb{C}^{n}} f(\xi)g(\xi)\overline{h(\xi)} d\mu(\xi)~.
 \end{equation}
 In fact,
 \begin{eqnarray*}
 \langle T_{f}g, h \rangle &=& \int_{\mathbb{C}^{n}} T_{f}g(w)\overline{h(w)} d\mu(w) \\
 &=& \int_{\mathbb{C}^{n}}\int_{\mathbb{C}^{n}} K(w,\xi) f(\xi)g(\xi) d\mu(\xi) \overline{h(w)} d\mu(w) \\
 &=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \int_{\mathbb{C}^{n}} K_{\xi}(w) \overline{h(w)} d\mu(w) d\mu(\xi) \\
 &=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \langle K_{\xi} , h \rangle d\mu(\xi) = \int_{\mathbb{C}^{n}} f(\xi)g(\xi) \overline{\langle h, K_{\xi} \rangle} d\mu(\xi) \\
 &=& \int_{\mathbb{C}^{n}} f(\xi)g(\xi)\overline{h(\xi)} d\mu(\xi)~.
 \end{eqnarray*}
 Therefore,
 \begin{eqnarray*}
 |\langle T_{f}k_{z}, k_{w} \rangle| &=& \left|\int_{\mathbb{C}^n}
 f(\xi)k_{z}(\xi) \overline{k_{w}(\xi)} d\mu(\xi)\right| \\
 &\le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n} \int_{\mathbb{C}^n} |k_{z}(\xi) k_{w}(\xi)| e^{-|\xi|^2} dV(\xi) \\
 & \le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n} e^{-\frac{1}{8}|z-w|^2} \int_{\mathbb{C}^n} e^{-\frac{1}{4}\left(|z-\xi|^2 + |w-\xi|^2\right)} dV(\xi) \\
 & \le & \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n} e^{-\frac{1}{8}|z-w|^2} \left(\int_{\mathbb{C}^n} e^{-\frac{1}{2}|z-\xi|^2} dV(\xi) \right)^{\frac{1}{2}} \left(\int_{\mathbb{C}^n} e^{-\frac{1}{2}|w-\xi|^2}dV(\xi) \right)^{\frac{1}{2}} \\
 &=& \frac{\left\lVert f \right\rVert_{\infty}}{\pi^n} e^{-\frac{1}{8}|z-w|^2} \int_{\mathbb{C}^n} e^{-\frac{1}{2}|\zeta|^2} dV(\zeta) \\
 &=& (\sqrt{2})^n \left\lVert f \right\rVert_{\infty} e^{-\frac{1}{8}|z-w|^2}~.
 \end{eqnarray*}
 This completes the proof.
 \end{proof}

\begin{defn}
 We denote by \(C^{*}(\mathcal{WL})\) the \(C^{*}\)-algebra generated by weakly localized operators on \(H^{2}(\mathbb{C}^n, d\mu)\). Also, \(C^{*}(\mathcal{WL})\) is actually the norm closure of \(\mathcal{WL}\) since \(\mathcal{WL} \) is a \(\star\)-algebra.
 \end{defn}

We will prove the following main result.

\begin{defn}
 We denote by \(C^{*}(\mathcal{WL})\) the \(C^{*}\)-algebra generated by weakly localized operators on \(H^{2}(\C^n, d\mu)\). Also, \(C^{*}(\mathcal{WL})\) is actually the norm closure of \(\mathcal{WL}\) since \(\mathcal{WL} \) is a \(\star\)-algebra.
 \end{defn}

The organization of this paper is as follows. In Section 2, we will give propositions in the case of the Fock space \(H^{2}(\C^n, d\mu)\) which are the analogue of those given by Xia in \cite{xia2015localization} in the Bergman space case of the unit ball. Later, using these propositions in Section 3, we establish the proof of Theorem \ref{XiaFock} and present a consequence.

\begin{defn}\label{normstar}
 For an entire function \(h\) in \(\C^{n}\), we write
 \begin{equation*}
 \norm{h}_{*} = \left(\int_{\C^{n}} |h(\zeta)|^{2} e^{-\frac{1}{2}|\zeta|^2} dV(\zeta)\right)^{\frac{1}{2}}~.
 \end{equation*}
 We denote by \(\mathcal{H}_{*}\) the collection of entire functions \(h\) on \(\C^n\) satisfying \(\norm{h}_{*}<\infty\).
 \end{defn}
 \begin{rem}
 The norm \(\norm{\cdot}_{*}\) is equivalent to the norm on the Fock space \(F^{2}_{1/2}\) given in relation (\ref{Fnorm1}) which is an Hilbert space. More precisely, \(\norm{\cdot}_{*} = (2\pi)^{n/2} \norm{\cdot}_{F^{2}_{1/2}}\). This ensures the continuity of \(\norm{\cdot}_{*}\).
 \end{rem}
 In what follows, we will use the operator \(U_{z}\) defined by
 \begin{equation*}
 U_{z}f(w) = f(z-w) k_{z}(w)~, \quad f \in H^{2}(\C^n, d\mu)~.
 \end{equation*}

\begin{pro}\label{proYz}
 Suppose \(\{c_{u}: u \in \Z^{2n}\}\) are complex numbers satisfying the condition \mbox{\(\sup_{u\in \Z^{2n}}|c_{u}| < \infty\)}. Then for each \(z\in \C^{n}\), the operator
 \begin{equation}\label{Yz}
 Y_{z} = \sum_{u \in \Z^{2n}} c_{u} k_{u-z} \otimes k_{u-z}
 \end{equation}
 is bounded on \(H^{2}(\C^n, d\mu)\). Moreover, the map \(z \mapsto Y_{z}\) from \(\C^{n}\) to \(\mathcal{B}(H^{2}(\C^{n},d\mu))\) is continuous with respect to the operator norm.
 \end{pro}
 \begin{proof}
 Let \(\{c_{u} : u \in \Z^{2n}\}\) be complex numbers such that \mbox{\(\sup_{u\in \Z^{2n}}|c_{u}| < \infty\)}. For \(z\in \C^{n}\), we define
 \begin{equation*}
 A_{z} = \sum_{u \in \Z^{2n}} c_{u} (U_{u}k_{z}) \otimes e_{u} \quad \text{ and } \quad B_{z} = \sum_{u \in \Z^{2n}}(U_{u}k_{z})\otimes e_{u}~.
 \end{equation*}
 Using the relation (\ref{Uu2}), we have \(Y_{z} = A_{z}B^{*}_{z}\). Set \(h_{u} = c_{u}k_{z}\), then using Definition \ref{normstar}, we have
 \begin{equation*}
 \norm{h_{u}}_{*} = |c_{u}|\norm{k_{z}}_{*} = (2\pi)^{n/2} e^{\frac{|z|^2}{2}} |c_{u}|~.
 \end{equation*}
 Therefore, \(\sup_{u\in \Z^{2n}} \norm{h_{u}}_{*} < \infty\). Applying Lemma \ref{LemUh} to \(h_{u} = c_{u}k_{z}\), we see that \(\norm{A_{z}} \le C \sup_{u\in \Z^{2n}}|c_{u}|\). Thus each \(A_{z}\) is bounded. Since \(B_z\) is just a special case of \(A_{z}\) (with \(c_{u}=1\) for all \(u \in \Z^{2n}\)), then it is also bounded and hence \(Y_{z} = A_{z}B^{*}_{z}\) is bounded.

\begin{lem}\label{Lim=0}
 Let \(B \in \mathcal{WL} \), then for every \(z,w \in \C^n \), we have
 \begin{equation*}
 \lim_{R\to \infty} \sup_{u\in \Z^{2n}} \sum_{\underset{|u-v|>R}{v\in \Z^{2n} }} |\langle B k_{u-z}, k_{v-w} \rangle| =0 \quad \text{ and } \quad \lim_{R\to \infty} \sup_{u\in \Z^{2n}} \sum_{\underset{|u-v|>R}{v\in \Z^{2n} }} |\langle k_{u-z}, Bk_{v-w}\rangle| = 0~.
 \end{equation*}
 \end{lem}
 \begin{proof}
 By \cite[Lemma 2.32]{zhu2012}, for any entire function \(f\) on \(\C^n\), we have
 \begin{equation*}
 \left|f(z)e^{-\frac{\alpha}{2}|z|^2}\right|^p \le C \int_{B(z,\delta)} |f(w)e^{-\frac{\alpha}{2}|w|^2}|^p dV(w) \quad \text{ for } z \in \C^n.
 \end{equation*}
 Hence for \(\alpha=p=1\) and \(\delta\) small such that the balls \(\{B(v-w,\delta): v \in \Z^{2n} \}\) are mutually disjoint, we have
 \begin{equation*}
 |\langle B k_{u-z}, k_{v-w} \rangle| = |Bk_{u-z} (v-w) |e^{-\frac{|v-w|^2}{2}} \le C \int_{B(v-w,\delta)} |Bk_{u-z}(\zeta) | e^{-\frac{|\zeta|^2}{2}} dV(\zeta).
 \end{equation*}
 Indeed, for \(\delta < \frac{1}{2} \), the balls \(\{B(v-w,\delta) : v \in \Z^{2n} \}\) are mutually disjoint. Otherwise, there would exist \(v,v' \in \Z^{2n} \) such that \(v \neq v' \), and a point \(\xi\) such that \(\xi \in B(v-w,\delta) \cap B(v'-w,\delta) \). In other words:
 \begin{equation*}
 |v-w-\xi| < \delta \quad \text{ and } \quad |v'-w - \xi| < \delta~.
 \end{equation*}
 This implies that
 \begin{equation*}
 |v-v'|=|(v-w-\xi)-(v'-w-\xi)|\le |v-w-\xi| + |v'-w-\xi| < \delta + \delta= 2 \delta < 1~.
 \end{equation*}
 That is \(|v-v'|<1\). This contradicts the well-known fact that \(|v-v'|\ge 1\). This result actually implies that there exists \(N\in \N\) such that each \(\zeta\in \C^n \) belongs to at most \(N\) balls in \(\{B(v-w,\delta): v\in \Z^{2n}\}\). That is \(\sum_{v \in \Z^{2n}} \chi_{B(v-w,\delta)}(\zeta) \le N \) for each \(\zeta \in \C^n \).

\begin{proof}[\textbf{Proof of Proposition \ref{EwBEzD} }]
 From (\ref{EwBEz}), we have
 \begin{equation*}
 E_{w} BE_{z} = \frac{1}{\pi^{2n}} \sum_{u,v \in \Z^{2n}} \langle Bk_{u-z}, k_{v-w} \rangle k_{v-w} \otimes k_{u-z}~.
 \end{equation*}
 Thus for any \(R>0\), we can write \(E_{w}BE_{z} = V_{R} + W_{R}\), where
 \begin{equation*}
 V_{R} = \frac{1}{\pi^{2n}} \sum_{\underset{|u-v|\le R}{u,v \in \Z^{2n}}} \langle Bk_{u-z}, k_{v-w} \rangle k_{v-w} \otimes k_{u-z} \quad \text{ and }
 \end{equation*}
 \begin{equation*}
 \quad W_{R} = \frac{1}{\pi^{2n}} \sum_{\underset{|u-v|>R}{u,v \in \Z^{2n}}} \langle Bk_{u-z}, k_{v-w} \rangle k_{v-w} \otimes k_{u-z}~.
 \end{equation*}
 To complete the proof, it suffices to prove that:
 \begin{enumerate}[label=(\alph*)]
 \item \label{itm:first1} \(\lim_{R\to \infty} \norm{W_{R}}=0\).
 \item \label{itm:second1} \(V_{R} \in \) span(\(\mathcal{D}_{0}\)) for every \(R>0\).
 \end{enumerate}

\begin{pro}\label{DoT1}
 We have \(\mathcal{D}_{0} \subset \mathcal{T}^{(1)}\).
 \end{pro}
 To establish the proof of this proposition, we will need the next three propositions.
 \begin{pro}\label{YzT1}
 Suppose that \(\{c_{u}: u \in \Z^{2n}\}\) is a bounded set of complex coefficients. Then for each \(z \in \C^{n}\), the operator \(Y_{z}\) defined in (\ref{Yz}) belongs to \(\mathcal{T}^{(1)}\).
 \end{pro}
 \begin{proof}
 \begin{enumerate}[label=(\alph*)]
 \item Let us first show that \(Y_{0} \in \mathcal{T}^{(1)}\). We have \(|u-v|\ge 1\) for all \(u \neq v \in \Z^{2n}\). Hence \(B(u,\frac{1}{2}) \cap B(v,\frac{1}{2}) = \emptyset \) for \(u\neq v\). For each \(0 < \varepsilon < \frac{1}{2}\), define the operator
 \begin{equation*}
 A_{\varepsilon} = \frac{1}{|B(0,\varepsilon)|} \int_{B(0,\varepsilon)} Y_{z} dV(z) .
 \end{equation*}
 From Proposition \ref{proYz}, we have the norm continuity of the map \(z\mapsto Y_{z}\) and it implies that
 \begin{equation*}
 \lim_{\varepsilon \to 0} \norm{Y_{0}-A_{\varepsilon}} = 0.
 \end{equation*}
 This comes from the fact that
 \begin{eqnarray*}
 \norm{Y_{0}-A_{\varepsilon}} &=& \norm{\frac{1}{|B(0,\varepsilon)|} \int_{B(0,\varepsilon)} (Y_{0}-Y_{z}) ~dV(z)} \le \frac{1}{|B(0,\varepsilon)|} \int_{B(0,\varepsilon)} \norm{Y_{0}-Y_{z}} ~dV(z)
 \end{eqnarray*}
 and \(\lim_{z\to 0}\norm{Y_{z}-Y_{0}}=0\) .

The organization of this paper is as follows. In Section 2, we will give propositions in the case of the Fock space \(H^{2}(\mathbb{C}^n, d\mu)\) which are the analogue of those given by Xia in \cite{xia2015localization} in the Bergman space case of the unit ball. Later, using these propositions in Section 3, we establish the proof of Theorem \ref{XiaFock} and present a consequence.

To establish the proof of Theorem~\ref{XiaFock}, the paper first "give[s] propositions in the case of the Fock space \(H^{2}(\mathbb{C}^n, d\mu)\)" in Section 2, as analogues of results of Xia \cite{xia2015localization} for the Bergman space of the unit ball. Then, "using these propositions in Section 3," it "establish[es] the proof of Theorem~\ref{XiaFock}."

To establish the proof of the main theorem, the paper first "give[s] propositions in the case of the Fock space \(H^{2}(\mathbb{C}^n, d\mu)\)" in Section 2, as analogues of results of Xia \cite{xia2015localization} for the Bergman space of the unit ball. Then, "using these propositions in Section 3," it "establish[es] the proof of the main theorem."

Question: Let $H^{2}(\mathbb{C}^{n},d\mu)$ be the classical Fock space, where $d\mu(z)=\pi^{-n}e^{-|z|^{2}}\,dV(z)$. Let $\mathcal{WL}$ denote the class of weakly localized operators on $H^{2}(\mathbb{C}^{n},d\mu)$, and let $C^{*}(\mathcal{WL})$ be the $C^{*}$-algebra generated by $\mathcal{WL}$. Let $\mathcal{T}^{(1)}$ denote the Toeplitz algebra on this Fock space, i.e. the norm-closed algebra generated by Toeplitz operators $T_{f}$ with bounded measurable symbols $f$. Which statement holds about these two operator algebras?

They are equal: $C^{*}(\mathcal{WL})=\mathcal{T}^{(1)}$.

Choices:
B. One only has a strict inclusion $C^{*}(\mathcal{WL})\subsetneq \mathcal{T}^{(1)}$.
C. There is a containment $C^{*}(\mathcal{WL})\subseteq \mathcal{T}^{(1)}$.
D. They coincide only after taking compact-operator quotients, namely $C^{*}(\mathcal{WL})/\mathcal{K}=\mathcal{T}^{(1)}/\mathcal{K}$, but not necessarily before quotienting.
E. They have the same strong-operator closure, but $C^{*}(\mathcal{WL})$ and $\mathcal{T}^{(1)}$ need not be equal as norm-closed algebras.