For $q\geq 0$, the poset     $\W_n^q$ is a lattice for any $n\geq 1$.

Let $q$ be a non-negative real number. A $q$-{\it decreasing word} of length $n\geq 0$ is a binary word of length $n$ satisfying the following constraint:

\centerline{\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\cdot a>b$.}
More precisely, whenever a block of zeros is followed by a block of ones, the length of the ones block must be strictly less than $q$ times the length of the preceding zeros block (see \cite{Barc,arXivBKV,bkv gray code,egecioglu irsic}). Therefore, such words can start with arbitrarily many 1's, and end with arbitrarily many 0's. Let $\mathcal{W}_n^q$ be the set of $q$-decreasing words of length $n$. For instance, we have $\mathcal{W}_4^1=\{0000,0001,0010,1000,1001,1100,1110,1111\}$, $\mathcal{W}_4^{\frac{1}{2}}=\{0000,0001,1000,1100,1110,1111\}$ and $\mathcal{W}_3^{\frac{\pi}{2}}=\{000,001,010,100,101,110,111\}$. It directly follows  that if $q<r$, then $\mathcal{W}_n^q\subseteq\mathcal{W}_n^r$ for any $n\geq 1$. Notice that if we define $\mathcal{W}_n^{q^+}=\bigcap_{r>q}\mathcal{W}_n^r$, or equivalently $\mathcal{W}_n^{q^+}$ is the set of binary words of length $n$ such that \\
\centerline{\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\cdot a\geq b$,}
then we have  $\mathcal{W}_n^{q}=\mathcal{W}_n^{q^+}$ for any $n$ when $q$ is irrational, and $\mathcal{W}_n^{q}\varsubsetneq\mathcal{W}_n^{q^+}$ when $q=c/d$, with $c/d$ an irreducible fraction, and $n$ sufficiently large ($n\geq c+d$). For instance, we have $$\mathcal{W}_4^{1^+}=\{0000,0001,0010,0100,1000,1001,1010,1100,0101,0011,1110,1101,1111\},$$ which contains strictly $\mathcal{W}_4^1$.

Any word $w\in\mathcal{W}_n^q$ can be written
        $$w=1^m0^{a_1}1^{b_1}\ldots0^{a_k}1^{b_k}0^\ell,$$
    with $m,\ell\geq 0$, and $q\cdot a_i>b_i\geq1$ for $1\leq i\leq k$ (with
$k$ possibly equal to zero). The maximal factors  of the form $0^a1^b$ with $q\cdot a>b\geq 1$ will be called {\it prime factors}.

The notion of $q$-decreasing words has recently attracted significant attention in the literature. This family of words exhibits a striking combinatorial property whenever $q$ is a positive integer. Indeed, they are in one-to-one correspondence with binary strings that avoid the pattern $1^{q+1}$, i.e. binary strings without $q+1$ consecutive 1 (see \cite{bkv gray code}). So, this implies that $q$-decreasing words of length $n$ (when $q$ is a positive integer) are enumerated by the $(q+1)$-generalized Fibonacci numbers $F_{n+1}^{q+1}$ where $F_n^q$ is defined by

$$F_n^{q}=F_{n-1}^{q}+F_{n-1}^{q}+\cdot +F_{n-q}^{q},$$ with initial conditions $F_n^q=0$ for $n<0$ and $F_0^q=1$ (see \cite{Fein,knuth,Mile}). It is well known that the generating function of these numbers is $$F_q(x)=\sum_{n\geq 0}F_n^qx^n=\frac{1}{1-x-x^2-\cdots -x^q}.$$

Recently, Barcucci, Bernini, Bilotta and Pinzani \cite{Barc} extended this bijection to $q$-decreasing words for any positive rational number $q$, showing that $\mathcal{W}^q$ is in one-to-one correspondence with binary words avoiding some patterns.

These words have also been studied from a generative prospective. Baril et al. \cite{bkv gray code} provide efficient algorithms for the generation of all $q$-decreasing words whenever $q$ is a positive integer. In particular, they construct a $3$‑Gray code for general $q$, and notably a $1$‑Gray code for the case $q=1$, thus resolving a conjecture posed in the context of interconnection networks by \cite{egecioglu irsic}. More recently, Wong et al. \cite{Wong} present a two-stage algorithm for generating cyclic $2$-Gray codes for $q$-decreasing words.

More generally, for any $q>0$, the generating function $W_q(x)$ for the number of $q$-decreasing words with respect to the length $n$ is given by
\begin{equation}\label{eq:w_q general}
W_q(x)=\frac{1}{(1-x)\left(1-\sum_{i=0}^{+\infty}x^{1+i+\left\lfloor\frac{i}{q}\right\rfloor}\right)},
\end{equation}
see \cite{sergey2,Serg}.
This expression can be simplified as follows when $q$ is rational, i.e. $q=c/d$ where $c$ and $d$ are positive integers:
\begin{equation}\label{eq:w_q rational}
W_q(x)=\frac{1-x^{c+d}}{(1-x)\left(1-x^{c+d}-\sum_{i=0}^{c-1}x^{1+i+\left\lfloor\frac{i}{q}\right\rfloor}\right)}.
\end{equation}
Note that when $q$ is an integer, i.e. when we fix $d=1$ and $c=q$ in the previous formula,  we obtain $W_q(x)=\frac{F_{q+1}(x)-1}{x}$.  Using (\ref{eq:w_q general}) and (\ref{eq:w_q rational}), Dovgal and Kirgizov \cite{sergey2} proved that for all real $q>0$, $[x^n]W_q(x)\underset{n\to\infty}{\sim}C_q\cdot\Phi(q)^n$, for a positive constant $C_q$, and a function $\Phi(q)$ that interpolates the $q$-bonacci numbers. In particular, when $q=c/d$ is a rational number, $\Phi(q)^{-1}$ is the smallest root in modulus of the polynomial $x^{c+d}+\sum_{i=0}^{c-1}x^{1+i+\left\lfloor\frac{i}{q}\right\rfloor}-1$. See \cite{sergey2} for additional properties of $\Phi(q)$.

To conclude this set of definitions, we introduce the main order-theoretic concepts used throughout this paper. These notions are standard and can be found, for instance, in \cite{Grat,stanley}. A {\it poset} $\mathcal{L}$ is a set endowed with a partial order
relation $\leq$. Given two elements $P,Q\in \mathcal{L}$, a {\it meet} (or
{\it greatest lower bound}) of $P$ and $Q$, denoted $P\wedge Q$, is an
element $R$ such that $R\leq P$, $R\leq Q$, and for any $S$ such that
$S\leq P$ and $S\leq Q$,  then we have $S\leq R$. Dually, a {\it join} (or
{\it least upper bound}) of $P$ and $Q$, denoted $P\vee Q$, is an
element $R$ such that $P\leq R$, $Q\leq R$, and for any $S$ such that
$P\leq S$ and $Q\leq S$, then we have $R\leq S$. Notice that join and meet
elements do not necessarily exist in a poset. A {\it lattice} is a
poset where any pair of elements admits a meet and a join.
An element $P\in\mathcal{L}$ is {\it join-irreducible} (resp. {\it
  meet-irreducible}) if $P=R\vee S$ (resp. $P=R\wedge S$) implies
$P=R$ or $P=S$.
  An {\it interval} $I$ in a poset $\mathcal{L}$ is a subset of
  $\mathcal{L}$ such that there exist $P,Q\in I$, $P\leq Q$, such that $I=\{R\in\mathcal{L}\ | \ P\leq R \mbox{ and } R\leq Q\}$. An  element $w$ is said to {\it cover} an element $v$  if $v<w$ and there is no $u\in\mathcal{L}$ such that $v<u<w$. In this case, we write  $v\lessdot w$ and the relation between $v$ and $w$ is called a {\it covering}.

In this paper, we equip $\mathcal{W}_n^q$ with the componentwise order. If $v=v_1v_2\ldots v_n$ and  $w=w_1w_2\ldots w_n$ are two $q$-decreasing words in $\mathcal{W}_n^q$ then
$$v\leq w\Longleftrightarrow v_i\leq w_i \mbox{ for all } 1\leq i\leq n.$$

Let  $\W_n^q:=(\mathcal{W}_n^q,\leq)$ be the poset defined by this order relation.  See Figure~\ref{fig:W_4^2} for an illustration of the poset $\W_5^1$.

\centerline{\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\cdot a>b$.}
More precisely, whenever a block of zeros is followed by a block of ones, the length of the ones block must be strictly less than $q$ times the length of the preceding zeros block (see \cite{Barc,arXivBKV,bkv gray code,egecioglu irsic}). Therefore, such words can start with arbitrarily many 1's, and end with arbitrarily many 0's. Let $\mathcal{W}_n^q$ be the set of $q$-decreasing words of length $n$. For instance, we have $\mathcal{W}_4^1=\{0000,0001,0010,1000,1001,1100,1110,1111\}$, $\mathcal{W}_4^{\frac{1}{2}}=\{0000,0001,1000,1100,1110,1111\}$ and $\mathcal{W}_3^{\frac{\pi}{2}}=\{000,001,010,100,101,110,111\}$. It directly follows that if $q<r$, then $\mathcal{W}_n^q\subseteq\mathcal{W}_n^r$ for any $n\geq 1$. Notice that if we define $\mathcal{W}_n^{q^+}=\bigcap_{r>q}\mathcal{W}_n^r$, or equivalently $\mathcal{W}_n^{q^+}$ is the set of binary words of length $n$ such that \\
\centerline{\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\cdot a\geq b$,}
then we have $\mathcal{W}_n^{q}=\mathcal{W}_n^{q^+}$ for any $n$ when $q$ is irrational, and $\mathcal{W}_n^{q}\varsubsetneq\mathcal{W}_n^{q^+}$ when $q=c/d$, with $c/d$ an irreducible fraction, and $n$ sufficiently large ($n\geq c+d$). For instance, we have $$\mathcal{W}_4^{1^+}=\{0000,0001,0010,0100,1000,1001,1010,1100,0101,0011,1110,1101,1111\},$$ which contains strictly $\mathcal{W}_4^1$.

Any word $w\in\mathcal{W}_n^q$ can be written
 $$w=1^m0^{a_1}1^{b_1}\ldots0^{a_k}1^{b_k}0^\ell,$$
 with $m,\ell\geq 0$, and $q\cdot a_i>b_i\geq1$ for $1\leq i\leq k$ (with
$k$ possibly equal to zero). The maximal factors of the form $0^a1^b$ with $q\cdot a>b\geq 1$ will be called {\it prime factors}.

$$F_n^{q}=F_{n-1}^{q}+F_{n-1}^{q}+\cdot +F_{n-q}^{q},$$ with initial conditions $F_n^q=0$ for $n<0$ and $F_0^q=1$ (see \cite{Fein,knuth,Mile}). It is well known that the generating function of these numbers is $$F_q(x)=\sum_{n\geq 0}F_n^qx^n=\frac{1}{1-x-x^2-\cdots -x^q}.$$

To conclude this set of definitions, we introduce the main order-theoretic concepts used throughout this paper. These notions are standard and can be found, for instance, in \cite{Grat,stanley}. A {\it poset} $\mathcal{L}$ is a set endowed with a partial order
relation $\leq$. Given two elements $P,Q\in \mathcal{L}$, a {\it meet} (or
{\it greatest lower bound}) of $P$ and $Q$, denoted $P\wedge Q$, is an
element $R$ such that $R\leq P$, $R\leq Q$, and for any $S$ such that
$S\leq P$ and $S\leq Q$, then we have $S\leq R$. Dually, a {\it join} (or
{\it least upper bound}) of $P$ and $Q$, denoted $P\vee Q$, is an
element $R$ such that $P\leq R$, $Q\leq R$, and for any $S$ such that
$P\leq S$ and $Q\leq S$, then we have $R\leq S$. Notice that join and meet
elements do not necessarily exist in a poset. A {\it lattice} is a
poset where any pair of elements admits a meet and a join.
An element $P\in\mathcal{L}$ is {\it join-irreducible} (resp. {\it
 meet-irreducible}) if $P=R\vee S$ (resp. $P=R\wedge S$) implies
$P=R$ or $P=S$.
 An {\it interval} $I$ in a poset $\mathcal{L}$ is a subset of
 $\mathcal{L}$ such that there exist $P,Q\in I$, $P\leq Q$, such that $I=\{R\in\mathcal{L}\ | \ P\leq R \mbox{ and } R\leq Q\}$. An element $w$ is said to {\it cover} an element $v$ if $v<w$ and there is no $u\in\mathcal{L}$ such that $v<u<w$. In this case, we write $v\lessdot w$ and the relation between $v$ and $w$ is called a {\it covering}.

In this paper, we equip $\mathcal{W}_n^q$ with the componentwise order. If $v=v_1v_2\ldots v_n$ and $w=w_1w_2\ldots w_n$ are two $q$-decreasing words in $\mathcal{W}_n^q$ then
$$v\leq w\Longleftrightarrow v_i\leq w_i \mbox{ for all } 1\leq i\leq n.$$

Let $\W_n^q:=(\mathcal{W}_n^q,\leq)$ be the poset defined by this order relation. See Figure~\ref{fig:W_4^2} for an illustration of the poset $\W_5^1$.

\label{fig:W_4^2}
\end{figure}

\noindent {\bf Outline of the paper.} In Section~\ref{sec:useful results}, we collect preliminary results that will be used throughout the paper. Many of these results

\centerline{\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\cdot a>b$.}
More precisely, whenever a block of zeros is followed by a block of ones, the length of the ones block must be strictly less than $q$ times the length of the preceding zeros block (see \cite{Barc,arXivBKV,bkv gray code,egecioglu irsic}). Therefore, such words can start with arbitrarily many 1's, and end with arbitrarily many 0's. Let $\mathcal{W}_n^q$ be the set of $q$-decreasing words of length $n$. For instance, we have $\mathcal{W}_4^1=\{0000,0001,0010,1000,1001,1100,1110,1111\}$, $\mathcal{W}_4^{\frac{1}{2}}=\{0000,0001,1000,1100,1110,1111\}$ and $\mathcal{W}_3^{\frac{\pi}{2}}=\{000,001,010,100,101,110,111\}$. It directly follows that if $q<r$, then $\mathcal{W}_n^q\subseteq\mathcal{W}_n^r$ for any $n\geq 1$. Notice that if we define $\mathcal{W}_n^{q^+}=\bigcap_{r>q}\mathcal{W}_n^r$, or equivalently $\mathcal{W}_n^{q^+}$ is the set of binary words of length $n$ such that \\
\centerline{\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\cdot a\geq b$,}
then we have $\mathcal{W}_n^{q}=\mathcal{W}_n^{q^+}$ for any $n$ when $q$ is irrational, and $\mathcal{W}_n^{q}\varsubsetneq\mathcal{W}_n^{q^+}$ when $q=c/d$, with $c/d$ an irreducible fraction, and $n$ sufficiently large ($n\geq c+d$). For instance, we have $$\mathcal{W}_4^{1^+}=\{0000,0001,0010,0100,1000,1001,1010,1100,0101,0011,1110,1101,1111\},$$ which contains strictly $\mathcal{W}_4^1$.

Any word $w\in\mathcal{W}_n^q$ can be written
 $$w=1^m0^{a_1}1^{b_1}\ldots0^{a_k}1^{b_k}0^\ell,$$
 with $m,\ell\geq 0$, and $q\cdot a_i>b_i\geq1$ for $1\leq i\leq k$ (with
$k$ possibly equal to zero). The maximal factors of the form $0^a1^b$ with $q\cdot a>b\geq 1$ will be called {\it prime factors}.

To conclude this set of definitions, we introduce the main order-theoretic concepts used throughout this paper. These notions are standard and can be found, for instance, in \cite{Grat,stanley}. A {\it poset} $\mathcal{L}$ is a set endowed with a partial order
relation $\leq$. Given two elements $P,Q\in \mathcal{L}$, a {\it meet} (or
{\it greatest lower bound}) of $P$ and $Q$, denoted $P\wedge Q$, is an
element $R$ such that $R\leq P$, $R\leq Q$, and for any $S$ such that
$S\leq P$ and $S\leq Q$, then we have $S\leq R$. Dually, a {\it join} (or
{\it least upper bound}) of $P$ and $Q$, denoted $P\vee Q$, is an
element $R$ such that $P\leq R$, $Q\leq R$, and for any $S$ such that
$P\leq S$ and $Q\leq S$, then we have $R\leq S$. Notice that join and meet
elements do not necessarily exist in a poset. A {\it lattice} is a
poset where any pair of elements admits a meet and a join.
An element $P\in\mathcal{L}$ is {\it join-irreducible} (resp. {\it
 meet-irreducible}) if $P=R\vee S$ (resp. $P=R\wedge S$) implies
$P=R$ or $P=S$.
 An {\it interval} $I$ in a poset $\mathcal{L}$ is a subset of
 $\mathcal{L}$ such that there exist $P,Q\in I$, $P\leq Q$, such that $I=\{R\in\mathcal{L}\ | \ P\leq R \mbox{ and } R\leq Q\}$. An element $w$ is said to {\it cover} an element $v$ if $v<w$ and there is no $u\in\mathcal{L}$ such that $v<u<w$. In this case, we write $v\lessdot w$ and the relation between $v$ and $w$ is called a {\it covering}.

Let $\W_n^q:=(\mathcal{W}_n^q,\leq)$ be the poset defined by this order relation. See Figure~\ref{fig:W_4^2} for an illustration of the poset $\W_5^1$.

\begin{figure}[h]
 \centering
\begin{tikzpicture}[scale=1.3]
 \node (11111) at (0,7.5) {$11111$};
 \node (11110) at (-1.5,6) {$11110$};
 \node (11001) at (0,4.5) {$11001$};
 \node (11100) at (-3,4.5) {$11100$};
 \node (00011) at (3,3) {$00011$};
 \node (10001) at (1,3) {$10001$};
 \node (10010) at (-1,3) {$10010$};
 \node (11000) at (-3,3) {$11000$};
 \node (00001) at (3,1.5) {$00001$};
 \node (00010) at (1,1.5) {$00010$};
 \node (10000) at (-3,1.5) {$10000$};
 \node (00100) at (-1,1.5) {$00100$};
 \node (00000) at (0,0) {$00000$};
 \draw (00000) -- (00100);\draw (00000) -- (10000);\draw (00000) -- (00001);\draw (00000) -- (00010);\draw (10000) -- (11000);\draw (10000) -- (10010);\draw (10000) -- (10001);\draw (00010) -- (10010);\draw (00010) -- (00011);\draw (00001) -- (10001);\draw (00001) -- (00011);\draw (00100) -- (11100);\draw (11000) -- (11001);\draw (11000) -- (11100);\draw (10010) -- (11110);\draw (10001) -- (11001);\draw (00011) -- (11111);\draw (11100) -- (11110);\draw (11001) -- (11111);\draw (11110) -- (11111);
\end{tikzpicture}
 \caption{The lattice $\W_5^1$. It contains 20 coverings (edges), 7 meet-irreducible elements, 5 join-irreducible elements and 56 intervals. }
 \label{fig:W_4^2}
\end{figure}

\noindent {\bf Outline of the paper.} In Section~\ref{sec:useful results}, we collect preliminary results that will be used throughout the paper. Many of these results are quite technical, due to the presence of floor and ceiling functions in generating functions related to $q$-decreasing words. Section~\ref{sec:join and coverings} is devoted to enumerative results concerning classical lattice parameters, specifically join-irreducible elements, as well as covering relations. For any rational number \( q > 0 \), we provide a closed-form expression for the generating function counting the number of coverings in \( \W^q_n \). When \( q \) is irrational, we present a formula that enables efficient computation of the initial terms of the series expansion (e.g., using \textsc{Maple}). We also prove that the asymptotic behavior of the number of coverings is connected to the function $\Phi(q)$ defined in the introduction above. In Section~\ref{sec:intervals}, we derive a closed-form expression for the generating function that enumerates the number of intervals in \( \W^q_n \) for any rational number \( q > 0 \). Finally, Section~\ref{sec:meet} presents the structure of meet-irreducible elements in $\W_n^q$ for any positive rational number $q$. This structure is the same as the one of words over an alphabet of $2\lceil q\rceil+1$ letters avoiding $\lceil q\rceil^2+2\lceil q\rceil-1$ consecutive patterns of length 2. Taking advantage of this classical structure, we present a method to obtain the closed form of the generating function enumerating the number of meet-irreducible elements in $\W_n^q$ for any rational number $q>0$.

In this section, we provide enumerative results for classical parameters of a lattice, namely the join-irreducible elements, as well as covering relations. The enumeration of meet-irreducible elements is a bit more intricate, so we treat it in Section \ref{sec:meet}.
We first give the enumeration of join-irreducible elements for any $q>0$, and then we conclude by giving closed form for the generating functions of the covering for positive rational numbers $q$, and a method for computing arbitrarily many terms of the generating functions for positive irrational numbers $q$.
\subsection{Join-irreducible elements}
\begin{thm}\label{join irreducible}
 For $q>0$ and $n\geq1$, there are exactly $n$ join-irreducible elements in $\W_n^q$.
\end{thm}
\begin{proof}
 In a finite lattice, an element is join-irreducible if and only if it covers exactly one element. We therefore count the words in $\W_n^q$ that cover exactly one element. Note that a factor $0^a1^b$, with $qa>b\geq 1$ covers only one element if and only if $b=1$ (otherwise it covers $0^a1^{b-1}0$ and $0^{a+1}1^{b-1}$). Now we investigate the elements covered by $1^m$. Given $i\geq0$, there is at most one word covered by $1^m$ with suffix $01^i$. So the words covered by $1^m$ are exactly

\begin{proof} For any words $v$ and $w$ in $\mathcal{W}_n^q$, we consider the binary word $a=a_1\ldots a_n$  where $a_i=1$ if and only if $v_i=w_i=1$. It is straightforward to see that $a\in \mathcal{W}_n^q$, and then $a$ is the greatest lower bound of $v$ and $w$,  which implies that $\W_n^q$ is a meet-semilattice. Since $1^n$ is the maximum element of $\W_n^q$, Proposition 3.3.1 in \cite{stanley} implies that $\W_n^q$ is a lattice.
\end{proof}

\begin{figure}[h]
    \centering
\begin{tikzpicture}[scale=1.3]
    \node (11111) at (0,7.5) {$11111$};
    \node (11110) at (-1.5,6) {$11110$};
    \node (11001) at (0,4.5) {$11001$};
    \node (11100) at (-3,4.5) {$11100$};
    \node (00011) at (3,3) {$00011$};
    \node (10001) at (1,3) {$10001$};
    \node (10010) at (-1,3) {$10010$};
    \node (11000) at (-3,3) {$11000$};
    \node (00001) at (3,1.5) {$00001$};
    \node (00010) at (1,1.5) {$00010$};
    \node (10000) at (-3,1.5) {$10000$};
    \node (00100) at (-1,1.5) {$00100$};
    \node (00000) at (0,0) {$00000$};
    \draw (00000) -- (00100);\draw (00000) -- (10000);\draw (00000) -- (00001);\draw (00000) -- (00010);\draw (10000) -- (11000);\draw (10000) -- (10010);\draw (10000) -- (10001);\draw (00010) -- (10010);\draw (00010) -- (00011);\draw (00001) -- (10001);\draw (00001) -- (00011);\draw (00100) -- (11100);\draw (11000) -- (11001);\draw (11000) -- (11100);\draw (10010) -- (11110);\draw (10001) -- (11001);\draw (00011) -- (11111);\draw (11100) -- (11110);\draw (11001) -- (11111);\draw (11110) -- (11111);
\end{tikzpicture}
    \caption{The lattice $\W_5^1$. It contains 20 coverings (edges), 7 meet-irreducible elements, 5 join-irreducible elements and 56 intervals. }
    \label{fig:W_4^2}
\end{figure}

\noindent {\bf Outline of the paper.} In Section~\ref{sec:useful results}, we collect preliminary results that will be used throughout the paper. Many of these results are quite technical, due to the presence of floor and ceiling functions in generating functions related to $q$-decreasing words. Section~\ref{sec:join and coverings} is devoted to enumerative results concerning classical lattice parameters, specifically join-irreducible elements, as well as covering relations. For any rational number \( q > 0 \), we provide a closed-form expression for the generating function counting the number of coverings in \( \mathbb{W}^q_n \). When \( q \) is irrational, we present a formula that enables efficient computation of the initial terms of the series expansion (e.g., using \textsc{Maple}). We also prove that the asymptotic behavior of the number of coverings is connected to the function $\Phi(q)$ defined in the introduction above. In Section~\ref{sec:intervals}, we derive a closed-form expression for the generating function that enumerates the number of intervals in \( \mathbb{W}^q_n \) for any rational number \( q > 0 \). Finally, Section~\ref{sec:meet} presents the structure of meet-irreducible elements in $\W_n^q$ for any positive rational number $q$. This structure is the same as the one of words over an alphabet  of $2\lceil q\rceil+1$ letters avoiding $\lceil q\rceil^2+2\lceil q\rceil-1$ consecutive patterns of length 2. Taking advantage of this classical structure, we present a method to obtain the closed form of the generating function enumerating the number of meet-irreducible elements in $\W_n^q$ for any rational number $q>0$.

For any words $v$ and $w$ in $\mathcal{W}_n^q$, consider the binary word $a=a_1\ldots a_n$ where $a_i=1$ iff $v_i=w_i=1$. It is \"straightforward to see\" that $a\in \mathcal{W}_n^q$, and then $a$ is the greatest lower bound of $v$ and $w$, so $\W_n^q$ is a meet-semilattice. Since $1^n$ is the maximum element of $\W_n^q$, Proposition 3.3.1 in \cite{stanley} implies that $\W_n^q$ is a lattice.

For any words $v$ and $w$ in $\mathcal{W}_n^q$, consider the binary word $a=a_1\ldots a_n$ where $a_i=1$ iff $v_i=w_i=1$. It is "straightforward to see" that $a\in \mathcal{W}_n^q$, and then $a$ is the greatest lower bound of $v$ and $w$, so $\W_n^q$ is a meet-semilattice. Since $1^n$ is the maximum element of $\W_n^q$, Proposition 3.3.1 in \cite{stanley} implies that $\W_n^q$ is a lattice.,

Question: Let \(q\ge 0\) and \(n\ge 1\). A binary word of length \(n\) is called \(q\)-decreasing if every maximal factor of the form \(0^a1^b\) satisfies either \(a=0\) or \(q\,a>b\). Let \(\mathcal{W}_n^q\) be the set of all such words, and equip it with the componentwise order: for \(v=v_1\cdots v_n\) and \(w=w_1\cdots w_n\), define \(v\le w\) iff \(v_i\le w_i\) for all \(1\le i\le n\). Write \(\W_n^q=(\mathcal{W}_n^q,\le)\). Which statement holds for every choice of \(q\ge 0\) and \(n\ge 1\)?

The poset \(\W_n^q\) is a lattice; equivalently, every pair of elements of \(\mathcal{W}_n^q\) has both a meet (greatest lower bound) and a join (least upper bound) in the componentwise order.

Choices:
B. The poset \(\W_n^q\) is a meet-semilattice but, for some choices of \(q\ge 0\) and \(n\ge 1\), it is not a lattice; in general, componentwise meets of elements of \(\mathcal{W}_n^q\) always exist, whereas componentwise joins need not belong to \(\mathcal{W}_n^q\).
C. For every \(q\ge 0\) and \(n\ge 1\), every pair of elements of \(\mathcal{W}_n^q\) has a meet (greatest lower bound) in the componentwise order; equivalently, \(\W_n^q\) is a meet-semilattice.
D. The poset \(\W_n^q\) is a lattice for every irrational \(q>0\), but this can fail for rational values of \(q\) because the distinction between the strict condition \(q\,a>b\) and the weak condition \(q\,a\ge b\) prevents componentwise joins from being well defined in general.
E. The poset \(\W_n^q\) is a lattice precisely for \(q>0\); when \(q=0\), the condition on maximal factors forces a boundary case in which \(\mathcal{W}_n^q\) is no longer closed under the meet and join operations in the componentwise order.

The stem defines the objects and order relation but does not explicitly or implicitly reveal that the poset is a lattice. No wording strongly privileges choice A over the others.

The item is close to a theorem-identification question: the correct option is essentially a clean theorem statement about \(\W_n^q\). It is not a pure tautology because the alternatives offer nearby competing claims, but it still largely tests recognition of the main result.

There is some reasoning pressure because the student must distinguish lattice vs. meet-semilattice, universal vs. exceptional-in-\(q\) behavior, and boundary cases such as \(q=0\). However, the item mainly rewards theorem recall or result recognition rather than substantial derivation from first principles.

The distractors are plausible and mathematically meaningful: one is a weaker true statement, others exploit common failure modes involving joins, rational/irrational distinctions, and the boundary case \(q=0\). They are distinct and well targeted.

A solid theorem-discrimination MCQ with good distractors and no answer leakage, but it leans more toward recall of the exact result than toward genuinely generative mathematical reasoning.


Real algebraic geometry can be thought of as a generalization of manifolds, where the continuous functions are replaced by polynomials with real coefficients. Application to physics also leads to a necessary generalization to associative algebraic geometry and a generalization of continuous Riemannian metrics, see the book of O.A. Laudal, \cite{Laudal21}. One of the main problems with this, is that Riemannian metrics are defined over the reals, and the polynomial algebra over the reals, $\mathbb R[x_1,\dots,x_n],$ contains more simple modules than $\mathbb R^n.$ Because the algebraic properties governing the simple modules are better controlled by an algebra over an algebraically closed field, the main result in this text is the construction of a $\mathbb C$-algebra $A_{\mathbb R}$ such that $\simp(A_{\mathbb R})\cong\mathbb R^n.$ Thus the points in $\mathbb R^n$ is in one-to-one correspondence with the simple $A_{\mathbb R}$-modules for which $\aspec(A_{\mathbb R})$ is a fine moduli, see the book \cite{S23} or the preprint \cite{S241}. For any real manifold $M$ we define an associative variety $(\mathcal M,\mathcal O^A_L)$ over $\mathbb C$ such that the points in $M$ is in bijective correspondence with the closed points in $\mathcal M,$ and such that the charts $U$ in $\mathcal M$ corresponds to $\mathcal O^A_L(U)=A_{\mathbb R}.$

In short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$

Because of the discovery of localization in associative rings, \cite{S252}, we include the general definition of associative schemes. Their purpose is to serve as moduli of algebraic objects which can be put in one-to-one correspondence with modules over associative schemes: That is, we need associative moduli to classify associative objects. This is most easily seen by the following.
Let $k$ be a field and consider the polynomial algebra in $n\geq 1$ variables $$A=k[x_1,\dots,x_n]=k[\underline x].$$ For a point $P=(p_1,\dots,p_n)\in k^n,$ we define the simple $A$-modules $$M(P)=A/(x_1-p_1,\dots,x_n-p_n).$$

\begin{abstract}
In the preprint \cite{S252} we proved that there exists a localizing ring $A_M$ for $A$ an associative ring with unit, and $M=\oplus_{i=1}^rM_i$ a direct sum of $r\geq 1$ simple right $A$-modules. For a homomorphism of associative rings $A\rightarrow B$ we define the contraction of a simple $B$-module to $A.$
Then we define the set of aprime right $A$-modules $\aspec A$ to be the set of simple $A$-modules together with contractions of such. When $A$ is commutative, $\aspec A=\spec A,$ and we define a topology on $\aspec A$ such that when $A$ is commutative, this is the Zariski topology. In the preprint \cite{S251}, we proved that when we have a topology and a localizing subcategory, there exists a sheaf of associative rings $\mathcal O_X$ on $\aspec A,$ agreeing with the usual sheaf of rings on $\spec A.$ In this text, we write out this construction, and we see that we can restrict the sheaf and topology to any subset $V\subseteq\aspec A.$ In particular, this
proves that we can use complex varieties in real algebraic geometry, by restricting in accordance with $\mathbb R\subseteq\mathbb C.$ Thus the theory of schemes over algebraically closed fields and its associative generalization can be applied to real (algebraic) geometry.
\end{abstract}

In short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$

\begin{proof} From \cite{S23} we know that $$\ext^1_A(M_1,M_2)\simeq\der_k(A,\hmm_k(M_1,M_2))/\inner$$ where $\hmm_k(M_1,M_2)$ is an $A-A$ bimodule by $a\phi(m)=\phi(am), \phi a(m)=\phi(m)a.$

When $P\neq Q$ we can consider $M_{\underline 0}, M_P$ with $p_1\neq 0.$
For $$\delta\in\der_k(A,\hmm_k(M_{\underline 0},M_P)$$ we have $$\begin{aligned}\delta(x_ix_1)&=\delta(x_1x_i)\\
&\Updownarrow\\
\delta(x_i)x_1+x_i\delta(x_1)&=\delta(x_1)x_i+x_1\delta(x_i)\\
&\Updownarrow\\
\delta(x_i)p_1&=\delta(x_1)p_i\\
&\Updownarrow\\
\delta(x_i)&=\delta(x_i)\frac{p_i}{p_1}.
\end{aligned}$$
This is also true for the inner derivations, proving that $\dim_k\inner=1$ so that $\ext^1_A(M_P,M_Q)=0$ when $P\neq Q.$
\end{proof}

\begin{proof} We have a homomorphism $f:A\rightarrow A/\mathfrak m$ such for every $s\in A\setminus\mathfrak m,$ $f(s)$ is a unit. By the universal property of localization, there exists a unique homomorphism $\phi:A_\mathfrak m\rightarrow A/\mathfrak m$ such that $\phi(\frac{a}{s})=\iota(a)\iota(s)^{-1}.$ This homomorphism is clearly surjective, and its kernel is $\mathfrak m A_\mathfrak m$ giving the wanted isomorphism.
\end{proof}

\begin{proof} We send the prime ideal $\mathfrak p\subset A$ to $A_\mathfrak p/\mathfrak p A_\mathfrak p$ which is $A$-prime by definition. On the other hand, let
$M$ be an aprime $A$-module, defined by $\iota_M: A\rightarrow B$ such that $M$ is a simple $B$-module. Let $\mathfrak m\subset B$ be the maximal ideal defining $M$ as a simple $B$-module. Then $\iota_M^{-1}(\mathfrak m)$ is a prime ideal in $A.$ That these two operations are inverses to each other follows from the fact that for a maximal ideal in a ring $B$ we have $B/\mathfrak m\simeq B_{\mathfrak m}/\mathfrak m B_\mathfrak m$ as proven i Lemma \ref{anotherloclemma}.
\end{proof}

\begin{definition} Define the subset of $k$-points in $\mathbb X$ by $$\tilde{\mathbb X}(k)=\{x\in X\subseteq\mathbb X|x\text{ is simple}\}\subseteq\mathbb X.$$ Then the induced associative subscheme $\mathbb X(k)$ is called the associative subscheme of $k$-points.
\end{definition}

\begin{proof} Because $k\subseteq\Bbbk$ is a sub-algebra, it follows that if $\phi\otimes\id$ is an isomorphism, then $\dim_k V_1=\dim_k V_2,$ and that choosing corresponding bases, $$0\neq\det(\phi\otimes\id)=\det\phi.$$
\end{proof}

\begin{proof} From \cite{S23} we know that $$\ext^1_A(M_1,M_2)\simeq\der_k(A,\hmm_k(M_1,M_2))/\inner$$ where $\hmm_k(M_1,M_2)$ is an $A-A$ bimodule by $a\phi(m)=\phi(am), \phi a(m)=\phi(m)a.$

When $P=Q$ we can assume $M_1=M_2=k[\underline x]/(\underline x).$ Then any inner derivation is on the form $\ad_\phi$ for which $\ad_\phi(x_i)=[\phi,x_i]=0.$ Thus the inner derivations is of dimension zero and $\ext^1_A(M_1,M_2)\simeq\der_k(A,k)$ where
$\operatorname d_i(x_i)=1,\ i=1,\dots,n$ gives a basis for the derivations.

When $P\neq Q$ we can consider $M_{\underline 0}, M_P$ with $p_1\neq 0.$
For $$\delta\in\der_k(A,\hmm_k(M_{\underline 0},M_P)$$ we have $$\begin{aligned}\delta(x_ix_1)&=\delta(x_1x_i)\\
&\Updownarrow\\
\delta(x_i)x_1+x_i\delta(x_1)&=\delta(x_1)x_i+x_1\delta(x_i)\\
&\Updownarrow\\
\delta(x_i)p_1&=\delta(x_1)p_i\\
&\Updownarrow\\
\delta(x_i)&=\delta(x_i)\frac{p_i}{p_1}.
\end{aligned}$$
This is also true for the inner derivations, proving that $\dim_k\inner=1$ so that $\ext^1_A(M_P,M_Q)=0$ when $P\neq Q.$
\end{proof}

It follows from the lemma that not all finite dimensional simple modules over a noncommutative $k$-algebra can be classified by a finitely generated commutative algebra. See \cite{S23} for a lot of examples.

From \cite{S23}: \(\ext^1_A(M_1,M_2)\simeq \der_k(A,\hmm_k(M_1,M_2))/\inner\), with \(\hmm_k(M_1,M_2)\) an \(A\)-\(A\) bimodule via \(a\phi(m)=\phi(am)\), \(\phi a(m)=\phi(m)a\).

- Case \(P=Q\): assume \(M_1=M_2=k[\underline x]/(\underline x)\). Any inner derivation is \(\ad_\phi\) and \(\ad_\phi(x_i)=[\phi,x_i]=0\), so \(\dim_k\inner=0\). Hence \(\ext^1_A(M_1,M_2)\simeq \der_k(A,k)\), and the derivations \(\operatorname d_i\) with \(\operatorname d_i(x_i)=1\) for \(i=1,\dots,n\) give a basis, so the dimension is \(n\).

- Case \(P\neq Q\): consider \(M_{\underline 0}, M_P\) with \(p_1\neq 0\). For \(\delta\in\der_k(A,\hmm_k(M_{\underline 0},M_P))\), comparing \(\delta(x_ix_1)=\delta(x_1x_i)\) yields
\[\delta(x_i)x_1+x_i\delta(x_1)=\delta(x_1)x_i+x_1\delta(x_i)\Rightarrow \delta(x_i)p_1=\delta(x_1)p_i\Rightarrow \delta(x_i)=\delta(x_1)\frac{p_i}{p_1}.\]
This relation also holds for inner derivations, and it is concluded that \(\dim_k\inner=1\), so \(\ext^1_A(M_P,M_Q)=0\) when \(P\neq Q\).

For two points $P,Q\in k^n$ we have that $$\dim_k\ext^1_A(M_P,M_Q)=\begin{cases}n,\ P=Q\\0,\ P\neq Q.\end{cases}$$,

Question:
Let $k$ be a field, let $A=k[x_1,\dots,x_n]$ be the polynomial algebra in $n\ge 1$ variables over $k$, and for each point $P=(p_1,\dots,p_n)\in k^n$ define the simple $A$-module
$$M_P:=A/(x_1-p_1,\dots,x_n-p_n).$$
Which statement holds for every pair of points $P,Q\in k^n$ about the $k$-dimension of $\operatorname{Ext}^1_A(M_P,M_Q)$?

$$\dim_k\operatorname{Ext}^1_A(M_P,M_Q)=\begin{cases}n,& P=Q,\\0,& P\ne Q.\end{cases}$$

Choices:
B. $$\dim_k\operatorname{Ext}^1_A(M_P,M_Q)=\begin{cases}n,& P=Q,\\1,& P\ne Q.\end{cases}$$
C. $$\dim_k\operatorname{Ext}^1_A(M_P,M_P)=n\qquad\text{for every }P\in k^n.$$
D. $$\dim_k\operatorname{Ext}^1_A(M_P,M_Q)=\begin{cases}n,& P=Q,\\0,& P\text{ and }Q\text{ differ in every coordinate}.\end{cases}$$
E. $$\dim_k\operatorname{Ext}^1_A(M_P,M_Q)=\begin{cases}n-1,& P=Q,\\0,& P\ne Q.\end{cases}$$

quotient by inner derivations in the off-diagonal case

use of a coordinate with nonzero difference after reducing to one point at the origin

dimension of derivations in the diagonal case

The stem does not state or strongly hint at the diagonal/off-diagonal Ext^1 dimensions; it only defines the modules and asks for the correct universal statement.

This is not a mere restatement of a theorem from the stem. The respondent must choose among several competing global descriptions, including a weaker true statement and subtly false variants.

Answering correctly requires genuine reasoning or recall of the Ext computation: one must identify both the diagonal value n and the off-diagonal vanishing, and also recognize that choice C is true but not the strongest statement asked for.

The distractors are plausible and mathematically targeted: B tests mistaken off-diagonal nonvanishing, C is a tempting weaker true statement, D reflects an over-specific coordinate-based vanishing claim, and E probes a dimension-count error.

A strong MCQ: no answer leakage, non-tautological framing, real reasoning pressure, and high-quality distractors that reflect common mathematical missteps.


\begin{theorem}\label{Teorema rigidez introduçao}
Let $x:M^n \rightarrow \mathbb{R}^{n+1}$ be a hypersurface with $n$ distinct, nonvanishing principal curvatures, let $Y:M^n \rightarrow \mathbb{C}^{n+3} \subset \mathbb{R}^{n+4}_2$ be its Laguerre position vector with Laguerre metric $g = \langle dY, dY \rangle$, and let $\lambda$ be the eigenvalues of the tensor $\mathbb{L}$. Suppose $M^n$ is connected and admits a parametrization by lines of curvature with respect to the metric $g$. If $x$ is an L-isotropic hypersurface, then $\lambda = 0$ and $x$ is L-isoparametric. Moreover, up to Laguerre transformation, this hypersurface is equivalent to the one described in \eqref{familia da dupin introdução} for $\phi = 0$.
\end{theorem}

\hspace{0.5cm}The theory of Laguerre geometry began in the 20th century with the study of Laguerre surfaces in $\mathbb{R}^3$, as presented in Blaschke’s book \cite{Blaschke}. Since then, several mathematicians have dedicated themselves to the study of Laguerre differential geometry, including Musso and Nicolodi in \cite{Musso}, \cite{Musso2}, and Palmer in \cite{Palmer}. The study of Laguerre geometry of surfaces in $\mathbb{R}^{3}$ continued  with the work of Li \cite{Li superficies}. Later, Song and Wang \cite{superficies minimas} studied minimal Laguerre surfaces in $\mathbb{R}^{3}$. The development of Laguerre geometry for surfaces naturally motivated a generalization to the case of hypersurfaces in $\mathbb{R}^{n+1}$. Subsequently, several papers were published addressing this more general case.

In Laguerre differential geometry, one studies the properties of Laguerre hypersurfaces that are invariant under the group of Laguerre transformations on the unit tangent bundle $U\mathbb{R}^{n+1}$. Li and Wang \cite{Li pequeno} studied this geometry for hypersurfaces in $\mathbb{R}^{n+1}$ using Cartan’s method of moving frames. Considering an immersion $x: M^n \rightarrow \mathbb{R}^{n+1}$ without umbilic points, with distinct nonzero principal curvatures, and $\xi: M^n \rightarrow \mathbb{S}^{n}$ a unit vector field normal to $x$, they defined the basic Laguerre invariants, namely: a Laguerre-invariant metric $g$, the Laguerre second fundamental form $\mathbb{B}$, the Laguerre form $\mathbb{C}$ and the Laguerre tensor $\mathbb{L}$.

Once these invariants were defined it was natural to undertake the task of classifying hypersurfaces, in the sense of Laguerre geometry, according to the properties satisfied by these invariants. The first results in this direction were obtained by Li, Li, and Wang \cite{Li paralela} where the authors introduced the concept of a hypersurface with parallel Laguerre second fundamental form $\mathbb{B}$, i.e., $\nabla \mathbb{B} = 0$, where $\nabla$ is the covariant derivative with respect to the Laguerre metric $g$. Moreover, they obtained a complete classification of such hypersurfaces, up to Laguerre transformations.

Subsequently, Li, Li, and Wang \cite{isotropicas} began to study a class of hypersurfaces in $\mathbb{R}^{n+1}$ with vanishing Laguerre form $\mathbb{C}$ and with all eigenvalues of the Laguerre tensor $\mathbb{L}$ constant equal to $\lambda$. We refer to hypersurfaces with this property as L-isotropic hypersurfaces. Li, Li, and Wang proved that $\lambda \geq 0$ and presented a characterization result for L-isotropic hypersurfaces. Moreover, they obtained an explicit classification of hypersurfaces with $\mathbb{C} = 0$ and constant eigenvalues of the tensor $\mathbb{L}$ that are not all equal.

Li and Sun \cite{Li e Sun em R4} defined another class of hypersurfaces, the L-isoparametric hypersurfaces, that are those for which the eigenvalues of the Laguerre second fundamental form $\mathbb{B}$ are constant and the Laguerre form $\mathbb{C}$ vanishes. They presented a classification of these hypersurfaces in the case of three-dimensional manifolds in $\mathbb{R}^{4}$. They also showed that an L-isoparametric hypersurface is a Dupin hypersurface. Conversely, a Dupin hypersurface, under the additional condition that the Laguerre form $\mathbb{C}$ vanishes, is an L-isoparametric hypersurface.

Song \cite{isop 2 distintas}, and Cezana and Tenenblat \cite{Miguel}, obtained the classification of L-isoparametric hypersurfaces in $\mathbb{R}^{n+1}$ with two distinct, nonzero principal curvatures.
Subsequently, Li and Shu \cite{3distintas e uma simples} obtained the classification of L-isoparametric hypersurfaces with three distinct principal curvatures, one of them being simple.

Shu published two other papers obtaining the classification of L-isoparametric hypersurfaces in $\mathbb{R}^5$ \cite{isop em R5} and in $\mathbb{R}^6$ \cite{isop em R6}, and he also proved that such hypersurfaces have parallel second fundamental, when they are not L-isotropic. More recently, Shu \cite{isop em Rn} studied L-isoparametric hypersurfaces and Dupin hypersurfaces in $\mathbb{R}^{n+1}$, presenting several classification theorems. In one of his results, considering an L-isoparametric hypersurface in $\mathbb{R}^{n+1}$, he proved that the eigenvalues of the Laguerre tensor $\mathbb{L}$ are all constant, possibly all equal or not.
When the eigenvalues of the Laguerre tensor $\mathbb{L}$ are all constant but not all equal, the classification was obtained by Li, Li, and Wang in \cite{isotropicas}.
Note that for the complete classification of L-isoparametric hypersurfaces, it is sufficient to determine which are the L-isotropic hypersurfaces.

Moreover, in \cite{isop em Rn}, using the classification theorem for L-isoparametric hypersurfaces, Shu presented the classification of proper Dupin hypersurfaces in $\mathbb{R}^{n+1}$ with the number of distinct principal curvatures $g \geq 2$. This result, in a certain sense, generalizes the theorem obtained by Cezana and Tenenblat \cite{Miguel dupin}, where they proved that proper Dupin hypersurfaces $M^n \subset \mathbb{R}^{n+1}$ with $n$ distinct, nonvanishing principal curvatures and constant Laguerre curvature, admitting a coordinate system by lines of curvature, are determined by $n$ constants. Moreover, they proved that any Dupin hypersurface under these conditions in $\mathbb{R}^{n+1}$ is described by
\begin{equation}\label{familia da dupin introdução}
x(u_1,\ldots,u_n) = (u_1,\ldots,u_n, 0) -
\dfrac{\sum_{i=1}^na_iu_i^2 + \phi}{\sum_{i=1}^na_i^2u_i^2 + 1}(a_1u_1,\ldots,a_nu_n,-1),
\end{equation}
where $\phi \in \mathbb{R}$ and the $a_i$ are distinct, nonzero constants.

This family of Dupin hypersurfaces was first obtained by Corro, Ferreira, and Tenenblat \cite{artigo do exemplo} in 1999 via a Ribaucour transformation of a hyperplane. Although this example was derived in a different context, it was applied to the Laguerre geometry setting, when $\phi = 0$, where it first appeared in \cite{Li paralela} as an example of an L-isoparametric hypersurface.

Since then, the family given by \eqref{familia da dupin introdução} has appeared in all classification results of L-isoparametric hypersurfaces. Furthermore, this family also provides an example of an L-isotropic hypersurface with all eigenvalues $\lambda$ of the Laguerre tensor $\mathbb{L}$ constant and equal to zero.

In this work, motivated by the family \eqref{familia da dupin introdução} and Shu’s article \cite{isop em Rn}, we investigate hypersurfaces that simultaneously possess both properties of being L-isotropic and L-isoparametric. More precisely, we obtain the following proposition:
\begin{proposition}\label{propo introducao....}
Let $x:M^n \rightarrow \mathbb{R}^{n+1}$, with $n \geq 3$, be a hypersurface without umbilic points, having distinct, nonvanishing principal curvatures. If $x$ is both L-isotropic and L-isoparametric, then all eigenvalues of the Laguerre tensor $\mathbb{L}$ are equal and zero, i.e., $\lambda = 0$.
\end{proposition}

We posed another natural question whether the family of hypersurfaces described in \eqref{familia da dupin introdução} is the only L-isotropic one when $\lambda = 0$. We partially answer this question positively by presenting a rigidity result for L-isotropic hypersurfaces in $\mathbb{R}^{n+1}$ parametrized by lines of curvature, up to Laguerre transformation.

Given a hypersurface $x:M^n \rightarrow \mathbb{R}^{n+1}$ with unit normal vector $\xi:M^n \rightarrow \mathbb{S}^n$, the Laguerre position vector $Y:M^n \rightarrow \mathbb{C}^{n+3} \subset \mathbb{R}^{n+4}_2$ is defined by
$Y=\rho (x\cdot\xi,-x\cdot\xi,\xi, 1)$,
where
\begin{equation}\label{oioiiiii....}
\rho = \sqrt{\sum_{i} (r - r_i)^2}
\end{equation}
is a function defined in terms of the curvature radii $r_i = \frac{1}{k_i}$ and the mean curvature radius $r = \frac{\sum_i r_i}{n}$ of $x$. We then prove our main result:

In Laguerre differential geometry, one studies the properties of Laguerre hypersurfaces that are invariant under the group of Laguerre transformations on the unit tangent bundle $U\mathbb{R}^{n+1}$. Li and Wang \cite{Li pequeno} studied this geometry for hypersurfaces in $\mathbb{R}^{n+1}$ using Cartan’s method of moving frames. Considering an immersion $x: M^n \rightarrow \mathbb{R}^{n+1}$ without umbilic points, with distinct nonzero principal curvatures, and $\xi: M^n \rightarrow \mathbb{S}^{n}$ a unit vector field normal to $x$, they defined the basic Laguerre invariants, namely: a Laguerre-invariant metric $g$, the Laguerre second fundamental form $\mathbb{B}$, the Laguerre form $\mathbb{C}$ and the Laguerre tensor $\mathbb{L}$.

Moreover, in \cite{isop em Rn}, using the classification theorem for L-isoparametric hypersurfaces, Shu presented the classification of proper Dupin hypersurfaces in $\mathbb{R}^{n+1}$ with the number of distinct principal curvatures $g \geq 2$. This result, in a certain sense, generalizes the theorem obtained by Cezana and Tenenblat \cite{Miguel dupin}, where they proved that proper Dupin hypersurfaces $M^n \subset \mathbb{R}^{n+1}$ with $n$ distinct, nonvanishing principal curvatures and constant Laguerre curvature, admitting a coordinate system by lines of curvature, are determined by $n$ constants. Moreover, they proved that any Dupin hypersurface under these conditions in $\mathbb{R}^{n+1}$ is described by
\begin{equation}\label{familia da dupin introdução}
x(u_1,\ldots,u_n) = (u_1,\ldots,u_n, 0) -
\dfrac{\sum_{i=1}^na_iu_i^2 + \phi}{\sum_{i=1}^na_i^2u_i^2 + 1}(a_1u_1,\ldots,a_nu_n,-1),
\end{equation}
where $\phi \in \mathbb{R}$ and the $a_i$ are distinct, nonzero constants.

Since then, the family given by \eqref{familia da dupin introdução} has appeared in all classification results of L-isoparametric hypersurfaces. Furthermore, this family also provides an example of an L-isotropic hypersurface with all eigenvalues $\lambda$ of the Laguerre tensor $\mathbb{L}$ constant and equal to zero.

In this work, motivated by the family \eqref{familia da dupin introdução} and Shu’s article \cite{isop em Rn}, we investigate hypersurfaces that simultaneously possess both properties of being L-isotropic and L-isoparametric. More precisely, we obtain the following proposition:
\begin{proposition}\label{propo introducao....}
Let $x:M^n \rightarrow \mathbb{R}^{n+1}$, with $n \geq 3$, be a hypersurface without umbilic points, having distinct, nonvanishing principal curvatures. If $x$ is both L-isotropic and L-isoparametric, then all eigenvalues of the Laguerre tensor $\mathbb{L}$ are equal and zero, i.e., $\lambda = 0$.
\end{proposition}

We posed another natural question whether the family of hypersurfaces described in \eqref{familia da dupin introdução} is the only L-isotropic one when $\lambda = 0$. We partially answer this question positively by presenting a rigidity result for L-isotropic hypersurfaces in $\mathbb{R}^{n+1}$ parametrized by lines of curvature, up to Laguerre transformation.

Given a hypersurface $x:M^n \rightarrow \mathbb{R}^{n+1}$ with unit normal vector $\xi:M^n \rightarrow \mathbb{S}^n$, the Laguerre position vector $Y:M^n \rightarrow \mathbb{C}^{n+3} \subset \mathbb{R}^{n+4}_2$ is defined by
$Y=\rho (x\cdot\xi,-x\cdot\xi,\xi, 1)$,
where
\begin{equation}\label{oioiiiii....}
\rho = \sqrt{\sum_{i} (r - r_i)^2}
\end{equation}
is a function defined in terms of the curvature radii $r_i = \frac{1}{k_i}$ and the mean curvature radius $r = \frac{\sum_i r_i}{n}$ of $x$. We then prove our main result:

\begin{theorem}(\cite{Li pequeno})
\label{teorema que caracteriza equivalente por t.laguerre com y}
Let $x,x^*: M^n\longrightarrow\mathbb{R}^{n+1}$ be two oriented hypersurfaces
with nonvanishing principal curvatures. Then $x$ and $x^*$ are
equivalent under Laguerre transformations if, and only if, there exists $T\in L\mathbb{G}$ such that
$Y^* = YT$, where $Y$ and $Y^*$ are the Laguerre positions vectors of $x$ and $x^*$ respectively.
\end{theorem}
Consider $y=Y/\rho$ given by \eqref{relacao entre Y e y}. Let
$III=\langle dy,dy \rangle$ be the
third fundamental form of $x$. Let
$\{E'_1, \ldots, E'_n \} $ be an orthonormal basis for
$III = \langle dy,dy\rangle= d\xi\cdot d\xi$, where the elements
of this basis can be written as $E'_{i} = r_ie_i $, $1\leq i\leq n $.

Consider an L-isotropic hypersurface $M^n\subset\mathbb{R}^{n+1}$, $n\geq 3$, parametrized by lines of curvature with $n$ distinct, nonvanishing principal curvatures. From \eqref{relacao entre os invariantes L, B e C} we can choose an orthonormal basis
$\{E_{1},E_{2},\ldots,E_{n}\}$ with respect to the Laguerre metric $g$ satisfying \eqref{tensores B e L comutando ingles..}.
Consider a coordinate system
$(u_1,\ldots,u_n)$ around $p\in M^n$ given by lines of curvature with respect to the Laguerre metric g. From now on, the index notation ``$,{i}$ '' denotes the partial derivative with respect to the coordinate $u_i$. Then
\begin{equation}\label{L metrica no sc}
 g\bigg(\dfrac{\partial}{\partial u_i},\dfrac{\partial}{\partial u_j}\bigg) = \bigg\langle dY\bigg(\dfrac{\partial}{\partial u_i}\bigg), dY\bigg(\dfrac{\partial}{\partial u_j}\bigg)\bigg\rangle = \langle Y_{,i} , Y_{,j} \rangle = \delta_{ij}g_{ii},
\end{equation}
In this coordinate system, the vector fields $\{E_1, \ldots, E_n \}$ are given by
\begin{equation}\label{campos Ei no sistema de coordenadas}
 E_i=\dfrac{1}{\sqrt{g_{ii}}}\dfrac{\partial}{\partial u_i},\, \, \, \, \, i=1,\ldots,n.
\end{equation}

\begin{proposition}\label{lema 1 arrumando o sistema}
Consider $x:M^n\rightarrow\mathbb{R}^{n+1}$ an L-isotropic hypersurface with $n$ distinct, nonvanishing principal curvatures,
$Y:M^n\rightarrow C^{n+3}\subset
\mathbb{R}^{n+4}_2$ the Laguerre position vector of $x$
with the Laguerre metric $g=\langle dY,dY \rangle$ and
$\lambda$ the eigenvalue of the tensor $\mathbb{L}$.
Suppose that $M^n$ is connected and admits a parameterization by lines of curvature $(u_1,u_2,\ldots,u_n)\in
U\subset\mathbb{R}^{n}$, where $U$ is open with respect to the Laguerre metric $g$.
Then the following system of equations is satisfied by $N$, $Y$ and $\eta$, given in
\eqref{definicao da Y},
\eqref{definicao do N}, \eqref{aplicacao normal de laguerre}, respectively
\begin{equation}\label{sistema final em coordenadas}
\begin{cases}
N_{,i} = \lambda Y_{,i}, \\
\\
\dfrac{Y_{,ii}}{g_{ii}} - \dfrac{g_{ii,i}Y_{,i}}{2g_{ii}^2} = \lambda Y + N + b_iP, \\
\\
Y_{,ij} = \displaystyle\sum_{\substack{l=1,\\l \neq i \neq j}}^{n}\Gamma^{l}_{ij}\dfrac{\sqrt{g_{ii}}\sqrt{g_{jj}}}{\sqrt{g_{ll}}}Y_{,l},\, \, \, \forall i\neq j=1,\ldots, n,\\
\\
\eta_{,i} =b_{i}Y_{,i},
\end{cases}
\end{equation}
where $P=(1, -1, \Vec{0},0)$, $\Vec{0}\in\mathbb{R}^{n+1}$.
\end{proposition}
\begin{proof}
Substituting
\eqref{tensores B e L comutando ingles..} and \eqref{campos Ei no sistema de coordenadas} into the structure equations
\eqref{equacoes de estrutura menos gerais}, we obtain the following system of differential equations,
\begin{equation}\label{equacoes de estrutura caso L-isotropica em coordenadas}
\begin{cases}
\dfrac{1}{\sqrt{g_{ii}}}N_{,i} = \dfrac{1}{\sqrt{g_{ii}}}\lambda Y_{,i},\, \, \, \forall i=1,\ldots, n,\\
\\
\dfrac{Y_{,ii}}{g_{ii}} - \dfrac{g_{ii,i}Y_{,i}}{2g_{ii}^2} = \lambda Y + N + \displaystyle\sum _{k}\Gamma^{k}_{ii}\dfrac{1}{\sqrt{g_{kk}}}Y_{,k} + b_{i}P,\, \, \, \forall i=1,\ldots, n,\\
\\
\dfrac{Y_{,ij}}{\sqrt{g_{ii}}\sqrt{g_{jj}}} - \dfrac{g_{ii,j}Y_{,i}}{2g_{ii}\sqrt{g_{ii}}\sqrt{g_{jj}}} = \displaystyle\sum _{k}\Gamma^{k}_{ij}\dfrac{1}{\sqrt{g_{kk}}}Y_{,k},\, \, \, \forall i\neq j=1,\ldots, n,\\
\\
\dfrac{1}{\sqrt{g_{ii}}}\eta_{,i} = \dfrac{1}{\sqrt{g_{ii}}}b_{i}Y_{,i},\, \, \, \forall i=1,\ldots, n,
\end{cases}
\end{equation}
where $b_i$ are the principal curvatures of Laguerre and $\Gamma^{k}_{ij}$ are the Christoffel symbols
in the metric $g$. We will prove that
\eqref{equacoes de estrutura caso L-isotropica em coordenadas} reduces to
\eqref{sistema final em coordenadas}. First, we want to prove that
$\Gamma^{i}_{ij} = 0$, for all $i,j=1,\ldots,n$. Furthermore, we will prove that
$\Gamma^{i}_{ij} = \Gamma^{i}_{ji} = 0$ is equivalent to $g_{ii,j} = 0$, for all $i\neq j=1,\ldots,n$.

\begin{proposition}\label{Lema da conta gigante das constantes}
Let $x:M^n\rightarrow\mathbb{R}^{n+1}$ be an
L-isotropic hypersurface with $n$ distinct nonzero principal curvatures. Consider
$Y:M^n\rightarrow C^{n+3}\subset
\mathbb{R}^{n+4}_2$ the Laguerre position vector of $x$,
with the Laguerre metric $g=\langle dY,dY \rangle$, and
$\eta:M^n\rightarrow C^{n+3}\subset
\mathbb{R}^{n+4}_2$ the Laguerre normal map.
Suppose that $M^n$ is connected and admits a parametrization by lines
of curvature with respect to the metric $g$. Then, the coordinate functions of
$Y$ are given by,

Finally, considering
$\langle Y, \eta \rangle = 0$ and using the relations $\eqref{F1i=-F2i}$,
$\eqref{F8 = 1+ F3}$, $\eqref{eta3=etan+4}$, and the fact that
$\psi^1 + \psi^2 = 1$, we have,
\begin{eqnarray} \label{eta e Y =0}
&&-Y^{1}+
\displaystyle\sum _{s=1}^{n}Y^{s+3}\eta^{s+3}
- \eta^3 =0.
\end{eqnarray}
Substituting
\eqref{Yl depois de todo ajeitamento} and
\eqref{etal depois da correcao resumido} into
\eqref{eta e Y =0} and using
\eqref{lembrando novamente que ciquadrado e 1}, we obtain,
\begin{eqnarray*}
&&\displaystyle\sum _{k,t=1}^{n}\Bigg[
2v_kv_tb_t\delta_{kt} +
\sqrt{2}v_kc^{t+3}_{t}(\beta_{t}^1-\beta_{t}^3b_{t})\delta_{kt}\Bigg] +\\
&&\, \, \, \, \, \, \, \, \, \, + \displaystyle\sum _{k=1}^{n}
\Bigg[\big(c^{k+3}_{k}\big)^2\beta^{3}_{k}
\beta^{1}_{k} - 2v_{k}^2b_{k} -
\sqrt{2}v_{k}c^{k+3}_{k}(\beta^{1}_{k}
- \beta^{3}_{k}b_{k}) - \gamma^{1}_{k}\Bigg]- \psi^3
=0.
\end{eqnarray*}
Thus,
\begin{equation}\label{relacao psi 3 depois da correcao}
\psi^3 = \displaystyle\sum _{k=1}^{n}
\Bigg[\big(c^{k+3}_{k}\big)^2\beta^{3}_{k}
\beta^{1}_{k} - \gamma^{1}_{k}\Bigg].
\end{equation}
Finally, substituting \eqref{relacao psi 3 depois da correcao} into \eqref{etal depois da correcao resumido}, we obtain the expression
\eqref{eta correcao resumido final}.
\end{proof}
In what follows, having obtained the vectors
$Y, \eta \in \mathbb{R}^{n+4}_2$, we can present an
explicit expression for $x$ and its unit normal vector $\xi$. Note that this is
equivalent to
obtaining, up to a Laguerre transformation,
the L-isotropic hypersurfaces in
$\mathbb{R}^{n+1}$ parametrized by lines of curvature.
More precisely, we have the following result.
\begin{theorem}
Let $x: M^n \rightarrow \mathbb{R}^{n+1}$ be a
hypersurface with $n$ distinct nonzero principal curvatures and
$Y: M^n \rightarrow C^{n+3} \subset
\mathbb{R}^{n+4}_2$ the Laguerre position vector of $x$
with Laguerre metric $g = \langle dY, dY \rangle$, and
$\lambda$ the eigenvalues of the tensor $\mathbb{L}$.
Suppose that $M^n$ is connected and admits a parametrization by lines
of curvature
with respect to the metric $g$. If $x$ is an
L-isotropic hypersurface, then $\lambda = 0$ and $x$ is
an $L$-isoparametric hypersurface. Moreover, up to a Laguerre transformation, this
hypersurface is equivalent to the one given by \eqref{definicao da HILF}.
\end{theorem}
\begin{proof}
From Lemma \ref{lema da aplicacao do FroFro}, we have
$\lambda = 0$ and $x$ is an
$L$-isoparametric hypersurface. It remains to
verify that $x$ is Laguerre equivalent
to the hypersurface described in
\eqref{definicao da HILF}. With this goal in mind, we will
explicitly determine $x$.

The proof of Theorem \ref{Teorema rigidez introduçao} clarifies that choosing the constant $\phi$ is equivalent to choosing the position vector $Y$, which implies an equivalence in the sense of Laguerre. Consequently, from the perspective of Laguerre geometry, the family of hypersurfaces described by \eqref{familia da dupin introdução} is equivalent, regardless of whether $\phi = 0$ or $\phi \neq 0$.

The classification of L-isotropic hypersurfaces is of significant importance, as it serves as a foundation for classifying all L-isoparametric hypersurfaces, based on the result proven by Shu in \cite{isop em Rn}. The case of L-isotropic hypersurfaces with $\lambda > 0$ remains an open area of investigation, as no examples are currently known. In this work, we establish necessary conditions for the existence of such hypersurfaces. Theorem \ref{Teorema rigidez introduçao} demonstrates that an L-isotropic hypersurface with $\lambda > 0$ cannot admit a parameterization by lines of curvature. Furthermore, we verify that any such hypersurface must possess at least three distinct, non-vanishing principal curvatures.

For the class of L-isotropic hypersurfaces with $\lambda > 0$, we also prove the following proposition:
\begin{proposition}\label{proposição 1 introdução}
Let $x:M^n \rightarrow \mathbb{R}^{n+1}$ be a hypersurface without umbilic points and with nonvanishing principal curvatures. If $x$ is an L-isotropic hypersurface with $\lambda > 0$, then
\begin{equation}
    0 < \rho^2 < \dfrac{1}{2\lambda},
\end{equation}
where $\rho$ is defined by \eqref{oioiiiii....}.
\end{proposition}

The post-theorem discussion says that “The proof of Theorem \ref{Teorema rigidez introduçao} clarifies that choosing the constant $\phi$ is equivalent to choosing the position vector $Y$, which implies an equivalence in the sense of Laguerre.” Hence, the argument identifies the parameter choice $\phi$ with a choice of Laguerre position vector, and concludes: “Consequently, from the perspective of Laguerre geometry, the family of hypersurfaces described by \eqref{familia da dupin introdução} is equivalent, regardless of whether $\phi = 0$ or $\phi \neq 0$.”

The post-theorem discussion says that “The proof given in establishing the main theorem clarifies that choosing the constant $\phi$ is equivalent to choosing the position vector $Y$, which implies an equivalence in the sense of Laguerre.” Hence, the argument identifies the parameter choice $\phi$ with a choice of Laguerre position vector, and concludes: “Consequently, from the perspective of Laguerre geometry, the family of hypersurfaces described by \begin{equation}\label{familia da dupin introdue7e3o}
x(u_1,\ldots,u_n) = (u_1,\ldots,u_n, 0) -
\dfrac{\sum_{i=1}^na_iu_i^2 + \phi}{\sum_{i=1}^na_i^2u_i^2 + 1}(a_1u_1,\ldots,a_nu_n,-1),
\end{equation} is equivalent, regardless of whether $\phi = 0$ or $\phi \neq 0$.”

\label{Teorema rigidez introduçao}
Let $x:M^n \rightarrow \mathbb{R}^{n+1}$ be a hypersurface with $n$ distinct, nonvanishing principal curvatures, let $Y:M^n \rightarrow \mathbb{C}^{n+3} \subset \mathbb{R}^{n+4}_2$ be its Laguerre position vector with Laguerre metric $g = \langle dY, dY \rangle$, and let $\lambda$ be the eigenvalues of the tensor $\mathbb{L}$. Suppose $M^n$ is connected and admits a parametrization by lines of curvature with respect to the metric $g$. If $x$ is an L-isotropic hypersurface, then $\lambda = 0$ and $x$ is L-isoparametric. Moreover, up to Laguerre transformation, this hypersurface is equivalent to the one described in
\begin{equation}\label{familia da dupin introdue7e3o}
x(u_1,\ldots,u_n) = (u_1,\ldots,u_n, 0) -
\dfrac{\sum_{i=1}^na_iu_i^2 + \phi}{\sum_{i=1}^na_i^2u_i^2 + 1}(a_1u_1,\ldots,a_nu_n,-1),
\end{equation}
for $\phi = 0$.

Question: Let x:M^n→R^{n+1} be a hypersurface with n distinct, nonzero principal curvatures. Let Y:M^n→C^{n+3}⊂R^{n+4}_2 be its Laguerre position vector, defined using a unit normal field ξ by Y=ρ(x·ξ,-x·ξ,ξ,1), where ρ=(Σ_i (r-r_i)^2)^{1/2}, r_i=1/k_i are the curvature radii, and r=(Σ_i r_i)/n is the mean curvature radius. Let g=<dY,dY> be the Laguerre metric, and let L be the Laguerre tensor with eigenvalues λ. Assume that M^n is connected, that it admits a parametrization by lines of curvature with respect to the metric g, and that x is L-isotropic. Which statement holds for every such hypersurface?

All eigenvalues of the Laguerre tensor L are zero, i.e. λ=0; the hypersurface x is L-isoparametric; and, up to a Laguerre transformation, x is Laguerre equivalent to a hypersurface of the form x(u_1,...,u_n)=(u_1,...,u_n,0)-((Σ_{i=1}^n a_i u_i^2)/(Σ_{i=1}^n a_i^2 u_i^2+1))(a_1u_1,...,a_nu_n,-1), for some distinct nonzero constants a_1,...,a_n (this is the family with φ=0).

Choices:
B.
All eigenvalues of the Laguerre tensor $L$ are zero, i.e. $\lambda=0$; and, up to a Laguerre transformation, $x$ is Laguerre equivalent to a hypersurface of the form
\[
x(u_1,\ldots,u_n)=(u_1,\ldots,u_n,0)-\frac{\sum_{i=1}^n a_i u_i^2+\phi}{\sum_{i=1}^n a_i^2 u_i^2+1}(a_1u_1,\ldots,a_nu_n,-1),
\]
for some distinct nonzero constants $a_1,\ldots,a_n$ and some arbitrary constant $\phi\in\mathbb{R}$; in particular, the value of $\phi$ is an essential geometric parameter.
C. All eigenvalues of the Laguerre tensor $L$ are zero, i.e. $\lambda=0$, and the hypersurface $x$ is $L$-isoparametric.
D.
The hypersurface $x$ is $L$-isoparametric and, up to a Laguerre transformation, is Laguerre equivalent to a hypersurface of the form
\[
x(u_1,\ldots,u_n)=(u_1,\ldots,u_n,0)-\frac{\sum_{i=1}^n a_i u_i^2}{\sum_{i=1}^n a_i^2 u_i^2+1}(a_1u_1,\ldots,a_nu_n,-1),
\]
for some distinct nonzero constants $a_1,\ldots,a_n$; moreover, the connectedness hypothesis may be omitted.
E.
There exists a Laguerre transformation after which $x$ is exactly a hypersurface of the form
\[
x(u_1,\ldots,u_n)=(u_1,\ldots,u_n,0)-\frac{\sum_{i=1}^n a_i u_i^2}{\sum_{i=1}^n a_i^2 u_i^2+1}(a_1u_1,\ldots,a_nu_n,-1),
\]
with distinct nonzero constants $a_1,\ldots,a_n$, and this representative is unique.

The stem does not explicitly state the conclusion or reveal the correct option. It gives hypotheses and asks for the resulting global statement, so there is no direct answer leakage.

This is essentially a theorem-recall item: the hypotheses in the stem are highly specific, and the correct choice restates the theorem's conclusion almost verbatim, including the classified model form.

Some reasoning is needed to distinguish the full classification statement from weaker or tampered variants (e.g., the role of φ, connectedness, uniqueness). However, solving it is driven more by recalling the exact theorem than by generating a conclusion from mathematical reasoning.

The distractors are plausible and mathematically targeted: one is a weaker true statement, others alter subtle classification details such as the essentiality of φ, omission of connectedness, or uniqueness of representatives. These reflect realistic failure modes.

A technically strong but theorem-recall-heavy MCQ: little answer leakage and good distractors, but it is close to a direct restatement of a classification theorem and only moderately tests generative reasoning.


\label{thm:main}
Let $E_0\in \mathbb R$ be a fixed energy for which $L_d(\omega,E_0)>0$.
For any $\omega\in \mathrm{DC}$, the integrated density of states of $H_{\omega,\theta}$ satisfies
    \begin{align}
        |\mathcal{N}(\omega,E)-\mathcal{N}(\omega,E')|\leq |E-E'|^{\beta},
    \end{align}
    for any $E,E'\in I_{E_0}$, a neighborhood of $E_0$, and any H\"older exponent $0<\beta<1/(2\kappa^d(\omega,E_0))$.

\begin{theorem}\label{thm:main}
Let $E_0\in \R$ be a fixed energy for which $L_d(\omega,E_0)>0$.
For any $\omega\in \mathrm{DC}$, the integrated density of states of $H_{\omega,\theta}$ satisfies
    \begin{align}
        |\mathcal{N}(\omega,E)-\mathcal{N}(\omega,E')|\leq |E-E'|^{\beta},
    \end{align}
    for any $E,E'\in I_{E_0}$, a neighborhood of $E_0$, and any H\"older exponent $0<\beta<1/(2\kappa^d(\omega,E_0))$.
\end{theorem}

\begin{theorem}\label{thm:acc=zeros}
Assume that for some $E\in \R$, $L^1(\omega,E)\geq \nu>0$.
Then for some $\gamma\in (0,1)$ and for $n$ large enough,
\begin{align}
\left| \frac{1}{2n} N_n (E,\delta/2)-\kappa^1(\omega,E)\right|\leq \delta^{-1} n^{-\gamma}.
\end{align}
\end{theorem}

\begin{aligned}\label{def:DC}
    \omega\in \mathrm{DC}:=&\bigcup_{a>0, A>1} \mathrm{DC}_{a,A}, \text{ where}\\
    \mathrm{DC}_{a,A}=&\left\{\omega\in \T:\, \|k\omega\|_{\T}\geq \frac{a}{|k|^A}\, \text{ for all } k\in \Z\setminus \{0\}\right\}.
\end{aligned}

\begin{lemma}\label{lem:numerator_diag}
    Let $\omega\in \mathrm{DC}$ and $x=y\in [2d,n-2d-1]$,
    Then for $\gamma>0$ as in Lemma \ref{lem:upperbd}, $n$ large, and $E'\in \C$ such that $|E'-E|\leq n^{-2}e^{-n^{\gamma}}$, uniformly in $\theta\in \T$:
    \begin{align}
        |\mu^{E'}_{[0,nd-1],x,x}(\theta)|\leq  e^{n(L^d(E)+\langle \log |\det B|\rangle+5n^{-\gamma})}.
    \end{align}
\end{lemma}

\begin{align}\label{eq:SDC}
 \mathrm{DC}_{\mathrm{strong}}:= \bigcup_{a>1, c>0} \left\{\omega\in \T:\, \|k\omega\|_{\T}\geq \frac{c}{|k|(\log |k|)^a}\, \text{ for all } k\in \Z\setminus \{0,\pm 1\}\right\}.
\end{align}

\begin{lemma}\label{lem:local_zero*}
Let $n$ be $\kappa_0$-admissible and large.
For each ball $B(z_0,r_n)$, with $z_0\in \kreis^1$ and $r\simeq r_n:=e^{-(\log n)^{C_0}}$ with $C_0>1$, there exists $|k|<(1-\epsilon)n/2$ such that $f_n(e^{2\pi ik\omega} z,E)$ has at most $2\kappa^d$ zeros in $B(z_0,r)$.
\end{lemma}

In this paper we study Schr\"odinger cocycles on a strip with Hamiltonian
\begin{equation}\label{eq:BVsys}
 (H_{\omega,\theta} \Phi)_n = B_{n+1}(\theta)\Phi_{n+1}+ B_n^{(*)}(\theta) \Phi_{n-1}+ V_n (\theta)\Phi_n
\end{equation}
where $F_n(\theta):=F(\theta+n\omega)$ for any $d\times d$-matrix valued function. We set $B^{(*)}(\theta)=(B(\theta))^*$ for $\theta\in \mathbb T$, and require it to be the analytic extension of $(B(\theta))^*$ off of the real torus. Here $\omega\in\mathbb T$ and we assume that $\omega$ is Diophantine, i.e.,
\begin{equation}\begin{aligned}\label{def:DC}
 \omega\in \mathrm{DC}:=&\bigcup_{a>0, A>1} \mathrm{DC}_{a,A}, \text{ where}\\
 \mathrm{DC}_{a,A}=&\left\{\omega\in \mathbb T:\, \|k\omega\|_{\mathbb T}\geq \frac{a}{|k|^A}\, \text{ for all } k\in \mathbb Z\setminus \{0\}\right\}.
\end{aligned}
\end{equation}
We further assume that $B,V\in C^{\omega}(\T_{\delta}, \mathrm{Mat}(d,\mathbb C))$ are analytic, where $$\T_{\delta}:=\{\theta+i\varepsilon:\, \theta\in \mathbb T,\, \varepsilon\in \mathbb R, \text{ and } |\varepsilon|\leq \delta\}$$with some positive $\delta>0$.
We assume throughout the paper that $V$ is Hermitian, and that~$B$ is invertible ($\det B(\theta)\neq 0$ for any $\theta\in \T_{\delta}$).
The difference equation $H_{\omega,\theta} \Phi=E\Phi$ is equivalent to the cocycle
\begin{align}\label{def:M_E}
 \mathcal{C} &\:: (\theta,\Psi)\in \mathbb T\times \mathbb C^{2d}\mapsto (\theta+\omega, M_E(\theta)\Psi), \notag \\
 \quad M_{E}(\theta) &= \left [ \begin{matrix}
 ( E-V(\theta))B(\theta)^{-1} & -B^{(*)}(\theta) \\
 B(\theta)^{-1} & 0
\end{matrix}\right]
\end{align}
in the sense that for $n\ge1$,
\begin{equation}\label{eq:MnE}
\mathcal{C}^n (\theta,\Psi)=(\theta+n\omega, M_{n,E}(\theta)\Psi), \quad M_{n,E}(\theta)=\prod_{j=n-1}^0 M_E(\theta+j\omega), \quad
\Psi_n:=\binom{B_{n}\Phi_n}{\Phi_{n-1}}
\end{equation}
satisfies $\Psi_n = M_{n,E} (\theta) \Psi_0$. The model~\eqref{eq:BVsys} was studied systematically by S.~Klein~\cite{SK_block}, and Duarte and Klein~\cites{DK1, DK2} within a very general framework, including the identically singular case when $\det B(\theta)\equiv 0$ on $\mathbb T^k$, $k\geq 1$. These models generalize the quasi-periodic Schr\"odinger Hamiltonians on a strip considered by Bourgain and Jitomirskaya~\cite{BJ}, who proved localization perturbatively for large disorder.

Since $M_E(\theta)$ is (complex) symplectic, see \eqref{def:symp}, for $\theta\in \mathbb T$, the Lyapunov exponents $\{L_j(\omega,M_E)\}_{j=1}^{2d}$, see definition in \eqref{def:LE}, satisfy $L_{2d+1-j}=-L_j$ for $1\le j\le d$.
Let $L^d(\omega,M_E)$ be the sum of the top $d$ Lyapunov exponents, and let $\kappa^d(\omega,M_E)$ be the acceleration, defined to be the right-derivative of $L^d(\omega,M_E(\cdot+i\varepsilon))$ in the imaginary direction $\varepsilon$, see \eqref{eq:acc}.

Throughout the paper we shall fix an energy $E_0\in \mathbb R$ such that $L_d(\omega,E_0)\geq \tau>0$. It is known, see \cite{AJS}, that $L^d(\omega,M_E(\cdot+i\varepsilon))$ and $L_d(\omega,M_E)$ are continuous in $E,\varepsilon$ and $\kappa^d(\omega,M_E)$ is upper semi-continuous in $E,\varepsilon$, hence we can assume that in a neighborhood $I_{E_0}$ of $E_0$, the following holds uniformly in $E\in I_{E_0}$ and $|\varepsilon|\leq \delta$, $\varepsilon\in \mathbb R$, for some $\delta=\delta(E_0)>0$:
\begin{align}\label{def:IE0}
\begin{cases}
L_d(\omega,M_E(\cdot+i\varepsilon))\geq \tau/2,\\
\kappa^d(\omega,M_E(\cdot+i\varepsilon))\leq \kappa^d(\omega,M_{E_0})\\
L^d(\omega,M_E(\cdot+i\varepsilon))\leq L^d(\omega,M_E)+2\pi \kappa^d(\omega,M_{E_0})\varepsilon.
\end{cases}
\end{align}
Note the third inequality is a corollary of the second one.

The density of states $\mathcal{N}(\omega,\cdot)$ is the limiting cumulative distribution function of the finite-volume eigenvalues.
Let $H_{\omega,\theta}|_{\Lambda}$ be the restriction of $H_{\omega,\theta}$ to the interval $\Lambda=[a,b]\subset \mathbb Z$.
Let $\{E_{\Lambda,j}(\omega,\theta)\}_{j=1}^{|\Lambda|}$ be the eigenvalues of $H_{\omega,\theta}|_{\Lambda}$.
Consider
\begin{align}
 N_{\Lambda}(\omega,E,\theta):=\frac{1}{|\Lambda|}\sum_{j=1}^{|\Lambda|}\chi_{(-\infty,E)}(E_{\Lambda,j}(\omega,\theta)).
\end{align}
It is well-known that the weak-limit
\begin{align}
 \lim_{a\to-\infty, b\to \infty}\mathrm{d} N_{\Lambda}(\omega,E,\theta)=:\mathrm{d} \mathcal{N}(\omega,E).
\end{align}
exists and is independent of $\theta$. In this paper we prove the following result.

\begin{align}\label{def:LE}
L_j(\omega, A,n):=\frac{1}{n}\int_{\T} \log \sigma_j(A_n(\omega,\theta))\, \mathrm{d}\theta, \text{ for } 1\leq j\leq k,
\end{align}

\begin{align}\label{def:symp}
    M^*\Omega M=\Omega,
\end{align}

\begin{equation}\label{eq:BVsys}
   (H_{\omega,\theta} \Phi)_n   = B_{n+1}(\theta)\Phi_{n+1}+ B_n^{(*)}(\theta) \Phi_{n-1}+ V_n (\theta)\Phi_n
\end{equation}

\begin{align}\label{eq:acc}
\kappa^1_{\varepsilon}(\omega, A):=\lim_{\varepsilon'\to 0^+} \frac{L^1_{\varepsilon+\varepsilon'}(\omega, A)-L^1_{\varepsilon}(\omega, A)}{2\pi \varepsilon'}.
\end{align}

In \cite{HS3}*{Lemma 2.8} we established the following large deviation theorem for $f_n$.
\begin{lemma}\label{lem:deno}
Let $\omega\in \mathrm{DC}$, and $\gamma>0$ be as in Lemma \ref{lem:LDTsig}.
Assume $L_d(\omega,M_E)\geq \nu>0$.
There exist $\gamma>0$, $N_0>1$ large and $0<\kappa_0\ll 1$ so that the {\it $\kappa_0$-admissible} sequence
\begin{align}\label{def:admissible}
\mathcal{N}:=\{n\geq N_0: \|n\omega\|_{\T}\leq \kappa_0\}
\end{align}
has the following property:
for any $|\varepsilon|\leq \delta/2$, and all large $\kappa_0$-admissible $n$, the following large deviation set
\begin{align}\label{def:B_fEn}
\mathcal{B}_{f,E,n,\varepsilon}:=
\big\{\theta\in \T: n^{-1}\log |f_{n}(\theta+i\varepsilon,E)|<\langle \log |\det B(\cdot+i\varepsilon)|\rangle +{L}^d_{\varepsilon}(\omega,M_E)- n^{-\gamma}\big\}
\end{align}
satisfies $\mathrm{mes}(\mathcal{B}_{f,E,n,\varepsilon})<e^{-n^{\gamma}}$.
\end{lemma}
\begin{remark}\label{rem:admissible}
 For every large integer $n>0$ there exists an admissible $\tilde n>0$ with $|n-\tilde n|\le C_*$ for some constant $C_*$.
\end{remark}

Recall that $E\in I_{E_0}$.
Our goal is to analyze the following for an arbitrary fixed $\theta$ as $N\to\infty$:
\begin{align}\label{eq:goal}
 &d_{\omega,N}(\theta,E-\eta,E+\eta)=\frac{1}{N}\mathrm{tr}(P_{[E-\eta, E+\eta)}(H_{\omega,\theta}|_{[0,N-1]})),
\end{align}
in which $H_{\omega,\theta}|_{[0,N-1]}$ is $H_{\omega,\theta}$ restricted to the interval $[0,N-1]$ with Dirichlet boundary condition and $P_{[E-\eta,E+\eta)}$ stands for the spectral projection.
Bounding \eqref{eq:goal} through the trace of the resolvent yields
\begin{align}\label{eq:dN<G}
 d_{\omega,N}(\theta,E-\eta,E+\eta)
 &\leq \frac{2\eta}{N} \mathrm{Im}(\mathrm{tr}(G_{[0,N-1]}^{E+i\eta}(\theta)))\\
 &=\frac{2\eta}{N}\sum_{k=0}^{N-1} \mathrm{Im} (G_{[0,N-1]}^{E+i\eta}(\theta;k,k)),
\end{align}
in which $G^{E+i\eta}_{[0,N-1]}(\theta)=(H_{\omega,\theta}|_{[0,N-1]}-(E+i\eta))^{-1}$ is the Green's function, i.e., the kernel of the resolvent.
Let an auxiliary parameter $\tau$ be defined as
\begin{align}\label{eq:choose_tau}
 \tau=\eta^{\frac{1}{2\kappa}}.
\end{align}
For any $n\in \N$, let
\begin{align}
 \mathcal{B}_{n,h}^{\eta}:=\{\theta\in \T: u_n(e^{2\pi i\theta},E+i\eta)<L(E,n)-h\cdot n^{-\gamma}\}.
\end{align}
be the large deviation set for $u_n(z,E+i\eta)=n^{-1}\log |D_n(z,E+i\eta)|$.
We choose $m\in \N$ such that
$e^{-(2m)^{\gamma_1}}\simeq \eta$.
The set $\mathcal{B}_{2m,2}^{\eta}$ satisfies a large deviation estimate
\begin{align}\label{eq:mes_B2m_2}
 \mes\;\mathcal{B}_{2m,2}^{\eta}\leq e^{-(2m)^{\gamma}}.
\end{align}
In fact, the large deviation estimate of Lemma \ref{lem:un_LDT} applies directly to $\mathcal{B}_{2m,1}^0$, implying that
\begin{align}\label{eq:mes_B2m_1}
 \mathrm{mes}\; \mathcal{B}_{2m,1}^0\leq e^{-(2m)^{\gamma}}.
\end{align}
If $\theta\in (\mathcal{B}_{2m,1}^0)^c$, $|D_{2m}(e^{2\pi i\theta},E)|>\exp(2m(L(E,2m)-(2m)^{-\gamma}))$. A telescoping argument, as in the proof of Lemma \ref{lem:un_lower_zeros_eta}, using that $\gamma_1>1-\gamma/2$, yields
\begin{align}
 |D_{2m}(e^{2\pi i\theta},E+i\eta)|>\frac{1}{2}|D_{2m}(e^{2\pi i\theta},E)|>\frac{1}{2}\exp(2m(L(E,2m)-(2m)^{-\gamma})).
\end{align}
This implies $(\mathcal{B}_{2m,1}^0)^c\subset (\mathcal{B}_{2m,2}^{\eta})^c$,
and hence \eqref{eq:mes_B2m_2}, when combining with \eqref{eq:mes_B2m_1}.

The proof is analogous to the scalar-valued case.
Our goal is to estimate the following expression for arbitrary fixed $E\in I_{E_0}$ and $\theta\in \T$ as $N\to\infty$:
\begin{align}\label{eq:goal*}
 &d_{\omega,dN}(\theta,E-\eta,E+\eta)=\frac{1}{dN}\mathrm{tr}(P_{[E-\eta, E+\eta)}(H_{\omega,\theta}|_{[0,dN-1]})),
\end{align}
cf.\ \eqref{eq:goal}. As before, we
bound~\eqref{eq:goal*} through the trace of the Green's function:
\begin{align}\label{eq:dN<G*}
 d_{\omega,dN}(\theta,E-\eta,E+\eta)
 &\leq \frac{2\eta}{dN} \mathrm{Im}(\mathrm{tr}(G_{[0,dN-1]}^{E+i\eta}(\theta)))\\
 &=\frac{2\eta}{dN}\sum_{k=0}^{dN-1} \mathrm{Im} (G_{[0,dN-1]}^{E+i\eta}(\theta;k,k)),
\end{align}
in which $G^{E+i\eta}_{[0,dN-1]}(\theta)=(H_{\omega,\theta}|_{[0,dN-1]}-(E+i\eta))^{-1}$ is the kernel of the resolvent.
Let the auxiliary parameter $\tau$ be defined as
\begin{align}\label{def:tau_d}
 \tau=\eta^{\frac{1}{2\kappa^d}},
\end{align}
in this block-valued case.
For any $\kappa_0$-admissible $n\in \N$, let
\begin{align}
 \mathcal{B}_{n,h}^{\eta}:=\{\theta\in \T: u_n(e^{2\pi i\theta},E+i\eta)<L^d(E,n)+\langle \log |\det B(\cdot)|\rangle-h\cdot n^{-\gamma}\}
\end{align}
be the large deviation set for $u_n(z,E+i\eta)=n^{-1}\log |f_n(z,E+i\eta)|$.
We choose $m\in \N$ such that $2m$ is $\kappa_0$-admissible and $e^{-(2m)^{\gamma_1}}\simeq \eta$.
The set $\mathcal{B}_{2m,2}^{\eta}$ satisfies a large deviation estimate
\begin{align}\label{eq:B2m_eta_block}
 \mes\,\mathcal{B}_{2m,2}^{\eta}\leq e^{-(2m)^{\gamma}}.
\end{align}
In fact, note $2m$ is $\kappa_0$ admissible, the large deviation estimate of Lemma \ref{lem:deno} applies to $\mathcal{B}_{2m,1}^0$ implying
\begin{align}\label{eq:B2m_0_block}
 \mathrm{mes}\,\mathcal{B}_{2m,1}^0\leq e^{-(2m)^{\gamma}}.
\end{align}
Suppose $\theta\in (\mathcal{B}_{2m,1}^0)^c$, then
\begin{align}
 |f_{2m}(e^{2\pi i\theta},E)|\geq \exp(2m(L^d(E,2m)+\langle \log |\det B(\cdot) |\rangle-(2m)^{-\gamma})).
\end{align}
By the same telescoping argument in the proof of Lemma \ref{lem:un_lower_zeros_eta*}, using, $\gamma_1>1-\gamma/2$, we have
\begin{align}
 |f_{2m}(e^{2\pi i\theta},E+i\eta)|\geq \frac{1}{2} |f_{2m}(e^{2\pi i\theta}, E)|\geq \exp(2m(L^d(E,2m)+\langle \log |\det B(\cdot) |\rangle-2(2m)^{-\gamma})).
\end{align}
This implies $(\mathcal{B}_{2m,1}^0)^c\subset (\mathcal{B}_{2m,2}^{\eta})^c$, hence the estimate \eqref{eq:B2m_eta_block} follows from \eqref{eq:B2m_0_block}.
Let $\{\xi_j\, e^{2\pi i\theta_j}\}_{j=1}^{J_0}$, $\xi_j>0$ and $\theta_j\in \T$, be the zeros of $f_{2m}(z,E+i\eta)$.
Cartan's estimate implies that
\begin{align}\label{eq:B2m_theta_pm_rm*}
\mathcal{B}_{2m,m^{\gamma/2}}^{\eta}\subset \bigcup_{j=1}^{J_0}(\theta_j-\tilde{r}_m, \theta_j+\tilde{r}_m),
\end{align}
with $\tilde{r}_m:=e^{-cm^{\gamma/2}}$ for some constant $c>0$, and $J_0\lesssim m$.
Applying Lemma~\ref{lem:un_lower_zeros_eta*} to $n=2m$ and $z_0=e^{2\pi i\theta_j}$, for each $1\leq j\leq J_0$, yields the existence of $|k_j|<(1-\varepsilon)m$, $C_j\in [1,2\kappa^d+1]$, and $\ell_j\in [1,2\kappa^d]$, such that $w_j(z):=f_{2m}(z e^{2\pi ik_j\omega},E+i\eta)$
has $\ell_j$ zeros in $B(e^{2\pi i\theta_j}, (4C_j+1)r_m)$, $r_m\simeq \exp(-(\log m)^{C_0})\gg \tilde{r}_m$.
Denoting the zeros of $f_{2m}(z e^{2\pi ik_j\omega},E+i\eta)$ by $\{\xi_{j,\ell}\, e^{2\pi i\theta_{j,\ell}}\}_{\ell=1}^{\ell_j}$ by $\xi_{j,\ell}>0$, $\theta_{j,\ell}\in \T$, one has
\begin{align}\label{eq:wj_lower_zeros*}
 \log |w_j(z)|\geq 2m(L^d(E,2m)+\langle \log |\det B(\cdot)|\rangle-m^{-\gamma/2})+\sum_{\ell=1}^{\ell_j}\log |z-\xi_{j,\ell}\, e^{2\pi i\theta_{j,\ell}}|,
\end{align}
for all $z\in B(e^{2\pi i\theta_j},(4C_j+2)r_m)$.
For each $1\leq j\leq J_0$ and $1\leq \ell\leq \ell_j$, let
 $ \tilde{I}_j,
 I_{j,\ell}$ be defined as in~\eqref{eq:Ijdef}.
With $\tau$ as in \eqref{eq:choose_tau} and $\tau\ll r_m$, we have for each $j,\ell$ that $I_{j,\ell}\subset \tilde{I}_j$.
As before, we let $\tilde{\mathcal{B}}_{2m}^{\eta}:=\bigcup_{j=1}^{J_0} \tilde{I}_j$.
Clearly, by \eqref{eq:B2m_theta_pm_rm*}, $\mathcal{B}_{2m,m^{\gamma/2}}^{\eta}\subset \tilde{\mathcal{B}}_{2m}^{\eta}$.
We again divide the analysis of $G^{E+i\eta}_{[0,dN-1]}(\theta; k,k)$ into three cases.

\begin{remark}
    The exponent $2\kappa^d(\omega,E)$ comes from the local zero count of the finite-volume characteristic polynomials, which is derived from the global zero count. When $d\geq 2$, this is the zero count of $f_n(\theta,E)$, see \eqref{def:fn}. If the blocks $B,V$ satisfy additional symmetries, then it is possible to deduce  smaller local zero counts, and hence improved H\"older exponents. We refer interested readers to \cite[Theorem 1.4 \& Theorem 1.6]{HS3} where the effects of additional symmetries are studied.
\end{remark}

As for the history of this type of regularity result: $\log$--H\"older continuity was established by Craig and Simon in \cite{CS} for general ergodic potentials. The stronger H\"older continuity of the integrated density of states associated with~\eqref{eq:BVsys} was first proved by Goldstein and the second author in \cite{GS1} for the scalar case $d=1$.
Later, \cite{GS2}*{Theorem 1.1} established that for a trigonometric polynomial $f$ of degree $k_0\ge1$ and assuming $\omega\in \mathrm{DC}_{\mathrm{strong}}$, see \eqref{eq:SDC}, and
$L(\omega,E)>0$  the IDS is $\beta$-H\"older continuous for any $\beta<1/(2k_0)$.
This result, for the same set of frequencies, was improved to $\beta<1/(2\kappa^1(\omega,E))$ in \cite{HS1}. Note for trigonometric $f$ one has $\kappa^1(\omega,E)\leq k_0$.
Theorem~\ref{thm:main}, when $d=1$, is an improvement of \cite{HS1}*{Theorem 1.3} in the sense that it covers more general Diophantine frequencies: $\mathrm{DC}_{\mathrm{strong}}\subset \mathrm{DC}$. When $d\geq 2$, Theorem~\ref{thm:main} is completely new.

For the almost Mathieu operator and $\alpha\in \mathrm{DC}$, the IDS was proved to be $1/2$-H\"older if $|\lambda|\neq 0,1$, see Avila, Jitomirskaya~\cite{AJ}.
The $1/2$-H\"older exponent was also proved for Diophantine $\alpha$ and $f=\lambda\, g$ with $g$ being analytic and the coupling constant $|\lambda|$ being small, first in the perturbative regime (smallness depends on $\alpha, g$) by Amor~\cite{Amor}, and then in the non-perturbative regime \cite{AJ} (with dependence on $\alpha$ removed).
For quasi-periodic long-range operators with large trigonometric polynomial potentials and Diophantine frequencies, the H\"older exponent in \cite{GS2} was improved recently in the perturbative regime~\cite{GYZ}.
These results are proved using the reducibility method.

For the proof of Theorem~\ref{thm:main} we first single out the Schr\"odinger operator case, where $d=1$, $B\equiv 1$ and $V=v$ is a non-constant analytic function. Sections~\ref{sec:sch1}, \ref{sec:sch2} present this case in details. The argument is variant of the one in~\cite{GS2}, but is simpler and more robust.
The difference lies in how we derive a favorable local zero count for a finite volume determinant relative to some interval $\Lambda\subset\mathbb Z$. In~\cite{GS2}, this is accomplished by slightly varying the size of~$\Lambda$, but not the position. Here we keep the size fixed but change the position. The strip case ($d>1$ in~\eqref{eq:BVsys}) follows the same outline but is more involved because it relies on zero counting techniques that apply to the Jacobi block case. Thankfully,  these were developed in~\cite{HS3}, so we can treat them as a black box, cf.~Theorem~\ref{thm:acc=zeros}.

We also point out that the Aubry dual of a subcritical Schr\"odinger operator $H$, with trigonometric potential with degree $d$, is a special family of $d\times d$ Jacobi block-valued matrix $\hat{H}$ with uniformly positive $L_d(\omega,E)>0$, see e.g.~\cite[Corollary 1.12]{HS2}. An easy computation shows that $\kappa^d(\omega,E)\leq d$, but one can show the local zero count is at most $2$, instead of $2d$. This is a small modification of Lemma \ref{lem:local_zero*} with considering not only $f_n(e^{2\pi ik\omega}z,E)$ with $|k|<(1-\varepsilon)n/2$, but also $f_n(e^{2\pi ik\omega/d}z,E)$ with $|k|<(1-\varepsilon)nd/2$. This is a feature due to the special form of $\hat{H}$, and has been explored in the proof of arithmetic Anderson localization of $\hat{H}$ in \cite[Theorem 1.7]{HS2} and \cite[Corollary 1.3]{HS3}. The local zero count being at most $2$ then implies almost $1/2$ H\"older continuity of the integrated density of states of the original subcritical Schr\"odinger with arbitrary trigonometric potentials with Diophantine frequencies.

Finally, we comment that the (non-)H\"older continuity of the integrated density of states for Liouville frequencies has also been studied \cite{YZ,HZ,ALSZ,HS4} when $d=1$,
and it would be natural to combine those considerations with the techniques developed here for $d\geq 2$.

Our proofs are essentially self-contained. The only ingredients we require are the large deviation estimates, developed in \cites{BG,GS1,HS3} and global zero counts of finite-volume determinants in \cites{HS1,HS3}.
As we mentioned, the Diophantine condition \eqref{def:DC} is weaker than that in \cites{GS2,HS1} where it was defined to be
\begin{align}\label{eq:SDC}
 \mathrm{DC}_{\mathrm{strong}}:= \bigcup_{a>1, c>0} \left\{\omega\in \mathbb T:\, \|k\omega\|_{\mathbb T}\geq \frac{c}{|k|(\log |k|)^a}\, \text{ for all } k\in \mathbb Z\setminus \{0,\pm 1\}\right\}.
\end{align}
The Anderson localization result of \cite{HS1} holds for the weaker Diophantine condition \eqref{def:DC}, see \cite{HS1}*{Remark 1.9}. However,  the H\"older continuity result of \cite{HS1} holds under \eqref{eq:SDC} due to the need for a sharp large deviation estimate as in \cite{GS2}.

We organize the paper as follows: Section~\ref{sec:Pre} contains the preliminary results. Sections~\ref{sec:sch1} and \ref{sec:sch2} prove Theorem~\ref{thm:main} in the Schr\"odinger case, more specifically, Sec.~\ref{sec:sch1} presents the crucial local factorization of the finite volume characteristic polynomials which lays the foundation of the proof of Theorem~\ref{thm:main} in Sec.~\ref{sec:sch2}. The Jacobi block-valued case is proved in Sections~\ref{sec:block1} and \ref{sec:block2}. Finally, the proof of Lemma~\ref{lem:numerator_diag} is contained in Sec.~\ref{sec:numerator}.

For the proof of Theorem~\ref{thm:main} the authors “first single out the Schr\"odinger operator case, where $d=1$, $B\equiv 1$ and $V=v$ is a non-constant analytic function.” Sections~\ref{sec:sch1}, \ref{sec:sch2} treat this case; “the argument is variant of the one in~\cite{GS2}, but is simpler and more robust.” The key difference is “how we derive a favorable local zero count for a finite volume determinant relative to some interval $\Lambda\subset\mathbb Z$”: in~\cite{GS2} this is done “by slightly varying the size of~$\Lambda$, but not the position,” whereas here “we keep the size fixed but change the position.”

The strip/block case (“$d>1$ in~\eqref{eq:BVsys}”) “follows the same outline but is more involved because it relies on zero counting techniques that apply to the Jacobi block case”; these zero counting tools were developed in~\cite{HS3} and are used “as a black box, cf.~Theorem~\ref{thm:acc=zeros}.”

Structurally, “Sec.~\ref{sec:sch1} presents the crucial local factorization of the finite volume characteristic polynomials which lays the foundation of the proof of Theorem~\ref{thm:main} in Sec.~\ref{sec:sch2},” and “the Jacobi block-valued case is proved in Sections~\ref{sec:block1} and \ref{sec:block2}.” The underlying inputs are stated to be “large deviation estimates” (from \cites{BG,GS1,HS3}) and “global zero counts of finite-volume determinants” (from \cites{HS1,HS3}); the remark also explains that the exponent $2\kappa^d(\omega,E)$ “comes from the local zero count of the finite-volume characteristic polynomials, which is derived from the global zero count.”

Question:
Consider the analytic quasi-periodic Schr\"odinger operator on a strip acting on sequences \(\Phi_n\in\mathbb C^d\) by
\[
(H_{\omega,\theta}\Phi)_n=B(\theta+(n+1)\omega)\Phi_{n+1}+B(\theta+n\omega)^*\Phi_{n-1}+V(\theta+n\omega)\Phi_n,
\]
where \(B,V\in C^{\omega}(\mathbb T_{\delta},\mathrm{Mat}(d,\mathbb C))\), \(V\) is Hermitian, and \(\det B(\theta)\neq 0\) on \(\mathbb T_{\delta}\). Assume the frequency \(\omega\) is Diophantine, i.e.
\[
\omega\in \mathrm{DC}:=\bigcup_{a>0,\,A>1}\left\{\omega\in\mathbb T:\ \|k\omega\|_{\mathbb T}\ge \frac{a}{|k|^A}\ \text{for all }k\in\mathbb Z\setminus\{0\}\right\}.
\]
For each energy \(E\in\mathbb R\), let the associated cocycle be
\[
M_E(\theta)=\begin{pmatrix}
(E-V(\theta))B(\theta)^{-1} & -B(\theta)^*\\
B(\theta)^{-1} & 0
\end{pmatrix}.
\]
Let \(L_d(\omega,E)\) denote the \(d\)-th Lyapunov exponent of this cocycle, let \(L^d\) be the sum of its top \(d\) Lyapunov exponents, and let \(\kappa^d(\omega,E)\) be the right derivative at \(\varepsilon=0\) of \(L^d(\omega,M_E(\cdot+i\varepsilon))\) with respect to the imaginary shift \(\varepsilon\). Let \(\mathcal N(\omega,E)\) be the integrated density of states, i.e. the phase-independent limiting cumulative distribution function of the finite-volume eigenvalues. Fix \(E_0\in\mathbb R\) such that \(L_d(\omega,E_0)>0\). Which statement holds for every such Diophantine \(\omega\)?

There exists a neighborhood \(I_{E_0}\) of \(E_0\) such that for every \(E,E'\in I_{E_0}\) and every H\"older exponent \(0<\beta<1/(2\kappa^d(\omega,E_0))\), one has
\[
|\mathcal N(\omega,E)-\mathcal N(\omega,E')|\le |E-E'|^{\beta}.
\]

Choices:
B.
There exists a neighborhood \(I_{E_0}\) of \(E_0\) such that for every \(E,E'\in I_{E_0}\) one has
\[
|\mathcal N(\omega,E)-\mathcal N(\omega,E')|\le |E-E'|^{\beta}
\]
for every H\"older exponent \(0<\beta<1/\kappa^d(\omega,E_0)\).
C.
There exists a neighborhood \(I_{E_0}\) of \(E_0\) and some H\"older exponent \(\beta>0\) such that for every \(E,E'\in I_{E_0}\),
\[
|\mathcal N(\omega,E)-\mathcal N(\omega,E')|\le |E-E'|^{\beta}.
\]
D.
For every H\"older exponent \(0<\beta<1/(2\kappa^d(\omega,E_0))\), there exists a neighborhood \(I_{E_0,\beta}\) of \(E_0\) such that for every \(E,E'\in I_{E_0,\beta}\),
\[
|\mathcal N(\omega,E)-\mathcal N(\omega,E')|\le |E-E'|^{\beta},
\]
and one may choose \(I_{E_0,\beta}\) independently of the Diophantine frequency \(\omega\in\mathrm{DC}\).
E.
There exists a neighborhood \(I_{E_0}\) of \(E_0\) such that for every \(E,E'\in I_{E_0}\) and every H\"older exponent \(0<\beta<1/(2\kappa^d(\omega,E))\), one has
\[
|\mathcal N(\omega,E)-\mathcal N(\omega,E')|\le |E-E'|^{\beta}.
\]

uniformity over all \(0<\beta<1/(2\kappa^d(\omega,E_0))\) replaced by existence of some positive \(\beta\)

dependence of the neighborhood on the fixed frequency \(\omega\)

acceleration evaluated at \(E_0\) replaced by acceleration at the variable energy \(E\)

The stem gives hypotheses and definitions but does not reveal the conclusion. The correct choice is not explicitly or trivially implied by wording in the prompt.

The item is largely a theorem-recall question: under a full list of assumptions, the student must identify the exact stated conclusion. The answer choices introduce subtle quantifier and parameter changes, so it is not a pure verbatim restatement, but it remains close to the source theorem.

Some reasoning is needed to compare the strength of the Hölder exponent bound, dependence on E versus E0, and neighborhood uniformity. However, the task mainly tests precise recall/discrimination of the theorem statement rather than deeper derivation or synthesis.

The distractors are mathematically plausible and target realistic failure modes: missing the factor 2, weakening to mere existence of some β, incorrectly claiming ω-uniform neighborhoods, and replacing κ^d(ω,E0) by κ^d(ω,E). They are distinct and well aligned with common misreadings.

A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it is still fairly close to a direct recall of the theorem rather than a genuinely generative reasoning task.