For any $d \geqslant 1$ and $j \in [0, d]$, we have $c_{d, 0}(j) = |s(d + 1, j + 1)|$.

\begin{equation} \label{eq:polynomial def}
  \sum_{n \geq 0} P_{d,k}(n)\, x^n = \frac{(1+x)^k}{(1-x)^{d+1}}
\end{equation}

\begin{remark}\label{ehrbij}
    The fact that the objects described above are counted by $d!P_{d, k}(n)$ also follows by the interpretation of $P_{d, k}(n)$ as an Ehrhart polynomial.
    More specifically, $P_{d, k}(n)$ counts points $\bm x \in \ZZ^d$, such that
    \begin{itemize}
        \item $|x_1| + \ldots +|x_k| + x_{k + 1} + \ldots + x_d \leq n$, and
        \item $x_{k + 1}, \ldots, x_d \geq 0$.
    \end{itemize}
    One way to represent a sequence of length $d$ of integers with sum of absolute values at most $n$ is as a sequence consisting of $n$ balls and $d$ dots.
    For example, with $n = 6$ and $d = 3$, the sequence $(2, 0, 3)$ (with only nonnegative terms) is represented as
    \begin{equation*}
        \emptycircle[0pt]\, \emptycircle[0pt]\, {\bullet}\, {\bullet}\, \emptycircle[0pt]\, \emptycircle[0pt]\, \emptycircle[0pt]\, {\bullet}\, \emptycircle[0pt].
    \end{equation*}
    The total number of balls is $n$, the first entry is the number of balls before the first dot, and for any $i \geq 2$, the $i$-th entry is the number of balls between the $(i - 1)$-th and $i$-th dots.
    If we want the $i$-th entry to be negative, we move the $i$-th dot into the ball immediately before it.
    For example, the sequence $(-1, 3, -2)$ is represented as
    \begin{equation*}
        \circled[0pt]{\bullet}\, \emptycircle[0pt]\, \emptycircle[0pt]\, \emptycircle[0pt]\, {\bullet}\, \emptycircle[0pt]\, \circled[0pt]{\bullet}.
    \end{equation*}
    Hence, $P_{d, k}(n)$ counts such configurations with $n$ balls and $d$ dots where only the first $k$ dots can be in balls.
    This interpretation can be used to show $d!P_{d, k}(n) = |W_{d, k}(n)|$.
\end{remark}

\label{sec:intro}

The Ehrhart polynomial \(\ehr_{\mathcal P}(n)\) of a lattice polytope \(\mathcal P\) counts the number of integer lattice points in the dilate \(n \mathcal P\). It is known that the Ehrhart polynomial of the cross-polytope
\begin{equation*}
  \mathcal P_d \coloneqq \left\{\bm x \in \mathbb{R}^d : |x_1| + |x_2| + \ldots + |x_d| \leqslant 1\right\}
\end{equation*}
is determined by the generating function
\begin{equation*}
  \sum_{n \geqslant 0} \ehr_{\mathcal P_d}(n)\, x^n = \frac{(1+x)^d}{(1-x)^{d+1}}.
\end{equation*}
More generally, if \(k \in [0,d]\), the polynomial \(P_{d,k}(n)\) defined by
\begin{equation} \label{eq:polynomial def}
  \sum_{n \geqslant 0} P_{d,k}(n)\, x^n = \frac{(1+x)^k}{(1-x)^{d+1}}
\end{equation}
is the Ehrhart polynomial of the \((d-k)\)-fold pyramid over \(\mathcal P_k\). The relevant background can be found for example in \cite{ccd}. The figure below illustrates \(\mathcal P_3\) (left), the pyramid over \(\mathcal P_2\) (middle) and \(\mathcal P_2\) itself (right).

\bigskip
\begin{center}
  \begin{tikzpicture}[scale=1.7]
    \coordinate (1) at (1,0,0);
    \coordinate (-1) at (-1,0,0);
    \coordinate (2) at (0,1,0);
    \coordinate (-2) at (0,-1,0);
    \coordinate (3) at (0,0,1);
    \coordinate (-3) at (0,0,-1);

ll[gray!20] (1) -- (3) -- (-1) -- (2) -- cycle;
ll[gray!40] (1) -- (3) -- (-1) -- (-2) -- cycle;

    \draw[rpath, dashed]
      (-3) -- (1)
      (-3) -- (2)
      (-3) -- (-1)
      (-3) -- (-2);
    \draw[rpath]
      (1) -- (3) -- (-1)
      (2) -- (3) -- (-2);
    \draw[rpath] (1) -- (2) -- (-1) -- (-2) -- cycle;

    \begin{scope}[shift={(2.5,0,0)}]
      \coordinate (1) at (1,0,0);
      \coordinate (-1) at (-1,0,0);
      \coordinate (2) at (0,1,0);
      \coordinate (3) at (0,0,1);
      \coordinate (-3) at (0,0,-1);

ll[gray!20] (1) -- (3) -- (-1) -- (2) -- cycle;

      \draw[rpath, dashed]
        (-3) -- (1)
        (-3) -- (-1)
        (-3) -- (2);
      \draw[rpath] (3) -- (2);
      \draw[rpath] (1) -- (3) -- (-1) -- (2) -- cycle;
    \end{scope}

    \begin{scope}[shift={(5,0,0)}]
      \coordinate (1) at (1,0,0);
      \coordinate (-1) at (-1,0,0);
      \coordinate (3) at (0,0,1);
      \coordinate (-3) at (0,0,-1);

ll[gray!20] (1) -- (3) -- (-1) -- (-3) -- cycle;
      \draw[rpath] (1) -- (3) -- (-1) -- (-3) -- cycle;
    \end{scope}
  \end{tikzpicture}
\end{center}

\noindent
The fact that \(P_{d,k}(n)\) is a polynomial is not immediately clear from \eqref{eq:polynomial def}, but one can check that
\begin{equation*}
  P_{d,k}(n) = \sum_{i=0}^k \binom{k}{i} \binom{n + d - i}{d}.
\end{equation*}

The coefficients of Ehrhart polynomials have been studied extensively, and they are usually hard to understand -- it is uncommon to be able to fully describe the coefficients for a given polytope. It is difficult even to determine if the coefficients are nonnegative, see \cite{ehrpos}. In this paper, we will examine these questions for (pyramids over) cross-polytopes. We define \(c_{d,k}(j)\) by
\begin{equation} \label{eqn}
  d! P_{d,k}(n) = \sum_{j=0}^d c_{d,k}(j) n^j,
\end{equation}
where the factor \(d!\) ensures that every coefficient is integral. Our main results are a combinatorial interpretation of the coefficients \(c_{d,k}(j)\), and a bijective proof of equation \eqref{eqn}. In particular, we prove combinatorially that the Ehrhart coefficients for (pyramids over) cross-polytopes are positive, which answers a question posed by Stanley \cite[Chapter 4, Exercise 1(b)]{ecnewex}.

The outline of the paper is as follows.
In Section \ref{sec:sri}, we construct a class of objects that are counted by the coefficients \(c_{d, k}(j)\), in particular showing that these coefficients are positive.
In Section \ref{sec:bij}, we give a combinatorial argument that proves \eqref{eqn} by constructing objects that are counted by both sides of the equation and exhibiting a bijection.
In Section \ref{sec:prop}, we use the combinatorial meaning we have for \(c_{d, k}(j)\) to obtain some properties of these coefficients.
Finally, in Section \ref{sec:poly}, we illustrate how the method we have used might be helpful for studying the Ehrhart coefficients of other polytopes.

\bigskip\noindent
Before proceeding with the general results, we will briefly describe a proof strategy for the simplest case \(k = 0\). The same ideas will serve as our guiding light for the rest of the paper.

The numbers \(c_{d, 0}(j)\) are just unsigned Stirling numbers of the first kind.
Recall that the (signed) Stirling number of the first kind \(s(m, i)\) is defined by
\begin{equation*}
    s(m, i) \coloneqq (-1)^{m - i} \cdot |\{\sigma \in S_m : \sigma \text{ has } i \text{ cycles}\}|.
\end{equation*}

The Ehrhart polynomial \(\ehr_{\mathcal P}(n)\) of a lattice polytope \(\mathcal P\) counts the number of integer lattice points in the dilate \(n \mathcal P\). It is known that the Ehrhart polynomial of the cross-polytope
\begin{equation*}
 \mathcal P_d \coloneqq \left\{\bm x \in \mathbb{R}^d : |x_1| + |x_2| + \ldots + |x_d| \leqslant 1\right\}
\end{equation*}
is determined by the generating function
\begin{equation*}
 \sum_{n \geqslant 0} \ehr_{\mathcal P_d}(n)\, x^n = \frac{(1+x)^d}{(1-x)^{d+1}}.
\end{equation*}
More generally, if \(k \in [0,d]\), the polynomial \(P_{d,k}(n)\) defined by
\begin{equation} \label{eq:polynomial def}
 \sum_{n \geqslant 0} P_{d,k}(n)\, x^n = \frac{(1+x)^k}{(1-x)^{d+1}}
\end{equation}
is the Ehrhart polynomial of the \((d-k)\)-fold pyramid over \(\mathcal P_k\). The relevant background can be found for example in \cite{ccd}. The figure below illustrates \(\mathcal P_3\) (left), the pyramid over \(\mathcal P_2\) (middle) and \(\mathcal P_2\) itself (right).

\bigskip
\begin{center}
 \begin{tikzpicture}[scale=1.7]
 \coordinate (1) at (1,0,0);
 \coordinate (-1) at (-1,0,0);
 \coordinate (2) at (0,1,0);
 \coordinate (-2) at (0,-1,0);
 \coordinate (3) at (0,0,1);
 \coordinate (-3) at (0,0,-1);

The coefficients of Ehrhart polynomials have been studied extensively, and they are usually hard to understand -- it is uncommon to be able to fully describe the coefficients for a given polytope. It is difficult even to determine if the coefficients are nonnegative, see \cite{ehrpos}. In this paper, we will examine these questions for (pyramids over) cross-polytopes. We define \(c_{d,k}(j)\) by
\begin{equation} \label{eqn}
 d! P_{d,k}(n) = \sum_{j=0}^d c_{d,k}(j) n^j,
\end{equation}
where the factor \(d!\) ensures that every coefficient is integral. Our main results are a combinatorial interpretation of the coefficients \(c_{d,k}(j)\), and a bijective proof of equation \eqref{eqn}. In particular, we prove combinatorially that the Ehrhart coefficients for (pyramids over) cross-polytopes are positive, which answers a question posed by Stanley \cite[Chapter 4, Exercise 1(b)]{ecnewex}.

The outline of the paper is as follows.
In Section \ref{sec:sri}, we construct a class of objects that are counted by the coefficients \(c_{d, k}(j)\), in particular showing that these coefficients are positive.
In Section \ref{sec:bij}, we give a combinatorial argument that proves \eqref{eqn} by constructing objects that are counted by both sides of the equation and exhibiting a bijection.
In Section \ref{sec:prop}, we use the combinatorial meaning we have for \(c_{d, k}(j)\) to obtain some properties of these coefficients.
Finally, in Section \ref{sec:poly}, we illustrate how the method we have used might be helpful for studying the Ehrhart coefficients of other polytopes.

\bigskip\noindent
Before proceeding with the general results, we will briefly describe a proof strategy for the simplest case \(k = 0\). The same ideas will serve as our guiding light for the rest of the paper.

The numbers \(c_{d, 0}(j)\) are just unsigned Stirling numbers of the first kind.
Recall that the (signed) Stirling number of the first kind \(s(m, i)\) is defined by
\begin{equation*}
 s(m, i) \coloneqq (-1)^{m - i} \cdot |\{\sigma \in S_m : \sigma \text{ has } i \text{ cycles}\}|.
\end{equation*}

\begin{equation} \label{eqn}
  d! P_{d,k}(n) = \sum_{j=0}^d c_{d,k}(j) n^j,
\end{equation}

\renewcommand{\epsilon}{\varepsilon}
 \renewcommand{\phi}{\varphi}
 \renewcommand{\emptyset}{\varnothing}
 \renewcommand{\leq}{\leqslant}
 \renewcommand{\geq}{\geqslant}
 }
lib.bib0000664000000000000000000000366315116032500011003 0ustar rootroot@article {bona_permutations_2000,
 AUTHOR = {B\'ona, Mikl\'os and McLennan, Andrew and White, Dennis},
 TITLE = {Permutations with roots},
 JOURNAL = {Random Struct. Algorithms},
 VOLUME = {17},
 YEAR = {2000},
 NUMBER = {2},
 PAGES = {157--167},
 ISSN = {1042-9832,1098-2418},
 DOI = {10.1002/1098-2418(200009)17:2<157::AID-RSA4>3.0.CO;2-2}
}

The numbers \(c_{d, 0}(j)\) are just unsigned Stirling numbers of the first kind.
Recall that the (signed) Stirling number of the first kind \(s(m, i)\) is defined by
\begin{equation*}
 s(m, i) \coloneqq (-1)^{m - i} \cdot |\{\sigma \in S_m : \sigma \text{ has } i \text{ cycles}\}|.
\end{equation*}

\begin{proof}
 From the definition of $P_{d, k}$, we see that $P_{d, 0}(n) = \binom{n + d}{d}$.
 The result now follows from the fact that $s(m, i)$ is the coefficient of $x^i$ in $x(x - 1) \cdots (x - m + 1)$ (see, for example, \cite[Proposition 1.3.7]{ec1}).
\end{proof}

Although we could stop here and say we have given combinatorial meaning to the coefficients $c_{d, 0}(j)$, we would ideally like to give combinatorial meaning to the fact that for any $n \geq 0$,
\begin{equation*}
 d!P_{d, 0}(n) = \sum_{j = 0}^d c_{d, 0}(j)n^j.
\end{equation*}
Especially since the polynomials $P_{d, k}$ are Ehrhart polynomials, it makes more sense to interpret $c_{d, k}(j)$ in this context rather than doing so independently.
Also note that if we show that $|s(d + 1, j + 1)|$ satisfies the above equation for all $n \geq 0$, then this implies that $c_{d, 0}(j) = |s(d + 1, j + 1)|$.

\begin{lemma}\label{lem:str}
 For any $d \geq 1$ and $k, j \in [0, d]$, we have
 \begin{equation*}
 c_{d, k}(j) = \sum_{i = 0}^k \binom{k}{i} \sum_{\ell = 0}^{d - i} |s(d - i + 1, \ell + 1)|s(i, j - \ell).
 \end{equation*}
\end{lemma}

\begin{proof}
 Note that for any permutation $\sigma \in S_m$ with $i$ cycles, the parity of $m - i$ is the same as the number of even cycles in $\sigma$.
 We now interpret the expression from Lemma \ref{lem:str} for $c_{d, k}(j)$ as the signed count of the following objects:
 Permutations of size $d + 1$ with $j + 1$ cycles where
 \begin{itemize}
 \item any cycle with all elements in $[k]$ is colored either red or blue,
 \item any cycle containing an element not in $[k]$ is uncolored, and
 \item the sign associated to the permutation is the number of even cycles that are colored red.
 \end{itemize}

\begin{lemma}
 For any $d \geq 1$ and $k \in [0, d]$, we have $c_{d, k}(0) = d!$ and $c_{d, k}(d) = 2^k$.
\end{lemma}

\begin{lemma}
 For any $d \geq 1$ and $k, j \in [0, d]$, we have $c_{d, k}(j) \leq c_{d + 1, k}(j)$.
\end{lemma}

\begin{proof}
    From the definition of $P_{d, k}$, we see that $P_{d, 0}(n) = \binom{n + d}{d}$.
    The result now follows from the fact that $s(m, i)$ is the coefficient of $x^i$ in $x(x - 1) \cdots (x - m + 1)$ (see, for example, \cite[Proposition 1.3.7]{ec1}).
\end{proof}

Although we could stop here and say we have given combinatorial meaning to the coefficients $c_{d, 0}(j)$, we would ideally like to give combinatorial meaning to the fact that for any $n \geqslant 0$,
\begin{equation*}
    d!P_{d, 0}(n) = \sum_{j = 0}^d c_{d, 0}(j)n^j.
\end{equation*}
Especially since the polynomials $P_{d, k}$ are Ehrhart polynomials, it makes more sense to interpret $c_{d, k}(j)$ in this context rather than doing so independently.
Also note that if we show that $|s(d + 1, j + 1)|$ satisfies the above equation for all $n \geqslant 0$, then this implies that $c_{d, 0}(j) = |s(d + 1, j + 1)|$.

To do this, we construct objects counted by $d!P_{d, 0}(n) = d!\binom{n + d}{d}$.
One choice is objects that consist of a permutation of size $d$ with $n$ balls distributed among the $d + 1$ slots created by the terms of the permutation.
For example, for $d = 9$ and $n = 6$, such an object is
\begin{equation*}
    \emptycircle\, 281\, \emptycircle\, 9\, \emptycircle\, \emptycircle\, 354\, \emptycircle\, 67\, \emptycircle\,.
\end{equation*}
It can also be shown that these objects are counted by $d!P_{d, 0}(n)$ using the interpretation of $P_{d, 0}(n)$ as an Ehrhart polynomial (see Remark \ref{ehrbij}).

We interpret $|s(d + 1, j + 1)|n^j$ as choosing a permutation of size $d + 1$ with $j + 1$ cycles as well as assigning the labels $1, 2, \ldots, j$ to $n$ balls.
Such a pair for $d = 8$, $j = 3$, and $n = 4$ is
\begin{equation*}
    (2) (7\ 3\ 5\ 4) (8\ 1) (9\ 6)\quad \text{and}\quad \emptycircle\, \overset{1,3}{\emptycircle}\, \emptycircle\, \overset{2}{\emptycircle}\,.
\end{equation*}

We write cycles in a permutation in \emph{canonical cycle form}: each cycle has its largest element first, and the cycles are ordered in increasing order of their largest elements.
The labels on balls indicate which cycles must be placed before it.
The label $1, 3$ on the second ball in the example above indicates that the first and third cycle should be placed before the second ball.
We do this by placing the cycles as they are into that slot and then removing the brackets.
This is just an instance of the fundamental bijection on permutations (see \cite[Proposition 1.3.1]{ec1}).
The cycle containing $d + 1$ is placed after the last ball and then the brackets and $d + 1$ are deleted.
Hence, the object corresponding to the example above is
\begin{equation*}
    \emptycircle\, 281\, \emptycircle\, \emptycircle\, 7354\, \emptycircle\, 6.
\end{equation*}

It can be checked that this is a bijection and gives us a proof that for all $n \geqslant 0$,
\begin{equation*}
    d!P_{d, 0}(n) = \sum_{j = 0}^d |s(d + 1, j + 1)| n^j
\end{equation*}
and in particular $c_{d, 0}(j) = |s(d + 1, j + 1)|$.
We will generalize the interpretation we have for $c_{d, 0}(j)$ in Section \ref{sec:sri} and then in Section \ref{sec:bij}, we will present a bijection generalizing the one we have just seen.

From the definition of $P_{d,k}$, $P_{d,0}(n)=\binom{n+d}{d}$. The theorem then follows since $s(m,i)$ is the coefficient of $x^i$ in $x(x-1)\cdots(x-m+1)$.

Equivalently, to show $c_{d,0}(j)=|s(d+1,j+1)|$, it suffices to prove that for all $n\ge 0$,
\[
 d!P_{d,0}(n)=\sum_{j=0}^d |s(d+1,j+1)|\,n^j.
\]
A combinatorial proof is outlined by:
(1) constructing objects counted by $d!P_{d,0}(n)=d!\binom{n+d}{d}$, e.g. “a permutation of size $d$ with $n$ balls distributed among the $d+1$ slots created by the terms of the permutation”;
(2) interpreting $|s(d+1,j+1)|n^j$ as “choosing a permutation of size $d+1$ with $j+1$ cycles as well as assigning the labels $1,2,\ldots,j$ to $n$ balls”, with cycles written in canonical cycle form and ball labels indicating “which cycles must be placed before it”;
(3) producing the permutation-with-slots object by placing the cycles into the indicated slots and “removing the brackets” (an instance of “the fundamental bijection on permutations”), then placing the cycle containing $d+1$ after the last ball and deleting the brackets and $d+1$.
It is stated that “It can be checked that this is a bijection,” yielding the identity above and hence $c_{d,0}(j)=|s(d+1,j+1)|$.

Question:
For integers \(d \geq 1\) and \(0 \leq j \leq d\), let the polynomial \(P_{d,0}(n)\) be defined by the generating function
\[
\sum_{n \geq 0} P_{d,0}(n)x^n=\frac{1}{(1-x)^{d+1}},
\]
and define integers \(c_{d,0}(j)\) by
\[
d!\,P_{d,0}(n)=\sum_{r=0}^d c_{d,0}(r)n^r.
\]
Also, for \(m\ge 0\) and \(0\le i\le m\), let the signed Stirling number of the first kind be
\[
s(m,i)=(-1)^{m-i}\,\bigl|\{\sigma\in S_m:\sigma\text{ has }i\text{ cycles}\}\bigr|.
\]
Which statement holds for every such pair \((d,j)\)?

Choices:
B. \(c_{d,0}(j)=|s(d,j)|\).
C. \(c_{d,0}(j)\ge 0\).
D. \(c_{d,0}(j)=|s(d+1,j)|\).
E. \(c_{d,0}(j)=(-1)^{d-j}|s(d+1,j+1)|\).


Classical Ramsey theory typically asks which structured subsets must appear in any finite colouring of the natural numbers. Raimi proposed a complementary point of view: he asked which partitions of $\mathbb{N}$ cannot be avoided by any finite colouring of $\mathbb{N}$, even after allowing a shift.

More precisely, given a partition $\mathbb{N}=E_1\cup E_2$, we say that $(E_1,E_2)$ is \emph{unavoidable} if for every finite partition $\mathbb{N}=\bigcup\limits_{j=1}^t F_j$ there exist $j\in\{1,\dots,t\}$ and $k\in\mathbb{N}$ such that both $(F_j+k)\cap E_1$ and $(F_j+k)\cap E_2$ are infinite. This formalises Raimi’s viewpoint on unavoidable partitions under shifts.

A classical theorem of Raimi \cite{Raimi} shows that such partitions do exist: there is a partition $\mathbb{N}=E_1\cup E_2$ with the property that for every finite partition $\mathbb{N}=\bigcup\limits_{j=1}^t F_j$ with $t\in\mathbb{N}$, one can find $j\in\{1,\dots,t\}$ and $k\in\mathbb{N}$ such that both $(F_j+k)\cap E_1$ and $(F_j+k)\cap E_2$ are infinite. Raimi’s original proof used topological methods. Hindman later gave an elementary proof \cite[p.~180, Theorem~11.15]{HM79} and showed that one may take $E_1$ to be the set of natural numbers whose last non-zero digit in the ternary expansion is $1$, and $E_2=\mathbb{N}\setminus E_1$.

Strengthenings of this phenomenon, imposing density conditions on the partition sets or guaranteeing positive densities in the conclusion, were obtained by Hegyv\'ari \cite{NH} and by Bergelson and Weiss \cite{Bergelson2}. More recently, Hegyv\'ari, Pach, and Pham \cite{HPP25} introduced a powerful and flexible framework, combining tools from harmonic analysis, additive combinatorics, and group theory, which yields polynomial and finite-group extensions of Raimi’s theorem and makes its connection to Ramsey theory explicit. Their beautiful construction in the finite-group setting has since been extended successfully to the continuous setting for circles by Kang, Koh, and the author in \cite{KKT25}, which is stated as follows.

A classical theorem of Raimi \cite{Raimi} shows that such partitions do exist: there is a partition $\mathbb{N}=E_1\cup E_2$ with the property that for every finite partition $\mathbb{N}=\bigcup\limits_{j=1}^t F_j$ with $t\in\mathbb{N}$, one can find $j\in\{1,\dots,t\}$ and $k\in\mathbb{N}$ such that both $(F_j+k)\cap E_1$ and $(F_j+k)\cap E_2$ are infinite. Raimi’s original proof used topological methods. Hindman later gave an elementary proof \cite[p.~180, Theorem~11.15]{HM79} and showed that one may take $E_1$ to be the set of natural numbers whose last non-zero digit in the ternary expansion is $1$, and $E_2=\mathbb{N}\setminus E_1$.

Before stating the main results, we need to introduce some notation.

For every finite measurable cover
\[
\mathbb{S}^{n-1} \subset F_1\cup\cdots\cup F_t,
\]
there exist an index $m_0\in\{1,\dots,t\}$ and $\theta_0\in C$ such that
\[
\sigma_{n-1}\big(R_{\theta_0}(F_{m_0})
 \cap E_i^{\mathbb{S}^{n-1}}\big)>0
\qquad\text{for all }1\le i\le r.
\]
where $R_\theta$ denotes the rotation defined in \eqref{definition-rotation} as restricted to $\mathbb{S}^{n-1}$.
\end{theorem}

\begin{theorem}[Rotational power surfaces]\label{thm:power-surface-Raimi}
Let $n\ge 3$ and $k>0$. Define
\[
\mathcal{S}_{k,R}
 := \big\{(x',x_n)\in\mathbb{R}^{n-1}\times\mathbb{R}
 : x_n = |x'|^{\,k},\ 0<|x'|\le R\big\}.
\]
Let
$\sigma_{k,R}$ be the normalized surface measure on $\mathcal{S}_{k,R}$.
Then there exists a measurable partition
$\{E_i^{\mathcal{S}_{k,R}}\}_{i=1}^r$ of $\mathcal{S}_{k,R}$ with the following property:
For every finite measurable cover
\[
\mathcal{S}_{k,R} \subset F_1\cup\cdots\cup F_t
\]
there exist an index $m_0\in\{1,\dots,t\}$ and $\theta_0\in C$ such that
\[
\sigma_{k,R}\big(R_{\theta_0}(F_{m_0})
 \cap E_i^{\mathcal{S}_{k,R}}\big)>0
\qquad\text{for all }1\le i\le r.
\]
In particular, $k=1$ yields the cone, and $k=2$ yields the paraboloid.
\end{theorem}

For every finite measurable cover
\[
\mathcal{C}_{R,\Omega} \subset F_1\cup\cdots\cup F_t
\]
there exist an index $m_0\in\{1,\dots,t\}$ and $\theta_0\in C$ such that
\[
\mu_{\text{\tiny $R,\Omega$}}\big(R_{\theta_0}(F_{m_0})\cap E_i^{\mathcal{C}_{R,\Omega}}\big)>0
\qquad\text{for all }1\le i\le r,
\]
where $R_\theta$ denotes the rotation defined in \eqref{definition-rotation} as restricted to $\mathcal{C}_{R,\Omega}$.
\end{theorem}

We now present the following general theorem, of which the spherical, cylindrical, and rotational power surface cases are specific instances.
\begin{theorem}\label{thm:general-circle-bundle}
Let $(X,\mu)$ be a probability space equipped with a measurable, measure-preserving action
\[
\{R_\theta\}_{\theta\in C},\qquad C:=\mathbb{R}/\mathbb{Z},
\]
of the circle group $C$. Assume there exist
\begin{itemize}
 \item a probability space $(Y,\nu)$,
 \item a measurable set $N\subset X$ with $\mu(N)=0$,
 \item a measurable bijection
 \[
 \Phi : C\times Y \longrightarrow X\setminus N,
 \]
\end{itemize}
satisfying:
\begin{enumerate}
 \item[(i)] (\emph{Equivariance}) For all $\theta,\alpha\in C$ and $y\in Y$,
 \[
 \Phi(\theta+\alpha,y) = R_\alpha(\Phi(\theta,y)).
 \]
 \item[(ii)] (\emph{Product disintegration}) For every bounded measurable function $f:X\to\mathbb{R}$,
 \[
 \int_X f(x)\,d\mu(x)
 = \int_C \int_Y f(\Phi(\theta,y))\,d\nu(y)\,d\mu_1(\theta),
 \]
 where $\mu_1$ is the normalized Lebesgue measure on $C$.
\end{enumerate}
Let $\{E_i^C\}_{i=1}^r$ be a measurable partition of $C$ with the Raimi property from Theorem~\ref{thm:KKT-circle}. Define
\begin{equation*}
 \begin{cases}
 E_1^X := \Phi(E_1^C\times Y) \cup N,\\
 E_i^X:=\Phi(E_i^C\times Y), ~~~~~~~~2 \leq i\leq r.
 \end{cases}
\end{equation*}
Then $\{E_i^X\}_{i=1}^r$ is a measurable partition of $X$ with the following property:

Now define a measurable partition of $C$ by
\[
 C_m := \{\theta\in C : m(\theta)=m\},\qquad 1\le m\le t.
\]
By construction, for every $\theta\in C_m$ we have
\begin{equation}\label{eq:Am-lower-bound}
 \nu(A_m(\theta))\ge \frac1t.
\end{equation}
By applying Theorem~ \ref{thm:KKT-circle} to the partition $\{C_m\}_{m=1}^t$ of $C$, there exist an index $m_0\in\{1,\dots,t\}$ and a rotation
\[
 R_{\theta_0}: C\to C,\qquad R_{\theta_0}(\theta)=\theta+\theta_0,
\]
such that
\begin{equation}\label{eq:KKT-circle-hit}
 \mu_1\big(R_{\theta_0}(C_{m_0})\cap E_i^C\big)>0,
 \qquad\, \forall\, 1\le i\le r.
\end{equation}

To complete the proof of the theorem, it remains to use \eqref{eq:KKT-circle-hit} to show that
\begin{align}\label{ineq:intersection-rotation-E_i^X}
 \mu\bigl(R_{\theta_0}(F_{m_0}) \cap E_i^X\bigr) > 0
 \qquad \text{for all } 1 \le i \le r.
\end{align}
Let
\[
 R_{\theta_0}(\theta,y):=(\theta+\theta_0,y)
\]
be the induced rotation on $C\times Y$. By equivariance, we have
\[
 R_{\theta_0}(F_{m_0})\cap (X\setminus N)
 = R_{\theta_0}\big(\Phi(F_{m_0}^{\mathrm e})\big)
 = \Phi\big(R_{\theta_0}(F_{m_0}^{\mathrm e})\big),
\]
which, together with the definition of $E_i^X$, gives
\[
 R_{\theta_0}(F_{m_0})\cap E_i^X
 = \Phi\Big(R_{\theta_0}(F_{m_0}^{\mathrm e})\cap (E_i^C\times Y)\Big)
 \cup \Big(R_{\theta_0}(F_{m_0})\cap N\cap E_i^X\Big).
\]
By the assumption on $N$, the definition of definition of $F_{m_0}^{\mathrm e}$, and the rotation on $C\times Y$, we have
\begin{align*}
 \mu\big(R_{\theta_0}(F_{m_0})\cap E_i^X\big)
 =& (\mu_1\times\nu)\big(R_{\theta_0}(F_{m_0}^{\mathrm e})\cap (E_i^C\times Y)\big) \\
 = &\int_{C}\int_Y
 \mathbf{1}_{R_{\theta_0}(F_{m_0}^{\mathrm e})}(\theta,y)
 \mathbf{1}_{E_i^C}(\theta)\,d\nu(y)\,d\mu_1(\theta) \\
 =& \int_{C}\int_Y
 \mathbf{1}_{A_{m_0}(\theta-\theta_0)}(y)\mathbf{1}_{E_i^C}(\theta)
 \,d\nu(y)\,d\mu_1(\theta) \\
 =& \int_{C} \nu\big(A_{m_0}(\theta-\theta_0)\big)\,\mathbf{1}_{E_i^C}(\theta)\,d\mu_1(\theta).
\end{align*}
By restricting the outer integral to those $\theta$ satisfying
$\theta-\theta_0\in C_{m_0}$, i.e.\ $\theta\in R_{\theta_0}(C_{m_0})$, and using \eqref{eq:Am-lower-bound} together with \eqref{eq:KKT-circle-hit}, we obtain
\begin{align*}
 \mu\big(R_{\theta_0}(F_{m_0})\cap E_i^X\big)
 \geq & \int_{R_{\theta_0}(C_{m_0})\cap E_i^C} \nu\big(A_{m_0}(\theta-\theta_0)\big)\,d\mu_1(\theta) \\
 &\ge \frac1t\,\mu_1\big(R_{\theta_0}(C_{m_0})\cap E_i^C\big)>0
\end{align*}
This is precisely \eqref{ineq:intersection-rotation-E_i^X}. This completes the proof.
\end{proof}

\begin{align}\label{definition-rotation}
R_\theta(x_1,x_2,x'',x_n)
 :=\bigl(\,
        x_1\cos(2\pi\theta) - x_2\sin(2\pi\theta),\
        x_1\sin(2\pi\theta) + x_2\cos(2\pi\theta),\
        x'',\ x_n
      \,\bigr),
\end{align}

\begin{theorem}[Kang--Koh--Tran {\cite{KKT25}}]\label{thm:KKT-circle}
Let $r,t\in\mathbb{N}$ with $r,t\ge2$. There exists a measurable partition
\[
C = \bigcup_{i=1}^r E_i
\]
such that for every finite measurable cover
\[
C \subset F_1 \cup \cdots \cup F_t,
\]
there exist an index $m \in \{1,\dots,t\}$ and a rotation $R_{\theta}$ satisfying
\begin{align*}
    \mu_1\big(R_{\theta}(F_m) \cap E_i\big) > 0
\quad\text{for all } 1\le i\le r.
\end{align*}
\end{theorem}

The theorem provides a partition of the circle with the property that every finite measurable cover admits a translate meeting each partition element in positive measure.  This paper shows that this phenomenon extends beyond compact abelian groups to a wide class of non-group geometric surfaces that still exhibit \textit{a hidden one-dimensional symmetry}. This answers a question raised in \cite{KKT25} concerning the extension
from the circle $C$ to the unit sphere $\mathbb{S}^{n-1}\subset \mathbb{R}^n$.

Before stating the main results, we need to introduce some notation.

For notational convenience, given a point
$x=(x_1,\dots,x_n)\in\mathbb{R}^n$, we write
\[
x' := (x_1,\dots,x_{n-1}) \in \mathbb{R}^{\,n-1},
\qquad
x'' := (x_3,\dots,x_{n-1}) \in \mathbb{R}^{\,n-3}.
\]
Throughout the paper, we denote by $C=\mathbb{R}/\mathbb{Z}$ the circle group. Let $\{E_i^C\}_{i=1}^r$ be the measurable partition of $C$ provided by Theorem~\ref{thm:KKT-circle}.

We also introduce the rotation
\begin{align}\label{definition-rotation}
R_\theta(x_1,x_2,x'',x_n)
 :=\bigl(\,
        x_1\cos(2\pi\theta) - x_2\sin(2\pi\theta),\
        x_1\sin(2\pi\theta) + x_2\cos(2\pi\theta),\
        x'',\ x_n
      \,\bigr),
\end{align}
that is, rotation by angle $2\pi\theta$ in the $(x_1,x_2)$–plane.

\begin{theorem}[Spheres]\label{thm:sphere-Raimi}
Let $n\ge 3$,
\[
\mathbb{S}^{n-1} := \{x\in\mathbb{R}^n : |x|=1\}
\]
be the unit sphere equipped with the normalized surface measure $\sigma_{n-1}$.
Then there exists a measurable partition $\{E_i^{\mathbb{S}^{n-1}}\}_{i=1}^r$ of $\mathbb{S}^{n-1}$ such that:

For every finite measurable cover
\[
\mathbb{S}^{n-1} \subset F_1\cup\cdots\cup F_t,
\]
there exist an index $m_0\in\{1,\dots,t\}$ and $\theta_0\in C$ such that
\[
\sigma_{n-1}\big(R_{\theta_0}(F_{m_0})
    \cap E_i^{\mathbb{S}^{n-1}}\big)>0
\qquad\text{for all }1\le i\le r.
\]
where $R_\theta$ denotes the rotation defined in \eqref{definition-rotation} as restricted to $\mathbb{S}^{n-1}$.
\end{theorem}

\begin{theorem}[Rotational power surfaces]\label{thm:power-surface-Raimi}
Let $n\ge 3$ and $k>0$. Define
\[
\mathcal{S}_{k,R}
 := \big\{(x',x_n)\in\mathbb{R}^{n-1}\times\mathbb{R}
        : x_n = |x'|^{\,k},\ 0<|x'|\le R\big\}.
\]
Let
$\sigma_{k,R}$ be the normalized surface measure on $\mathcal{S}_{k,R}$.
Then there exists a measurable partition
$\{E_i^{\mathcal{S}_{k,R}}\}_{i=1}^r$ of $\mathcal{S}_{k,R}$ with the following property:
For every finite measurable cover
\[
\mathcal{S}_{k,R} \subset F_1\cup\cdots\cup F_t
\]
there exist an index $m_0\in\{1,\dots,t\}$ and $\theta_0\in C$ such that
\[
\sigma_{k,R}\big(R_{\theta_0}(F_{m_0})
       \cap E_i^{\mathcal{S}_{k,R}}\big)>0
\qquad\text{for all }1\le i\le r.
\]
In particular, $k=1$ yields the cone, and $k=2$ yields the paraboloid.
\end{theorem}

\begin{theorem}[Cylindrical surface]\label{thm:cylinder-Raimi}
Let $n\ge 3$, $R>0$, and $\Omega\subset\mathbb{R}^{n-2}$ be a
bounded Borel set. Define the cylindrical surface
\[
\mathcal{C}_{R,\Omega}
:= \big\{(x_1,x_2, x'', x_n)\in\mathbb{R}^2\times\mathbb{R}^{n-2}
 : x_1^2+x_2^2=R^2,\ (x'', x_n)\in\Omega\big\}.
\]
Let $\mu_{\text{\tiny $R,\Omega$}}$ be the normalized surface measure on $\mathcal{C}_{R,\Omega}$.
Then there exists a measurable partition $\{E_i^{\mathcal{C}_{R,\Omega}}\}_{i=1}^r$ of $\mathcal{C}_{R,\Omega}$ with the following property:

For every finite measurable cover
\[
\mathcal{C}_{R,\Omega} \subset F_1\cup\cdots\cup F_t
\]
there exist an index $m_0\in\{1,\dots,t\}$ and $\theta_0\in C$ such that
\[
\mu_{\text{\tiny $R,\Omega$}}\big(R_{\theta_0}(F_{m_0})\cap E_i^{\mathcal{C}_{R,\Omega}}\big)>0
\qquad\text{for all }1\le i\le r,
\]
where $R_\theta$ denotes the rotation defined in \eqref{definition-rotation} as restricted to $\mathcal{C}_{R,\Omega}$.
\end{theorem}

\medskip
\paragraph{Sketch of proof.}
The essential observation is that spheres, rotational power surfaces, and cylinders all carry a natural measure-preserving action of the circle $C$ given by rotation in the $(x_1,x_2)$-plane.  Moreover, each of these surfaces admits a measurable, measure-preserving trivialization
\[
\Phi : C\times Y \longrightarrow X\setminus N,
\]
where $Y$ is a suitable parameter space and $N$ is a lower-dimensional set of measure zero.  This allows the surface measure on $X$ to disintegrate as
\[
d\mu_X(x)
 = d\mu_1(\theta)\, d\nu_Y(y),
\]
mirroring the product structure of $C\times Y$.
The equivariance identity
\[
\Phi(\theta+\alpha,y)=R_\alpha(\Phi(\theta,y))
\]
then enables the measurable Raimi partition on the base circle to be lifted directly to a partition of $X$.

The proofs of our three geometric theorems reduce to verifying the structural hypotheses of the general circle-bundle theorem introduced in Section~\ref{section-main-result}.  Once this framework is in place, the partition constructed in the previous work of Kang, Koh, and the author on the circle $C$ automatically induces the desired Raimi-type partitions on $\mathbb{S}^{n-1}$, on rotational power surfaces, and on circular cylindrical surfaces.

Our approach naturally leads to the following open question:
Let $M$ be a compact hyperbolic surface with its normalized area measure.
Does $M$ admit a measurable Raimi--type partition (in the sense of this paper), even though it has no circle action and therefore lies outside our circle--bundle framework?

{\bf The paper is organized as follows.}
In Section~\ref{section-main-result}, we establish the general circle-bundle theorem, which serves as the main structural tool of the paper.  Sections~\ref{section-sphere}, \ref{section-power-surface}, and \ref{section-cylinder} are devoted to the proofs of the three principal applications: the sphere, the rotational power surfaces, and the circular cylindrical surfaces, respectively.  Each result follows by verifying the hypotheses of the general theorem and applying the measurable Raimi partition on the base circle.

The post-theorem text does not sketch a proof of Theorem~\ref{thm:KKT-circle}; it only states that Theorem~\ref{thm:KKT-circle} provides a partition of the circle with a translate/rotation intersection property and then uses it as an input for later results. The only explicit \paragraph{Sketch of proof.} concerns the later geometric theorems (sphere/power surfaces/cylinders): it observes these surfaces have a measure-preserving circle action by rotation in the $(x_1,x_2)$-plane, admits a measurable measure-preserving trivialization \(\Phi: C\times Y\to X\setminus N\) leading to disintegration \(d\mu_X=d\mu_1(\theta)\,d\nu_Y(y)\), uses equivariance \(\Phi(\theta+\alpha,y)=R_\alpha(\Phi(\theta,y))\) to lift the partition from the base circle, and reduces the proofs to verifying hypotheses of a general circle-bundle theorem and then applying the partition from Kang--Koh--Tran on \(C\).

The post-theorem text does not sketch a proof of the main theorem; it only states that the main theorem provides a partition of the circle with a translate/rotation intersection property and then uses it as an input for later results. The only explicit \paragraph{Sketch of proof.} concerns the later geometric theorems (sphere/power surfaces/cylinders): it observes these surfaces have a measure-preserving circle action by rotation in the $(x_1,x_2)$-plane, admits a measurable measure-preserving trivialization \(\Phi: C\times Y\to X\setminus N\) leading to disintegration \(d\mu_X=d\mu_1(\theta)\,d\nu_Y(y)\), uses equivariance \(\Phi(\theta+\alpha,y)=R_\alpha(\Phi(\theta,y))\) to lift the partition from the base circle, and reduces the proofs to verifying hypotheses of a general circle-bundle theorem and then applying the partition from the main theorem on \(C\).

Question: Let $C:=\mathbb{R}/\mathbb{Z}$ be the circle group, let $\mu_1$ denote its normalized Haar (Lebesgue) probability measure, and for $\theta\in C$ let the rotation $R_\theta:C\to C$ be given by $R_\theta(x)=x+\theta\pmod 1$. Fix integers $r,t\in\mathbb{N}$ with $r,t\ge 2$. Which statement holds?

There exists a measurable partition $C=\bigcup_{i=1}^r E_i$ such that for every measurable cover $C\subset F_1\cup\cdots\cup F_t$, there exist an index $m\in\{1,\dots,t\}$ and some rotation $R_\theta$ with $\mu_1\big(R_\theta(F_m)\cap E_i\big)>0$ for every $1\le i\le r$.

Choices:
B. There exists a measurable partition $C=\bigcup_{i=1}^r E_i$ such that for every measurable cover $C\subset F_1\cup\cdots\cup F_t$, there exists a single rotation $R_\theta$ for which, for every $m\in\{1,\dots,t\}$, one has $\mu_1\big(R_\theta(F_m)\cap E_i\big)>0$ for every $1\le i\le r$.
C. There exists a measurable partition $C=\bigcup_{i=1}^r E_i$ such that for every measurable cover $C\subset F_1\cup\cdots\cup F_t$, there exist an index $m\in\{1,\dots,t\}$ and some rotation $R_\theta$ with $\mu_1\big(R_\theta(F_m)\cap E_i\big)>0$ for at least one $i\in\{1,\dots,r\}$.
D. For every measurable partition $C=\bigcup_{i=1}^r E_i$, and for every measurable cover $C\subset F_1\cup\cdots\cup F_t$, there exist an index $m\in\{1,\dots,t\}$ and some rotation $R_\theta$ with $\mu_1\big(R_\theta(F_m)\cap E_i\big)>0$ for every $1\le i\le r$.
E. There exists a measurable partition $C=\bigcup_{i=1}^r E_i$ such that for every measurable cover $C\subset F_1\cup\cdots\cup F_t$, there exist an index $m\in\{1,\dots,t\}$ and some rotation $R_\theta$ with $\mu_1\big(R_\theta(F_m)\cap E_i\big)=\mu_1(E_i)$ for every $1\le i\le r$.

existential choice of a single favorable cover element

dropped the requirement that all partition cells be hit positively

positive-measure intersection replaced by full-measure capture of each partition cell


\begin{lemma}[see \cite{lorenz}]\label{lemalorenz}
	Let $a>0$, $\sigma> 1$ and $s\in[0,\frac{3}{2})$. Then, there exists a positive constant $C_{a,\sigma,s}$ such that, for all $f,g\in \dot{H}_{a,\sigma}^{s}(\mathbb{R}^3)$, we have
	$$\|fg\|_{\dot{H}_{a,\sigma}^{s}}\leq C_{a,\sigma,s}	\|f\|_{\dot{H}_{a,\sigma}^{s}}\|g\|_{\dot{H}_{a,\sigma}^{s}}.$$
\end{lemma}

\begin{theorem}\label{teoremaexistenciaB}
Assume that $a>0$, $\sigma>1$, $\alpha>\frac{1}{2}$, $\beta>\frac{1}{2}$ and $0\leq s<\frac{3}{2}$. Let $(u_0,\theta_0)\in \dot{H}_{a,\sigma}^s$ such that $\hbox{\emph{div}}\,u_0=0$. Then,
  there exist an instant $T=T(a,\sigma,s,\alpha,\beta,u_0,\theta_0)>0$ and a unique mild solution $(u,\theta)\in C_T(\dot{H}_{a,\sigma}^s)$ for the Boussinesq equations \emph{(\ref{micropolar})} that satisfies
    \begin{align*}
  &\|(u,\theta)\|_{L^\infty_T(\dot{H}_{a,\sigma}^{s})}
\lesssim \frac{1-T}{T^{1-\frac{1}{2\alpha}}+T^{1-\frac{1}{2\beta}}}.
  \end{align*}
 \end{theorem}

\begin{lemma}[see \cite{pontofixo}]\label{pontofixo}
		Let $(X,\|\cdot\|)$ be a Banach space, $L:X\rightarrow X$ a continuous linear operator and
 $B:X\times X\rightarrow X$ a continuous bilinear operator, i.e., there exist positive constants $C_1$ and $C_2$ such that
 $$\|L(x)\|\leq C_1 \|x\|,\quad \|B(x,y)\|\leq C_2 \|x\|\|y\|,\quad \forall x,y\in X.$$
 Then, for each $C_1\in (0,1)$ and $x_0\in X$ that satisfy $4C_2\|x_0\| <(1-C_1)^2$, one has that the equation
 $$a=x_0+B(a,a)+L(a), \quad a\in X,$$
 admits solution $x\in X$. Moreover, $x$ satisfies the inequality $\|x\|\leq \frac{2\|x_0\|}{1-C_1}$ and this is the only one such that $\|x\|\leq  \frac{1-C_1}{2C_2}.$
	\end{lemma}

\begin{align}\label{lwfinal}
\displaystyle\|L(w,v)\|_{X}\leq  T \|(w,v)\|_{X},\quad\forall (w,v)\in X.
\end{align}

\begin{lemma}[see \cite{robert}]\label{lemanovow1}
Let $a\geq0$, $\sigma\geq1$, $s\in\mathbb{R}$ and $\theta\geq1$. The following inequality holds:
$$\|f\|_{\dot{H}_{a,\sigma}^{s+1}}\leq \|f\|_{\dot{H}_{a,\sigma}^{s}}^{1-\frac{1}{\theta}}\|f\|_{\dot{H}_{a,\sigma}^{s+\theta}}^{\frac{1}{\theta}}.$$
  \end{lemma}

\begin{align}\label{wilber17}
\nonumber\displaystyle\|B_1((w,v),(\gamma,\phi))(t)\|_{\dot{H}_{a,\sigma}^s}&\leq  \int_0^t\Big(\int_{\mathbb{R}^3} |\xi|^{2}e^{-2(t-\tau)|\xi|^{2\alpha}}|\xi|^{2s} e^{2a|\xi|^{\frac{1}{\sigma}}} |\mathcal{F}(\gamma\otimes w)(\xi)|^2\,d\xi\Big)^{\frac{1}{2}}\,d\tau\\
\nonumber&\leq C_{\alpha} \int_0^t (t-\tau)^{-\frac{1}{2\alpha}}\Big(\int_{\mathbb{R}^3} |\xi|^{2s} e^{2a|\xi|^{\frac{1}{\sigma}}} |\mathcal{F}(\gamma\otimes w)(\xi)|^2\,d\xi\Big)^{\frac{1}{2}}\,d\tau\\
&\leq C_{\alpha} \int_0^t (t-\tau)^{-\frac{1}{2\alpha}} \|(\gamma\otimes w)(\tau)\|_{\dot{H}_{a,\sigma}^{s}}\,d\tau.
\end{align}

\begin{align}\label{wilber25}
 \frac{1}{2}\partial_t|\widehat{u}(t)|^2+|\xi|^{2\alpha}|\widehat{ u}|^2 &\leq
|\widehat{u}\cdot \widehat{u\cdot\nabla \displaystyle u}|+|\widehat{u}_3 \widehat{\theta}|.
\end{align}

\begin{align}
\nonumber\displaystyle\|B_1((w,v),(\gamma,\phi))(t)\|_{\dot{H}_{a,\sigma}^s}
&\leq C_{a,\sigma,s,\alpha}\|\gamma\|_{L_T^{\infty}(\dot{H}_{a,\sigma}^{s})} \|w\|_{L_T^{\infty}(\dot{H}_{a,\sigma}^{s})}\int_0^t (t-\tau)^{-\frac{1}{2\alpha}} d\tau\\
\label{w1}&\leq C_{a,\sigma,s,\alpha}T^{1-\frac{1}{2\alpha}}\|(w,v)\|_{X} \|(\gamma,\phi)\|_{X},
\end{align}

\begin{align}\label{wilber19}
\nonumber &I_2=(I_{21}^{(1)}+I_{21}^{(2)},I_{22}):= \Big(\int_{0}^{\kappa_m}[e^{- (\kappa_m-\tau)(-\Delta)^\alpha}-e^{- (\kappa_n-\tau)(-\Delta)^\alpha}] \mathbb{P}[u\cdot \nabla u]\,d\tau+\\ &
\int_{0}^{\kappa_m}[e^{-  (\kappa_n-\tau)(-\Delta)^\alpha}-e^{- (\kappa_m-\tau)(-\Delta)^\alpha}] \mathbb{P}[\theta e_3]\,d\tau,\int_{0}^{\kappa_m}[e^{- (\kappa_m-\tau)(-\Delta)^\beta}-e^{- (\kappa_n-\tau)(-\Delta)^\beta}] (u\cdot \nabla \theta)\,d\tau\Big),
\end{align}

\begin{align}\label{3}
\frac{1}{2}\frac{d}{dt}\|u(t)\|_{\dot{H}_{a,\sigma}^s}^2+\|u(t)\|_{\dot{H}_{a,\sigma}^{s+\alpha}}^2 &\leq
|\langle u, u\cdot\nabla u\rangle_{\dot{H}_{a,\sigma}^s}|+  |\langle u_3,\theta\rangle_{\dot{H}_{a,\sigma}^s} |.
\end{align}

\hspace{0.5cm}
In this paper, we consider the following three-dimensional Boussinesq equations:
\begin{equation}\label{micropolar}
\left\{
\begin{array}{l}
u_t
\;\!+\,
u \cdot \nabla u
\,+\,
\nabla \;\!p \:\!\,+\,  \:\!(-\Delta)^\alpha u
\;=\;
\theta e_3, \quad x\in \mathbb{R}^3, \quad t > 0,\\
\theta_t
\;\!+\,
u \cdot \nabla \theta\,+\, \,(-\Delta)^\beta \theta
\;=\;
0, \quad x\in \mathbb{R}^3, \quad t > 0,\\
\mbox{div}\:u  \;=\; 0, \quad x\in \mathbb{R}^3, \quad t > 0,\\
u(x,0) \,=\, u_0(x),
\;\,
\theta(x,0) \,=\, \theta_0(x), \quad x\in \mathbb{R}^3,
\end{array}
\right.
\end{equation}
where $u(x,t) = \left(u_{1}(x,t), u_{2}(x,t), u_{3}(x,t)\right) \in \mathbb{R}^3$ denotes the incompressible velocity field, and
$\theta(x,t) \in \mathbb{R}$ represents the temperature of the fluid. The parameters $\alpha$ and $\beta$ belong to
$(\tfrac{1}{2},\infty)$, and we set $e_{3} = (0,0,1)$. The initial velocity field $u_{0}$ in \eqref{micropolar} is assumed to be divergence-free, that is,
$$
\operatorname{div} u_{0} = 0,
$$
and we denote by $(-\Delta)^{\gamma}$ the fractional Laplacian, defined for a suitable function $f$ by
$$
\mathcal{F}[(-\Delta)^{\gamma} f](\xi) = |\xi|^{2\gamma} \widehat{f}(\xi) .
$$

Observe that, in the case $\theta \equiv 0$ and $\alpha = 1$, the system \eqref{micropolar} reduces to the classical incompressible Navier-Stokes equations
\begin{equation}\label{NS}
\left\{
\begin{array}{l}
u_t
\;\!+\,
u \cdot \nabla u
\,+\,
\nabla \;\!p
\;=\;
\Delta u, \quad x\in \mathbb{R}^3, \quad t > 0,\\

\mbox{div}\:u  \;=\; 0, \quad x\in \mathbb{R}^3, \quad t > 0.

\end{array}
\right.
\end{equation}

The Boussinesq equations (\ref{micropolar}) are simplified yet powerful models widely used in the study of oceanic and atmospheric dynamics. By incorporating buoyancy effects while filtering out sound waves, they provide a mathematically tractable framework for describing stratified fluid flows under the influence of gravity. These equations also arise in several other areas of Physics, including thermal convection, geophysical fluid dynamics, and plasma modeling; see, for instance, \cite{MR2245751,MR2540168} for more details.

It is worth noting, however, that although \eqref{micropolar} with fractional dissipation may initially appear to be a purely mathematical generalization, there are geophysical scenarios in which the Boussinesq equations  with a fractional Laplacian (\ref{micropolar}) naturally arise. A typical example occurs in the middle atmosphere, where upward-travelling flows experience changes due to variations in atmospheric properties, even though the incompressibility and Boussinesq approximations remain valid. In this regime, the effects of kinematic and thermal diffusion are attenuated by the thinning of the atmosphere, an anomalous behavior that can be effectively modeled using a spatial fractional Laplacian. See \cite{MR2379269, Gill1982} for further discussions.

There is an extensive literature, in both two and three dimensions, concerning the well-posedness and qualitative properties of the system \eqref{micropolar} in a variety of functional settings (see, for instance, \cite{MR4884564,MR3503190,MR4795499,MR3808344,MR3008326,MR3759571,cilon1} and the references therein).

In this work, we are concerned with the time evolution of solutions. More precisely, we aim to establish finite-time blow-up criteria for mild solutions of \eqref{micropolar} in Sobolev-Gevrey type spaces. For $a \ge 0$, $\sigma \ge 1$, and $s \in \mathbb{R}$, the (homogeneous) Sobolev-Gevrey space is defined by
$$
\dot{H}_{a,\sigma}^s=\dot{H}_{a,\sigma}^s(\mathbb{R}^3)
    = \left\{ f \in \mathcal{S}' : \widehat{f} \in L^1_{\operatorname{loc}}\,\, \text{and} \,\, \
       \|f\|_{\dot{H}_{a,\sigma}^s}:=\Big[\int_{\mathbb{R}^3} |\xi|^{2s} e^{2a |\xi|^{1/\sigma}} |\widehat{f}(\xi)|^2\, d\xi\Big]^{\frac{1}{2}} < \infty
      \right\},
$$
where $\widehat{f}$ denotes the Fourier transform
$$
\mathcal{F}(f)(\xi)=\widehat{f}(\xi):=\int_{\mathbb{R}^3}e^{-i\xi\cdot x}f(x)\,dx.
$$
Essentially, $\dot{H}_{a,\sigma}^s \equiv L^{2}\!(|\xi|^{s} e^{a|\xi|^{1/\sigma}}\, d\xi)$ and, in particular, $\dot{H}_{0,\sigma}^s \equiv \dot{H}^s$, \textit{i.e.}, the classical (homogeneous) Sobolev space.  This class of functions plays a crucial role because, according to  Paley--Wiener Theorem (see \cite{HormanderALPDO1}, Chapter~7, for further details), for $\sigma=1$ a function $ f $ belongs to the space $ \dot{H}_{a,1}^s(\mathbb{R}^d) $ if, and only if, it admits a holomorphic extension $ F $ to the strip

$$
S_a = \{ x + i y  \in \mathbb{C}^d: x, y \in \mathbb{R}^d,\ |y| < a \},
$$
such that
$$
    \sup_{|y| < a} \| F(x + i y) \|_{\dot{H}^s} < \infty.
$$
In other words, the parameter $ a \geq 0 $ determines the width of the complex strip to which
functions in $ \dot{H}_{a,1}^s(\mathbb{R}^d) $ can be analytically extended. For $\sigma>1$, the regularity falls into a non-analytic Gevrey regime, meaning that derivatives grow in a controlled but non-analytic manner, which is equivalently reflected by a subexponential decay at high frequencies.

Consequently, the study of
differential equations in such analyticity-based function classes has attracted significant attention in recent years. See, for instance, \cite{MR3504420, MR4632081, MR4369830, MR2169876, MR1026858, MR2265624} and references therein for more details on this topic.

The first main result of this work establishes the local well-posedness of the Cauchy problem \eqref{micropolar} in  Sobolev-Gevrey class. Recall that, we say that $(u,\theta)=(u,\theta)(x,t)$ is a mild solution to the Boussinesq equations (\ref{micropolar}) if this application satisfies the associated integral formulations (\ref{wilber15}) and (\ref{wilber16}), which are established via the fractional heat semigroup.

\hspace{0.5cm}
In this paper, we consider the following three-dimensional Boussinesq equations:
\begin{equation}\label{micropolar}
\left\{
\begin{array}{l}
u_t
\;\!+\,
u \cdot \nabla u
\,+\,
\nabla \;\!p \:\!\,+\, \:\!(-\Delta)^\alpha u
\;=\;
\theta e_3, \quad x\in \mathbb{R}^3, \quad t > 0,\\
\theta_t
\;\!+\,
u \cdot \nabla \theta\,+\, \,(-\Delta)^\beta \theta
\;=\;
0, \quad x\in \mathbb{R}^3, \quad t > 0,\\
\mbox{div}\:u \;=\; 0, \quad x\in \mathbb{R}^3, \quad t > 0,\\
u(x,0) \,=\, u_0(x),
\;\,
\theta(x,0) \,=\, \theta_0(x), \quad x\in \mathbb{R}^3,
\end{array}
\right.
\end{equation}
where $u(x,t) = \left(u_{1}(x,t), u_{2}(x,t), u_{3}(x,t)\right) \in \mathbb{R}^3$ denotes the incompressible velocity field, and
$\theta(x,t) \in \mathbb{R}$ represents the temperature of the fluid. The parameters $\alpha$ and $\beta$ belong to
$(\tfrac{1}{2},\infty)$, and we set $e_{3} = (0,0,1)$. The initial velocity field $u_{0}$ in \eqref{micropolar} is assumed to be divergence-free, that is,
$$
\operatorname{div} u_{0} = 0,
$$
and we denote by $(-\Delta)^{\gamma}$ the fractional Laplacian, defined for a suitable function $f$ by
$$
\mathcal{F}[(-\Delta)^{\gamma} f](\xi) = |\xi|^{2\gamma} \widehat{f}(\xi) .
$$

Observe that, in the case $\theta \equiv 0$ and $\alpha = 1$, the system \eqref{micropolar} reduces to the classical incompressible Navier-Stokes equations
\begin{equation}\label{NS}
\left\{
\begin{array}{l}
u_t
\;\!+\,
u \cdot \nabla u
\,+\,
\nabla \;\!p
\;=\;
\Delta u, \quad x\in \mathbb{R}^3, \quad t > 0,\\

In this work, we are concerned with the time evolution of solutions. More precisely, we aim to establish finite-time blow-up criteria for mild solutions of \eqref{micropolar} in Sobolev-Gevrey type spaces. For $a \ge 0$, $\sigma \ge 1$, and $s \in \mathbb{R}$, the (homogeneous) Sobolev-Gevrey space is defined by
$$
\dot{H}_{a,\sigma}^s=\dot{H}_{a,\sigma}^s(\mathbb{R}^3)
 = \left\{ f \in \mathcal{S}' : \widehat{f} \in L^1_{\operatorname{loc}}\,\, \text{and} \,\, \
 \|f\|_{\dot{H}_{a,\sigma}^s}:=\Big[\int_{\mathbb{R}^3} |\xi|^{2s} e^{2a |\xi|^{1/\sigma}} |\widehat{f}(\xi)|^2\, d\xi\Big]^{\frac{1}{2}} < \infty
 \right\},
$$
where $\widehat{f}$ denotes the Fourier transform
$$
\mathcal{F}(f)(\xi)=\widehat{f}(\xi):=\int_{\mathbb{R}^3}e^{-i\xi\cdot x}f(x)\,dx.
$$
Essentially, $\dot{H}_{a,\sigma}^s \equiv L^{2}\!(|\xi|^{s} e^{a|\xi|^{1/\sigma}}\, d\xi)$ and, in particular, $\dot{H}_{0,\sigma}^s \equiv \dot{H}^s$, \textit{i.e.}, the classical (homogeneous) Sobolev space. This class of functions plays a crucial role because, according to Paley--Wiener Theorem (see \cite{HormanderALPDO1}, Chapter~7, for further details), for $\sigma=1$ a function $ f $ belongs to the space $ \dot{H}_{a,1}^s(\mathbb{R}^d) $ if, and only if, it admits a holomorphic extension $ F $ to the strip

$$
S_a = \{ x + i y \in \mathbb{C}^d: x, y \in \mathbb{R}^d,\ |y| < a \},
$$
such that
$$
 \sup_{|y| < a} \| F(x + i y) \|_{\dot{H}^s} < \infty.
$$
In other words, the parameter $ a \geq 0 $ determines the width of the complex strip to which
functions in $ \dot{H}_{a,1}^s(\mathbb{R}^d) $ can be analytically extended. For $\sigma>1$, the regularity falls into a non-analytic Gevrey regime, meaning that derivatives grow in a controlled but non-analytic manner, which is equivalently reflected by a subexponential decay at high frequencies.

Consequently, the study of
differential equations in such analyticity-based function classes has attracted significant attention in recent years. See, for instance, \cite{MR3504420, MR4632081, MR4369830, MR2169876, MR1026858, MR2265624} and references therein for more details on this topic.

The first main result of this work establishes the local well-posedness of the Cauchy problem \eqref{micropolar} in Sobolev-Gevrey class. Recall that, we say that $(u,\theta)=(u,\theta)(x,t)$ is a mild solution to the Boussinesq equations (\ref{micropolar}) if this application satisfies the associated integral formulations (\ref{wilber15}) and (\ref{wilber16}), which are established via the fractional heat semigroup.

\begin{equation}\label{micropolar}
\left\{
\begin{array}{l}
u_t
\;\!+\,
u \cdot \nabla u
\,+\,
\nabla \;\!p \:\!\,+\,  \:\!(-\Delta)^\alpha u
\;=\;
\theta e_3, \quad x\in \mathbb{R}^3, \quad t > 0,\\
\theta_t
\;\!+\,
u \cdot \nabla \theta\,+\, \,(-\Delta)^\beta \theta
\;=\;
0, \quad x\in \mathbb{R}^3, \quad t > 0,\\
\mbox{div}\:u  \;=\; 0, \quad x\in \mathbb{R}^3, \quad t > 0,\\
u(x,0) \,=\, u_0(x),
\;\,
\theta(x,0) \,=\, \theta_0(x), \quad x\in \mathbb{R}^3,
\end{array}
\right.
\end{equation}

\begin{align}\label{wilber15}
u(t)= e^{- t(-\Delta)^{\alpha}}u_0 - \int_{0}^te^{- (t-\tau)(-\Delta)^{\alpha}} \mathbb{P}(u\cdot\nabla u)\,d\tau + \int_{0}^te^{- (t-\tau)(-\Delta)^{\alpha}} \mathbb{P}(\theta e_3)\,d\tau,\quad\forall t>0.
\end{align}

\begin{align}\label{wilber16}
\theta(t)= e^{- t(-\Delta)^{\beta}}\theta_0 - \int_{0}^te^{-  (t-\tau)(-\Delta)^{\beta}} [u\cdot \nabla\theta]\,d\tau ,\quad\forall t>0.
\end{align}

The first main result of this work establishes the local well-posedness of the Cauchy problem \eqref{micropolar} in Sobolev-Gevrey class. Recall that, we say that $(u,\theta)=(u,\theta)(x,t)$ is a mild solution to the Boussinesq equations (\ref{micropolar}) if this application satisfies the associated integral formulations (\ref{wilber15}) and (\ref{wilber16}), which are established via the fractional heat semigroup.

Next, we investigate the qualitative behavior of the local solution provided by Theorem \ref{teoremaexistenciaB}, assuming that its maximal lifespan is finite.

First of all, apply Helmholtz's projector $\mathbb{P}$, defined via Fourier transform (see \cite{PN} for more details)
$$
\mathcal{F}[\mathbb{P}f](\xi) := \widehat{f}(\xi) - \frac{\widehat{f}(\xi)\cdot \xi}{|\xi|^2}\, \xi,
$$
the heat semigroup $e^{-(t-\tau)(-\Delta)^{\alpha}}$ to the first equation in (\ref{micropolar}), and integrate the result obtained over $[0,t]$ to obtain
\begin{align}\label{wilber15}
u(t)= e^{- t(-\Delta)^{\alpha}}u_0 - \int_{0}^te^{- (t-\tau)(-\Delta)^{\alpha}} \mathbb{P}(u\cdot\nabla u)\,d\tau + \int_{0}^te^{- (t-\tau)(-\Delta)^{\alpha}} \mathbb{P}(\theta e_3)\,d\tau,\quad\forall t>0.
\end{align}
Similarly, by using the heat semigroup $e^{- (t-\tau)(-\Delta)^{\beta}}$ in the second equation of (\ref{micropolar}), and integrating over $[0,t]$, we deduce
\begin{align}\label{wilber16}
\theta(t)= e^{- t(-\Delta)^{\beta}}\theta_0 - \int_{0}^te^{- (t-\tau)(-\Delta)^{\beta}} [u\cdot \nabla\theta]\,d\tau ,\quad\forall t>0.
\end{align}
Hence, we are able to write the following equality:
\begin{align}\label{estimativaprojecao3}
(u,\theta)(t)&= (e^{- t(-\Delta)^{\alpha}}u_0,e^{- t (-\Delta)^{\beta}}\theta_0) + B((u,\theta),(u,\theta))(t)+L(u,\theta)(t),\quad\forall t>0.
\end{align}
where
\begin{align}\label{bilinearB}
B((w,v),(\gamma,\phi))(t)=(B_1((w,v),(\gamma,\phi)),B_2((w,v),(\gamma,\phi)))(t),\quad\forall t>0,
\end{align}
and also
\begin{align}\label{linearL}
L(w,v)(t)=(L_1(w,v)(t),L_2(w,v)(t)),\quad\forall t>0,
\end{align}
with
\begin{align}\label{bilinearB1}
B_1((w,v),(\gamma,\phi))(t)= - \int_{0}^te^{- (t-\tau)(-\Delta)^{\alpha}} \mathbb{P}(w\cdot\nabla \gamma)\,d\tau,
\end{align}
\begin{align}\label{bilinearB2}
B_2((w,v),(\gamma,\phi))(t)= - \int_{0}^te^{- (t-\tau)(-\Delta)^{\beta}} [w\cdot \nabla \phi]\,d\tau,
\end{align}
\begin{align}\label{linearL1eL2}
L_1(w,v)(t)= \int_{0}^te^{- (t-\tau)(-\Delta)^{\alpha}} \mathbb{P}[ve_3]\,d\tau \quad \hbox{ and }\quad L_2(w,v)(t)= 0,
\end{align}
for all $w,v,\gamma,\phi\in C_T(\dot{H}_{a,\sigma}^s)$ ($T>0$ will be revealed below). Denote $X= [C_T (\dot{H}_{a,\sigma}^{s})]^3$ $\times C_T (\dot{H}_{a,\sigma}^{s}) (\equiv C_T (\dot{H}_{a,\sigma}^{s})$ throughout this work)$\footnote{Here, $\|(f,g)\|_{X}:=[\|f\|^2_{L^\infty_T(\dot{H}_{a,\sigma}^s)}+\|g\|^2_{L^\infty_T(\dot{H}_{a,\sigma}^s)}]^{\frac{1}{2}}$}$ and notice that $B$ and $L$ are bilinear and linear operators on $X\times X$ and $X$, respectively.

At last, notice that
\begin{align}\label{mudanca1}
\displaystyle\|e^{- t(-\Delta)^{\alpha}} u_0\|^2_{\dot{H}_{a,\sigma}^s}&=\int_{\mathbb{R}^3} e^{-2 t|\xi|^{2\alpha}}|\xi|^{2s} e^{2a|\xi|^{\frac{1}{\sigma}}} |\widehat{u}_0(\xi)|^2\,d\xi\leq \|u_0\|_{\dot{H}_{a,\sigma}^{s}}^2,\quad\forall t\in [0,T],
\end{align}
and also that
\begin{align}\label{mudanca2}
\displaystyle\|e^{- t(-\Delta)^{\beta}} \theta_0\|_{\dot{H}_{a,\sigma}^s}\leq \|\theta_0\|_{\dot{H}_{a,\sigma}^{s}},\quad\forall t\in [0,T].
\end{align}
From (\ref{mudanca1}) and (\ref{mudanca2}), it follows that
\begin{align}\label{dadoincial}
\|(e^{-t(-\Delta)^{\alpha}}u_0,e^{-t (-\Delta)^{\beta}}\theta_0)\|_{X}\leq\|(u_0,\theta_0)\|_{\dot{H}_{a,\sigma}^{s}}.
\end{align}
Thus, we are able to determine $T>0$ as follows:
$$T< \min\Big\{\Big[\sqrt{8C_{a,\sigma,s,\alpha,\beta}\|(u_0,\theta_0)\|_{\dot{H}_{a,\sigma}^{s}}}+1\Big]^{-\frac{4\alpha}{2\alpha-1}},\Big[\sqrt{8C_{a,\sigma,s,\alpha,\beta}\|(u_0,\theta_0)\|_{\dot{H}_{a,\sigma}^{s}}}+1\Big]^{-\frac{4\beta}{2\beta-1}}\Big\},$$
where $C_{a,\sigma,s,\alpha,\beta}$ is established in (\ref{w5}). Therefore, by Lemma \ref{pontofixo}, there exists a unique solution $(u,\theta)\in X$ for the equation (\ref{estimativaprojecao3}) (it is enough to take a look at (\ref{lwfinal}), (\ref{w5}) and (\ref{dadoincial})) that satisfies the following inequality:
$$\|(u,\theta)\|_{X}\leq \frac{1-T}{2C_{a,\sigma,s,\alpha,\beta}[T^{1-\frac{1}{2\alpha}}+T^{1-\frac{1}{2\beta}}]}.$$

Now, let us assume that $\displaystyle \limsup_{t\nearrow T^*} \|(u,\theta)(t)\|_{\dot{H}_{a,\sigma}^{s}} < \infty$. Thus, apply Theorem \ref{teoremaexistenciaB} to obtain a positive constant $C_{a,\sigma,s}$ such that
\begin{align}\label{limitacao}
\displaystyle \|(u,\theta)(t)\|_{\dot{H}_{a,\sigma}^{s}} \leq C_{a,\sigma,s},\quad\forall\,t\in[0,T^*).
\end{align}
Thereby, by integrating over $[0,t]$ the inequality (\ref{esqueci}) and using (\ref{limitacao}), we can write the following inequality:
\begin{align*}
&\nonumber\|(u,\theta)(t)\|_{\dot{H}_{a,\sigma}^s}^2+\int_0^t\|u(\tau)\|_{\dot{H}_{a,\sigma}^{s+\alpha}}^2+\int_0^t\|\theta(\tau)\|_{\dot{H}_{a,\sigma}^{s+\beta}}^2\,d\tau \leq \|(u_0,\theta_0)\|_{\dot{H}_{a,\sigma}^s}^2+C_{a,\sigma,s,\alpha,\beta}T^*,
\end{align*}
for all $t\in[0,T^*).$ As a result, particularly, one has
\begin{align}\label{e4}
\int_0^t\|u(\tau)\|_{\dot{H}_{a,\sigma}^{s+\alpha}}^2d\tau+\int_0^t\|\theta(\tau)\|_{\dot{H}_{a,\sigma}^{s+\beta}}^2\,d\tau \leq C_{a,\sigma,s,\alpha,\beta,u_0,\theta_0,T^*},\quad\forall\,t\in[0,T^*).
\end{align}

On the other hand, take $L^2$-inner product of the Boussinesq equations (\ref{micropolar}), with $u$ and $\theta$, respectively, and integrate the results obtained over $[0,t]$ in order to be capable of writing the following inequality:
\begin{align}\label{wilber32}
\frac{1}{2}\frac{d}{dt}\|(u,\theta)(t)\|_{L^2}^2+\|(-\Delta)^{\frac{\alpha}{2}}u(t)\|_{L^2}^2+\|(-\Delta)^{\frac{\beta}{2}}\theta(t)\|_{L^2}^2 &\leq \|(u,\theta)(t)\|_{L^2}^{2},\quad\forall t\in [0,T^*).
\end{align}
By Gronwall's Lemma, we have
\begin{align}\label{normal2}
 \|(u,\theta)(t)\|_{L^2}\leq e^{T^*}\|(u_0,\theta_0)\|_{L^2},\quad\forall t\in [0,T^*).
\end{align}
Thereby, by using (\ref{wilber12}) and (\ref{normal2}), one infers
\begin{align*}
\frac{C_{s,\alpha,u_0,\theta_0,T^*}}{[e^{C_\alpha(T^*-t)}-1]^{\frac{2s(2\alpha-1)}{3\alpha}}}\left(\frac{C'_{\sigma,s,\alpha,u_0,\theta_0,T^*}}{[e^{C_\alpha(T^*-t)}-1]^{\frac{2\alpha-1}{3\alpha\sigma}}}\right)^k \leq \|(u,\theta)(t)\|_{\dot{H}^{s+\frac{k}{2\sigma}}}^2,\quad\forall t\in [0,T^*).
\end{align*}
As a consequence, we have the inequality below:
\begin{align}\label{wnovo1}
\frac{C_{s,\alpha,u_0,\theta_0,T^*}}{[e^{C_\alpha(T^*-t)}-1]^{\frac{2s(2\alpha-1)}{3\alpha}}}\frac{\left(\frac{2aC'_{\sigma,s,\alpha,u_0,\theta_0,T^*}}{[e^{C_\alpha(T^*-t)}-1]^{\frac{2\alpha-1}{3\alpha\sigma}}}\right)^k }{k!} &\leq \int_{\mathbb{R}^3} \frac{(2a|\xi|^{\frac{1}{\sigma}})^k}{k!}|\xi|^{2s}|(\widehat{u},\widehat{\theta})(t)|^{2}\,d\xi,\quad\forall t\in [0,T^*).
\end{align}
Hence, by using Monotone Convergence Theorem in the inequality (\ref{wnovo1}), we deduce
\begin{align}\label{wilber14}
\nonumber\frac{C_{s,\alpha,u_0,\theta_0,T^*}}{[e^{C_\alpha(T^*-t)}-1]^{\frac{2s(2\alpha-1)}{3\alpha}}}\sum_{k=2\sigma_0+1}^{\infty}\frac{\left(\frac{2aC'_{\sigma,s,\alpha,u_0,\theta_0,T^*}}{[e^{C_\alpha(T^*-t)}-1]^{\frac{2\alpha-1}{3\alpha\sigma}}}\right)^k }{k!} &\leq \int_{\mathbb{R}^3} \sum_{k=2\sigma_0+1}^{\infty}\frac{(2a|\xi|^{\frac{1}{\sigma}})^k}{k!}|\xi|^{2s}|(\widehat{u},\widehat{\theta})(t)|^{2}\,d\xi\\
&\leq \|(u,\theta)(t)\|_{\dot{H}_{a,\sigma}^s}^2,
\end{align}
for all $t\in [0,T^*)$, where $2\sigma_0$ is the integer part of $2\sigma\mu$.

The proof of this result relies on a fixed-point argument. However, for both the classical and the fractional Boussinesq equations, a major difficulty arises in handling the coupling terms $\theta e_{3}$ and $(u \cdot \nabla)\theta$, which demand delicate estimates within the chosen functional framework.

Next, we investigate the qualitative behavior of the local solution provided by Theorem \ref{teoremaexistenciaB}, assuming that its maximal lifespan is finite.

\begin{theorem}\label{teoremaB}
Assume that $a>0$, $\sigma>1$, $\alpha\geq 1, \beta \geq 1, 0\leq s<\frac{3}{2}$ and $n\in \mathbb{N}$. Let $(u_0,\theta_0)\in \dot{H}_{a,\sigma}^s$ such that $\hbox{\emph{div}}\,u_0=0$. Consider that $(u,\theta)\in C([0,T^*);\dot{H}_{a,\sigma}^s)$ is the  solution
for the Boussinesq equations \emph{(\ref{micropolar})} in the maximal time interval $0\leq t < T^*$  given in Theorem \emph{\ref{teoremaexistenciaB}}. If $T^*<\infty$, then the following statements hold:
      \begin{enumerate}
  \item[\textbf{\emph{i)}}]  $\displaystyle \limsup_{t\nearrow T^*} \|(u,\theta)(t)\|_{\dot{H}_{\frac{a}{(\sqrt{\sigma})^{(n-1)}},\sigma}^s}=+\infty$;
  \item[\textbf{\emph{ii)}}] $\displaystyle\int_{t}^{T^*} [\|e^{\frac{a}{\sigma(\sqrt{\sigma})^{(n-1)}}|\cdot|^{\frac{1}{\sigma}}}(\widehat{u},\widehat{\theta})(\tau)\|_{L^1}
^{\frac{2\alpha}{2\alpha-1}}+\|e^{\frac{a}{\sigma(\sqrt{\sigma})^{(n-1)}}|\cdot|^{\frac{1}{\sigma}}}(\widehat{u},\widehat{\theta})(\tau)\|_{L^1}^{\frac{2\beta}{2\beta-1}}]\,d\tau=\infty$;
 \item[\textbf{\emph{iii)}}]  $\displaystyle \|e^{\frac{a}{\sigma(\sqrt{\sigma})^{(n-1)}}|\cdot|^{\frac{1}{\sigma}}}(\widehat{u},\widehat{\theta})(t)\|_{L^1}^{\frac{2\alpha}{2\alpha-1}}
    +\|e^{\frac{a}{\sigma(\sqrt{\sigma})^{(n-1)}}|\cdot|^{\frac{1}{\sigma}}}(\widehat{u},\widehat{\theta})(t)\|_{L^1}^{\frac{2\beta}{2\beta-1}}\geq  (e^{C(T^*-t)}-1)^{-1};$
  \item [\textbf{\emph{iv)}}] $\displaystyle \|(u,\theta)(t)\|_{\dot{H}_{\frac{a}{(\sqrt{\sigma})^n},\sigma}^s}^{\frac{2\alpha}{2\alpha-1}}+
\|(u,\theta)(t)\|_{\dot{H}_{\frac{a}{(\sqrt{\sigma})^n},\sigma}^s}^{\frac{2\beta}{2\beta-1}}\gtrsim(e^{C(T^{*}-t)}-1)^{-1}.$
  \end{enumerate}
for all $t\in [0,T^{*})$ and $C>0$ is a positive constant.
\end{theorem}

 As a corollary, we obtain that the local solutions provided by Theorem \ref{teoremaexistenciaB} exhibit exponential growth, which in turn ensures finite-time blow-up with an explicit exponential rate.

\begin{corollary}\label{corolario}
Assume that $a>0$, $\sigma>1$, $\alpha=\beta\geq 1 \hbox{  and  } 0\leq s<\frac{3}{2}$. Let $(u_0,\theta_0)\in \dot{H}_{a,\sigma}^s\cap L^2$ such that $\hbox{\emph{div}}\,u_0=0$. Consider that $(u,\theta)\in C([0,T^*);\dot{H}_{a,\sigma}^s)$ is the  solution
for the Boussinesq equations \emph{(\ref{micropolar})} in the maximal time interval $0\leq t < T^*$  given in Theorem \emph{\ref{teoremaexistenciaB}}. If $T^*<\infty$, then the following statement holds:
   $$
   \displaystyle
\|(u,\theta)(t)\|_{\dot{H}_{a,\sigma}^s} \gtrsim  {(e^{C_1(T^*-t)}-1)^{\varrho_1}}
\exp\{C_2(e^{C_1(T^*-t)}-1)^{\varrho_2}\},
$$
for all \(t \in [0, T^{*})\), where $\varrho_1 := \frac{(1 - 2\alpha)\,[\,2(s\sigma + \sigma_{0}) + 1\,]}{6\alpha\sigma}<0$, $\varrho_2:=\frac{1-2\alpha}{3\alpha\sigma}<0$ and \(2\sigma_{0}\) denotes the integer part of \(2\sigma\mu\) \emph{(}with \(\mu > \tfrac{3}{2}\)\emph{)}.
Here, $C_1=C_1(\alpha)$ and $C_2=C_2(a,\alpha,\sigma,s,u_0,\theta_0,T^*)$ are constants. In particular
$$
\lim_{t \nearrow T^*}\|(u,\theta)(t)\|_{\dot{H}_{a,\sigma}^s} = +\infty,
$$
exponentially.
\end{corollary}

\bigskip

\begin{remark}
Let us point out  relevant  improvements presented by our main  results.
\begin{enumerate}
  \item \underline{Solution spaces}: In this text, we have approached Gevrey classes as solution spaces. Thus, we have decided to compare our contributions to some results established by R.~Guterres, W.~G. Melo, N.~F. Rocha and T.~S.~R. Santos \cite{robert} (see also \cite{wilberr2,wilberr3,wilberr4,wilberr5,wilberr6,wilberr7,MR3504420,MR4632081,MR2169876,MR4369830} and references therein), which ones are similar to ours. First of all, Theorem \ref{teoremaexistenciaB} presents weaker regularities for the mild solutions than Theorem $1.1$ obtained in \cite{robert} (check also Theorem $1.4$ in \cite{robert}, which is related to Theorem \ref{teoremaB} and Corollary \ref{corolario}). This fact occurs because we have applied Lemma \ref{lemalorenz} instead of Lemma $2.3$  presented in \cite{robert}, and have worked in a slightly different way as well  (see (\ref{wilber17}) and (\ref{w1}) below for more details). However, this choice led us to not consider the usual homogenous Sobolev spaces (see the proof of Lemma $2.9$ in \cite{lorenz}). In addition, at last, it is worth to notice that $s<\frac{3}{2}$ is our assumption due to the completeness of Sobolev-Gevrey space (see also Lemma \ref{lemalorenz}) and, furthermore, $\alpha,\beta\geq 1$ in Theorem \ref{teoremaB} and Corollary \ref{corolario} because of the use of Lemma \ref{lemanovow1} (it is a technical issue).
  \item \underline{Boussinesq equations}: The term $\theta e_3$ given in (\ref{micropolar}) is the main mathematical difference between the Boussinesq equations and the generalized MHD equations studied by \cite{robert} (and, consequently, the Navier-Stokes equations (\ref{NS})) in this paper. Because of this term, we have made the use of a different fixed point theorem (see Lemma \ref{pontofixo} and Lemma $2.1$ in \cite{lorenz}). More specifically, we were supposed to add new equalities and estimates  in order to obtain our conclusions (see, for example, (\ref{linearL}), (\ref{linearL1eL2})--(\ref{lwfinal}), (\ref{3}), (\ref{esqueci}), (\ref{wilber19})--(\ref{wilber23}), (\ref{wilber4}), (\ref{wilber24}), (\ref{wilber25}), (\ref{wilber26})--(\ref{wilber9}), (\ref{i)n=1}), (\ref{wilber32}), (\ref{normal2}) and (\ref{wilber33})).
  \item \underline{New lower bounds}: As a consequence of the results discussed in the last item above, we have proved new blow-up criteria for local mild solutions of the Boussinesq equations (\ref{micropolar}) as, for instance, Theorem \ref{teoremaB} \textbf{iii)} and \textbf{iv)}, and Corollary \ref{corolario} (compare these results to Theorem $1.4$ iii), iv) and v) in \cite{robert}). Lastly, it is important to point out that other information has been showed in this work, see Theorem \ref{teoremaexistenciaB} and Theorem \ref{teoremaB} \textbf{i)} and \textbf{ii)}.
\end{enumerate}

\end{remark}

\bigskip

\noindent\textbf{Notations. }For  $(X,\|\cdot\|_{X})$ be a normed space and $I \subset \mathbb{R}$ an interval. We define
$$
C(I;X)=\{f:I\to X \text{ continuous}\}, \qquad
\|f\|_{L^{\infty}(I;X)}=\sup_{t\in I}\|f(t)\|_{X},
$$
and for $T>0$, we  write $C_{T}(X)=C([0,T];X)$. For $1 \le p < \infty$, the Bochner space $L^{p}(I;X)$ is equipped with the norm
$$
\|f\|_{L^{p}(I;X)}=\left(\int_{I}\|f(t)\|_{X}^{p}\,dt\right)^{1/p},
$$
and we denote $L_{T}^{p}(X)=L^{p}([0,T];X)$ with$\footnote{Here, $\|(f,g)\|_{L^1}:=\|f\|_{L^1}+\|g\|_{L^1}$  and $\|(f,g)\|_{\dot{H}_{a,\sigma}^s}:=[\|f\|^{2}_{\dot{H}_{a,\sigma}^s}+\|g\|^{2}_{\dot{H}_{a,\sigma}^s}]^{\frac{1}{2}}.$}$ $1\leq p \leq \infty$. Furthermore, for the functions $f$ and $g=(g_1,g_2,g_3)$, the tensor product is  defined by $f \otimes g := (g_{1}f, g_{2}f, g_{3}f)$.

\bigskip

\noindent\textbf{Organization of the paper.}
The structure of the paper is as follows.
In Section~\ref{secaonotacoes}, we present the auxiliary lemmas and technical tools required for the proofs of our main results.
Section~$3$ is devoted to the proofs of Theorems~\ref{teoremaexistenciaB} and~\ref{teoremaB}, as well as Corollary~\ref{corolario}, each of which is addressed in a separate subsection.

The post-theorem text states that “The proof of this result relies on a fixed-point argument.” It also highlights the main technical obstacle: “a major difficulty arises in handling the coupling terms $\theta e_{3}$ and $(u \cdot \nabla)\theta$, which demand delicate estimates within the chosen functional framework.”

\label{teoremaexistenciaB}
Assume that $a>0$, $\sigma>1$, $\alpha>\frac{1}{2}$, $\beta>\frac{1}{2}$ and $0\leq s<\frac{3}{2}$. Let $(u_0,\theta_0)\in \dot{H}_{a,\sigma}^s$ such that $\hbox{\emph{div}}\,u_0=0$. Then,
  there exist an instant $T=T(a,\sigma,s,\alpha,\beta,u_0,\theta_0)>0$ and a unique mild solution $(u,\theta)\in C_T(\dot{H}_{a,\sigma}^s)$ for the Boussinesq equations
\begin{equation}\label{micropolar}
\left\{
\begin{array}{l}\nu_t
\;\!+\,\nu \cdot \nabla u
\,+\,\nabla \;\!p \:\!\,+\,  \:\!(-\Delta)^\alpha u
\;=\;\ntheta e_3, \quad x\in \mathbb{R}^3, \quad t > 0,\\
\theta_t
\;\!+\,\nu \cdot \nabla \theta\,+\, \,(-\Delta)^\beta \theta
\;=\;\n0, \quad x\in \mathbb{R}^3, \quad t > 0,\\
\mbox{div}\:u  \;=\; 0, \quad x\in \mathbb{R}^3, \quad t > 0,\\\nu(x,0) \,=\, u_0(x),
\;\,\ntheta(x,0) \,=\, \theta_0(x), \quad x\in \mathbb{R}^3,
\end{array}
\right.
\end{equation}
that satisfies
    \begin{align*}
  &\|(u,\theta)\|_{L^\infty_T(\dot{H}_{a,\sigma}^{s})}
\lesssim \frac{1-T}{T^{1-\frac{1}{2\alpha}}+T^{1-\frac{1}{2\beta}}}.
  \end{align*}

Question:
Let
\[
\dot H_{a,\sigma}^s(\mathbb R^3)=\left\{f\in\mathcal S' : \widehat f\in L^1_{\mathrm{loc}},\ \|f\|_{\dot H_{a,\sigma}^s}:=\left(\int_{\mathbb R^3}|\xi|^{2s}e^{2a|\xi|^{1/\sigma}}|\widehat f(\xi)|^2\,d\xi\right)^{1/2}<\infty\right\},
\]
with the pair \((u_0,\theta_0)\) understood componentwise in this Sobolev-Gevrey class. Consider the three-dimensional Boussinesq system on \(\mathbb R^3\times(0,\infty)\):
\[
\begin{cases}
 u_t+u\cdot\nabla u+\nabla p+(-\Delta)^\alpha u=\theta e_3,\\
 \theta_t+u\cdot\nabla\theta+(-\Delta)^\beta\theta=0,\\
 \operatorname{div}u=0,\\
 u(x,0)=u_0(x),\quad \theta(x,0)=\theta_0(x),
\end{cases}
\]
where \(e_3=(0,0,1)\) and \(\mathcal F[(-\Delta)^\gamma f](\xi)=|\xi|^{2\gamma}\widehat f(\xi)\). A mild solution means a solution satisfying the associated Duhamel integral formulation with the semigroups \(e^{-t(-\Delta)^\alpha}\) and \(e^{-t(-\Delta)^\beta}\). Assume that \(a>0\), \(\sigma>1\), \(\alpha>\tfrac12\), \(\beta>\tfrac12\), \(0\le s<\tfrac32\), and \(\operatorname{div}u_0=0\). Which existence statement holds? (Here \(X\lesssim Y\) means \(X\) is bounded by a constant multiple of \(Y\).)

There exists a time \(T=T(a,\sigma,s,\alpha,\beta,u_0,\theta_0)>0\) and a unique mild solution \((u,\theta)\in C([0,T];\dot H_{a,\sigma}^s)\) of the above Boussinesq Cauchy problem such that
\[
\|(u,\theta)\|_{L_T^\infty(\dot H_{a,\sigma}^s)}\lesssim \frac{1-T}{T^{1-\frac{1}{2\alpha}}+T^{1-\frac{1}{2\beta}}}.
\]

Choices:
B.
There exists a time \(T=T(a,\sigma,s,\alpha,\beta,\|u_0\|_{\dot H_{a,\sigma}^s},\|\theta_0\|_{\dot H_{a,\sigma}^s})>0\) and a unique mild solution \((u,\theta)\in C([0,T];\dot H_{a,\sigma}^s)\) of the above Boussinesq Cauchy problem for every \(s\ge 0\) such that
\[
\|(u,\theta)\|_{L_T^\infty(\dot H_{a,\sigma}^s)}\lesssim \frac{1-T}{T^{1-\frac{1}{2\alpha}}+T^{1-\frac{1}{2\beta}}}.
\]
C. There exists a time \(T=T(a,\sigma,s,\alpha,\beta,u_0,\theta_0)>0\) and a mild solution \((u,\theta)\in C([0,T];\dot H_{a,\sigma}^s)\) of the above Boussinesq Cauchy problem.
D.
There exists a time \(T=T(a,\sigma,s,\alpha,\beta)>0\), depending only on the parameters \(a,\sigma,s,\alpha,\beta\) and not on the initial data, and a unique mild solution \((u,\theta)\in C([0,T];\dot H_{a,\sigma}^s)\) of the above Boussinesq Cauchy problem such that
\[
\|(u,\theta)\|_{L_T^\infty(\dot H_{a,\sigma}^s)}\lesssim \frac{1-T}{T^{1-\frac{1}{2\alpha}}+T^{1-\frac{1}{2\beta}}}.
\]
E.
For every \(T>0\), there exists a unique mild solution \((u,\theta)\in C([0,T];\dot H_{a,\sigma}^s)\) of the above Boussinesq Cauchy problem such that
\[
\|(u,\theta)\|_{L_T^\infty(\dot H_{a,\sigma}^s)}\lesssim \frac{1-T}{T^{1-\frac{1}{2\alpha}}+T^{1-\frac{1}{2\beta}}}.
\]

local-in-time existence replaced by global-in-time existence


\label{thm:main}
    Let $r \in \{2,3,4\}$, let $\W_{\underline a}$ be the fully reduced web basis for $\mathsf{Inv}(V_q^{\underline a})$, and let $\underline{C}$ be a clasp sequence for $\underline{a}$ with weights $\lambda_1,\ldots,\lambda_m$. Then the following are equivalent for $[W]_q \in \W_{\underline a}$:
    \begin{enumerate}
        \item $[W]_q\notin \ker\pi_{\underline C}$,
        \item $W$ is non-convex (see \Cref{def:nonconvex}),
        \item $W$ has no trips that start and end in the same clasp.
    \end{enumerate}
    Moreover, the clasped web invariants for these webs $W$ form a basis for $\mathsf{Inv}\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$.

\begin{definition}\label{def:latticeword}
    Given a sequence of sorted clasps $\underline C = (\underline c_1,\dots, \underline c_m)$, a lattice word $L$ has no \emph{$\underline C$-descents} if $L = L_1\cdots L_m$ with $\type(L_i) = \underline c_i, \forall i$ and each $L_i$ has no descents.
    The set of all lattice words with no $\underline C$-descents is denoted $\mathsf L(\underline C)$, and the set of all balanced lattice words with no $\underline C$-descents is denoted $\BL(\underline C)$.
\end{definition}

\begin{thm}
    \label{thm:main}
    Let $r \in \{2,3,4\}$, let $\W_{\underline a}$ be the fully reduced web basis for $\inv(V_q^{\underline a})$, and let $\underline{C}$ be a clasp sequence for $\underline{a}$ with weights $\lambda_1,\ldots,\lambda_m$. Then the following are equivalent for $[W]_q \in \W_{\underline a}$:
    \begin{enumerate}
        \item $[W]_q\notin \ker\pi_{\underline C}$,
        \item $W$ is non-convex (see \Cref{def:nonconvex}),
        \item $W$ has no trips that start and end in the same clasp.
    \end{enumerate}
    Moreover, the clasped web invariants for these webs $W$ form a basis for $\inv\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$.
\end{thm}

\subsection{Webs and web bases}

Consider a tensor product $V_q^{\underline{a}} = \bigotimes_{i=1}^n V_q(\omega_{a_i})$ of irreducible complex representations of $U_q(\mathfrak{sl}_r)$ indexed by fundamental weights $\omega_1,\ldots, \omega_{r-1}$. \emph{Webs}, introduced by Kuperberg \cite{Kuperberg} as a tool for the computation of quantum link invariants, give a diagrammatic calculus for $\hom_{U_q(\mathfrak{sl}_r)}(V_q^{\underline{a}'},V_q^{\underline{a}''})$ (and its quantum group deformation). By dualizing and moving tensor factors, we may equally well choose to study the space of tensor invariants:
\[
\mathsf{Inv}(V_q^{\underline{a}}) \coloneqq \hom_{U_q(\mathfrak{sl}_r)}(V_q^{\underline{a}},\mathbb{C}(q)).
\]

Webs $W$ representing elements $[W]_q \in \mathsf{Inv}(V_q^{\underline{a}})$ are planar bipartite graphs, whose edges are colored by fundamental weights, and which are embedded in a disk whose boundary vertices $b_1,\ldots,b_n$ are each incident to a single edge, colored $\omega_{a_i}$. For this reason, we call $\underline{a}$ the \emph{boundary conditions} of the web. We follow the conventions (of e.g. \cite{Sikora} and \cite{Fraser-Lam-Le}) that the sum of the indices of the fundamental weights coloring the edges incident to each internal vertex is $r$.

There are in general many relations (see \cite{Cautis-Kamnitzer-Morrison}) between the invariants $[W]_q$ of webs with boundary conditions $\underline{a}$. A \emph{web basis} is a subset of the web invariants forming a basis. Kuperberg gave a $U_q(\mathfrak{sl}_3)$ web basis consisting of the \emph{non-elliptic} webs. Beyond early applications for quantum link invariants \cite{Khovanov}, the non-elliptic basis has also found application to skein modules \cite{Le.Sikora}, dimer models \cite{Douglas-Kenyon-Shi}, and dynamical algebraic combinatorics \cite{Petersen-Pylyavskyy-Rhoades}, among other areas.

Since Kuperberg's work, much effort has gone into constructing higher-rank web bases, with additional special properties. In \cite{GPPSS-sl4}, the first \emph{rotation-invariant} $U_q(\mathfrak{sl}_4)$ web basis was constructed using \emph{hourglass plabic graphs}. This is the basis of \emph{(top) fully reduced} web invariants $\W_{\underline a}$. Fully reduced hourglass plabic graphs also recover Kuperberg's basis for $U_q(\mathfrak{sl}_3)$.

An obvious limitation of the fully reduced web bases is that they are heretofore only known for tensor products of irreducibles indexed by fundamental weights. To extend this basis to arbitrary tensor products of irreducibles, we need to understanding the \emph{clasping} of these webs. Kuperberg gave clasping rules for $U_q(\mathfrak{sl}_3)$, and these clasped webs have appeared, for example, in well-known conjectures of Fomin--Pylyavskyy \cite{Fomin-Pylyavskyy-advances} relating basis webs to cluster algebras.

\subsection{Clasped webs}
\emph{Clasped webs} provide for an extension of the diagrammatic calculus to spaces of morphisms between tensor products of general finite-dimensional (type-$1$) irreducible representations $V_q(\lambda)$.

Fix a partition of $[n]= I_1\sqcup \cdots \sqcup I_m$, with each $I_i$ an interval. This defines the \emph{clasp sequence} $\underline C = (\underline c_1,\dots, \underline c_m)$ where $\underline c_i = (a_j)_{j\in I_i} \in [r-1]^{|I_i|}$ is the $i$-th \emph{clasp}. Oftentimes we will refer to the corresponding set of boundary vertices $\{b_j\ |\ j\in I_i\}$ also as the $i$-th clasp, with $\underline c_i$ recording the tuple of boundary conditions of the vertices in the clasp. The weight of the clasp $\underline c_i$ is defined to be $\mathsf{wt}(\underline c_i) = \sum_{j\in I_i}\omega_{a_j}\in \Lambda^+,$ where $\Lambda^+$ is the set of dominant integral weights for $G=\SL_r(\mathbb{C})$.

Fix a clasp sequence $\underline C = (\underline c_1,\dots, \underline c_m)$ for boundary conditions $\underline a$, and let $\lambda_i = \mathsf{wt}(\underline c_i)$. Recall that for each $i$ there is an inclusion (unique up to scaling) of $U_q(\mathfrak{sl}_r)$-representations $V_q(\lambda_i)\xhookrightarrow{} V_q^{\underline c_i}.$ This induces an inclusion of $U_q(\mathfrak{sl}_r)$-representations
\[
\bigotimes_{i=1}^m V_q(\lambda_i) \xhookrightarrow{} \bigotimes_{i=1}^m V_q^{\underline c_i} = V_q^{\underline a}.
\]
Therefore, we obtain a surjective map of invariant spaces
\[\pi_{\underline C}:\mathsf{Inv}(V_q^{\underline a}) \to \mathsf{Inv}\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right).\]
The image $\pi_{\underline{C}}([W]_q)$ of a web invariant $[W]_q \in \mathsf{Inv}(V_q^{\underline{a}})$ is a \emph{clasped web invariant}.

\subsection{Webs as hourglass plabic graphs}

\emph{Hourglass plabic graphs} are a combinatorial manifestation of webs, introduced by G.--Pechenik--Pfannerer--Striker--Swanson \cite{two-column, GPPSS-sl4}; see \Cref{sec:hpg-background}. In these, edges colored by $\omega_{a}$ are drawn as twisted ``hourglass" edges of $a$ strands. The fundamental combinatorial data associated to an hourglass plabic graph are the \emph{trips}: certain walks along its edges. These trips can be used to characterize the graphs appearing in the fully reduced web basis. As we see in our main result\footnote{The equivalence of (1) and (2) for $r=2,3$ was shown by Kuperberg \cite{Kuperberg}. The equivalence of these with (3) for $r=2,3$ will appear in independent forthcoming work of Catania, Kim, and Pfannerer \cite{catania-kim-pfannerer}.} below, they remarkably also exactly characterize the kernel of the clasping map $\pi_{\underline C}$.

Consider a tensor product $V_q^{\underline{a}} = \bigotimes_{i=1}^n V_q(\omega_{a_i})$ of irreducible complex representations of $U_q(\mathfrak{sl}_r)$ indexed by fundamental weights $\omega_1,\ldots, \omega_{r-1}$. \emph{Webs}, introduced by Kuperberg \cite{Kuperberg} as a tool for the computation of quantum link invariants, give a diagrammatic calculus for $\hom_{U_q(\mathfrak{sl}_r)}(V_q^{\underline{a}'},V_q^{\underline{a}''})$ (and its quantum group deformation). By dualizing and moving tensor factors, we may equally well choose to study the space of tensor invariants:
\[
\mathsf{Inv}(V_q^{\underline{a}}) \coloneqq \hom_{U_q(\mathfrak{sl}_r)}(V_q^{\underline{a}},\mathbb{C}(q)).
\]

Webs $W$ representing elements $[W]_q \in \mathsf{Inv}(V_q^{\underline{a}})$ are planar bipartite graphs, whose edges are colored by fundamental weights, and which are embedded in a disk whose boundary vertices $b_1,\ldots,b_n$ are each incident to a single edge, colored $\omega_{a_i}$. For this reason, we call $\underline{a}$ the \emph{boundary conditions} of the web. We follow the conventions (of e.g. \cite{Sikora} and \cite{Fraser-Lam-Le}) that the sum of the indices of the fundamental weights coloring the edges incident to each internal vertex is $r$.

Fix a partition of $[n]= I_1\sqcup \cdots \sqcup I_m$, with each $I_i$ an interval. This defines the \emph{clasp sequence} $\underline C = (\underline c_1,\dots, \underline c_m)$ where $\underline c_i = (a_j)_{j\in I_i} \in [r-1]^{|I_i|}$ is the $i$-th \emph{clasp}. Oftentimes we will refer to the corresponding set of boundary vertices $\{b_j\ |\ j\in I_i\}$ also as the $i$-th clasp, with $\underline c_i$ recording the tuple of boundary conditions of the vertices in the clasp. The weight of the clasp $\underline c_i$ is defined to be $\mathsf{wt}(\underline c_i) = \sum_{j\in I_i}\omega_{a_j}\in \Lambda^+,$ where $\Lambda^+$ is the set of dominant integral weights for $G=\SL_r(\mathbb{C})$.

Fix a clasp sequence $\underline C = (\underline c_1,\dots, \underline c_m)$ for boundary conditions $\underline a$, and let $\lambda_i = \mathsf{wt}(\underline c_i)$. Recall that for each $i$ there is an inclusion (unique up to scaling) of $U_q(\mathfrak{sl}_r)$-representations $V_q(\lambda_i)\xhookrightarrow{} V_q^{\underline c_i}.$ This induces an inclusion of $U_q(\mathfrak{sl}_r)$-representations
\[
\bigotimes_{i=1}^m V_q(\lambda_i) \xhookrightarrow{} \bigotimes_{i=1}^m V_q^{\underline c_i} = V_q^{\underline a}.
\]
Therefore, we obtain a surjective map of invariant spaces
\[\pi_{\underline C}:\mathsf{Inv}(V_q^{\underline a}) \to \mathsf{Inv}\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right).\]
The image $\pi_{\underline{C}}([W]_q)$ of a web invariant $[W]_q \in \mathsf{Inv}(V_q^{\underline{a}})$ is a \emph{clasped web invariant}.

\subsection{Webs as hourglass plabic graphs}

\emph{Hourglass plabic graphs} are a combinatorial manifestation of webs, introduced by G.--Pechenik--Pfannerer--Striker--Swanson \cite{two-column, GPPSS-sl4}; see \Cref{sec:hpg-background}. In these, edges colored by $\omega_{a}$ are drawn as twisted ``hourglass" edges of $a$ strands. The fundamental combinatorial data associated to an hourglass plabic graph are the \emph{trips}: certain walks along its edges. These trips can be used to characterize the graphs appearing in the fully reduced web basis. As we see in our main result\footnote{The equivalence of (1) and (2) for $r=2,3$ was shown by Kuperberg \cite{Kuperberg}. The equivalence of these with (3) for $r=2,3$ will appear in independent forthcoming work of Catania, Kim, and Pfannerer \cite{catania-kim-pfannerer}.} below, they remarkably also exactly characterize the kernel of the clasping map $\pi_{\underline C}$.

\begin{definition}\label{def:nonconvex}
    A fully reduced hourglass plabic graph $W$ with a clasp sequence $\underline C = (\underline c_1,\dots, \underline c_m)$ is \emph{non-convex} (with respect to $\underline C$) if for each $i\in [m]$ and every $\underline c_i$-cut path $\gamma$, $\weight(\gamma)\ge \weight(\underline c_i)$ in the dominance ordering on $\Lambda^+$. We say that $W$ is \emph{partially convex} if it is not non-convex.
\end{definition}

\emph{Hourglass plabic graphs} are a combinatorial manifestation of webs, introduced by G.--Pechenik--Pfannerer--Striker--Swanson \cite{two-column, GPPSS-sl4}; see \Cref{sec:hpg-background}. In these, edges colored by $\omega_{a}$ are drawn as twisted ``hourglass" edges of $a$ strands. The fundamental combinatorial data associated to an hourglass plabic graph are the \emph{trips}: certain walks along its edges. These trips can be used to characterize the graphs appearing in the fully reduced web basis. As we see in our main result\footnote{The equivalence of (1) and (2) for $r=2,3$ was shown by Kuperberg \cite{Kuperberg}. The equivalence of these with (3) for $r=2,3$ will appear in independent forthcoming work of Catania, Kim, and Pfannerer \cite{catania-kim-pfannerer}.} below, they remarkably also exactly characterize the kernel of the clasping map $\pi_{\underline C}$.

\begin{remark}
 The fact that the web invariants not killed by the projection $\pi_{\underline C}$ form a basis in $\inv\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$ is notable. This property is one of several special properties (also including rotation-invariance) that the fully reduced web bases share with Lusztig's dual canonical basis \cite{Lusztig:canonical}. It was initially hoped that Kuperberg's basis agreed with the dual canonical basis until this was disproven by Khovanov--Kuperberg \cite{Khovanov-Kuperberg}, however the two bases do seem to share many distinguished properties.
\end{remark}

\begin{remark}
 \label{rem:schur}
 For any sequence of weights $\underline \lambda = (\lambda_1,\dots, \lambda_m)$, the Littlewood--Richardson rule implies that the coefficient of $s_\mu$ when $s_{\lambda_1}\cdots s_{\lambda_m}$ when expressed in the Schur polynomial basis is equal to the number of $\underline \lambda$-Littlewood--Richardson tableaux of shape $\mu$. In particular, if $\lambda_i=\weight(\underline c_i)$, then
 \[|\RT(\underline C)| = \dim \inv_G\left(\bigotimes_{i=1}^m V(\lambda_i)\right).\]
\end{remark}

We are now ready to prove the theorem for sorted clasps.
\begin{thm}\label{thm:sorted}
 Fix boundary conditions $\underline a \in [3]^n$ and a \emph{sorted} clasp sequence $\underline C = (\underline c_1,\dots, \underline c_m)$, and let $\lambda_i = \weight(\underline c_i)$. Then a basis for $\inv_G\left(\bigotimes_{i=1}^m V(\lambda_i)\right)$ is given by
 \[\mathcal W_{\underline C} \coloneqq \{\pi_{\underline C}([W]) \mid [W]\in \mathcal W_{\underline a}\setminus \ker\pi_{\underline C}\}.\]
 Moreover, the following are equivalent for $W\in \mathcal W_{\underline a}$:
 \begin{enumerate}
 \item $[W] \notin \ker\pi_{\underline C}$.
 \item $W$ is non-convex.
 \item There are no local boundary configurations from \Cref{fig:bad_configs} occurring within any clasp of $W$.
 \item The lattice word $\L(W) = \partial\sep_W$ has no $\underline C$-descents.
 \item $W$ has no trips that start and end in the same clasp.
 \end{enumerate}

\begin{thm}
 Let $\underline C = (\underline c_1, \dots, \underline c_m)$ be a clasp sequence on $\underline a\in [3]^n$, with $\weight(\underline c_i) = \lambda_i$. Then a basis for $\inv_G\left(\bigotimes_{i=1}^m V(\lambda_i)\right)$ is given by
 \[\mathcal W_{\underline C}\coloneqq \{\pi_{\underline C}([W]): [W]\in \mathcal W_{\underline a}\setminus \ker \pi_{\underline C}\}.\]
 Moreover, the following are equivalent:
 \begin{enumerate}
 \item $[W]\notin \ker\pi_{\underline C}$,
 \item $W$ is non-convex,
 \item $W$ has no trips that start and end in the same clasp.
 \end{enumerate}
 \label{thm:main_general}
\end{thm}

\begin{thm}\label{thm:sorted3}
 Fix boundary conditions $\underline a \in [2]^n$ and a {sorted} clasp sequence $\underline C = (\underline c_1,\dots, \underline c_m)$, and let $\lambda_i = \weight(\underline c_i)$. Then a basis for $\inv_G\left(\bigotimes_{i=1}^m V(\lambda_i)\right)$ is given by
 \[\mathcal W_{\underline C} \coloneqq \{\pi_{\underline C}([W]): [W]\in \mathcal W_{\underline a}\setminus \ker\pi_{\underline C}\}.\]
 Moreover, the following are equivalent for $W\in \mathcal W_{\underline a}$:
 \begin{enumerate}
 \item $[W] \notin \ker\pi_{\underline C}$.
 \item $W$ is non-convex.
 \item There are no local boundary configurations from \Cref{fig:badconf_3} occurring within any clasp of $W$.
 \item The lattice word $\L(W) = \partial\sep_W$ has no $\underline C$-descents.
 \item $W$ has no trips that start and end in the same clasp.
 \end{enumerate}

\begin{proof}
 By \Cref{prop:nonconvex}, $\W_{\underline C}$ spans the invariant space, and we show that $|\W_{\underline C}|\le \dim\inv_G\left(\bigotimes_{i=1}^m V(\lambda_i)\right)$ while simultaneously proving the equivalence of (1), (2), (3) and (4).(1)$\implies$ (2) is \Cref{prop:nonconvex}.(2)$\implies$(3) is seen directly from \Cref{fig:badconf_3}. (3)$\implies$(4) is proven exactly like \Cref{prop:descent}, using the KK-growth rules instead. There are far fewer cases here, since each descent is of one of the following types:
 \begin{itemize}
 \item $ab$ with $a<b$,
 \item $\overline{ab}$ with $a>b$, or
 \item $1\overline 1$.
 \end{itemize}

\begin{thm}
 Let $\underline C = (\underline c_1, \dots, \underline c_m)$ be a clasp sequence on $\underline a\in [2]^n$, with $\weight(\underline c_i) = \lambda_i$. Then a basis for $\inv_G\left(\bigotimes_{i=1}^m V(\lambda_i)\right)$ is given by
 \[\mathcal W_{\underline C}\coloneqq \{\pi_{\underline C}([W]): [W]\in \mathcal W_{\underline a}\setminus \ker \pi_{\underline C}\}.\]
 Moreover, the following are equivalent:
 \begin{enumerate}
 \item $[W]\notin \ker\pi_{\underline C}$,
 \item $W$ is non-convex,
 \item $W$ has no trips that start and end in the same clasp.
 \end{enumerate}
\end{thm}

\begin{definition}\label{def:nonconvex}
    A fully reduced hourglass plabic graph $W$ with a clasp sequence $\underline C = (\underline c_1,\dots, \underline c_m)$ is \emph{non-convex} (with respect to $\underline C$) if for each $i\in [m]$ and every $\underline c_i$-cut path $\gamma$, $\weight(\gamma)\ge \weight(\underline c_i)$ in the dominance ordering on $\Lambda^+$. We say that $W$ is \emph{partially convex} if it is not non-convex.
\end{definition}

\begin{remark}
    The fact that the web invariants not killed by the projection $\pi_{\underline C}$ form a basis in $\mathsf{Inv}\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$ is notable. This property is one of several special properties (also including rotation-invariance) that the fully reduced web bases share with Lusztig's dual canonical basis \cite{Lusztig:canonical}. It was initially hoped that Kuperberg's basis agreed with the dual canonical basis until this was disproven by Khovanov--Kuperberg \cite{Khovanov-Kuperberg}, however the two bases do seem to share many distinguished properties.
\end{remark}

In the case when $\underline C$ is \emph{sorted}, i.e., each $\underline c_i$ is a weakly increasing tuple, the following theorem gives us more criteria to easily check for non-convexity.

\begin{thm}
    \label{thm:intro-sorted}
    In the setting of \Cref{thm:main}, assume further that $\underline C$ is sorted. Then the following additional conditions are equivalent to \Cref{thm:main}(1)-(3):
    \begin{enumerate}
        \item[(4)] No clasp contains a ``bad" local configuration (see \Cref{fig:bad_configs,fig:badconf_3,fig:u_turn}).
        \item[(5)] The lattice word $\mathcal{L}(W) = \partial\mathrm{sep}(W)$ has no $\underline C$-descents (see \Cref{def:latticeword}).
    \end{enumerate}
\end{thm}

The map $\partial\mathrm{sep}$ appearing in \Cref{thm:intro-sorted}(5) is the bijection from \cite{GPPSS-sl4} between elements of $\W_{\underline a}$ and (lattice words of) the associated \emph{fluctuating tableaux} \cite{fluctuating-paper}, certain generalizations of standard Young tableaux (which correspond to the case $\underline{a}=(1,1,\ldots,1)$).

A key tool in the proof of \Cref{thm:main} is the \emph{swap} map of \Cref{sec:sorting}, which is of interest even outside the context of clasping. It gives bijections between the fully reduced web bases of \cite{GPPSS-sl4} for any ordering of the same tensor factors.

\begin{thm}
\label{thm:intro-swap}
    The map $\mathsf{swap}$ gives bijections between the fully reduced web bases $\W_{\underline a}$ for all permutations of the boundary conditions $\underline{a}$.
\end{thm}

For ease of exposition, in most of the paper we deal only with the classical groups $\SL_r$. All of our results apply also to invariants of the quantum group $U_q(\mathfrak{sl}_r)$, but this extension will be straightforward.

\subsection{Outline}

In \Cref{sec:prelim} we recall background on web invariants and on the combinatorial constructions introduced in \cite{GPPSS-sl4}. In \Cref{sec:clasped-webs} we set up some machinery for clasping webs. In \Cref{sec:descents} we establish a correspondence between certain descents in a lattice word and bad local configurations in the corresponding basis web. In \Cref{Sec:Fund} we show that lattice words avoiding these descents (and therefore webs avoiding the bad configurations) have the right number to form a basis for the clasped invariant space. This is applied in \Cref{sec:main-sorted-proof} to prove \Cref{thm:intro-sorted}. \Cref{sec:sorting} introduces the swap map which is used to prove \Cref{thm:intro-swap} and thereby \Cref{thm:main}, after some verifications for $r=2$ and $3$, which take place in \Cref{sec:r-2-3}.

A proof outline for \Cref{thm:main} is indicated as follows. First, the paper recalls background on web invariants and the combinatorial constructions of \cite{GPPSS-sl4} (\Cref{sec:prelim}). Next it develops “some machinery for clasping webs” (\Cref{sec:clasped-webs}). Then it “establish[es] a correspondence between certain descents in a lattice word and bad local configurations in the corresponding basis web” (\Cref{sec:descents}). After that, it shows that “lattice words avoiding these descents (and therefore webs avoiding the bad configurations) have the right number to form a basis for the clasped invariant space” (\Cref{Sec:Fund}); this is used to prove the sorted case \Cref{thm:intro-sorted} (\Cref{sec:main-sorted-proof}). Finally, a “key tool in the proof of \Cref{thm:main} is the \emph{swap} map” (\Cref{sec:sorting}), which yields bijections between fully reduced web bases for permutations of boundary conditions (proving \Cref{thm:intro-swap}) and “thereby \Cref{thm:main}, after some verifications for $r=2$ and $3$” carried out in \Cref{sec:r-2-3}.

A proof outline for the main theorem is indicated as follows. First, the paper recalls background on web invariants and the combinatorial constructions of \cite{GPPSS-sl4} (Next it proves these preliminary results.). Next it develops “some machinery for clasping webs” (Next it proves these results.). Then it “establish[es] a correspondence between certain descents in a lattice word and bad local configurations in the corresponding basis web” (Next it proves these results.). After that, it shows that “lattice words avoiding these descents (and therefore webs avoiding the bad configurations) have the right number to form a basis for the clasped invariant space” (Next it proves these results.); this is used to prove the following theorem.
\begin{thm}
    \label{thm:intro-sorted}
    In the setting of \Cref{thm:main}, assume further that $\underline C$ is sorted. Then the following additional conditions are equivalent to \Cref{thm:main}(1)-(3):
    \begin{enumerate}
        \item[(4)] No clasp contains a ``bad" local configuration (see \Cref{fig:bad_configs,fig:badconf_3,fig:u_turn}).
        \item[(5)] The lattice word $\L(W) = \partial\mathrm{sep}(W)$ has no $\underline C$-descents (see \Cref{def:latticeword}).
    \end{enumerate}
\end{thm} (Next it proves these results.). Finally, a “key tool in the proof of the main theorem is the \emph{swap} map” (Next it proves these results.), which yields bijections between fully reduced web bases for permutations of boundary conditions (proving the following theorem.
\begin{thm}
\label{thm:intro-swap}
    The map $\swap$ gives bijections between the fully reduced web bases $\W_{\underline a}$ for all permutations of the boundary conditions $\underline{a}$.
\end{thm} ) and “thereby the main theorem, after some verifications for $r=2$ and $3$” carried out in Next it proves these results..

\label{thm:main}
    Let $r \in \{2,3,4\}$, let $\W_{\underline a}$ be the fully reduced web basis for $\mathsf{Inv}(V_q^{\underline a})$, and let $\underline{C}$ be a clasp sequence for $\underline{a}$ with weights $\lambda_1,\ldots,\lambda_m$. Then the following are equivalent for $[W]_q \in \W_{\underline a}$:
    \begin{enumerate}
        \item $[W]_q\notin \ker\pi_{\underline C}$,
        \item $W$ is non-convex, where:
        \begin{definition}\label{def:nonconvex}
    A fully reduced hourglass plabic graph $W$ with a clasp sequence $\underline C = (\underline c_1,\dots, \underline c_m)$ is \emph{non-convex} (with respect to $\underline C$) if for each $i\in [m]$ and every $\underline c_i$-cut path $\gamma$, $\weight(\gamma)\ge \weight(\underline c_i)$ in the dominance ordering on $\Lambda^+$. We say that $W$ is \emph{partially convex} if it is not non-convex.
\end{definition}
        \item $W$ has no trips that start and end in the same clasp.
    \end{enumerate}
    Moreover, the clasped web invariants for these webs $W$ form a basis for $\mathsf{Inv}\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$.

Biconditional or Equivalence
Classification or Bijection

Question: Let $r\in\{2,3,4\}$. Let $\mathcal W_{\underline a}$ be the fully reduced web basis of $\mathsf{Inv}(V_q^{\underline a})$, and let $\underline C=(\underline c_1,\dots,\underline c_m)$ be a clasp sequence for the boundary conditions $\underline a$, with $\lambda_i=\mathrm{wt}(\underline c_i)$. For a fully reduced hourglass plabic graph $W$, say that $W$ is non-convex with respect to $\underline C$ if for every $i\in[m]$ and every $\underline c_i$-cut path $\gamma$, one has $\mathrm{wt}(\gamma)\ge \mathrm{wt}(\underline c_i)$ in the dominance order. Which statement exactly describes the web basis elements whose clasped images under $\pi_{\underline C}$ give the invariant space $\mathsf{Inv}\!\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$?

They are exactly the basis elements $[W]_q\in \mathcal W_{\underline a}$ satisfying any, equivalently all, of the following conditions: $[W]_q\notin \ker \pi_{\underline C}$; $W$ is non-convex; $W$ has no trip that starts and ends in the same clasp. Moreover, the clasped web invariants $\pi_{\underline C}([W]_q)$ for precisely these webs form a basis of $\mathsf{Inv}\!\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$.

Choices:
B. They are exactly the basis elements $[W]_q\in \mathcal W_{\underline a}$ satisfying any, equivalently all, of the following conditions: $[W]_q\notin \ker \pi_{\underline C}$; $W$ is non-convex; for each clasp $\underline c_i$ there exists a $\underline c_i$-cut path $\gamma$ with $\mathrm{wt}(\gamma)\ge \mathrm{wt}(\underline c_i)$ in the dominance order. Moreover, the clasped web invariants $\pi_{\underline C}([W]_q)$ for precisely these webs form a basis of $\mathsf{Inv}\!\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$.
C. The clasped web invariants $\pi_{\underline C}([W]_q)$ coming from basis elements $[W]_q\in \mathcal W_{\underline a}$ with $[W]_q\notin \ker \pi_{\underline C}$ span $\mathsf{Inv}\!\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$; in particular, every web satisfying the equivalent conditions $[W]_q\notin \ker \pi_{\underline C}$, non-convexity, and having no trip that starts and ends in the same clasp contributes such a spanning vector.
D. They are exactly the basis elements $[W]_q\in \mathcal W_{\underline a}$ satisfying any, equivalently all, of the following conditions: $[W]_q\notin \ker \pi_{\underline C}$; $W$ is non-convex; $W$ has no trip whose two endpoints lie in adjacent clasps. Moreover, the clasped web invariants $\pi_{\underline C}([W]_q)$ for precisely these webs form a basis of $\mathsf{Inv}\!\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$.
E. They are exactly the basis elements $[W]_q\in \mathcal W_{\underline a}$ satisfying any, equivalently all, of the following conditions: $[W]_q\notin \ker \pi_{\underline C}$; $W$ is partially convex; $W$ has no trip that starts and ends in the same clasp. Moreover, the clasped web invariants $\pi_{\underline C}([W]_q)$ for precisely these webs form a basis of $\mathsf{Inv}\!\left(\bigotimes_{i=1}^m V_q(\lambda_i)\right)$.

forall-cut-path quantifier in non-convexity

basis conclusion weakened to spanning conclusion

trip obstruction changed from same clasp to adjacent clasps