\begin{theorem}\label{thm:main1}
For each $n \ge 10$, the Torelli group of $X_n = 2\mathbb{CP}^2 \# n \overline{\mathbb{CP}^2}$ admits a surjective homomorphism $\Phi : T(X_n) \to \mathbb{Z}^\infty$.
\end{theorem}

Let $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The {\em mapping class group} $M(X)$ of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The {\em Torelli group} $T(X) \subseteq M(X)$ is the subgroup of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. Non-trivial elements of $T(X)$ represent exotic diffeomorphisms in the sense that they are continuously isotopic to the identity but not smoothly isotopic. Thus, the Torelli group $T(X)$ can be seen as a measure of the difference between continuous and smooth isotopy.

It was first established by Ruberman that the Torelli group of a compact, simply-connected $4$-manifold can be non-trivial \cite{rub1,rub2}. Ruberman constructed versions of the Donaldson and Seiberg--Witten invariants for $1$-parameter families of $4$-manifolds and used these invariants to detect non-triviality of elements of the Torelli group. More specifically, his invariant can be used to show that for every $n \ge 2$ and $m \ge 10n+1$, the Torelli group of $2n \mathbb{CP}^2 \# m \overline{\mathbb{CP}^2}$, surjects to $\mathbb{Z}^\infty$ (where $\mathbb{Z}^\infty$ denotes a free abelian group of countably infinite rank). Baraglia--Konno proved a more general gluing formula for the families Seiberg--Witten invariants \cite{bk1}, which for example can be used to show a similar result for the Torelli groups of $2n \mathbb{CP}^2 \# 10n \overline{\mathbb{CP}^2}$ for each odd $n \ge 3$ and also for for $n(S^2 \times S^2) \# nK3$ for each $n \ge 1$. In all of these cases the parity of $b_+$ must be even due to a dimension constraint and there is a further constraint that $b_+ > 2$ so as to avoid the presence of any chamber structure for the families Seiberg--Witten invariants. Thus the smallest possible value of $b_+$ allowed by these methods is $b_+ = 4$.

The main goal of this paper is to extend the methods of \cite{rub1,rub2,bk1} to the case $b_+ = 2$, using $1$-parameter Seiberg--Witten invariants in the presence of chambers. In a series of papers \cite{tom1,tom2}, the second author has proven a general gluing formula for the families Seiberg--Witten invariants which can be applied even for low values of $b_+$ where chambers are present. The results in this paper are obtained by applying this more general gluing formula to the case of $1$-parameter families. Our first main result is that the Torelli groups of certain simply-connected $4$-manifolds with $b_+ = 2$ can be infinitely generated:

\begin{abstract}
Let $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The {\em mapping class group} of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The {\em Torelli group} of $X$ is the subgroup of the mapping class group consisting of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. We prove that for each $n \ge 10$, the Torelli group of $2\mathbb{CP}^2 \# n \overline{\mathbb{CP}^2}$ surjects to $\mathbb{Z}^\infty$. We also prove that the mapping class group of $2 \mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ is not finitely generated. Our proofs of these results makes use of Seiberg--Witten invariants for $1$-parameter familes of $4$-manifolds and in particular a gluing formula for connected sum families. Since the manifolds we consider have $b_+ = 2$, the chamber structure of the $1$-parameter Seiberg--Witten invariants plays an important role.
\end{abstract}

It was first established by Ruberman that the Torelli group of a compact, simply-connected $4$-manifold can be non-trivial \cite{rub1,rub2}. Ruberman constructed versions of the Donaldson and Seiberg--Witten invariants for $1$-parameter families of $4$-manifolds and used these invariants to detect non-triviality of elements of the Torelli group. More specifically, his invariant can be used to show that for every $n \ge 2$ and $m \ge 10n+1$, the Torelli group of $2n \mathbb{CP}^2 \# m \overline{\mathbb{CP}^2}$, surjects to $\mathbb{Z}^\infty$ (where $\mathbb{Z}^\infty$ denotes a free abelian group of countably infinite rank). Baraglia--Konno proved a more general gluing formula for the families Seiberg--Witten invariants \cite{bk1}, which for example can be used to show a similar result for the Torelli groups of $2n \mathbb{CP}^2 \# 10n \overline{\mathbb{CP}^2}$ for each odd $n \ge 3$ and also for for $n(S^2 \times S^2) \# nK3$ for each $n \ge 1$. In all of these cases the parity of $b_+$ must be even due to a dimension constraint and there is a further constraint that $b_+ > 2$ so as to avoid the presence of any chamber structure for the families Seiberg--Witten invariants. Thus the smallest possible value of $b_+$ allowed by these methods is $b_+ = 4$.

The main goal of this paper is to extend the methods of \cite{rub1,rub2,bk1} to the case $b_+ = 2$, using $1$-parameter Seiberg--Witten invariants in the presence of chambers. In a series of papers \cite{tom1,tom2}, the second author has proven a general gluing formula for the families Seiberg--Witten invariants which can be applied even for low values of $b_+$ where chambers are present. The results in this paper are obtained by applying this more general gluing formula to the case of $1$-parameter families. Our first main result is that the Torelli groups of certain simply-connected $4$-manifolds with $b_+ = 2$ can be infinitely generated:

Prior to this result it had been shown by Konno--Mallick--Taniguchi \cite{kmt} that $2\mathbb{CP}^2 \# m \overline{\mathbb{CP}^2}$ has an exotic diffeomorphism for each $m \ge 11$. Qiu proves that $2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ admits an exotic diffeomorphism with infinite order in the Torelli group \cite{qiu}. Qiu's method is similar to the one used in this paper, although the conclusion is not as strong.

\begin{theorem}\label{thm:main2}
The mapping class group of $X = 2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ is not finitely generated. More precisely, there is an index $2$-subgroup $M_+(X) \subseteq M(X)$ and a surjective homomorphism $\Phi : M_+(X) \to \mathbb{Z}^\infty$.
\end{theorem}

\begin{theorem}
For each $n \ge 10$, the Torelli group of $X_n = 2\mathbb{CP}^2 \# n \overline{\mathbb{CP}^2}$ admits a surjective homomorphism $\Phi : T(X_n) \to \mathbb{Z}^\infty$ to a free abelian group of countably infinite rank.
\end{theorem}
\begin{proof}
Let $\mathbb{Z}[\mathcal{S}(X_n)] = \bigoplus_{\mathfrak{s} \in \mathcal{S}(X_n)} \mathbb{Z}$. Elements of $\mathbb{Z}[\mathcal{S}(X_n)]$ can be regarded as functions $\mathcal{S}(X_n) \to \mathbb{Z}$ with finite support. Define a homomorphism $\Phi' : T(X_n) \to \bigoplus_{\mathfrak{s} \in \mathcal{S}(X_n) } \mathbb{Z}$ by taking $(\Phi'(f))(\mathfrak{s}) = SW^c_{X_n , \mathfrak{s} , \mathbb{Z}}(f)$. The function $\mathfrak{s} \mapsto SW^c_{X_n , \mathfrak{s} , \mathbb{Z}}(f)$ has finite support because of the compactness properties of the Seiberg--Witten equations.

The image $im(\Phi')$ of $\Phi'$ is a subgroup of the free abelian group $\mathbb{Z}[\mathcal{S}(X_n)]$ and hence is free abelian. Let $\Phi : T(X_n) \to im(\Phi')$ be the homomorphism obtained by factoring $\Phi'$ through its image. The theorem will follow if we can show that $im(\Phi')$ has infinite rank (since $\mathbb{Z}[\mathcal{S}(X_n)]$ has countably infinite rank, the rank of $im(\Phi')$ can be at most countably infinite). For $n=10$, Lemma \ref{lem:torelli} immediately implies that $im(\Phi')$ has countably infinite rank. For $n > 10$ we use induction on $n$ and the blow-up formula. Write $X_n = X_{n-1} \# \overline{\mathbb{CP}^2}$. Let $f \in T(X_{n-1})$. We can isotopy $f$ so that it is the identity in a neighbourhood of a point. Then we can form the connected sum diffeomorphism $f \# id_{\overline{\mathbb{CP}^2}} \in T(X_n)$. Let $\kappa$ be a spin$^c$-structure on $\overline{\mathbb{CP}^2}$ with $c(\kappa)^2 = -1$. Let $\mathfrak{s} \in \mathcal{S}(X_{n-1})$. Then $\mathfrak{s} \# \kappa \in \mathcal{S}(X_n)$. The blowup formula gives
\[
SW^c_{X_n , \mathfrak{s} \# \kappa , \mathbb{Z}}(f \# id_{\overline{\mathbb{CP}^2}}) = SW^c_{X_{n-1} , \mathfrak{s} , \mathbb{Z}}(f).
\]
Hence there are infinitely many spin$^c$-structures on $X_n$ for which $SW^c_{X_n , \mathfrak{s} , \mathbb{Z}} : T(X_n) \to \mathbb{Z}$ is non-zero and hence $\Phi'( T(X_n) )$ has infinite rank.
\end{proof}

\begin{theorem}
The mapping class group of $X = 2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ is not finitely generated. More precisely there is a surjective homomorphism $\Phi : M_+(X) \to \mathbb{Z}^\infty$.
\end{theorem}
\begin{proof}
Define $\Phi' : M_+(X) \to \mathbb{Z}^\infty$ to be given by
\[
\Phi' = \bigoplus_{n=1}^{\infty} SW_{X , \mathcal{O}_{2n-1}} : M_+(X) \to \bigoplus_{n=1}^{\infty} \mathbb{Z}.
\]
To prove the result, it suffices to show that the image of $\Phi'$ is not finitely generated. Then we let $\Phi$ be the homomorphism obtained by replacing the codomain of $\Phi'$ with the image of $\Phi'$. The argument is almost identical to the proof of \cite[Theorem 3.1]{bar}, except that now we use $E(1)_{2,2n+1}$ in place of $E(n)_q$.
\end{proof}

Prior to this result it had been shown by Konno--Mallick--Taniguchi \cite{kmt} that $2\mathbb{CP}^2 \# m \overline{\mathbb{CP}^2}$ has an exotic diffeomorphism for each $m \ge 11$. Qiu proves that $2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ admits an exotic diffeomorphism with infinite order in the Torelli group \cite{qiu}. Qiu's method is similar to the one used in this paper, although the conclusion is not as strong.

Our second main result concerns infinite generation of the mapping class group $M(X)$ (by infinite generation, we mean to say that $M(X)$ is not finitely generated). Note that infinite generation of $M(X)$ does not follow from infinite generation of $T(X)$, because a finitely generated group can have infinitely generated subgroups.

\begin{theorem}\label{thm:main2}
The mapping class group of $X = 2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$ is not finitely generated. More precisely, there is an index $2$-subgroup $M_+(X) \subseteq M(X)$ and a surjective homomorphism $\Phi : M_+(X) \to \mathbb{Z}^\infty$.
\end{theorem}

Note that by Schreier's lemma \cite{ser} every finite index subgroup of a finitely generated group is finitely generated. So the infinite generation of $M_+(X)$ implies the infinite generation of $M(X)$.

The first examples of compact, simply-connected $4$-manifolds whose mapping class groups are not finitely generated were given by Baraglia \cite{bar} and Konno \cite{kon}, namely $M(X)$ is not finitely generated for $X = 2n \mathbb{CP}^2 \# 10n \overline{\mathbb{CP}^2}$ where $n \ge 3$ is odd and also for $X = n (S^2 \times S^2) \# nK3$, $n \ge 1$. Note that these are precisely the manifolds of the form $E(m) \# (S^2 \times S^2)$, $m \ge 2$. Theorem \ref{thm:main2} says the same result is true for the case $m=1$, that is, for $X = E(1) \# (S^2 \times S^2) = 2\mathbb{CP}^2 \# 10 \overline{\mathbb{CP}^2}$.

\subsection{Outline of the proofs of the main results}
To each diffeomorphism $f \in M(X)$, one can form the mapping cylinder $E(f)$. This is the $1$-parameter family of $4$-manifolds obtained from $[0,1] \times X$ by identifying the ends via $f$. If $\mathfrak{s}$ is a spin$^c$-structure which is preserved by $f$ and for which the expected dimension of the families Seiberg--Witten moduli space for $E(f)$ is zero, then one obtains a numerical invariant by counting the number of solutions of the Seiberg--Witten equations for the family $E(f)$. When $b_+ = 2$, this invariant depends on the choice of chamber and so is not strictly an invariant of $f$ alone. However under certain circumstances we find that a distinguished choice of chamber exists, and so we obtain invariants. More specifically, there are two cases that we consider:
\begin{itemize}
\item[(1)]{{\bf The constant chamber:} assume that $f \in T(X)$. Then $f$ acts trivially on $H^2(X ; \mathbb{R})$ and so the local system over $S^1$ whose fibres are $H^2$ of the fibres of $E(f)$ has trivial monodromy. This leads to a trivialisation (unique up to homotopy) of the bundle $\mathcal{H}^+ \to S^1$ whose fibres are $H^+$ of the fibres of $E(f)$. The constant chamber is the chamber which corresponds to the homotopy class of a constant section of $\mathcal{H}^+$ under the above trivialisation.}
\item[(2)]{{\bf The zero chamber:} assume that $c(\mathfrak{s})^2 \ge 0$ and $c(\mathfrak{s})$ is not torsion. Then there is a well-defined chamber corresponding to taking the self-dual $2$-form perturbation of the Seiberg--Witten equations to be zero.}
\end{itemize}

These two chambers are shown to coincide when they are both defined. Corresponding to the constant and zero chambers are families Seiberg--Witten invariants $SW^c_{X , \mathfrak{s}}(f)$ and $SW^0_{X , \mathfrak{s}}(f)$ depending only on $(X , \mathfrak{s})$ and the isotopy class of $f$. In particular, the constant chamber invariants define homomorphisms $SW^c_{X , \mathfrak{s}} : T(X) \to \mathbb{Z}$. Compactness properties of the Seiberg--Witten equations implies that for any given $f \in T(X)$, the invariants $SW^c_{X , \mathfrak{s}}(f)$ are non-zero for only finitely many spin$^c$-structures. Thus we obtain a homomorphism
\[
\Phi : T(X) \to \bigoplus_{\mathfrak{s}} \mathbb{Z}
\]
where the sum is over spin$^c$-structures for which the corresponding families Seiberg--Witten moduli space is zero dimensional. The proof of Theorem \ref{thm:main1} follows by showing that the image of $\Phi$ has infinite rank. For this we need to construct an infinite sequence of spin$^c$-structures $\{ \mathfrak{s}_n\}$ and diffeomorphisms $\{ t_n \}$ for which $SW^c_{X , \mathfrak{s}_n}( t_n ) \neq 0$. The diffeomorphisms $t_n$ are constructed in a similar fashion to \cite{rub1}, \cite{bk1}, making use of diffeomorphisms $E(1)_{p,q} \# (S^2 \times S^2) \cong E(1) \# (S^2 \times S^2)$.

\noindent{\bf Acknowledgments.} D. Baraglia was financially supported by an Australian Research Council Future Fellowship, FT230100092.

To prove Theorem~\ref{thm:main1}, the paper associates to each diffeomorphism $f\in M(X)$ its mapping cylinder $E(f)$ (a $1$-parameter family over $S^1$), and—when $f$ preserves a spin$^c$ structure $\mathfrak{s}$ with zero expected families Seiberg--Witten dimension—defines a numerical invariant by “counting the number of solutions of the Seiberg--Witten equations for the family $E(f)$.” Since $b_+=2$ this count depends on a chamber, but the authors identify circumstances giving a distinguished chamber:

(1) **Constant chamber:** if $f\in T(X)$, then $f$ acts trivially on $H^2(X;\mathbb R)$ so the local system over $S^1$ has trivial monodromy, yielding a (homotopically unique) trivialisation of $\mathcal H^+\to S^1$; the constant chamber corresponds to a constant section.

(2) **Zero chamber:** if $c(\mathfrak{s})^2\ge 0$ and $c(\mathfrak{s})$ is non-torsion, there is a chamber given by taking the self-dual $2$-form perturbation to be zero.

They show these chambers “coincide when they are both defined,” producing invariants $SW^c_{X,\mathfrak{s}}(f)$ and $SW^0_{X,\mathfrak{s}}(f)$ depending only on $(X,\mathfrak{s})$ and the isotopy class of $f$. In particular, the constant-chamber invariants give homomorphisms $SW^c_{X,\mathfrak{s}}:T(X)\to\mathbb Z$. By compactness, for fixed $f\in T(X)$ only finitely many $\mathfrak{s}$ contribute nontrivially, so one gets a combined homomorphism
\[
\Phi:T(X)\to\bigoplus_{\mathfrak{s}}\mathbb Z,
\]
(summing over zero-dimensional families moduli spaces). The proof of Theorem~\ref{thm:main1} then “follows by showing that the image of $\Phi$ has infinite rank,” by constructing an infinite sequence of spin$^c$-structures $\{\mathfrak{s}_n\}$ and diffeomorphisms $\{t_n\}$ with $SW^c_{X,\mathfrak{s}_n}(t_n)\ne 0$. The $t_n$ are built “in a similar fashion to \cite{rub1}, \cite{bk1},” using diffeomorphisms $E(1)_{p,q}\#(S^2\times S^2)\cong E(1)\#(S^2\times S^2)$.

To prove the main theorem, the paper associates to each diffeomorphism $f\in M(X)$ its mapping cylinder $E(f)$ (a $1$-parameter family over $S^1$), and—when $f$ preserves a spin$^c$ structure $\mathfrak{s}$ with zero expected families Seiberg--Witten dimension—defines a numerical invariant by “counting the number of solutions of the Seiberg--Witten equations for the family $E(f)$.” Since $b_+=2$ this count depends on a chamber, but the authors identify circumstances giving a distinguished chamber:

(1) **Constant chamber:** if $f\in T(X)$, then $f$ acts trivially on $H^2(X;\mathbb R)$ so the local system over $S^1$ has trivial monodromy, yielding a (homotopically unique) trivialisation of $\mathcal H^+\to S^1$; the constant chamber corresponds to a constant section.

(2) **Zero chamber:** if $c(\mathfrak{s})^2\ge 0$ and $c(\mathfrak{s})$ is non-torsion, there is a chamber given by taking the self-dual $2$-form perturbation to be zero.

They show these chambers “coincide when they are both defined,” producing invariants $SW^c_{X,\mathfrak{s}}(f)$ and $SW^0_{X,\mathfrak{s}}(f)$ depending only on $(X,\mathfrak{s})$ and the isotopy class of $f$. In particular, the constant-chamber invariants give homomorphisms $SW^c_{X,\mathfrak{s}}:T(X)\to\mathbb Z$. By compactness, for fixed $f\in T(X)$ only finitely many $\mathfrak{s}$ contribute nontrivially, so one gets a combined homomorphism
\[
\Phi:T(X)\to\bigoplus_{\mathfrak{s}}\mathbb Z,
\]
(summing over zero-dimensional families moduli spaces). The proof of the main theorem then “follows by showing that the image of $\Phi$ has infinite rank,” by constructing an infinite sequence of spin$^c$-structures $\{\mathfrak{s}_n\}$ and diffeomorphisms $\{t_n\}$ with $SW^c_{X,\mathfrak{s}_n}(t_n)\ne 0$. The $t_n$ are built “in a similar fashion to \cite{rub1}, \cite{bk1},” using diffeomorphisms $E(1)_{p,q}\#(S^2\times S^2)\cong E(1)\#(S^2\times S^2)$.,

Question: For each integer $n \ge 10$, let $X_n = 2\mathbb{CP}^2 \# n\overline{\mathbb{CP}^2}$. Let the mapping class group $M(X_n)$ be the group of smooth isotopy classes of diffeomorphisms of $X_n$, and let the Torelli group $T(X_n) \subseteq M(X_n)$ be the subgroup consisting of smooth isotopy classes of diffeomorphisms that are continuously isotopic to the identity. Here $\mathbb{Z}^\infty$ denotes a free abelian group of countably infinite rank. Which existence statement holds for these manifolds?

For every $n \ge 10$, there exists a surjective homomorphism $\Phi : T(X_n) \to \mathbb{Z}^\infty$.

Choices:
B. For every $n \ge 10$, there exists a surjective homomorphism $\Phi : M(X_n) \to \mathbb{Z}^\infty$.
C. For every $n \ge 10$, there exists a homomorphism $\Phi : T(X_n) \to \mathbb{Z}^\infty$ with infinite image.
D. For every $n \ge 10$, there exists a single homomorphism $\Phi : T(X_n) \to \mathbb{Z}^\infty$ that is defined uniformly for all $n$ and is surjective for each $X_n$.
E. For every $n \ge 9$, there exists a surjective homomorphism $\Phi : T(X_n) \to \mathbb{Z}^\infty$.

constant-chamber homomorphisms require Torelli trivial action on cohomology

dropped surjectivity onto all of \mathbb{Z}^\infty, keeping only infinite image

dependence of the construction on the individual manifold X_n

The stem does not explicitly reveal the correct option. It only sets up the objects and asks which existence statement is true; the exact surjectivity/domain/range claim is not given away.

This is essentially a theorem-recall item: the correct choice appears to be the target result stated nearly verbatim, with distractors formed by small perturbations of that statement.

There is some reasoning pressure in checking quantifiers, the group involved, surjectivity versus infinite image, and the boundary value of n. However, the item mainly tests recognition of the precise theorem statement rather than deeper generative mathematical reasoning.

The distractors are plausible and mathematically meaningful: they vary the ambient group, weaken surjectivity, alter quantifier uniformity, or shift the threshold n >= 10 to n >= 9. These align with common recall and overgeneralization errors.

A solid theorem-recognition MCQ with strong distractors and no direct answer leakage, but it is close to a verbatim restatement and only moderately tests reasoning.


\begin{proposition}\label{pr:q.discont.act.join}
  The free diagonal right action of $\left(\bQ,+\right)$ on $\bQ*\bQ$ is discontinuous if the join is equipped with the quotient topology.
\end{proposition}

\begin{theorem}\label{th:lb.cc}
  For Hausdorff topological groups $\bG$ with $\bG^n$ countably-compact the $\bG$-action on the universal space $\left(E_n\bG,\tau_{\varinjlim}\right)$ is continuous.
\end{theorem}

\begin{theorem}\label{th:disc.cone}
  If a Hausdorff topological group $\bG$
  \begin{itemize}[wide]
  \item acts discontinuously on its cone $\cC \bG$ with the quotient topology;

  \item or has distinct quotient and product topologies on $\bG\times \cC \bG$
  \end{itemize}
  then it acts discontinuously on its quotient-topologized join $\bG*\bG$, and hence also on $E_n \bG$, $1\le n\le \infty$.  \qedhere
\end{theorem}

\label{se:colim.top}

Specializing \Cref{pr:princ.bdl.ext} back to trivial $\bU$, the total space $E\bG=E_{\alpha}\bG$ featuring in that proof for a (here, compact) topological group $\bG$ is variousl

\begin{equation}\label{eq:x.ast.y}
  X*Y
  :=
  X\times Y\times [0,1]
  \bigg/
  \left(
    \begin{aligned}
      (x,y,0)&\sim (x,y',0)\\
      (x,y,1)&\sim (x',y,1)\\
    \end{aligned}
  \right)
\end{equation}

\emph{Bundles} feature below in two related but distinct senses, to be distinguished by context:
\begin{itemize}[wide]
\item On the one hand there are the familiar (vector/fiber/principal)-bundle, always, here, \emph{locally trivial} \cite[Definition 4.7.2]{hus_fib}, background on which will be referenced variously from \cite{hjjm_bdle,hus_fib,ms-cc,steen_fib} and a number of other sources. Some matters of convenience:
  \begin{itemize}[wide]
  \item We freely (and sometimes tacitly) pass back and forth \cite[Assertion 18.3.4]{hjjm_bdle} between rank-$q$ vector or $q\times q$-matrix algebra bundles with their corresponding \emph{principal bundles} \cite[Definition 5.2.2]{hjjm_bdle} over $GL(q)$ or the projective general liner group $PGL(q):=GL(q)/\left(\text{central }\bC^{\times}\right)$ respectively.

  \item It is harmless by the selfsame \cite[Assertion 18.3.4]{hjjm_bdle} to furthermore work with \emph{Hermitian} vector bundles, i.e. \cite[\S 14.1]{ms-cc} those equipped with appropriately compatible Hilbert-space structures on their fibers. In principal-bundle language this has the effect of reducing the groups to $U(q)<GL(q)$ and $PU(q)<PGL(q)$ respectively.
  \end{itemize}

\item On the other hand, there are the \emph{Banach bundles}
  \begin{equation*}
    \cA
    \xrightarrowdbl[\quad\text{continuous open}\quad]{\quad\pi\quad}
    X
    ,\quad
    \text{Banach-space fibers }\left(\cA_x:=\pi^{-1}x,\ \|\cdot\|_x\right)
  \end{equation*}
  of \cite[Definition II.13.4]{fd_bdl-1}, \cite[Definition 1.1]{dg_ban-bdl}, \cite[\S 3]{zbMATH03539905} and recalled below in \Cref{def:eqv.ban.bdl}. They \emph{can} be locally trivial in the sense of \cite[Remark II.13.9]{fd_bdl-1} but need not be. Throughout the paper they will for the most part be \emph{continuous} in the sense that the globally-defined norm
  \begin{equation}\label{eq:glob.nrm}
    \cA
    \xrightarrow{\quad\|\cdot\|\quad}
    \bR_{\ge 0}
    ,\quad
    \|\cdot\||_{\cA_x}
    :=
    \|\cdot\|_x
  \end{equation}
  is continuous (\emph{(F)} Banach bundles in \cite[post Definition 1.1, p.8]{dg_ban-bdl} terminology).

  In full generality (the \emph{(H)} Banach bundles of loc. cit.) \Cref{eq:glob.nrm} would be only \emph{upper semicontinuous} \cite[\S 1.3]{hk_shv-bdl}. Bundle base spaces and topological groups are always assumed Hausdorff absent explicit warnings.
\end{itemize}

Per standard conventions (e.g. \cite[Definition 2.2]{dupre_hilbund-1} or \cite[Definition IV.1.4.1]{blk} in a slightly different but related setting), Banach bundles are \emph{(sub-)homogeneous} if all fibers have the same finite dimension (or, respectively, there is an upper finite bound on fiber dimensions). For occasional added specificity, \emph{$n$-homogeneity} and \emph{$n$-sub-homogeneity} refer to the fibers being $n$ or $(\le n)$-dimensional respectively.

By way of a recollection before stating some of the main results, we reprise the usual Banach-bundle definitions in a slightly expanded framework to afford \emph{equivariance} with respect to a group operating on the entire package, as this will be the framework of interest throughout the paper (cf. \cite[Definition 1.1]{dg_ban-bdl}).

\begin{definition}\label{def:eqv.ban.bdl}
  For a topological (usually compact) group $\bU$ a \emph{$\bU$-equivariant Banach bundle} over $\Bbbk\in \{\bC,\bR\}$ (shorthand\footnote{The term is intended to avoid the confusion that would likely be caused by \emph{$\bU$-bundle}: the latter might be understood as having $\bU$ for its \emph{structure group} \cite[Definition 4.5.1]{hus_fib}.}: \emph{Banach bundle$_{\bU}$}) consists of
  \begin{itemize}[wide]
  \item a continuous open $\bU$-equivariant map $\cE\xrightarrowdbl{\pi}X$ for topological spaces $\cE$ and $X$ equipped with continuous $\bU$-actions;

  \item with the \emph{fibers} $\cE_x:=\pi^{-1}x$, $x\in X$ carrying Banach-space structures compatible with the $\bU$-actions (which thus restrict to jointly-continuous representations $\bU\times \cE_x\to \cE_x$);

  \item so that scalar multiplication and addition are continuous;

  \item and the norm on $\cE$ is continuous (for \emph{continuous} or \emph{(F)} Banach bundles) or upper semicontinuous (for \emph{(H)} Banach bundles).
  \end{itemize}
  Equivariant Hilbert-space or Banach-algebra or $C^*$-algebra bundles are defined similarly, with the fibers $\cE_x$ carrying those richer structures instead.
\end{definition}

The common theme unifying the various threads of the discussion below is the extent to which singularities of Banach/Banach-algebra/$C^*$ bundles preclude the existence of better-behaved (e.g. locally-trivial) larger ambient bundles or, alternatively, subbundles. Several situations come to mind where understanding Banach-bundle (mis)behavior at and around fiber-dimension jumps is at the very least useful.
\begin{enumerate}[(a),wide]
\item\label{item:motiv.brnch} The \emph{non-commutative branched covers} introduced in \cite[Deﬁnition 1.2]{pt_brnch} are unital $C^*$ embeddings $B\le A$ which satisfying convenient finiteness conditions (being of \emph{finite index} \cite[Definition 2]{fk_fin-ind}) not recalled here explicitly. Specialized to commutative $B$, they will be subhomogeneous $C^*$ bundles studied in \cite{bg_cx-exp,2409.17807v1,2409.03531v2}.

  It is in that setting that the embeddability of a subhomogeneous $C^*$ bundle into a homogeneous one becomes relevant: homogeneity (hence also homogeneous embeddability) will ensure \cite[Proposition 3.4]{bg_cx-exp} the desired finiteness constraints hold, whereas they need not if the subhomogeneous bundle's fiber-dimension jump loci are sufficiently ill-behaved (per \cite[Theorem 1.3]{2409.17807v1} addressing \cite[Problem 3.11]{bg_cx-exp}, say).

\item\label{item:motiv.sph} An adjacent theme to embeddability into homogeneous bundles is their \emph{prolongation} from closed subsets of the base space; this is what the title's \emph{germs} refer to: the term usually refers to objects (sheaf sections \cite[p.2]{bred_shf_2e_1997}, functions \cite[\S 27.18]{km_glob}) defined locally around a target (point, closed subset, etc.), identified if in agreement on a sufficiently small neighborhood of that target.

  One instance of the problem (which did in fact provide motivation for the present work) arises in relation to studying the \emph{non-commutative spheres} $\bS_{\theta}^{2n-1}$ introduced in \cite[Definition 2.1]{no_sph} as objects dual to the unital $C^*$-algebras generated universally by
  \begin{equation*}
    \text{normal }T_k,\quad k\in \left\{1..n\right\}
    \quad:\quad
    \left[
    \begin{aligned}
      T_{\ell} T_k
      &=
        \exp\left(2\pi i \theta_{k\ell}\right) T_k T_{\ell},\quad\forall k,\ell\\
      \sum_{k=1}^n T_k^* T_k
      &=1
    \end{aligned}
    \right.
  \end{equation*}
  for skew-symmetric $\theta\in M_n(\bR)$. The latter is the same type of deformation parameter featuring in the definition of the \emph{non-commutative tori} $\bT^n_{\theta}$ (\cite[p.193]{rief_case}, \cite[\S 12.2]{gbvf_ncg}), the difference being that for these the generators $U_k$ would be unitary rather than normal.

  For rational deformation parameters $\theta\in M_n(\bQ)$ the $\bS^{2n-1}_{\theta}$ are non-commutative branched covers over classical spheres $\bS^{2n-1}$ in the sense of \Cref{item:motiv.brnch} (e.g. by a conjunction of \cite[Theorem B]{2508.04922v1} and \cite[Theorem A]{2409.03531v2}), particularly pleasant for $n=2$ \cite[Proposition 4.5]{cp_cont-equiv_xv3}:
  \begin{equation*}
    \cA\xrightarrowdbl{\quad \pi\quad}\bS^3
    ,\quad
    \cA_x\cong
    \begin{cases}
      M_q&x\not\in h^{-1}\left(p_{\pm}\right)\\
      \bC^q&\text{otherwise},
    \end{cases}
    ,\quad
    \begin{aligned}
      \bS^3
      &\xrightarrowdbl[\ \text{Hopf fibration}\ ]{\quad h\quad}\bS^2\\
      p_{\pm}&:=\text{antipodal pair}
    \end{aligned}
  \end{equation*}
  where $q$ is the lowest-terms denominator of the rational $\theta_{12}$.

  It is useful, in computing the dimension invariants attached by \cite[Definitions 3.1 and 3.20]{hajacindex_xv2} (also \cite[Deﬁnition 2.3]{cpt_dim}) to finite-group actions on $\bS^{2n-1}_{\theta}$, to extend the $\bC^q$-fibered restriction of $\cA$ from one of the exceptional circles to a neighborhood thereof, equivariantly, as a piece of auxiliary scaffolding; \Cref{th:prlng.homog.abdls} affords this.
\end{enumerate}

To proceed to an outline of the paper's contents, the first extensibility result considered below reads as follows.

\emph{Bundles} feature below in two related but distinct senses, to be distinguished by context:
\begin{itemize}[wide]
\item On the one hand there are the familiar (vector/fiber/principal)-bundle, always, here, \emph{locally trivial} \cite[Definition 4.7.2]{hus_fib}, background on which will be referenced variously from \cite{hjjm_bdle,hus_fib,ms-cc,steen_fib} and a number of other sources. Some matters of convenience:
 \begin{itemize}[wide]
 \item We freely (and sometimes tacitly) pass back and forth \cite[Assertion 18.3.4]{hjjm_bdle} between rank-$q$ vector or $q\times q$-matrix algebra bundles with their corresponding \emph{principal bundles} \cite[Definition 5.2.2]{hjjm_bdle} over $GL(q)$ or the projective general liner group $PGL(q):=GL(q)/\left(\text{central }\bC^{\times}\right)$ respectively.

\item On the other hand, there are the \emph{Banach bundles}
 \begin{equation*}
 \cA
 \xrightarrowdbl[\quad\text{continuous open}\quad]{\quad\pi\quad}
 X
 ,\quad
 \text{Banach-space fibers }\left(\cA_x:=\pi^{-1}x,\ \|\cdot\|_x\right)
 \end{equation*}
 of \cite[Definition II.13.4]{fd_bdl-1}, \cite[Definition 1.1]{dg_ban-bdl}, \cite[\S 3]{zbMATH03539905} and recalled below in \Cref{def:eqv.ban.bdl}. They \emph{can} be locally trivial in the sense of \cite[Remark II.13.9]{fd_bdl-1} but need not be. Throughout the paper they will for the most part be \emph{continuous} in the sense that the globally-defined norm
 \begin{equation}\label{eq:glob.nrm}
 \cA
 \xrightarrow{\quad\|\cdot\|\quad}
 \bR_{\ge 0}
 ,\quad
 \|\cdot\||_{\cA_x}
 :=
 \|\cdot\|_x
 \end{equation}
 is continuous (\emph{(F)} Banach bundles in \cite[post Definition 1.1, p.8]{dg_ban-bdl} terminology).

\begin{definition}\label{def:eqv.ban.bdl}
 For a topological (usually compact) group $\bU$ a \emph{$\bU$-equivariant Banach bundle} over $\Bbbk\in \{\bC,\bR\}$ (shorthand\footnote{The term is intended to avoid the confusion that would likely be caused by \emph{$\bU$-bundle}: the latter might be understood as having $\bU$ for its \emph{structure group} \cite[Definition 4.5.1]{hus_fib}.}: \emph{Banach bundle$_{\bU}$}) consists of
 \begin{itemize}[wide]
 \item a continuous open $\bU$-equivariant map $\cE\xrightarrowdbl{\pi}X$ for topological spaces $\cE$ and $X$ equipped with continuous $\bU$-actions;

The common theme unifying the various threads of the discussion below is the extent to which singularities of Banach/Banach-algebra/$C^*$ bundles preclude the existence of better-behaved (e.g. locally-trivial) larger ambient bundles or, alternatively, subbundles. Several situations come to mind where understanding Banach-bundle (mis)behavior at and around fiber-dimension jumps is at the very least useful.
\begin{enumerate}[(a),wide]
\item\label{item:motiv.brnch} The \emph{non-commutative branched covers} introduced in \cite[Deﬁnition 1.2]{pt_brnch} are unital $C^*$ embeddings $B\le A$ which satisfying convenient finiteness conditions (being of \emph{finite index} \cite[Definition 2]{fk_fin-ind}) not recalled here explicitly. Specialized to commutative $B$, they will be subhomogeneous $C^*$ bundles studied in \cite{bg_cx-exp,2409.17807v1,2409.03531v2}.

It is useful, in computing the dimension invariants attached by \cite[Definitions 3.1 and 3.20]{hajacindex_xv2} (also \cite[Deﬁnition 2.3]{cpt_dim}) to finite-group actions on $\bS^{2n-1}_{\theta}$, to extend the $\bC^q$-fibered restriction of $\cA$ from one of the exceptional circles to a neighborhood thereof, equivariantly, as a piece of auxiliary scaffolding; \Cref{th:prlng.homog.abdls} affords this.
\end{enumerate}

To proceed to an outline of the paper's contents, the first extensibility result considered below reads as follows.

\begin{definition}\label{def:eqv.ban.bdl}
  For a topological (usually compact) group $\bU$ a \emph{$\bU$-equivariant Banach bundle} over $\Bbbk\in \{\bC,\bR\}$ (shorthand\footnote{The term is intended to avoid the confusion that would likely be caused by \emph{$\bU$-bundle}: the latter might be understood as having $\bU$ for its \emph{structure group} \cite[Definition 4.5.1]{hus_fib}.}: \emph{Banach bundle$_{\bU}$}) consists of
  \begin{itemize}[wide]
  \item a continuous open $\bU$-equivariant map $\cE\xrightarrowdbl{\pi}X$ for topological spaces $\cE$ and $X$ equipped with continuous $\bU$-actions;

  \item with the \emph{fibers} $\cE_x:=\pi^{-1}x$, $x\in X$ carrying Banach-space structures compatible with the $\bU$-actions (which thus restrict to jointly-continuous representations $\bU\times \cE_x\to \cE_x$);

  \item so that scalar multiplication and addition are continuous;

  \item and the norm on $\cE$ is continuous (for \emph{continuous} or \emph{(F)} Banach bundles) or upper semicontinuous (for \emph{(H)} Banach bundles).
  \end{itemize}
  Equivariant Hilbert-space or Banach-algebra or $C^*$-algebra bundles are defined similarly, with the fibers $\cE_x$ carrying those richer structures instead.
\end{definition}

\begin{theorem}\label{th:prlng.homog.abdls}
  \begin{enumerate}[(1),wide]
  \item\label{item:th:prlng.homog.abdls:ban} Let $\bU$ be a compact Lie group, $\cA\xrightarrowdbl{\hspace{0pt}} X$ a continuous (unital) Banach-algebra bundle$_{\bU}$ over a paracompact base space, $Z\subseteq X$ a closed $\bU$-subset, and $\cB\le \cA|_Z$ a locally-trivial algebra subbundle$_{\bU}$ with finite-dimensional semisimple fibers.

    There are
    \begin{itemize}[wide]
    \item a closed $\bU$-invariant neighborhood $W\supseteq Z$ in $X$;

    \item and a locally-trivial subbundle$_{\bU}$ $\wt{\cB}\xrightarrowdbl{\hspace{0pt}} W$ of $\cA|_W$;

    \item with $\wt{\cB}|_Z=\cB$.
    \end{itemize}

  \item\label{item:th:prlng.homog.abdls:cast} The analogous result holds also for continuous $C^*$ bundles$_{\bU}$.
  \end{enumerate}
\end{theorem}

\begin{abstract}
 Consider a continuous bundle $\mathcal{E}\to X$ of Banach/Hilbert spaces or Banach/$C^*$-algebras over a paracompact base space, equivariant for a compact Lie group $\mathbb{U}$ operating on all structures involved. We prove that in all cases homogeneous equivariant subbundles extend equivariantly from $\mathbb{U}$-invariant closed subsets of $X$ to closed invariant neighborhoods thereof (provided the fibers are semisimple in the Banach-algebra variant). This extends a number of results in the literature (due to Fell for non-equivariant local extensibility around a single point for $C^*$-algebras and the author for semisimple Banach algebras). The proofs are based in part on auxiliary results on (a) the extensibility of equivariant compact-Lie-group principal bundles locally around invariant closed subsets of paracompact spaces, as a consequence of equivariant-bundle classifying spaces being absolute neighborhood extensors in the relevant setting and (b) an equivariant-bundle version of Johnson's approximability of almost-multiplicative maps from finite-dimensional semisimple Banach algebras with Banach morphisms.
\end{abstract}

To proceed to an outline of the paper's contents, the first extensibility result considered below reads as follows.

The multiplicative counterpart to \Cref{th:prlng.homog.bdls} is as follows.

\begin{theorem}\label{th:prlng.homog.abdls}
 \begin{enumerate}[(1),wide]
 \item\label{item:th:prlng.homog.abdls:ban} Let $\bU$ be a compact Lie group, $\cA\xrightarrowdbl{\hspace{0pt}} X$ a continuous (unital) Banach-algebra bundle$_{\bU}$ over a paracompact base space, $Z\subseteq X$ a closed $\bU$-subset, and $\cB\le \cA|_Z$ a locally-trivial algebra subbundle$_{\bU}$ with finite-dimensional semisimple fibers.

The proofs of \Cref{th:prlng.homog.bdls,th:prlng.homog.abdls} rely on a few technical asides presumably of some interest in their own right:
\begin{enumerate}[(a),wide]
\item\label{item:intro.equiv.bdls} There is, first, the ability to extend $\bU$-equivariant principal $\bG$-bundles locally around $\bU$-invariant closed subspaces $Z\subseteq X$ of paracompact $\bU$-spaces. This forms the object of \Cref{pr:princ.bdl.ext}, and is reminiscent of familiar results on he extensibility of (bundle or sheaf) \emph{sections} from closed subspaces: \cite[Theorem I.5.10]{kar_k_1978}, \cite[Th\'eor\`eme 3.3.1]{god_faisc_1958}, \cite[Proposition 1.1]{seg}, etc.

\item whereupon we can average against the Haar probability measure of $\bU$ to obtain a $\bU$-equivariant morphism $\cG\to \cE$ agreeing with the original embedding over $Z$ (much as in \cite[Proposition 1.1]{seg}, for instance).
 \end{itemize}
 Such an extension will still be an embedding
 \begin{equation*}
 \cG|_W
 \lhook\joinrel\xrightarrow{\quad \iota\quad}
 \cE|_W
 ,\quad
 \text{closed neighborhood }W\supseteq Z,
 \end{equation*}
 providing the desired \emph{Banach} bundle. In the more rigid setup of Hilbert bundles, an additional step will replace $\iota$ with the isometric component $\iota_{\mathrm{iso}}$ in the \emph{polar decomposition} \cite[\S I.5.2.2]{blk} $\iota=\iota_{\mathrm{iso}}|\iota|$.
\end{th:prlng.homog.bdls}

\begin{remarks}\label{res:gen.triv}
 \begin{enumerate}[(1),wide]
 \item For $\bU=\{1\}$ and \emph{trivial} (rather than locally trivial) $\cF$ \Cref{th:prlng.homog.bdls} reflects the fact (e.g. \cite[Corollary 2.10]{laz_bb-sel} or \cite[p.15]{dg_ban-bdl}) that bounded sections of (even (H), rather than (F)) Banach bundles extend from closed subsets of paracompact base spaces:
 \begin{itemize}[wide]
 \item having selected $n$ sections witnessing triviality across all of $Z$;

\begin{enumerate}[label={},wide]
 \item\textbf{\Cref{item:th:prlng.homog.abdls:ban}} In first instance, \Cref{pr:princ.bdl.ext} applied the automorphism group $\bG$ of some finite-dimensional $C^*$-algebra, i.e. a semidirect product of the form
 \begin{equation*}
 \left(\prod_{i=1}^k PU(n_i)\right)\rtimes \left(\text{finite group}\right),
 \end{equation*}
 will extend $\cB$ over a closed neighborhood (so we may as well assume globally) to a locally-trivial, $B$-fibered bundle$_{\bU}$ $\cM\xrightarrowdbl{\hspace{0pt}}X$ with $\cM|_A\cong \cB$. Precisely as in the proof of \Cref{th:prlng.homog.bdls}, we have a $\bU$-equivariant embedding
 \begin{equation*}
 \cM|_{W}
 \lhook\joinrel\xrightarrow{\quad\iota\quad}
 \cA|_W
 ,\quad
 \text{some closed neighborhood }W\supseteq Z
 \end{equation*}
 of \emph{vector} bundles, (unital and) multiplicative over $Z$. As a matter of convenience, a slight perturbation
 \begin{equation*}
 \iota
 \xmapsto{\quad}
 \bigg(
 \cM_x\ni a
 \xmapsto{\quad}
 \iota(a)+1_{\cA_x}-\iota\left(1_{\cM_x}\right)
 \in \cA_x
 \bigg)
 ,\quad
 x\in W
 \end{equation*}
 over sufficiently small $W$ followed by averaging over $\bU$ will further ensure $\iota$ is unital throughout $W$ (observe that this process will not have altered $\iota$ over the original smaller subspace $Z\subseteq X$).

\begin{proposition}\label{pr:princ.bdl.ext}
  Consider
  \begin{itemize}[wide]
  \item a compact Lie group $\bU$ acting via $\alpha:\bU\circlearrowright \bG$ on another;

  \item a closed $\bU$-equivariant embedding $Z\subseteq X$ into a paracompact $\bU$-space;

  \item and a principal $\bG$-bundle$_{\alpha}$ $\cP\xrightarrowdbl{\pi}Z$.
  \end{itemize}
  For some closed $\bU$-neighborhood $W\supseteq Z$ there is a principal $\bG$-bundle$_{\alpha}$ $\wt{\cP}\xrightarrowdbl{\hspace{0pt}}W$ with $\wt{\cP}|_Z\cong \cP$.
\end{proposition}

\begin{theorem}\label{th:prlng.homog.bdls}
  \begin{enumerate}[(1),wide]
  \item Let $\bU$ be a compact Lie group, $\cE\xrightarrowdbl{\hspace{0pt}} X$ a continuous Banach bundle$_{\bU}$ over a paracompact base space, $Z\subseteq X$ a closed $\bU$-subset, and $\cF\le \cE|_Z$ a locally-trivial $n$-homogeneous subbundle$_{\bU}$.

    There are
    \begin{itemize}[wide]
    \item a closed $\bU$-invariant neighborhood $W\supseteq Z$ in $X$;

    \item and a locally-trivial $n$-homogeneous subbundle$_{\bU}$ $\wt{\cF}\xrightarrowdbl{\hspace{0pt}} W$ of $\cE|_W$;

    \item with $\wt{\cF}|_Z=\cF$.
    \end{itemize}

  \item The analogous result holds also for continuous Hilbert bundles$_{\bU}$.
  \end{enumerate}
\end{theorem}

The multiplicative counterpart to \Cref{th:prlng.homog.bdls} is as follows.

\begin{theorem}\label{th:prlng.homog.abdls}
  \begin{enumerate}[(1),wide]
  \item\label{item:th:prlng.homog.abdls:ban} Let $\bU$ be a compact Lie group, $\cA\xrightarrowdbl{\hspace{0pt}} X$ a continuous (unital) Banach-algebra bundle$_{\bU}$ over a paracompact base space, $Z\subseteq X$ a closed $\bU$-subset, and $\cB\le \cA|_Z$ a locally-trivial algebra subbundle$_{\bU}$ with finite-dimensional semisimple fibers.

    There are
    \begin{itemize}[wide]
    \item a closed $\bU$-invariant neighborhood $W\supseteq Z$ in $X$;

    \item and a locally-trivial subbundle$_{\bU}$ $\wt{\cB}\xrightarrowdbl{\hspace{0pt}} W$ of $\cA|_W$;

    \item with $\wt{\cB}|_Z=\cB$.
    \end{itemize}

  \item\label{item:th:prlng.homog.abdls:cast} The analogous result holds also for continuous $C^*$ bundles$_{\bU}$.
  \end{enumerate}
\end{theorem}

\Cref{th:prlng.homog.abdls} generalizes \cite[Theorem 3.1]{fell_struct} (in the $C^*$ setting) and \cite[Proposition 2.8]{2409.03531v2} (for both $C^*$ and Banach-algebra bundles) by allowing equivariance and extensibility locally around arbitrary ($\bU$-invariant) closed subsets rather than single points.

The proofs of \Cref{th:prlng.homog.bdls,th:prlng.homog.abdls} rely on a few technical asides presumably of some interest in their own right:
\begin{enumerate}[(a),wide]
\item\label{item:intro.equiv.bdls} There is, first, the ability to extend $\bU$-equivariant principal $\bG$-bundles locally around $\bU$-invariant closed subspaces $Z\subseteq X$ of paracompact $\bU$-spaces. This forms the object of \Cref{pr:princ.bdl.ext}, and is reminiscent of familiar results on he extensibility of (bundle or sheaf) \emph{sections} from closed subspaces: \cite[Theorem I.5.10]{kar_k_1978}, \cite[Th\'eor\`eme 3.3.1]{god_faisc_1958}, \cite[Proposition 1.1]{seg}, etc.

\item\label{item:intro.amnm} Secondly, in order to leverage the ``purely-linear'' \Cref{th:prlng.homog.bdls} into the ``multiplicative'' \Cref{th:prlng.homog.abdls}, we employ an equivariant bundle version of Johnson's principle \cite[Theorem 3.1, Corollary 3.2, Theorem 7.2]{john_approx} that almost-multiplicative linear ($*$-)maps from semisimple finite-dimensional Banach/$C^*$-algebras into arbitrary Banach/$C^*$-algebras are close to morphisms in the apposite category. This is one instance of what has since come to be termed \emph{Hyers-Ulam-Rassias stability} \cite[Introduction]{jung_hur-stab} and the family-wide, equivariant adaptation of Johnson's arguments can be regarded as fitting in that broad framework.
\end{enumerate}

In reference to item \Cref{item:intro.equiv.bdls} above it becomes topical to contrast the various topologies one typically equips the total space of the \emph{(Milnor-model) universal bundle} $E\bG$ \cite[Definition 7.2.7]{hjjm_bdle} of a(n always Hausdorff) topological group. \Cref{se:colim.top} revolves around that issue:
\begin{itemize}[wide]
\item \Cref{th:disc.cone} observes that the principal action $E\bG\times \bG\to E\bG$ is discontinuous for the colimit topology on $E\bG$ whenever, say, $\bG\times -$ fails to preserve the topological quotient
  \begin{equation*}
    \bG\times [0,1]
    \xrightarrowdbl{\quad}
    \text{\emph{cone} }
    \cC \bG:=\bG\times [0,1]\big/\bG\times \{0\}.
  \end{equation*}
  Indeed, such groups fail to act continuously even on the truncated total space
  \begin{equation*}
    E_1\bG=\bG*\bG
    \underset{\text{closed}}{\subseteq}
    E\bG,
  \end{equation*}
  with `*' denoting the \emph{topological join} \cite[p.9]{hatch_at} recalled in \Cref{eq:x.ast.y}. As simple a group as $(\bQ,+)$ will exhibit this behavior: \Cref{pr:q.discont.act.join}.

\item while \Cref{th:lb.cc} notes that by contrast, a group with \emph{countably-compact} power $\bG^n$ always acts continuously on its colimit-topologized truncated universal space $E_n \bG=\bG^{*(n+1)}$.
\end{itemize}

\subsection*{Acknowledgments}

I am grateful for illuminating comments and suggestions from B. Badzioch, M. Megrelishvili and M. Ramachandran.

The proofs of \Cref{th:prlng.homog.bdls,th:prlng.homog.abdls} are described as relying on two technical ingredients:
\begin{enumerate}[(a),wide]
\item the ability to extend $\bU$-equivariant principal $\bG$-bundles locally around $\bU$-invariant closed subspaces $Z\subseteq X$ of paracompact $\bU$-spaces (\Cref{pr:princ.bdl.ext}), in a way "reminiscent" of extensibility results for sections from closed subspaces;
\item to pass from the "purely-linear" \Cref{th:prlng.homog.bdls} to the "multiplicative" \Cref{th:prlng.homog.abdls}, an equivariant bundle version of Johnson's principle is used: almost-multiplicative linear ($*$-)maps from semisimple finite-dimensional Banach/$C^*$-algebras into arbitrary Banach/$C^*$-algebras are "close to morphisms" in the appropriate category (viewed as an instance of Hyers-Ulam-Rassias stability), with an equivariant, family-wise adaptation of Johnson's arguments.
\end{enumerate}

The proofs of To prove the main theorem, and of the following theorem are described as relying on two technical ingredients:
\begin{theorem}\label{th:prlng.homog.abdls}
  \begin{enumerate}[(1),wide]
  \item\label{item:th:prlng.homog.abdls:ban} Let $\bU$ be a compact Lie group, $\cA\xrightarrowdbl{\hspace{0pt}} X$ a continuous (unital) Banach-algebra bundle$_{\bU}$ over a paracompact base space, $Z\subseteq X$ a closed $\bU$-subset, and $\cB\le \cA|_Z$ a locally-trivial algebra subbundle$_{\bU}$ with finite-dimensional semisimple fibers.

    There are
    \begin{itemize}[wide]
    \item a closed $\bU$-invariant neighborhood $W\supseteq Z$ in $X$;

    \item and a locally-trivial subbundle$_{\bU}$ $\wt{\cB}\xrightarrowdbl{\hspace{0pt}} W$ of $\cA|_W$;

    \item with $\wt{\cB}|_Z=\cB$.
    \end{itemize}

  \item\label{item:th:prlng.homog.abdls:cast} The analogous result holds also for continuous $C^*$ bundles$_{\bU}$.
  \end{enumerate}
\end{theorem}
\begin{enumerate}[(a),wide]
\item the ability to extend $\bU$-equivariant principal $\bG$-bundles locally around $\bU$-invariant closed subspaces $Z\subseteq X$ of paracompact $\bU$-spaces.
We first prove the following proposition.
\begin{proposition}\label{pr:princ.bdl.ext}
  Consider
  \begin{itemize}[wide]
  \item a compact Lie group $\bU$ acting via $\alpha:\bU\circlearrowright \bG$ on another;

  \item a closed $\bU$-equivariant embedding $Z\subseteq X$ into a paracompact $\bU$-space;

  \item and a principal $\bG$-bundle$_{\alpha}$ $\cP\xrightarrowdbl{\pi}Z$.
  \end{itemize}
  For some closed $\bU$-neighborhood $W\supseteq Z$ there is a principal $\bG$-bundle$_{\alpha}$ $\wt{\cP}\xrightarrowdbl{\hspace{0pt}}W$ with $\wt{\cP}|_Z\cong \cP$.
\end{proposition}
\item to pass from the ``purely-linear'' To prove the main theorem, to the ``multiplicative'' theorem above, an equivariant bundle version of Johnson's principle is used: almost-multiplicative linear ($*$-)maps from semisimple finite-dimensional Banach/$C^*$-algebras into arbitrary Banach/$C^*$-algebras are ``close to morphisms'' in the appropriate category (viewed as an instance of Hyers-Ulam-Rassias stability), with an equivariant, family-wise adaptation of Johnson's arguments.
\end{enumerate},

Question: Let \(\mathbf U\) be a compact Lie group. A continuous \(\mathbf U\)-equivariant Banach bundle over a space \(X\) is a Banach bundle \(\pi:\mathcal E\to X\) equipped with continuous \(\mathbf U\)-actions on \(\mathcal E\) and \(X\) such that \(\pi\) is \(\mathbf U\)-equivariant and the norm on \(\mathcal E\) is continuous. A closed \(\mathbf U\)-subset \(Z\subseteq X\) means a closed \(\mathbf U\)-invariant subset. A locally trivial \(n\)-homogeneous subbundle\(_{\mathbf U}\) means a \(\mathbf U\)-equivariant locally trivial subbundle whose fibers all have dimension \(n\). Under these assumptions, which existence statement holds for Banach bundles and also for Hilbert bundles?

If \(\mathcal E\to X\) is a continuous Banach bundle\(_{\mathbf U}\) over a paracompact base space, \(Z\subseteq X\) is a closed \(\mathbf U\)-subset, and \(\mathcal F\le \mathcal E|_Z\) is a locally trivial \(n\)-homogeneous subbundle\(_{\mathbf U}\), then there exist a closed \(\mathbf U\)-invariant neighborhood \(W\supseteq Z\) in \(X\) and a locally trivial \(n\)-homogeneous subbundle\(_{\mathbf U}\) \(\widetilde{\mathcal F}\to W\) of \(\mathcal E|_W\) such that \(\widetilde{\mathcal F}|_Z=\mathcal F\). The same conclusion also holds when \(\mathcal E\to X\) is a continuous Hilbert bundle\(_{\mathbf U}\).

Choices:
B. If \(\mathcal E\to X\) is a continuous Banach bundle\(_{\mathbf U}\) over a paracompact base space, \(Z\subseteq X\) is a closed \(\mathbf U\)-subset, and \(\mathcal F\le \mathcal E|_Z\) is a locally trivial \(n\)-homogeneous subbundle\(_{\mathbf U}\), then there exist a \(\mathbf U\)-invariant open neighborhood \(W\supseteq Z\) in \(X\) and a locally trivial \(n\)-homogeneous subbundle\(_{\mathbf U}\) \(\widetilde{\mathcal F}\to W\) of \(\mathcal E|_W\) such that \(\widetilde{\mathcal F}|_Z=\mathcal F\). The same conclusion also holds when \(\mathcal E\to X\) is a continuous Hilbert bundle\(_{\mathbf U}\).
C. If \(\mathcal E\to X\) is a continuous Banach bundle\(_{\mathbf U}\) over a paracompact base space, \(Z\subseteq X\) is a closed \(\mathbf U\)-subset, and \(\mathcal F\le \mathcal E|_Z\) is a locally trivial \(n\)-homogeneous subbundle\(_{\mathbf U}\), then there exist a closed \(\mathbf U\)-invariant neighborhood \(W\supseteq Z\) in \(X\) and a locally trivial subbundle\(_{\mathbf U}\) \(\widetilde{\mathcal F}\to W\) of \(\mathcal E|_W\) such that \(\widetilde{\mathcal F}|_Z=\mathcal F\).
D. If \(\mathcal E\to X\) is a continuous Banach bundle\(_{\mathbf U}\) over a paracompact base space, \(Z\subseteq X\) is a closed \(\mathbf U\)-subset, and \(\mathcal F\le \mathcal E|_Z\) is an \(n\)-homogeneous subbundle\(_{\mathbf U}\), then there exist a closed \(\mathbf U\)-invariant neighborhood \(W\supseteq Z\) in \(X\) and a locally trivial \(n\)-homogeneous subbundle\(_{\mathbf U}\) \(\widetilde{\mathcal F}\to W\) of \(\mathcal E|_W\) such that \(\widetilde{\mathcal F}|_Z=\mathcal F\). The same conclusion also holds when \(\mathcal E\to X\) is a continuous Hilbert bundle\(_{\mathbf U}\).
E. If \(\mathcal E\to X\) is a continuous Banach bundle\(_{\mathbf U}\) over a paracompact base space, \(Z\subseteq X\) is a closed \(\mathbf U\)-subset, and \(\mathcal F\le \mathcal E|_Z\) is a locally trivial \(n\)-homogeneous subbundle\(_{\mathbf U}\), then there exists a closed \(\mathbf U\)-invariant neighborhood \(W\supseteq Z\) in \(X\) such that for every continuous Banach bundle\(_{\mathbf U}\) structure on \(\mathcal E|_W\) extending the given one on \(\mathcal E|_Z\), there is a locally trivial \(n\)-homogeneous subbundle\(_{\mathbf U}\) \(\widetilde{\mathcal F}\to W\) of \(\mathcal E|_W\) with \(\widetilde{\mathcal F}|_Z=\mathcal F\). The same conclusion also holds when \(\mathcal E\to X\) is a continuous Hilbert bundle\(_{\mathbf U}\).

preservation of n-homogeneous fiber dimension and Hilbert-bundle analogue

initial local triviality of the subbundle on Z

existential quantifier for the ambient restriction and extension neighborhood

The stem does not explicitly reveal the correct choice or uniquely encode it. It only sets up definitions and asks for the valid existence statement.

The item is close to a theorem-recall question: the correct option is essentially the theorem statement with small perturbations in the alternatives. It is not a pure verbatim restatement, but it is only mildly non-tautological.

Selecting the answer requires careful comparison of subtle logical differences (closed vs. open neighborhood, preservation of n-homogeneity, local triviality assumptions, quantifier strength), but it mainly tests precise recall/parsing rather than substantial generative mathematical reasoning.

Several distractors are plausible and target realistic confusions, but choice C appears to be a weaker true statement, which makes the distractor set flawed unless the prompt had asked for the strongest conclusion. This weakens single-answer validity.

A moderately good but flawed theorem-recognition MCQ: no major answer leakage and some subtle distractors, but it is largely recall-based and likely ambiguous because a weaker true option is included among the distractors.


\label{thm0}
Let $d\geq 2$ be an integer and let $K\subset \mathbb{R}^d$ be compact, uniformly disconnected, and admit a Lipschitz capture. Then there exists a Lipschitz capture $F=f([0,1])$ of $K$ such that for every $x\in K$, $\Psi-\wtan(F,x)=\mathfrak{C}_U(\mathbb{R}^d;{\bf 0})$.

\begin{ex}\label{ex1}[A set of Hausdorff dimension $1$ which attains every pseudotangent along a Lipschitz curve]

For natural numbers $k\geq 2$, let $C_k\subset [0,1/k^2]$ be a self-similar Cantor set of Hausdorff dimension $\log(k)/\log(k+1)$. Then by self-similarity, $\dim_A(C_k)=\log(k)/\log(k+1)$ as well. In $\R^d$, for $d\geq 2$, let
\[K:=\{{\bf 0}\}\cup\left(\bigcup_{k\geq 2} \{0\}^{d-2}\times \{1/k\}\times C_k\right).\]
Then $K\subset \R^d$ is compact, $\dimh(K)=1$, $\mathcal{H}^1(K)=0$, and $K$ admits a Lipschitz capture $F$ by applying Theorem \ref{thm0} to each $C_k$ in such a way that the curves do not intersect and such that the sum of the lengths of the curves is finite, then adding also the segment $[0,1]\times\{0\}$. Then for all $x\in K$, $\Psi-\wtan (F,x)=\mathfrak{C}_U(\R^d;{\bf 0})$.

\end{ex}

\label{prelim}
\begin{definition}
Given a set $K\subset \R^d$, we say that a map $f:[0,1]\to \R^d$ or its image $F=f([0,1])$ is a \emph{Lipschitz capture of} $K$ if $f$ is a Lipschitz function and $K\

\begin{ex}\label{ex2}[A Lipschitz curve with non-unique pseudotangents at every point]

Let $\{q_j\}_{j\in\N}$ be an enumeration of $\Q\cap [0,1]$, let $K_0:=[0,1]\times\{0\}\subset \R^2$, and let \[K_1:=K_0\cup\left(\bigcup_{j\in\N}\{q_j\}\times [0,\frac{1}{4(j+1)^2}]\right).\]
We proceed by recursion.

For $n\geq 1$, given finite-length continua $K_{n-1}\subset K_n$, satisfying that the components of $K_n\setminus K_{n-1}$ are either all parallel to the $x$-axis or all parallel to the $y$-axis, and such that $K_{n-1}\subset \overline{K_n\setminus K_{n-1}}$, we construct $K_{n+1}$ as follows. Let $\{(x_j^n,y_j^n)\}_{j\in\N}$ be a countable dense subset of $K_n\setminus K_{n-1}$. If the components of $K_n\setminus K_{n-1}$ are all parallel to the $x$-axis, then let
\[K_{n+1}:=K_n\cup\left(\bigcup_{j\in\N} \{x_j^n\}\times[y_j^n,y_j^n+\frac{1}{(j+1)^2(n+1)^2}]\right),\]
and if the components of $K_n\setminus K_{n-1}$ are instead all parallel to the $y$-axis, then let
\[K_{n+1}:=K_n\cup \left(\bigcup_{j\in\N}[x_j^n,x_j^n+\frac{1}{(j+1)^2(n+1)^2}]\times\{y_n^j\}\right).\]

The Hausdorff limit $K_\infty$ of these sets $K_n$ exists and by  \cite[Theorem 4.4]{AO} and by Go{\l}ab's Semicontinuity Theorem, it is a continuum of finite length, and so a Lipschitz curve. It is straightforward to see that since points with a unique tangent line are dense in $K_\infty$ and since points admitting a tangent which is not a line are also dense in $K_\infty$, every point of $K_\infty$ admits at least two distinct pseudotangents.
\end{ex}

\begin{theorem}\label{thm0}
Let $d\geq 2$ be an integer and let $K\subset \R^d$ be compact, uniformly disconnected, and admit a Lipschitz capture. Then there exists a Lipschitz capture $F=f([0,1])$ of $K$ such that for every $x\in K$, $\Psi-\wtan(F,x)=\mathfrak{C}_U(\R^d;{\bf 0})$.
\end{theorem}

Rademacher's Theorem, a classical theorem in geometric measure theory, states that if $f:\mathbb{R}^m\to \mathbb{R}^n$ is Lipschitz, then $f$ is differentiable at $\mathscr{L}^m$-almost every point $x\in \mathbb{R}^m$, where $\mathscr{L}^m$ denotes the $m$-dimensional Lebesgue measure on $\mathbb{R}^m$ (see for example \cite[Theorem 7.3]{Mattila}). Significant work over the years has gone into investigating potential converses and extensions to this theorem, for instance one could ask whether there exists a Lebesgue-null set $N\subset \mathbb{R}^m$ such that for every Lipschitz function $f:\mathbb{R}^m\to \mathbb{R}^n$, there exists a point $x\in N$ such that $f$ is differentiable at $x$. An answer to this question has been obtained: such a Lebesgue-null set $N\subset\mathbb{R}^m$ exists if and only if $m>n$; see the work of Preiss and Speight in \cite{PS15} for the proof and for a more complete history of this problem.

In addition to the classical version of Rademacher's Theorem, there are many notions of ``tangents to sets" which also admit a version of this theorem. If $\Gamma\subset \mathbb{R}^d$ is a Lipschitz curve, meaning that it is the image $f([0,1])$ of a Lipschitz map $f:[0,1]\to\mathbb{R}^d$, then $\Gamma$ admits a tangent line at $\mathcal{H}^1$-almost all of its points. This statement holds for approximate tangent planes (see \cite[Theorem 15.19]{Mattila}), for approximate tangent cones (see \cite[Theorem 3.8]{falc}), and most recently for tangents in the sense of Badger and Lewis \cite[Definition 3.1]{BL} (see \cite[Theorem 1.1]{QSS}). Here and throughout this article, $\mathcal{H}^1$ denotes the $1$-dimensional Hausdorff measure. Furthermore, with Vellis we proved in \cite[Appendix A]{QSS} that in every $\mathbb{R}^d$ with $d\geq 2$, there exists a Lipschitz curve $H\subset \mathbb{R}^d$ containing the origin ${\bf 0}$ such that every possible tangent is attained for $H$ at ${\bf 0}$. The goal of this paper is to expand on the work in this direction with the following result.

\maketitle
\section{Introduction}
Rademacher's Theorem, a classical theorem in geometric measure theory, states that if $f:\R^m\to \R^n$ is Lipschitz, then $f$ is differentiable at $\mathscr{L}^m$-almost every point $x\in \R^m$, where $\mathscr{L}^m$ denotes the $m$-dimensional Lebesgue measure on $\R^m$ (see for example \cite[Theorem 7.3]{Mattila}). Significant work over the years has gone into investigating potential converses and extensions to this theorem, for instance one could ask whether there exists a Lebesgue-null set $N\subset \R^m$ such that for every Lipschitz function $f:\R^m\to \R^n$, there exists a point $x\in N$ such that $f$ is differentiable at $x$. An answer to this question has been obtained: such a Lebesgue-null set $N\subset\R^m$ exists if and only if $m>n$; see the work of Preiss and Speight in \cite{PS15} for the proof and for a more complete history of this problem.

Here and throughout, we use the notation $\wtan (X,x)$ to denote the set of \emph{tangents} to the set $X$ at the point $x$, $\Psi-\wtan (X,x)$ to denote the set of \emph{pseudotangents} to the set $X$ at the point $x$, and $\mathfrak{C}_U(\R^d;{\bf 0})$ to denote the collection of all closed subsets of $\R^d$ which contain the origin and have only unbounded components, see Section \ref{prelim} for the details of these sets. In Proposition \ref{prop:alltan}, we prove that as long as $X\subset \R^d$ is a nondegenerate continuum, it holds that $\Psi-\wtan(X,x)\subset \mathfrak{C}_U(\R^d;{\bf 0})$ for each point $x\in X$. Note that in Theorem \ref{thm0}, we obtain a result in terms of \emph{pseudotangents}, not tangents. The difference is articulated precisely in Section \ref{prelim}, but for the time being the reader should notice that Lipschitz curves may behave very differently with respect to pseudotangents as opposed to with respect to tangents. In particular, in Example \ref{ex2} we show that there is a nondegenerate Lipschitz curve where \emph{no point} admits a unique pseudotangent, let alone a unique pseudotangent line, while by contrast it is known that a Lipschitz curve must, $\mathcal{H}^1$-almost everywhere, admit a unique \emph{tangent}, and that the unique tangent must be a line. In light of this, we are led to the following question, to which we (perhaps ambitiously) conjecture that the answer is ``yes".
\begin{question}\label{q1}
For each integer $d\geq 2$, does there exist a Lipschitz curve $\Gamma\subset \R^d$ such that for every point $x\in\Gamma$, $\Psi-\wtan(\Gamma,x)=\mathfrak{C}_U(\R^d;{\bf 0})$?
\end{question}

Our strategy for proving Theorem \ref{thm0} involves constructing a Lipschitz capture $G=g([0,1])$ of the set $K$ such that there is a countable set $\{z_k\}_{k\in\N}\subset G\setminus K$ for which for all $k\in\N$, $\wtan(G,z_k)=\mathfrak{C}_U(\R^d;{\bf 0})$ and such that $K\subset \overline{\{z_k\}}_{k\in\N}$. We then prove as a consequence that all of the points $z\in K$, $\Psi-\wtan(G,z)=\mathfrak{C}_U(\R^d;{\bf 0})$; indeed the proof of this fact does not rely on the point $z$ being in $K$, it depends only on $z$ being contained in the closure $\overline{\{z_k\}}_{k\in\N}$. By this argument, we do obtain a stronger result about sets of points in Lipschitz curves on which every tangent is attained than we found in \cite[Appendix A]{QSS}; we improve from showing that there is a Lipschitz curve with \emph{a} point that admits every possible tangent to showing that there is a Lipschitz curve with \emph{a countable set} of points admitting every possible tangent, and that furthermore this countable set can accumulate to an uncountable set.

\begin{rem}\label{tanrem}
Note the following two facts about tangents and pseudotangents, the proofs of which are left to the interested reader.
\begin{enumerate}
\item For any non-empty closed set $K$ and any point $x\in K$, $\wtan(K,x)\subset\Psi-\wtan (K,x)$.
\item If $(x_i)_{i\in\N}\subset K$ converges to a point $x\in K$ and if $T\in\wtan(K,x_i)$ for each $i$, then $T\in\Psi-\wtan(K,x)$ as well.
\end{enumerate}
\end{rem}
Throughout this article, $H$ will denote the set of the same name as in \cite[Appendix A]{QSS}, to which we refer the reader for the details of its construction. The critical properties of $H$ for this article are that $H$ is a finite-length continuum, so by \cite[Theorem 4.4]{AO} it is a Lipschitz curve, which contains ${\bf 0}$, is contained in $[-1,1]^d$, and for which $\wtan(H,{\bf 0})=\mathfrak{C}_U(\R^d;{\bf 0})$.
\section{Results}\label{results}

Further, note that the following result holds, with proof exactly the same as the proof of \cite[Lemma 2.5]{QSS}, up to replacing instances of $x$ with $x_n$ instead (to move from tangents to pseudotangents). Structurally the proofs are identical and all of the arguments still hold \emph{mutatis mutandis}.
\begin{proposition}\label{prop:alltan}
Let $d\geq 2$ be an integer and let $X\subset \R^d$ be a nondegenerate continuum. Then for every point $x\in X$, we have that $\Psi-\wtan(X,x)\subset \mathfrak{C}_U(\R^d;{\bf 0})$.
\end{proposition}
This means that the pseudotangent collection given in Theorem \ref{thm0} is maximal at all of the points of $K$. The proof of Theorem \ref{thm0} follows after several lemmata about the geometry of Lipschitz captures of the set $K$. Fix, for the remainder of this section, a compact, uniformly disconnected, perfect set $K\subset\R^d$ which admits a Lipschitz capture $G_0=g_0([0,1])$. By uniform disconnectedness, let $\lambda>0$ such that for every pair of distinct points $x,y\in K$, if $\{p_0,p_1,\dots,p_n\}\subset K$ with $p_0=x$ and $p_n=y$, then there exists some $i\in\{0,1,\dots,n-1\}$ for which $|p_{i+1}-p_i|>\lambda|x-y|$.

\begin{lemma}\label{lem2}
For every $x\in K$, there exists a Lipschitz capture $F=f([0,1])$ of $K$ such that
\begin{equation}\label{eqn2}
\Psi-\wtan(F,x)=\mathfrak{C}_U(\R^d;{\bf 0}).
\end{equation}
Moreover, if $G=g([0,1])$ is any Lipschitz capture of $K$ and $r,\delta>0$ are any positive numbers, then we can find a Lipschitz capture $F=f([0,1])$ of $K$ satisfying (\ref{eqn2}) and further satisfying $F\setminus B(x,r)=G\setminus B(x,r)$ and $\mathcal{H}^1(F)-\mathcal{H}^1(G)< \delta$.
\end{lemma}
\begin{proof}
Fix a point $x\in K$ and let $(y_n)_{n\in\N}\subset K\setminus\{x\}$ be a sequence of (distinct) points with $\lim_{n\to\infty} y_n=x$ and with $\sum_{n\in\N}|x-y_n|<\infty$. Let $G=g([0,1])$ be a Lipschitz capture of $K$. For each $n\in\N$, note that $\dist(g^{-1}(\{x\}),g^{-1}(\{y_n\}))=:\eta_n>0$ with $\eta_n\to 0$ as $n\to\infty$, so let $a_n\in g^{-1}(\{x\})$ and $b_n\in g^{-1}(\{y_n\})$ realize $\eta_n=|a_n-b_n|$. Then by Lemma \ref{lem1}, for each $n\in\N$, there exists a value $\zeta_n\in(\min(\{a_n,b_n\}),\max(\{a_n,b_n\}))$ with $\dist(g(\zeta_n),K)> \frac{1}{4}\lambda|x-y_n|$ and such that $g(\zeta_n)$ is a point of $\mathcal{H}^1$ density $1$ in $G$.

First, observe that the Hausdorff limit $\lim_{n\to\infty} G_n=:G$ exists, $K\subset G$, and $G$ is a continuum of finite length by Go{\l}ab's Semicontinuity Theorem, so it is a Lipschitz capture of $K$ by \cite[Theorem 4.4]{AO} (see this same paper for discussion of Go{\l}ab's Semicontinuity Theorem as well). We now claim that for every $n\in\N$, $\Psi-\wtan (G,x_n)=\mathfrak{C}_U(\R^d;{\bf 0})$.

For natural numbers $k\geq 2$, let $C_k\subset [0,1/k^2]$ be a self-similar Cantor set of Hausdorff dimension $\log(k)/\log(k+1)$. Then by self-similarity, $\dim_A(C_k)=\log(k)/\log(k+1)$ as well. In $\R^d$, for $d\geq 2$, let
\[K:=\{{\bf 0}\}\cup\left(\bigcup_{k\geq 2} \{0\}^{d-2}\times \{1/k\}\times C_k\right).\]
Then $K\subset \R^d$ is compact, $\dimh(K)=1$, $\mathcal{H}^1(K)=0$, and $K$ admits a Lipschitz capture $F$ by applying Theorem \ref{thm0} to each $C_k$ in such a way that the curves do not intersect and such that the sum of the lengths of the curves is finite, then adding also the segment $[0,1]\times\{0\}$. Then for all $x\in K$, $\Psi-\wtan (F,x)=\mathfrak{C}_U(\R^d;{\bf 0})$.

Here and throughout, we use the notation $\wtan (X,x)$ to denote the set of \emph{tangents} to the set $X$ at the point $x$, $\Psi-\wtan (X,x)$ to denote the set of \emph{pseudotangents} to the set $X$ at the point $x$, and $\mathfrak{C}_U(\mathbb{R}^d;{\bf 0})$ to denote the collection of all closed subsets of $\mathbb{R}^d$ which contain the origin and have only unbounded components, see Section \ref{prelim} for the details of these sets. In Proposition \ref{prop:alltan}, we prove that as long as $X\subset \mathbb{R}^d$ is a nondegenerate continuum, it holds that $\Psi-\wtan(X,x)\subset \mathfrak{C}_U(\mathbb{R}^d;{\bf 0})$ for each point $x\in X$. Note that in Theorem \ref{thm0}, we obtain a result in terms of \emph{pseudotangents}, not tangents. The difference is articulated precisely in Section \ref{prelim}, but for the time being the reader should notice that Lipschitz curves may behave very differently with respect to pseudotangents as opposed to with respect to tangents. In particular, in Example \ref{ex2} we show that there is a nondegenerate Lipschitz curve where \emph{no point} admits a unique pseudotangent, let alone a unique pseudotangent line, while by contrast it is known that a Lipschitz curve must, $\mathcal{H}^1$-almost everywhere, admit a unique \emph{tangent}, and that the unique tangent must be a line. In light of this, we are led to the following question, to which we (perhaps ambitiously) conjecture that the answer is ``yes".
\begin{question}\label{q1}
For each integer $d\geq 2$, does there exist a Lipschitz curve $\Gamma\subset \mathbb{R}^d$ such that for every point $x\in\Gamma$, $\Psi-\wtan(\Gamma,x)=\mathfrak{C}_U(\mathbb{R}^d;{\bf 0})$?
\end{question}

Our strategy for proving Theorem \ref{thm0} involves constructing a Lipschitz capture $G=g([0,1])$ of the set $K$ such that there is a countable set $\{z_k\}_{k\in\mathbb{N}}\subset G\setminus K$ for which for all $k\in\mathbb{N}$, $\wtan(G,z_k)=\mathfrak{C}_U(\mathbb{R}^d;{\bf 0})$ and such that $K\subset \overline{\{z_k\}}_{k\in\mathbb{N}}$. We then prove as a consequence that all of the points $z\in K$, $\Psi-\wtan(G,z)=\mathfrak{C}_U(\mathbb{R}^d;{\bf 0})$; indeed the proof of this fact does not rely on the point $z$ being in $K$, it depends only on $z$ being contained in the closure $\overline{\{z_k\}}_{k\in\mathbb{N}}$. By this argument, we do obtain a stronger result about sets of points in Lipschitz curves on which every tangent is attained than we found in \cite[Appendix A]{QSS}; we improve from showing that there is a Lipschitz curve with \emph{a} point that admits every possible tangent to showing that there is a Lipschitz curve with \emph{a countable set} of points admitting every possible tangent, and that furthermore this countable set can accumulate to an uncountable set.

\subsection{Acknowledgments}
We would like to thank Jeremy Tyson for many helpful conversations giving this project direction, as well as for his commentary on an early draft. Additionally, we thank Vyron Vellis for suggesting Example \ref{ex1} and its inclusion in the final manuscript.

The post-theorem introduction gives the following strategy to prove Theorem~\ref{thm0}. Construct a Lipschitz capture $G=g([0,1])$ of $K$ with a countable set $\{z_k\}_{k\in\mathbb{N}}\subset G\setminus K$ such that for all $k$, one has $\wtan(G,z_k)=\mathfrak{C}_U(\mathbb{R}^d;{\bf 0})$, and such that $K\subset \overline{\{z_k\}}_{k\in\mathbb{N}}$. Then show that for every point $z$ in the closure $\overline{\{z_k\}}_{k\in\mathbb{N}}$ (in particular for all $z\in K$), it follows that $\Psi-\wtan(G,z)=\mathfrak{C}_U(\mathbb{R}^d;{\bf 0})$; the argument "does not rely on the point $z$ being in $K$, it depends only on $z$ being contained in the closure."

The post-theorem introduction gives the following strategy to prove the main theorem. Construct a Lipschitz capture $G=g([0,1])$ of $K$ with a countable set $\{z_k\}_{k\in\mathbb{N}}\subset G\setminus K$ such that for all $k$, one has $\wtan(G,z_k)=\mathfrak{C}_U(\mathbb{R}^d;{\bf 0})$, and such that $K\subset \overline{\{z_k\}}_{k\in\mathbb{N}}$. Then show that for every point $z$ in the closure $\overline{\{z_k\}}_{k\in\mathbb{N}}$ (in particular for all $z\in K$), it follows that $\Psi-\wtan(G,z)=\mathfrak{C}_U(\mathbb{R}^d;{\bf 0})$; the argument "does not rely on the point $z$ being in $K$, it depends only on $z$ being contained in the closure."

Question: Let \(d\ge 2\) be an integer, and let \(K\subset \mathbb{R}^d\) be compact and uniformly disconnected, meaning that there exists \(\lambda>0\) such that for every distinct \(x,y\in K\) and every finite chain \(\{p_0,p_1,\dots,p_n\}\subset K\) with \(p_0=x\) and \(p_n=y\), there is some \(i\in\{0,\dots,n-1\}\) for which \(|p_{i+1}-p_i|>\lambda |x-y|\). Assume also that \(K\) admits a Lipschitz capture, i.e. there exists a Lipschitz map \(f:[0,1]\to \mathbb{R}^d\) whose image contains \(K\). For a set \(X\subset \mathbb{R}^d\) and a point \(x\in X\), let \(\Psi\!-\mathrm{tan}(X,x)\) denote the set of pseudotangents to \(X\) at \(x\) (pointed blow-up limits with base points tending to \(x\)), and let \(\mathfrak{C}_U(\mathbb{R}^d;\mathbf{0})\) be the collection of all closed subsets of \(\mathbb{R}^d\) that contain the origin and whose connected components are all unbounded. Under these assumptions, which existence statement holds?

There exists a Lipschitz capture \(F=f([0,1])\) of \(K\) such that for every \(x\in K\), \(\Psi\!-\mathrm{tan}(F,x)=\mathfrak{C}_U(\mathbb{R}^d;\mathbf{0})\).

Choices:
B. There exists a Lipschitz capture \(F=f([0,1])\) of \(K\) such that for every \(x\in F\), \(\Psi\!-\mathrm{tan}(F,x)=\mathfrak{C}_U(\mathbb{R}^d;\mathbf{0})\).
C. There exists a Lipschitz capture \(F=f([0,1])\) of \(K\) such that for every \(x\in K\), one has \(\mathfrak{C}_U(\mathbb{R}^d;\mathbf{0})\subseteq \Psi\!-\mathrm{tan}(F,x)\).
D. For every Lipschitz capture \(F=f([0,1])\) of \(K\) and every \(x\in K\), \(\Psi\!-\mathrm{tan}(F,x)=\mathfrak{C}_U(\mathbb{R}^d;\mathbf{0})\).
E. There exists a Lipschitz capture \(F=f([0,1])\) of \(K\) such that for every \(x\in K\), \(\mathrm{tan}(F,x)=\mathfrak{C}_U(\mathbb{R}^d;\mathbf{0})\).

The stem does not reveal the correct option explicitly or through obvious wording cues. The correct answer must be identified by comparing subtle differences in quantifiers, scope, and notions of tangent versus pseudotangent.

This is essentially a theorem-statement recognition item: the stem gives the hypotheses and asks which existence statement holds, with the correct choice closely matching the theorem’s conclusion.

Some reasoning is needed to distinguish equality from inclusion, existence from universality, and points in K versus all points in F. However, the item mainly tests precise recall of the theorem rather than deeper generative mathematical reasoning.

The distractors are strong and mathematically meaningful: they vary quantifier scope, weaken the conclusion, overgeneralize the capture, or confuse tangents with pseudotangents. These reflect realistic failure modes.

Well-constructed in terms of distractor quality and lack of answer leakage, but it is largely a theorem-recall question and therefore only moderately tests genuine reasoning.


\begin{prop}[{cf.~\cite[Corollary~3.4]{S78}}]\label{prop:ci}
Let $R$ be a standard graded $\kk$-algebra.
If $R$ is a complete intersection, then the $h$-polynomial of $R$ is of type~CI, that is, $h_R(t)=\prod_{i=1}^r\Psi_{m_i}(t)$ for some non-negative integer $r$ and positive integers $m_1, \dots, m_r$.
\end{prop}

Throughout this paper, let $\Bbbk$ denote a field. We say that $R$ is a \emph{standard graded} $\Bbbk$-algebra
if $R = \bigoplus_{n \ge 0} R_n$ is a commutative graded $\Bbbk$-algebra with $R_0 = \Bbbk$ and finitely generated in degree one.
(A standard graded $\Bbbk$-algebra is also called a \emph{homogeneous} $\Bbbk$-algebra in other literature.)

\subsection{Background}
The Hilbert series of a (not necessarily commutative) graded $\Bbbk$-algebra is one of the most fundamental invariants in the study of graded $\Bbbk$-algebras.
It provides a concise analytic expression encoding the dimensions of the homogeneous components and thus captures the growth and structural features of the $\Bbbk$-algebra as a $\Bbbk$-vector space.
It is a classical and well-known fact that if a standard graded $\Bbbk$-algebra is Cohen-Macaulay, then the numerator polynomial of its Hilbert series, called the \textit{$h$-polynomial}, has all positive coefficients.
This fact is one of the cornerstones in graded commutative algebra.
Furthermore, several algebraic properties can be effectively characterized in terms of the Hilbert series.
A prominent theorem is Stanley's characterization of the Gorenstein property via the symmetry of the Hilbert series (\cite{S78}),
which has inspired subsequent studies establishing necessary conditions for generalized notions of Gorensteinness,
such as almost Gorensteinness (\cite{H}), nearly Gorensteinness (\cite{M}), and Artin-Schelter Gorensteinness (\cite{JZ}).

It is also known that when a standard graded $\Bbbk$-algebra is a complete intersection, its Hilbert series takes a particularly simple form.
In fact, for a positive integer $m$, let \[\Psi_m(t)=1+t+\cdots+t^{m-1}.\]
Then the $h$-polynomial of a standard graded complete intersection is of the form of a product of a finite collection of $\Psi_m(t)$'s (see Proposition~\ref{prop:ci}).
In addition, the Hilbert series also serves as an essential tool in investigating noncommutative graded complete intersections (see \cite{KKZ, KKZ2}).

\subsection{Cyclotomic standard graded $\Bbbk$-algebras and polynomials of type CI}

In recent years, increasing attention has been directed to the finer analytic behavior of the Hilbert series of standard graded $\Bbbk$-algebras, particularly the roots of their $h$-polynomials.
These roots encode subtle structural information and sometimes admit striking combinatorial interpretations
(see, e.g., the survey \cite{Br} for combinatorial consequences of the real-rootedness of the numerator polynomials of generating series).

Motivated by this perspective, we focus on the following notion for standard graded $\Bbbk$-algebras:

\begin{dfn}[cf. \cite{BD, KKZ}]
A standard graded $\Bbbk$-algebra $R$ is called \textit{cyclotomic} if all roots of its $h$-polynomial $h_R(t)$ lie on the unit circle in the complex plane.
\end{dfn}

Monic polynomials with integer coefficients whose roots all lie on the unit circle are sometimes called \textit{Kronecker polynomials}.
It is known that every such polynomial can be expressed as a product of cyclotomic polynomials.
Note that the $h$-polynomials of cyclotomic $\Bbbk$-algebras are \textit{not necessarily cyclotomic polynomials themselves}, but rather products of cyclotomic polynomials.
We say that a polynomial is \textit{of type CI} if it can be written as a product of $\Psi_m(t)$'s.
In particular, the $h$-polynomials of standard graded complete intersections are of type CI.
However, the converse does not hold in general; see, for example, \cite[Example~3.7]{S78}.
Borzi and D'Al\`i showed that the converse does hold when $R$ is Koszul~\cite[Theorem~3.9]{BD}.
Moreover, it was proved that a cyclotomic polynomial $\Phi_m(t)$ coincides with $h_R(t)$ of some standard graded $\Bbbk$-algebra $R$ if and only if $m$ is prime and $R$ is a hypersurface~\cite[Theorem~4.3]{BD}.

Despite these results, in general, one cannot expect the ring-theoretic structure of $R$ to be determined by the cyclotomicity of its $h$-polynomial.
This naturally leads to the following question:
\begin{q}
Which polynomials can occur as the $h$-polynomials of cyclotomic standard graded $\Bbbk$-algebras, other than those of type CI?
\end{q}
For example, \cite[Theorem~4.3]{BD} indicates that a single cyclotomic polynomial $\Phi_m(t)$ is rarely equal to the $h$-polynomial of a standard graded $\Bbbk$-algebra.

\subsection{Main Results}

The aim of this paper is to make a contribution to this question.
Our main result is stated as follows.

\subsection{Background}
The Hilbert series of a (not necessarily commutative) graded $\Bbbk$-algebra is one of the most fundamental invariants in the study of graded $\Bbbk$-algebras.
It provides a concise analytic expression encoding the dimensions of the homogeneous components and thus captures the growth and structural features of the $\Bbbk$-algebra as a $\Bbbk$-vector space.
It is a classical and well-known fact that if a standard graded $\Bbbk$-algebra is Cohen-Macaulay, then the numerator polynomial of its Hilbert series, called the \textit{$h$-polynomial}, has all positive coefficients.
This fact is one of the cornerstones in graded commutative algebra.
Furthermore, several algebraic properties can be effectively characterized in terms of the Hilbert series.
A prominent theorem is Stanley's characterization of the Gorenstein property via the symmetry of the Hilbert series (\cite{S78}),
which has inspired subsequent studies establishing necessary conditions for generalized notions of Gorensteinness,
such as almost Gorensteinness (\cite{H}), nearly Gorensteinness (\cite{M}), and Artin-Schelter Gorensteinness (\cite{JZ}).

\begin{dfn}[cf. \cite{BD, KKZ}]
A standard graded $\Bbbk$-algebra $R$ is called \textit{cyclotomic} if all roots of its $h$-polynomial $h_R(t)$ lie on the unit circle in the complex plane.
\end{dfn}

Monic polynomials with integer coefficients whose roots all lie on the unit circle are sometimes called \textit{Kronecker polynomials}.
It is known that every such polynomial can be expressed as a product of cyclotomic polynomials.
Note that the $h$-polynomials of cyclotomic $\Bbbk$-algebras are \textit{not necessarily cyclotomic polynomials themselves}, but rather products of cyclotomic polynomials.
We say that a polynomial is \textit{of type CI} if it can be written as a product of $\Psi_m(t)$'s.
In particular, the $h$-polynomials of standard graded complete intersections are of type CI.
However, the converse does not hold in general; see, for example, \cite[Example~3.7]{S78}.
Borzi and D'Al\`i showed that the converse does hold when $R$ is Koszul~\cite[Theorem~3.9]{BD}.
Moreover, it was proved that a cyclotomic polynomial $\Phi_m(t)$ coincides with $h_R(t)$ of some standard graded $\Bbbk$-algebra $R$ if and only if $m$ is prime and $R$ is a hypersurface~\cite[Theorem~4.3]{BD}.

Despite these results, in general, one cannot expect the ring-theoretic structure of $R$ to be determined by the cyclotomicity of its $h$-polynomial.
This naturally leads to the following question:
\begin{q}
Which polynomials can occur as the $h$-polynomials of cyclotomic standard graded $\Bbbk$-algebras, other than those of type CI?
\end{q}
For example, \cite[Theorem~4.3]{BD} indicates that a single cyclotomic polynomial $\Phi_m(t)$ is rarely equal to the $h$-polynomial of a standard graded $\Bbbk$-algebra.

\subsection{Main Results}

The aim of this paper is to make a contribution to this question.
Our main result is stated as follows.

\begin{abstract}
We call a standard graded commutative $\kk$-algebra \textit{cyclotomic} if its $h$-polynomial has all its roots on the unit circle in the complex plane.
Complete intersections provide typical examples of cyclotomic algebras,
since the $h$-polynomial of any standard graded complete intersection is a product of polynomials of the form $1 + t + \cdots + t^{m-1}$.
We refer to such polynomials as being \textit{of type CI}.
A natural question is whether there exists a cyclotomic standard graded $\kk$-algebra whose $h$-polynomial is not of type CI.
In this paper, we give a partial answer to this question.
We show that the $h$-polynomial $h_R(t)$ of a cyclotomic standard graded $\kk$-algebra $R$ is of type CI whenever $h_R(1) \in \{1, 4, 6\}$ or $h_R(1)$ is prime.
On the other hand, if $n \ge 8$ and $n$ is not prime, then there exists a cyclotomic standard graded $\kk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1) = n$.
\end{abstract}

Monic polynomials with integer coefficients whose roots all lie on the unit circle are sometimes called \textit{Kronecker polynomials}.
It is known that every such polynomial can be expressed as a product of cyclotomic polynomials.
Note that the $h$-polynomials of cyclotomic $\kk$-algebras are \textit{not necessarily cyclotomic polynomials themselves}, but rather products of cyclotomic polynomials.
We say that a polynomial is \textit{of type CI} if it can be written as a product of $\Psi_m(t)$'s.
In particular, the $h$-polynomials of standard graded complete intersections are of type CI.
However, the converse does not hold in general; see, for example, \cite[Example~3.7]{S78}.
Borzi and D'Al\`i showed that the converse does hold when $R$ is Koszul~\cite[Theorem~3.9]{BD}.
Moreover, it was proved that a cyclotomic polynomial $\Phi_m(t)$ coincides with $h_R(t)$ of some standard graded $\kk$-algebra $R$ if and only if $m$ is prime and $R$ is a hypersurface~\cite[Theorem~4.3]{BD}.

The aim of this paper is to make a contribution to this question.
Our main result is stated as follows.

Note that $h_R(1)$ is nothing but the multiplicity of $R$ (see \cite[Proposition 4.1.9]{BH}), which is one of the important invariants of graded modules.

\section{Proof of Theorem~\ref{thm:main} (2)}\label{sec:(2)}
The goal of this section is to give a proof of Theorem~\ref{thm:main} (2).
Specifically, for any non-prime $n$ with $n \geq 8$, we construct a cyclotomic standard graded $\kk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$.
Such $\kk$-algebras are realized as Stanley-Reisner rings of certain simplicial complexes.

We prove the following theorem, which implies Theorem~\ref{thm:main} (2).
\begin{thm}\label{thm:p=3}
{\em (1)} Let $q \ge 5$ be an odd integer.
Then there exists a cyclotomic standard graded $\kk$-algebra $R$ such that $h_R(t)$ is not of type CI with $h_R(1)=2q$.

\noindent
{\em (3)} Let $5 \le p \le q$ be odd integers.
Then there exists a cyclotomic standard graded $\kk$-algebra $R$ such that $h_R(t)$ is not of type CI with $h_R(1)=pq$.
\end{thm}

We postpone the proof of this theorem. So far, assume Theorem~\ref{thm:p=3} holds. Then we can prove the desired result. In fact,
let $R$ be a cyclotomic standard graded $\kk$-algebra whose $h$-polynomial is not of type CI.
Then, for any positive integer $a$, the standard graded $\kk$-algebra $R'=R[x]/(x^a)$ has the $h$-polynomial $h_{R'}(t)$ which is not of type CI and $h_{R'}(1)=a\cdot h_R(1)$.
(In fact, we have $h_{R'}(t)=h_R(t)\Psi_a(t)$.)

\begin{thm}[Main Result]\label{thm:main}
{\em (1)} The $h$-polynomial $h_R(t)$ of any cyclotomic standard graded $\kk$-algebra $R$ is of type CI if $h_R(1) \in \{1,4,6\}$ or $h_R(1)$ is prime.

\noindent
{\em (2)} There exists a cyclotomic standard graded $\kk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1)=n$ if $n$ is a non-prime number at least $8$.
\end{thm}

Note that $h_R(1)$ is nothing but the multiplicity of $R$ (see \cite[Proposition 4.1.9]{BH}), which is one of the important invariants of graded modules.

\subsection{Structure of the paper}
We briefly describe the structure of this paper.
In Section~\ref{sec:pre}, we fix our notation (Hilbert series, $h$-polynomials, simplicial complexes, etc.),
review fundamental notions (cyclotomic polynomials, Stanley-Reisner theory, and related topics),
and recall classical results, namely Macaulay's theorem and the $g$-theorem, which will be used in the proof of Theorem~\ref{thm:main}.
Sections~\ref{sec:(1)} and~\ref{sec:(2)} are devoted to the proofs of parts~(1) and~(2) of Theorem~\ref{thm:main}, respectively.

The post-theorem introduction gives only a high-level roadmap: it says that Section~\ref{sec:pre} recalls classical results—"namely Macaulay's theorem and the $g$-theorem"—"which will be used in the proof of Theorem~\ref{thm:main}," and that Sections~\ref{sec:(1)} and~\ref{sec:(2)} contain the proofs of parts~(1) and~(2) of Theorem~\ref{thm:main}, respectively.

The post-theorem introduction gives only a high-level roadmap: it says that next it recalls classical results—"namely Macaulay's theorem and the $g$-theorem"—"which will be used in establishing the main theorem," and that later it contains the proofs of parts~(1) and~(2) of the main theorem, respectively.

[Main Result]\label{thm:main}
{\em (1)} The $h$-polynomial $h_R(t)$ of any cyclotomic standard graded $\Bbbk$-algebra $R$ is of type CI if $h_R(1) \in \{1,4,6\}$ or $h_R(1)$ is prime.

\noindent
{\em (2)} There exists a cyclotomic standard graded $\Bbbk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1)=n$ if $n$ is a non-prime number at least $8$.,

Question: Let $R$ be a standard graded commutative $\Bbbk$-algebra with Hilbert series of the form $H_R(t)=\dfrac{h_R(t)}{(1-t)^d}$, where $h_R(t)$ is its $h$-polynomial. Call $R$ cyclotomic if all roots of $h_R(t)$ lie on the unit circle in $\mathbb{C}$. A polynomial is said to be of type CI if it can be written as a product of polynomials $\Psi_m(t)=1+t+\cdots+t^{m-1}$. Which statement holds for such algebras and $h$-polynomials?

For every cyclotomic standard graded $\Bbbk$-algebra $R$, if $h_R(1)\in\{1,4,6\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI. Moreover, for every non-prime integer $n\ge 8$, there exists a cyclotomic standard graded $\Bbbk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$.

Choices:
B. For every cyclotomic standard graded $\Bbbk$-algebra $R$, if $h_R(1)\in\{1,4,6,8\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI. Moreover, for every non-prime integer $n\ge 9$, there exists a cyclotomic standard graded $\Bbbk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$.
C. For every cyclotomic standard graded $\Bbbk$-algebra $R$, if $h_R(1)\in\{1,4,6\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI.
D. For every cyclotomic standard graded $\Bbbk$-algebra $R$, if $h_R(1)\in\{1,4,6\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI. Moreover, there exists a non-prime integer $N\ge 8$ such that for every non-prime integer $n\ge N$, there exists a cyclotomic standard graded $\Bbbk$-algebra $R$ with $h_R(t)$ not of type CI and $h_R(1)=n$.
E. For every cyclotomic standard graded $\Bbbk$-algebra $R$, if $h_R(1)\in\{1,4,6\}$ or $h_R(1)$ is composite, then $h_R(t)$ is of type CI. Moreover, for every prime integer $n\ge 8$, there exists a cyclotomic standard graded $\Bbbk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$.

sharp exceptional values and threshold $n\ge 8$

dropped the existence clause in part (2)

uniform bound replaced exact range of all non-prime $n\ge 8$

prime/non-prime dichotomy in both directions

The stem only defines the terms and asks which statement is true; it does not reveal the specific exceptional values, quantifiers, or existence claim appearing in the correct option.

The item is largely theorem-recognition: the correct answer is essentially the exact statement of a result. However, it is not a pure tautology because the options introduce meaningful competing variants with altered thresholds and quantifiers.

Selecting the best answer requires moderate reasoning or careful recall, especially to track the sharp set {1,4,6}, the prime condition, and the exact existence clause for every non-prime n >= 8. Still, it does not require constructing a proof or applying the theorem in a new setting.

The distractors are strong: they are close to the target statement, differ in mathematically meaningful ways, and reflect common failure modes such as weakening the conclusion, shifting thresholds, or confusing prime/composite cases.

A solid MCQ with little answer leakage and strong distractors, but it mainly tests precise theorem recall/comparison rather than deeper generative mathematical reasoning.