Question:
Consider the Wythoff-type game on positions (x,y) ∈ ℕ² in which a move from (x,y) consists of replacing it by (x-n,y), (x,y-n), or (x-n,y-n), where n is any positive integer allowed by the coordinates. The terminal positions are exactly those with x+y ≤ 2, and under normal play a P-position means a losing position. Let φ=(1+√5)/2, and define g:ℕ→ℕ by g(0)=1, g(1)=0, and for n≥0,

 g(n)=1-g(m) if floor(nφ)=floor(m(φ+1))+1 for some m≥0, and g(n)=1 otherwise.

Which statement gives exactly the set of P-positions of this game?

Choices:
B.
The set of P-positions is exactly

{ (⌊nφ⌋+g(n), ⌊nφ²⌋+g(n)) : n≥0 } ∪ { (⌊nφ²⌋+g(n), ⌊nφ⌋+g(n)) : n≥0 }.
C.
Every position of the form

(⌊nφ⌋+g(n)-1, ⌊nφ²⌋+g(n))

with n≥0 is a P-position.
D.
The set of P-positions is exactly

{ (⌊nφ⌋+g(n)-1, ⌊nφ²⌋+g(n)) : n≥1 } ∪ { (⌊nφ²⌋+g(n), ⌊nφ⌋+g(n)-1) : n≥1 }.
E.
The set of P-positions is exactly

{ (⌊nφ⌋+g(n)-1, ⌊nφ²⌋+g(n)) : n≥0 }.

The stem does not explicitly state the correct option or uniquely reveal choice A. It gives the game and the auxiliary function g, but the exact P-position description still has to be identified from the alternatives.

This is close to a theorem-statement recognition question: the task is essentially to pick the exact characterization of the P-positions. However, it is not a pure restatement, since the options differ in meaningful ways such as symmetry, boundary indexing, and offset terms.

There is some reasoning pressure because the student must distinguish an exact characterization from weaker true statements and near-miss formulas. Still, the item mainly rewards theorem recall/recognition rather than substantial generative derivation.

The distractors are mathematically plausible and target realistic failure modes: dropping the converse, missing symmetry, mishandling the n=0 boundary case, and omitting the offset adjustment. They are distinct and well aligned with common mistakes.

A solid MCQ with strong distractors and no answer leakage, but it leans toward theorem recognition rather than deep generative reasoning.


\begin{thm}[Kleiner--Leeb] \label{thm-klsph}
Let $B$ be a spherical building, let $C\subset B$ be a closed, convex, spherical subset. Then there exists an apartment $A\subset B$ which contains $C$.
\end{thm}

Euclidean buildings were introduced by Jacques Tits in \cite{Tits} and play an important role in different areas of mathematics. Their metric structure played a central role in super-rigidity results starting with \cite{KleinerLeeb1}, \cite{GS}. We refer to \cite{Linus},
 \cite{KLM} for further investigations of their metric properties and to \cite{BM} and the literature therein for super-rigidity questions related to Euclidean buildings.

The following result, possibly known to Bruce Kleiner and Bernhard Leeb, and verified in \cite{KleinerLeeb1} in a special case, has been discussed in other special cases in several independent works. In this note we provide a proof of this result:

The following result, possibly known to Bruce Kleiner and Bernhard Leeb, and verified in \cite{KleinerLeeb1} in a special case, has been discussed in other special cases in several independent works. In this note we provide a proof of this result:

All Euclidean buildings in this note, in particular, in Theorem \ref{thm-main}, are assumed to be complete, cf. Remark \ref{rem: non-complete} below for a comment about possible generalizations.

\begin{rem} \label{rem: non-complete}
 Here is the only place in this paper, where we discuss non-complete buildings, we refer to \cite{Parreau}, \cite{Str} for their discussion.
 In any non-complete Euclidean building, Theorem \ref{thm-main} cannot hold as stated. Indeed, any such building $X$ contains a (half-open) geodesic $\gamma:[0,1) \to X$, whose image $C$ is closed as a subset of $X$. Such $C$ cannot be contained in an apartment. However, an extension of Theorem \ref{thm-main} to non-complete Euclidean buildings and to
 $\Lambda$-buildings (cf. \cite{Schwer}) seems possible, we hope to discuss it in a forthcoming work.
\end{rem}

\begin{cor}
Let $C\subset B$ be a closed, convex, spherical subset in a spherical building $B$. Let $x\in C$ be arbitrary and let $A_x\subset \Sigma _x B$ be an apartment in the space of directions such that $\Sigma _xC\subset A_x$. Then there exists an apartment $A\subset B$ which contains $C$ and such that $\Sigma _xA=A_x$.
\end{cor}

\begin{cor} \label{cor: cone}
Let $C$ be a flat, closed, convex subset in a Euclidean building $X$. Assume that $C$ is a Euclidean cone with its tip at the point $x$. Let $A_x$ be an apartment in the tangent cone $T_xX$, which contains $T_xC$. Then $C$ is contained in an apartment $A$, such that $T_xA=A_x$.
\end{cor}

\begin{cor}
Let $F\subset X$ be a convex subset isometric to a Euclidean space in a Euclidean building $X$. Then for every $x\in F$ the tangent space $T_x(P(F))$ at $x$ to the parallel set $P(F)$ of $F$ coincides with the parallel set $P(T_xF)$ of the flat $T_xF$ in the building $T_xX$.
\end{cor}

\begin{lem} \label{lem: final}
 Let $X$ be a CAT(0) space. Let $Y_1,Y_2 $ be closed, convex subsets of $X$ isometric to convex subsets of some Euclidean spaces. Let $x\in Y_1\cap Y_2$ be a point and assume that
 \begin{enumerate}
 \item $T:={\operatorname{Log}}_x (Y_1) \cup {\operatorname{Log}}_x (Y_2) \subset T_xX \,$ is a convex subset of $T_xX$,
 \item ${\operatorname{Log}}_x (Y_1) \cap {\operatorname{Log}}_x (Y_2) ={\operatorname{Log}}_x (Y_1\cap Y_2)\,.$
 \end{enumerate}
 Then $Y:=Y_1\cup Y_2$ is a convex subset of $X$ isometric to $T$.
\end{lem}

Choose an arbitrary interior point $x$ of the facet $\mathcal F$. Then $T_x \mathcal P$ is a $k$-dimensional, flat halfspace in the tangent space $T_xX$. By Corollary \ref{cor: cone}, we find an apartment $A^x$ in the (infinitesimal) Euclidean building $T_xX$ with $T_x \mathcal P \subset A^x$.

\begin{rem} \label{rem: non-complete}
	Here is the only place in this paper, where we discuss non-complete buildings, we refer to  \cite{Parreau}, \cite{Str} for their discussion.
	In any non-complete Euclidean building, Theorem \ref{thm-main} cannot hold as stated. Indeed, any such building $X$ contains a (half-open) geodesic $\gamma:[0,1) \to X$, whose image $C$ is closed as a subset of $X$.  Such $C$ cannot be contained in an apartment. 	However, an extension  of Theorem \ref{thm-main} to non-complete Euclidean buildings and to
	 $\Lambda$-buildings (cf. \cite{Schwer})  seems possible, we hope to discuss it in a forthcoming work.
\end{rem}

\begin{thm} \label{thm-main}
	Let $X$ be a Euclidean  building.  Let $C\subset X$ be a convex subset of $X$
	which is isometric to a convex subset of some Euclidean space.  Then $C$ is contained in an apartment $A$  of $X$.

	Moreover, if for some $x\in C$ we choose an apartment  $A^x$  in the infinitesimal Euclidean building  $T_xX$ with $T_xC\subset A^x$ then $A$ can be chosen so that $T _xA=A^x$.
\end{thm}

All Euclidean buildings in this note, in particular, in Theorem \ref{thm-main}, are assumed to be complete, cf. Remark \ref{rem: non-complete} below for a comment about possible generalizations.

The analogue of the first part of this result  for spherical buildings is contained in   \cite{KleinerLeeb1}, see Theorem \ref{thm-klsph} below.
In some special cases,  Theorem \ref{thm-main} is contained in \cite{KleinerLeeb1} as well.

A different proof of the spherical theorem  was obtained later in \cite[Proposition 1.3]{BL} in a slightly more general setting.  Moreover, this approach also covered the case of Theorem \ref{thm-main} for discrete Euclidean buildings, \cite[Remark 1.4]{BL}.

For some convex subsets in discrete Euclidean buildings,  Theorem \ref{thm-main} is presented in \cite[Theorem 11.53]{Abram}. The case of convex triangles $C$ in Bruhat--Tits buildings appears in \cite[Exercise 2.3.10.3]{Rousseau}.  Several  special cases of  Theorem \ref{thm-main} have been obtained and used in \cite{BM}.

 \begin{rem} \label{rem: non-complete}
	Here is the only place in this paper, where we discuss non-complete buildings, we refer to  \cite{Parreau}, \cite{Str} for their discussion.
	In any non-complete Euclidean building, Theorem \ref{thm-main} cannot hold as stated. Indeed, any such building $X$ contains a (half-open) geodesic $\gamma:[0,1) \to X$, whose image $C$ is closed as a subset of $X$.  Such $C$ cannot be contained in an apartment. 	However, an extension  of Theorem \ref{thm-main} to non-complete Euclidean buildings and to
	 $\Lambda$-buildings (cf. \cite{Schwer})  seems possible, we hope to discuss it in a forthcoming work.
\end{rem}

  In Section \ref{sec: sph} we follow \cite{KleinerLeeb1}, in order to  reprove and  to extend the spherical version of our main theorem.   One major difference between the spherical and the Euclidean case is  the  finiteness of the simplicial decomposition of spherical apartments, which allows for a choice of a potential enlargement $A$  of the convex subset $C$.  In Section 4,  we prove Theorem \ref{thm-main} by enlarging the convex set step by step.  In order to make this enlargement procedure  successful, we deal with the case of convex polyhedra $C$ first,
  and then deduce the result for  general convex subsets  by an appropriate approximation.

{\bf Acknowledgments.} Alexander Lytchak is grateful to the authors of \cite{BM} for their interest in this topic, especially  to Christine Breiner, for several related questions, which gave rise to this project.    Raphael Appenzeller would like to thank Isobel Davies for discussions. Raphael Appenzeller and Auguste H\'ebert acknowledge support from the Procope project
"Buildings, galleries and beyond" and would like to thank its members, St\'ephane Gaussent, Zijun Li, Bianca Marchionna, Paul Philippe and Petra Schwer for helpful discussions.

In Section 4, the authors "prove Theorem \ref{thm-main} by enlarging the convex set step by step." To make this enlargement procedure work, they "deal with the case of convex polyhedra $C$ first, and then deduce the result for general convex subsets by an appropriate approximation." They also note (contrasting with the spherical case) that in the Euclidean case one cannot rely on "the finiteness of the simplicial decomposition of spherical apartments," which in the spherical setting "allows for a choice of a potential enlargement $A$ of the convex subset $C$."

Next, the authors prove the main theorem by enlarging the convex set step by step. To make this enlargement procedure work, they deal with the case of convex polyhedra $C$ first, and then deduce the result for general convex subsets by an appropriate approximation. They also note (contrasting with the spherical case) that in the Euclidean case one cannot rely on the finiteness of the simplicial decomposition of spherical apartments, which in the spherical setting allows for a choice of a potential enlargement $A$ of the convex subset $C$.

Question: Let $X$ be a complete Euclidean building, and let $C\subset X$ be a convex subset that is isometric to a convex subset of some Euclidean space. For a point $x\in C$, let $T_xX$ denote the tangent cone at $x$ (the infinitesimal Euclidean building at $x$), and let $T_xC\subset T_xX$ denote the tangent cone of $C$ at $x$. Suppose that, for some chosen $x\in C$, an apartment $A^x\subset T_xX$ satisfies $T_xC\subset A^x$. Which conclusion holds under these hypotheses?

There exists an apartment $A\subset X$ such that $C\subset A$. Moreover, if $x\in C$ and an apartment $A^x\subset T_xX$ is chosen with $T_xC\subset A^x$, then $A$ can be chosen so that $T_xA=A^x$.

Choices:
B. There exists an apartment $A\subset X$ such that $C\subset A$. Moreover, for every $x\in C$ and every apartment $A^x\subset T_xX$ with $T_xC\subset A^x$, the same apartment $A$ can be chosen so that $T_xA=A^x$.
C. There exists an apartment $A\subset X$ such that $C\subset A$.
D. There exists an apartment $A\subset X$ such that $C\subset A$ provided that $C$ is a convex polyhedron. Moreover, if $x\in C$ and an apartment $A^x\subset T_xX$ is chosen with $T_xC\subset A^x$, then $A$ can be chosen so that $T_xA=A^x$ in this polyhedral case.
E. There exists an apartment $A\subset X$ such that $C\subset A$. Moreover, if $x\in C$ and an apartment $A^x\subset T_xX$ is chosen with $T_xC\subset A^x$, then every apartment $A\subset X$ containing $C$ satisfies $T_xA=A^x$.

dependence of the global apartment on the chosen tangent apartment and point

dropped the prescribed tangent-space matching condition $T_xA=A^x$

approximation step from convex polyhedra to arbitrary convex Euclidean subsets

existential choice of apartment containing $C$ replaced by a uniqueness-type condition on all such apartments

The stem does not explicitly reveal the correct option or uniquely signal choice A. It states only the hypotheses and asks which existence statement is valid.

This is essentially a multiple-choice restatement of a theorem: the correct answer is the theorem's conclusion almost verbatim, including the tangent-apartment refinement.

There is some reasoning pressure because the distractors vary by quantifier strength, locality vs globality, and uniqueness vs existence, and choice C is a weaker true statement. Still, the item mainly tests theorem recall rather than genuine derivation.

The distractors are mathematically plausible and well-targeted: B overstates uniqueness, C is a weaker true statement, D imposes an overly strong simultaneous condition, and E weakens global containment to a local one.

Good distractor design, but the item is largely theorem-recall and close to tautological rather than a strong test of generative reasoning.


Question: Let $X$ be a complete Euclidean building, and let $C\subset X$ be a convex subset that is isometric to a convex subset of some Euclidean space. For a point $x\in C$, let $T_xX$ denote the tangent cone at $x$ (the infinitesimal Euclidean building at $x$), and let $T_xC\subset T_xX$ denote the tangent cone of $C$ at $x$. Suppose that, for some chosen $x\in C$, an apartment $A^x\subset T_xX$ satisfies $T_xC\subset A^x$. Which conclusion holds under these hypotheses?

Choices:
B. There exists an apartment $A\subset X$ such that $C\subset A$. Moreover, for every $x\in C$ and every apartment $A^x\subset T_xX$ with $T_xC\subset A^x$, the same apartment $A$ can be chosen so that $T_xA=A^x$.
C. There exists an apartment $A\subset X$ such that $C\subset A$.
D. There exists an apartment $A\subset X$ such that $C\subset A$ provided that $C$ is a convex polyhedron. Moreover, if $x\in C$ and an apartment $A^x\subset T_xX$ is chosen with $T_xC\subset A^x$, then $A$ can be chosen so that $T_xA=A^x$ in this polyhedral case.
E. There exists an apartment $A\subset X$ such that $C\subset A$. Moreover, if $x\in C$ and an apartment $A^x\subset T_xX$ is chosen with $T_xC\subset A^x$, then every apartment $A\subset X$ containing $C$ satisfies $T_xA=A^x$.

The stem states hypotheses only and does not explicitly reveal the correct conclusion. Although the mention of a chosen tangent apartment foreshadows a conclusion involving matching tangent data, several options exploit that same feature, so the correct answer is not leaked.

This is essentially a theorem-recall item: the correct option is the precise theorem conclusion under the stated hypotheses. The task is largely to recognize the exact statement rather than derive a new consequence.

There is some reasoning pressure because the student must distinguish the exact existential conclusion from stronger quantifier claims, weaker true statements, and false uniqueness/conditional variants. But the item still mainly tests recognition of the theorem's exact wording rather than substantial generative reasoning.

The distractors are mathematically plausible and target common failure modes: strengthening existential to universal dependence, dropping part of the conclusion, adding an unnecessary polyhedral restriction, and replacing existence by uniqueness-type behavior. They are distinct and well-aligned with likely misunderstandings.

A solid theorem-recognition MCQ with strong distractors, but it is fairly tautological and only moderately probes reasoning beyond recall of the exact statement.


\begin{theorem}\label{thm:square}
We have $|\cS_2(n)|=(\frac{1}{2}-o(n^{-1/15})) 2^{2n}$.
\end{theorem}

Let $[k]^{*}$ be the collection of words of finite length over the finite alphabet $[k] = \{1,\ldots, k\}$, and say $s'\in [k]^*$ is a \textit{subword} of $s \in [k]^*$ if $s'$ is a (not necessarily contiguous) subsequence of $s$. A \textit{shuffle square} is a word $s\in [k]^*$ that can be partitioned into two disjoint, identical subwords. Shuffle squares were first studied by Henshall, Rampersad, and Shallit~\cite{henshall}, in the theory of formal languages. In this area, Boulteau and Vialette~\cite{binarynphard} proved that recognizing binary shuffle squares is NP-complete, building on work of Rizzi and Vialette~\cite{unshufflelargek} and Buss and Soltys~\cite{unshuffle7buss} on larger alphabets.

As shuffle squares must have even length, it is convenient to say that a word of length $2n$ has \textit{semi-length} $n$. Let $\cS_k(n)$ be the set of shuffle squares of semi-length $n$ over the alphabet $[k]$. Henshall, Rampersad, and Shallit asked for the asymptotics of $|\cS_k(n)|$, and gave conjectural asymptotic formulas for $n$ fixed and $k\rightarrow\infty$. These formulas were verified by the first author, Huang, Nam, and Thaper~\cite{HHNT}, who also posed the following surprising conjecture in the opposite regime, where $k=2$ is fixed and $n\rightarrow \infty$, supported by numerical evidence from the OEIS~\cite{oeis}.

\begin{conjecture}[\cite{HHNT}]
Asymptotically half of all even-length binary words are shuffle squares, that is, $|\cS_2(n)| = (\frac{1}{2}-o(1)) 2^{2n}$.
\end{conjecture}

Note that shuffle squares must have an even count of each letter, so this conjecture states that almost all words satisfying this obvious parity constraint are actually shuffle squares. Towards this, they proved using a greedy algorithm that $|\cS_2(n)| \ge \binom{2n}{n} = \Theta(n^{-1/2}2^{2n})$. Our main theorem resolves this conjecture in the affirmative with an explicit polynomial error term.

\begin{conjecture}[\cite{HHNT}]
Asymptotically half of all even-length binary words are shuffle squares, that is, $|\cS_2(n)| = (\frac{1}{2}-o(1)) 2^{2n}$.
\end{conjecture}

Note that shuffle squares must have an even count of each letter, so this conjecture states that almost all words satisfying this obvious parity constraint are actually shuffle squares. Towards this, they proved using a greedy algorithm that $|\cS_2(n)| \ge \binom{2n}{n} = \Theta(n^{-1/2}2^{2n})$. Our main theorem resolves this conjecture in the affirmative with an explicit polynomial error term.

The exponent $-1/15$ is not optimized, but within a modest factor of the best possible using our arguments. It would be interesting to decide if the error term is actually polynomial or exponential.

While previous work dealt with the worst-case behavior of $\rm{LT}(s)$, \cref{thm:square} implies the best possible bound for its average-case behavior.
\begin{corollary}\label{corollary:lt} For almost all $s\in\{0,1\}^{n}$, $\rm{LT}(s)\geq \sceil{\frac{n}{2}}-1$.
\end{corollary}

\begin{lemma}\label{lem:pt_decomposition} Let $s\in\{0,1\}^\omega$ be uniformly random. Let $\cev q_t(w;v)=q_t(\rev(w);\rev(v))$.
Then for all $v\in\{0,1\}^*$,
\begin{equation}\label{eq:pt_as_qt}
\PP[v\in B_n(s)]=\cev q_n(\emptyword;v)+\frac 12\sum_{t=0}^{n-1}\sum_{w\in \{0,1\}^*}\Big(\delta_t(1w1,w)\cev q_{n-t-1}(w1;v)+\delta_t(0w0,w)\cev q_{n-t-1}(w0;v)\Big).
\end{equation}
In particular, we have
\begin{equation}\label{eq:pt_as_qt_beta}
\PP[\emptyword\in B_n(s)]=q_n(\emptyword)+\sum_{t=0}^{n-1}\sum_{w1\in \Sigma_2^{(01)}}\delta_t(1w1,w) q_{n-t-1}(\rev(w1)).
\end{equation}
\end{lemma}
Roughly speaking, \eqref{eq:pt_as_qt_beta} captures the fact that any successful buffer thread $(w_j)_j$ has a critical first time $n-t-1$ at word $w_{n-t-1}=1w$ where it diverges from the greedy algorithm.
\begin{proof}
We prove \eqref{eq:pt_as_qt} by induction on $n$. By construction, $\PP[v\in B_0]=\bbone_{v=\emptyword}=q_0(\emptyword;\rev(v))$. Observe that for any word $v1$ ending in $1$, we have
\begin{align*}
\PP[v1\in B_{n+1}]&=\PP[s(n+1)=1]\PP[(v\in B_n)\cup (1v1\in B_n)]+\PP[s(n+1)=0]\PP[0v1\in B_n]\\
&=\frac 12\Big(\PP[v\in B_n]+\delta_n(1v1,v)+\PP[0v1\in B_n]\Big).
\end{align*}
Observe that $q_n$ satisfies a similar recurrence: for each initializiation of the form $1v'$, we can condition on the first bit of $s$ to get
\begin{align*}
q_n(w;1v')&=\PP[s(1)=1]\PP[\greedy(s[2,n],v')=w]+\PP[s(1)=0]\PP[\greedy(s[2,n],1v'0)=w]\\
&=\frac 12(q_{n-1}(w;v')+q_{n-1}(w;1v'0)).
\end{align*}
Reversing the words, we obtain
\[
\cev q_n(w;v1)=\frac 12(\cev q_{n-1}(w;v)+\cev q_{n-1}(w;0v1)).
\]
Denote the RHS of~\eqref{eq:pt_as_qt} by $Q_n(v)$. Since the above is a linear recurrence relation, $Q_n$ obeys the same equation, plus the added terms at time $n+1$:
\[
Q_{n+1}(v1)=\frac 12(Q_n(v)+Q_n(0v1))+\frac 12\sum_{w\in \{0,1\}^*}\Big(\delta_n(1w1,w)\cev q_{0}(w1;v1)+\delta_n(0w0,w)\cev q_{0}(w0;v1)\Big)
\]
Recall that $\cev q_0(w;v)=\bbone_{w=v}$, so the summation evaluates to $\frac 12\delta_n(1v1,v)$. Thus by induction, $\PP[v1\in B_{n+1}]=Q_{n+1}(v1)$. By symmetry, $\PP[v0\in B_{n+1}]=Q_{n+1}(v0)$, showing \eqref{eq:pt_as_qt} holds for all nonempty words $v$ at time $n+1$.

We prove that $X^*_1$ has the desired statistics and $X^*_2$ is a negligible perturbation. First, consider $X^*_1$. Since $X,Z$ are independent, we immediately have $\Ex[(X-1)\bbone_{Z=0}]=\frac12 (k-1)$, and $\Ex[(X-1-k)^2\bbone_{Z=0}]=\frac 12(2k+1)$. Next we control the second term in $X_1^*$; both technical claims below are proven in Section~\ref{section:numericalclaims}.
\begin{claim}\label{claim:Ybounds} (Proved in~\cref{section:numericalclaims}.) With $X,Y,Z$ as defined above, $\Ex[(Y-Z)\bbone_{Z>0}]=\frac 12(k-3+2^{-k})$ and $\Ex[(Y-Z-k)^2\bbone_{Z>0}]<4k+9$.
\end{claim}
Together, we have
\begin{align*}
\Ex[X^*_1-k]&=\frac 12(k-1)+\frac 12(k-3+2^{-k})-k=-2+2^{-k-1},\quad\text{and}\\
\Ex[(X^*_1-k)^2]&=\Ex[(X^*_1-k)^2\bbone_{Z=0}]+\Ex[(X^*_1-k)^2\bbone_{Z>0}]<\frac 12(2k+1)+\frac 12(4k+9)=3k+5.
\end{align*}
Next, we control $X^*_2$. Because $\PP[X=0]
=2^{-k}$, it is simple to check that $\Ex[\max(Z,1)\bbone_{X=0}]=\frac 32\cdot 2^{-k}$ and $\Ex[\max(Z,1)^2\bbone_{X=0}]=\frac 72\cdot 2^{-k}$, which are very small. The next claim bounds the $2C'$ term.
\begin{claim}\label{claim:C'bounds}(Proved in~\cref{section:numericalclaims}.) With $M,Z,C'$ as defined above, $\Ex[C']\leq 2(7/8)^{k'}$ and $\Ex[(C')^2]\leq 6(7/8)^{k'}$.
\end{claim}
We conclude that $\Ex[X^*_2],\Ex[(X^*_2)^2]=O((7/8)^k)$. Therefore,
\[
\Ex[X^*-k]=\Ex[X^*_1-k]+\Ex[X^*_2]=-2+O((7/8)^k),
\]
and similarly by the Minkowski inequality,
\[
\Ex[(X^*-k)^2]\leq \Big(\sqrt{\Ex[(X^*_1-k)^2]}+\sqrt{\Ex[(X^*_2)^2]}\Big)^2=3k+5+o(1)
\]
This proves~\eqref{eq:algorithmstep2}. To obtain tail bounds~\eqref{eq:algorithmstep3} and~\eqref{eq:algorithmstep4}, first note that
\begin{equation}\label{eq:T*expection}
T^*\leq T_3= (k+X)+(Z+1)+(X-1+Y),
\end{equation}
and thus $\Ex[T^*]\leq 4k$. By considering cases, we have
\[\PP[T^* \geq (4+5\eps)k]\leq \PP[X\geq (1+\eps)k]+\PP[Y\geq (1+2\eps)k\mid X<(1+\eps)k]+\PP[Z\geq \eps k]\]
By the definition of $Z$, we have $\PP[Z\geq \eps k]=2^{-\eps k}$. For the other two terms, recall that $X\sim \rm{NB}(k)$, and the conditional distribution $\big(Y\mid X< (1+\eps)k\big)$ is stochastically dominated by $\rm{NB}((1+\eps)k)$. Both terms can be bounded by Proposition \ref{prop:nbchernoff}, so
\[
\PP[T^*\geq (4+5\eps)k]\leq 3e^{-\Omega(\eps^2k/(1+\eps))},
\]
which gives~\eqref{eq:algorithmstep3}. We obtain tail bounds on $X^*$ similarly. Using~\eqref{eq:X*expression}, we concentrate $X^*$ by bounding $X,Y$, and $Z$, noting that $|2C'-Z|\leq |Z|$:
\begin{multline*}
\PP[|X^*-k|\geq 3\eps k]\leq \PP[Z\geq \eps k]+\PP[|X-k-1|\geq \eps k]+
\PP\big[|Y-k|\geq 2\eps k\ \big|\ |X-k-1|<\eps k\big].
\end{multline*}
Verifying that each of these is bounded by $e^{-\Omega(\eps^2k/(1+\eps))}$, we conclude~\eqref{eq:algorithmstep4}.
\end{proof}

\begin{problem}\label{prob:ternary} Is it true that $|\cS_3(n)|=(9-o(1))^n$?
\end{problem}
\noindent This problem is closely related to the extremal problem of determining whether $\rm{LT}(3,n)=(\frac 12-o(1))n$. The following one directional implication follows from work of Bukh and Zhou.
\begin{theorem}[{\cite[Theorem 4]{bukhzhou}}]\label{prop:alphatwins} If $\cS_k(n)\leq (b_k)^{2n+o(n)}$ and there exists $\alpha_k\in (2/k,1)$ such that
\[
\alpha_k\log\Big(\frac{b_k}{\alpha_kk}\Big)+(1-\alpha_k)\log \Big(\frac{k-1}{k(1-\alpha_k)}\Big)<0,
\]
then $\rm{LT}(k,n)< \frac 12\alpha_kn$ for sufficiently large $n$. In particular, if $b_k<k$, then we can take $\alpha_k<1$.
\end{theorem}
\noindent This can be obtained by a union bound over all ``leftmost" embeddings of shuffle squares in a random word $s$. Though we do not offer a conjecture on Problem~\ref{prob:ternary}, numerical evidence suggests that $|\cS_3(n)|=o(9^n)$.

The exponent $-1/15$ is not optimized, but within a modest factor of the best possible using our arguments. It would be interesting to decide if the error term is actually polynomial or exponential.

Closely related to the shuffle square problem is the ``twins'' problem introduced by Axenovich, Person, and Puzynina~\cite{Axenovich}. For a word $s\in [k]^*$, let $\rm{LT}(s)$ be the maximum semi-length of a subword of $s$ that is a shuffle square, so $s\in \cS_k(n)$ if and only if $\rm{LT}(s)=n$. Equivalently, $\rm{LT}(s)$ is the largest $m$ such that $s$ contains ``twins'' of length $m$, i.e. disjoint identical subwords of length $m$. Axenovich, Person, and Puzynina proved that for any $s\in \{0,1\}^{2n}$, we have $\rm{LT}(s)=(1-o(1))n$. More progress has been made on twins for various alphabets~\cite{basu,bukhzhou}, and the twins problem is related to regularity lemmas for words, the longest common subsequence problem \cite{bukhhogenson,bukhma,ChvatalSankoff,He-Li}, and codes for correcting deletion errors \cite{Guruswami-He-Li}.

While previous work dealt with the worst-case behavior of $\rm{LT}(s)$, \cref{thm:square} implies the best possible bound for its average-case behavior.
\begin{corollary}\label{corollary:lt} For almost all $s\in\{0,1\}^{n}$, $\rm{LT}(s)\geq \sceil{\frac{n}{2}}-1$.
\end{corollary}

\begin{proof} Let $s\in \{0,1\}^{2n}$ be uniformly random among words with an even number of $0$'s and $1$'s. If $\rm{LT}(s)=n$, then by deleting one or two bits, we immediately have $\rm{LT}(s[1,2n-1])=n-1$ and $\rm{LT}(s[1,2n-2])\geq n-2$. Thus by \cref{thm:square},
\[
\PP[\rm{LT}(s[1,2n-2])\geq n-2]\geq \PP[\rm{LT}(s[1,2n-1])=n-1]\geq \PP[s\in \cS_2(n)]=1-o(1).
\]
Moreover, $s[1,2n-1]$ is uniformly random in $\{0,1\}^{2n-1}$ and $s[1,2n-2]$ is uniformly random in $\{0,1\}^{2n-2}$, implying the corollary.
\end{proof}

In \cite{HHNT}, it was conjectured that almost every binary word can be made into a shuffle square by deleting at most three bits - \cref{corollary:lt} shows that it actually suffices to delete two (which is best possible).

\par{\bf Organization.} In Section~\ref{section:preliminaries}, we introduce the notion of a \textit{buffer} for a binary word, thereby reformulating the problem as a discrete stochastic process on subsets of $\{0,1\}^*$. We define and analyze the greedy algorithm using buffers and obtain asymptotic results on its behavior. In Section~\ref{section:buffersetanalysis}, we analyze the evolution of the set of all buffers to show that a random word $s$ can ``absorb" small defects in an almost-perfect partition. In Section~\ref{section:boostedgreedy}, we give an improved algorithm which produces a shuffle square with constant probability. This boosted greedy algorithm stores local information about the word and occasionally backtracks for an improvement. Together with the buffer set analysis, this proves~\cref{thm:square}. In Section~\ref{section:conclusion}, we discuss the shuffle square problem on larger alphabets, and we provide a boosted greedy algorithm for $k$-ary words which gives a new lower bound for $|\cS_k(n)|$.

In the ".\par{\bf Organization.}" paragraph, the authors outline how they prove Theorem~\ref{thm:square}: (1) they "introduce the notion of a \textit{buffer} for a binary word, thereby reformulating the problem as a discrete stochastic process on subsets of $\{0,1\}^*$"; (2) they "define and analyze the greedy algorithm using buffers and obtain asymptotic results on its behavior"; (3) they "analyze the evolution of the set of all buffers to show that a random word $s$ can \`\`absorb\" small defects in an almost-perfect partition"; (4) they "give an improved algorithm which produces a shuffle square with constant probability"—a "boosted greedy algorithm" that "stores local information about the word and occasionally backtracks for an improvement"; (5) "[t]ogether with the buffer set analysis, this proves~\cref{thm:square}."

In the ".\par{\bf Organization.}" paragraph, the authors outline how they prove the main theorem: (1) they "introduce the notion of a \textit{buffer} for a binary word, thereby reformulating the problem as a discrete stochastic process on subsets of $\{0,1\}^*$"; (2) they "define and analyze the greedy algorithm using buffers and obtain asymptotic results on its behavior"; (3) they "analyze the evolution of the set of all buffers to show that a random word $s$ can \`\`absorb\" small defects in an almost-perfect partition"; (4) they "give an improved algorithm which produces a shuffle square with constant probability"—a "boosted greedy algorithm" that "stores local information about the word and occasionally backtracks for an improvement"; (5) "[t]ogether with the buffer set analysis, this proves the main theorem."

Question: Let $[2]^*$ be the set of all finite binary words over $\{0,1\}$. A word $s'\in[2]^*$ is a subword of $s\in[2]^*$ if $s'$ is a subsequence of $s$ (not necessarily contiguous). A binary word $s$ of length $2n$ is called a shuffle square if its positions can be partitioned into two disjoint subwords that are identical as words. Let $\mathcal S_2(n)$ denote the set of shuffle squares of semi-length $n$, i.e. binary shuffle squares of length $2n$. Which asymptotic estimate holds for $|\mathcal S_2(n)|$ as $n\to\infty$?

$|\mathcal S_2(n)|=\left(\frac{1}{2}-o\!ig(n^{-1/15}\big)\right)2^{2n}$.

Choices:
B. $|\mathcal S_2(n)|=\left(\frac{1}{2}+o\!\big(n^{-1/15}\big)\right)2^{2n}$.
C. $|\mathcal S_2(n)|=\left(\frac{1}{2}-o(1)\right)2^{2n}$.
D. $|\mathcal S_2(n)|=\left(\frac{1}{2}-O\!\big(n^{-1/15}\big)\right)2^{2n}$.
E. $|\mathcal S_2(n)|\geq \left(\frac{1}{2}-o\!\big(n^{-1/15}\big)\right)2^{2n}$.

defect-absorption yields a deficit from $1/2$, not an excess

explicit polynomial error term $o(n^{-1/15})$

non-effective little-o error replaced by stronger big-O control

exact asymptotic equality replaced by one-sided lower bound at the same scale

The stem gives only the definition of binary shuffle squares and asks for the asymptotic count; it does not reveal the specific asymptotic form or strongly hint at the correct sign/error term.

The item is close to a direct theorem recall question: it asks which asymptotic statement holds, and the correct option is essentially the theorem's conclusion. The nearby variants add some discrimination, but the structure is still largely a restatement.

There is some reasoning pressure because the student must distinguish the exact result from a weaker true statement and from stronger/false variants. However, the question mainly tests recall of a specific asymptotic theorem rather than generating or deriving a conclusion from principles.

The distractors are well-designed: one flips the sign of the correction, one gives a weaker true asymptotic, one offers an unjustifiably stronger rate, and one confuses the density ceiling. These are plausible and mathematically distinct failure modes.

A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it only moderately avoids tautology and does limited work in testing genuinely generative reasoning.


[Low-degree structure of $H^*(B\mathcal{G}_k)$]\label{thm:main}
Work $2$--locally and fix an integer $k$.  Consider the Serre spectral
sequence \eqref{eq:Serre-SS} for the fibration
$\Omega^3_0G_2\to B\mathcal{G}_k\to B\G$, and let
$M=H^*(\Omega^3_0G_2;\mathbb{F}_2)$.
\begin{enumerate}[label=\textup{(\roman*)}]
  \item In total degree $j=s+t\le 10$ the $E_2$--term is naturally
  isomorphic, as a bigraded $H^*(B\G)$--module, to
  \[
  E_2^{*,*}(k)^{\le 10} \;\cong\;
  H^*(B\G)\otimes M^{\le 10},
  \]
  with $E_2^{s,t}=H^s(B\G)\otimes M^t$.

  \item Using the low-degree homotopy groups of $G_2$ and the Hurewicz
  theorem, one has
  \[
  M^j = H^j(\Omega^3_0G_2;\mathbb{F}_2)=0\quad\text{for }1\le j\le 4,
  \qquad
  M^5\neq 0.
  \]
  In particular, there exists a nonzero class
  \[
  u_5 \in M^5\cong E_2^{0,5}(k)
  \]
  and $u_5$ is the first positive-degree generator of
  $H^*(\Omega^3_0G_2)$.

  \item In total degree $j\le 10$ the differentials
  \[
  d_r\colon E_r^{0,t}(k)\longrightarrow E_r^{r,t-r+1}(k)
  \]
  with $2\le r\le 5$ vanish on all classes in bidegree $(0,t)$, and
  the only page on which a differential with source in $E_r^{0,5}(k)$
  can be nonzero is $r=6$.  After choosing a generator
  $x_6\in H^6(B\G)\cong E_2^{6,0}(k)$ and $u_5\in M^5\cong E_2^{0,5}(k)$
  there is a uniquely determined scalar $\varepsilon(k)\in\mathbb{F}_2$ such that
  \[
  d_6(u_5) = \varepsilon(k)\, x_6
  \]
  in $E_6^{6,0}(k)$.

  \item Let $U\subset E_2^{*,*}(k)$ denote the $H^*(B\G)$--submodule
  generated by $u_5$.  In total degree $j\le 10$ all differentials
  $d_r$ with $2\le r\le 5$ vanish on $U$, and every differential
  \[
  d_6\colon E_6^{s,5}(k)\longrightarrow E_6^{s+6,0}(k)
  \]
  with $s+5\le 10$ satisfies
  \[
  d_6(x\cdot u_5) = x\cdot d_6(u_5)
  \qquad\text{for }x\in H^s(B\G).
  \]
  In particular, within $U$ the entire effect of the Serre
  differentials in total degree $\le 10$ is determined by the single
  scalar $\varepsilon(k)\in\mathbb{F}_2$.

  \item For each $j\le 10$ the spectral sequence induces a finite
  filtration of $H^j(B\mathcal{G}_k)$ whose associated graded
  contains, as a direct summand, the image of $U$ in
  $E_\infty^{*,*}(k)^j$.  This summand is explicitly computable from
  $H^*(B\G)\otimes M^{\le 10}$ together with $\varepsilon(k)$, while
  the remaining extension term depends only on $M^{\le j}$ and the
  multiplicative structure of the Serre spectral sequence.
\end{enumerate}
In particular, in total degree $j\le 10$ the dependence of
$H^j(B\mathcal{G}_k;\mathbb{F}_2)$ on the bundle class $k$ that is visible
through the first nontrivial fibre class $u_5$ is completely encoded
in the single Serre differential $d_6(u_5)=\varepsilon(k)x_6$.

\begin{theorem}[Low-degree structure of $H^*(B\mathcal{G}_k)$]\label{thm:main}
Work $2$--locally and fix an integer $k$.  Consider the Serre spectral
sequence \eqref{eq:Serre-SS} for the fibration
$\Omega^3_0\G\to B\mathcal{G}_k\to \BG$, and let
$M=H^*(\Omega^3_0\G;\Ftwo)$.
\begin{enumerate}[label=\textup{(\roman*)}]
  \item In total degree $j=s+t\le 10$ the $E_2$--term is naturally
  isomorphic, as a bigraded $H^*(\BG)$--module, to
  \[
  E_2^{*,*}(k)^{\le 10} \;\cong\;
  H^*(\BG)\otimes M^{\le 10},
  \]
  with $E_2^{s,t}=H^s(\BG)\otimes M^t$.

  \item Using the low-degree homotopy groups of $\G$ and the Hurewicz
  theorem, one has
  \[
  M^j = H^j(\Omega^3_0\G;\Ftwo)=0\quad\text{for }1\le j\le 4,
  \qquad
  M^5\neq 0.
  \]
  In particular, there exists a nonzero class
  \[
  u_5 \in M^5\cong E_2^{0,5}(k)
  \]
  and $u_5$ is the first positive-degree generator of
  $H^*(\Omega^3_0\G)$.

  \item In total degree $j\le 10$ the differentials
  \[
  d_r\colon E_r^{0,t}(k)\longrightarrow E_r^{r,t-r+1}(k)
  \]
  with $2\le r\le 5$ vanish on all classes in bidegree $(0,t)$, and
  the only page on which a differential with source in $E_r^{0,5}(k)$
  can be nonzero is $r=6$.  After choosing a generator
  $x_6\in H^6(\BG)\cong E_2^{6,0}(k)$ and $u_5\in M^5\cong E_2^{0,5}(k)$
  there is a uniquely determined scalar $\varepsilon(k)\in\Ftwo$ such that
  \[
  d_6(u_5) = \varepsilon(k)\, x_6
  \]
  in $E_6^{6,0}(k)$.

  \item Let $U\subset E_2^{*,*}(k)$ denote the $H^*(\BG)$--submodule
  generated by $u_5$.  In total degree $j\le 10$ all differentials
  $d_r$ with $2\le r\le 5$ vanish on $U$, and every differential
  \[
  d_6\colon E_6^{s,5}(k)\longrightarrow E_6^{s+6,0}(k)
  \]
  with $s+5\le 10$ satisfies
  \[
  d_6(x\cdot u_5) = x\cdot d_6(u_5)
  \qquad\text{for }x\in H^s(\BG).
  \]
  In particular, within $U$ the entire effect of the Serre
  differentials in total degree $\le 10$ is determined by the single
  scalar $\varepsilon(k)\in\Ftwo$.

  \item For each $j\le 10$ the spectral sequence induces a finite
  filtration of $H^j(B\mathcal{G}_k)$ whose associated graded
  contains, as a direct summand, the image of $U$ in
  $E_\infty^{*,*}(k)^j$.  This summand is explicitly computable from
  $H^*(\BG)\otimes M^{\le 10}$ together with $\varepsilon(k)$, while
  the remaining extension term depends only on $M^{\le j}$ and the
  multiplicative structure of the Serre spectral sequence.
\end{enumerate}
In particular, in total degree $j\le 10$ the dependence of
$H^j(B\mathcal{G}_k;\Ftwo)$ on the bundle class $k$ that is visible
through the first nontrivial fibre class $u_5$ is completely encoded
in the single Serre differential $d_6(u_5)=\varepsilon(k)x_6$.
\end{theorem}

\subsection{Gauge groups and their homotopy types}

Let $G$ be a compact, connected Lie group, let $X$ be a pointed CW
complex, and let $P\to X$ be a principal $G$--bundle.  The \emph{gauge
group} $\mathcal{G}(P)$ is the topological group of $G$--equivariant
bundle automorphisms of $P$ that cover the identity on $X$.  When $X$
is finite, Crabb--Sutherland showed that there are only finitely many
homotopy types among the gauge groups $\mathcal{G}(P)$ as $P$ ranges
over all principal $G$--bundles over $X$~\cite{CrabbSutherland}.  In
the fundamental case $X=S^4$ and $G$ simply connected and simple,
isomorphism classes of $G$--bundles are classified by
$\pi_4(BG)\cong \mathbb{Z}$, and we write $\mathcal{G}_k(G)$ for the
gauge group of the bundle classified by $k\in \mathbb{Z}$.

The problem of determining the homotopy types of these gauge groups
for specific Lie groups $G$ has received sustained attention; see
for example \cite{KonoSU2,TheriaultSp2,KishimotoTheriaultTsutayaG2}.  In many cases
the classification is governed by the order of a fundamental Samelson
product in~$G$.

The case $G=G_2$ is particularly interesting because of the exceptional
nature of $G_2$.  Kishimoto, Theriault and Tsutaya proved that the
classification for $G_2$--gauge groups reduces to determining the
order of the Samelson product
\[
\langle i_3,1\rangle\in [\Sigma^3G_2,G_2],
\]
where $i_3\colon S^3\to G_2$ is the inclusion of the bottom cell and
$1$ denotes the identity map~\cite{KishimotoTheriaultTsutayaG2}.

Very recently, Kameko~\cite{KamekoG2gauge} resolved this issue at the prime $2$ and
completed the $p$--local classification of $G_2$--gauge groups.  He
proved that the order of $\langle i_3,1\rangle$ is exactly $84$
\cite[Theorem~1.1]{KamekoG2gauge}.
Consequently, the $p$--local homotopy type of $\mathcal{G}_k(G_2)$ depends only on
the greatest common divisor $(k,84)$.  Thus the homotopy types of the
gauge groups are completely understood up to $p$--localization.  It is
natural to ask how far this information is reflected in the
(co)homology of their classifying spaces.

\subsection{Cohomology of $B\mathcal{G}_k$ and the Steenrod algebra}

Gottlieb identified the classifying space of $\mathcal{G}_k$ with a
component of a mapping space~\cite{GottliebGauge}: specifically,
$B\mathcal{G}_k$ is homotopy equivalent to $\mathrm{Map}_k(S^4,BG)$,
the connected component of the mapping space $\mathrm{Map}(S^4,BG)$
containing the classifying map $k\colon S^4\to BG$.
This identification yields a fundamental evaluation fibration
\[
\Omega^3_0 G \;\longrightarrow\; B\mathcal{G}_k
\xrightarrow{\;\mathrm{ev}\;} BG.
\]
For $G=G_2$, this expresses $B\mathcal{G}_k$ as a twisted
mapping space over the base $B\G$.

The mod~$2$ cohomology of $B\G$ is classically known: Borel showed that
\[
H^*(B\G;\mathbb{F}_2) \cong \mathbb{F}_2[x_4,x_6,x_7]
\]
is a polynomial algebra on generators in degrees $4,6,7$
\cite{BorelCohomology}.  The action of the Steenrod algebra $\mathcal{A}$ on
$H^*(B\G)$ can be described explicitly via Wu formulas, and from the
viewpoint of the hit problem one is interested in minimal
generating sets for the unstable $\mathcal{A}$--module $\mathbb{F}_2[x_4,x_6,x_7]$.

By contrast, there seems to be no published description of the full
mod~$2$ cohomology ring $H^*(B\mathcal{G}_k;\mathbb{F}_2)$ for any
nontrivial $k$, nor of its structure as an unstable $\mathcal{A}$--module.
Our aim here is to give a precise structural description of
$H^*(B\mathcal{G}_k;\mathbb{F}_2)$ in low degrees, in terms of the evaluation
fibration and the first nontrivial cohomology of the triple loop space
$\Omega^3_0G_2$.

\subsection{Main result: a low-degree description}

We work $2$--locally throughout and write $H^*(-)$ for
$H^*(-;\mathbb{F}_2)$.  Consider the Serre spectral sequence of the
evaluation fibration
\begin{equation}\label{eq:Serre-SS}
E_2^{s,t}(k)\;\cong\; H^s(B\G;H^t(\Omega^3_0G_2))
\;\Longrightarrow\; H^{s+t}(B\mathcal{G}_k).
\end{equation}
The $E_2$--term is a module over $H^*(B\G)$ and an unstable module
over the Steenrod algebra.  The twisting of this spectral sequence
depends on $k$ through the connecting map
\[
\partial_k\colon G_2 \longrightarrow \Omega^3_0 G_2,
\]
which is the triple adjoint of the Samelson product $\langle
k\cdot i_3,1\rangle$ by work of
Kishimoto--Theriault--Tsutaya~\cite{KishimotoTheriaultTsutayaG2}.
Kameko's calculation of the $2$--primary order of $\langle i_3,1\rangle$
implies that, after localization at $2$, the map $\partial_k$ depends
only on $k\bmod 4$.

The cohomology of the fibre $\Omega^3_0 G$ can in principle be
computed by iterated Eilenberg--Moore spectral sequences applied to
the path-loop fibrations of $BG$ and $G$.
Choi and Yoon conjectured that, for any simply connected finite
$H$--space $X$, the Eilenberg--Moore spectral sequences converging
to the mod $p$ homology of $\Omega^2 X$ and $\Omega^3 X$ collapse at
$E_2$~\cite{ChoiYoonEMSS}.  This conjecture was subsequently proved by
Lin~\cite{LinEMSS}; in particular, for any compact simple Lie group
$G$ and any prime $p$, the Eilenberg--Moore spectral sequences computing
$H_*(\Omega^2 G;\mathbb F_p)$ and $H_*(\Omega^3 G;\mathbb F_p)$
collapse at $E_2$.  Combined with the classical description of
$H^*(B\G)$, this implies that $H^*(\Omega^3_0G_2)$ is of finite type in
each degree and that its generators occur in bounded degrees.  For our
purposes, however, we only need to know the very first nontrivial
cohomology group of $\Omega^3_0G_2$.

Let
\[
M \;=\; H^*(\Omega^3_0G_2;\mathbb{F}_2)
\]
and write $M^{\le N}=\bigoplus_{j\le N}M^j$ for its truncation in
degrees $\le N$.  Our main theorem describes the Serre spectral
sequence \eqref{eq:Serre-SS} in total degree $\le 10$ and singles out
a distinguished scalar $\varepsilon(k)\in\mathbb{F}_2$ which already
captures the effect of $k$ on the first nontrivial fibre class.

Let $G$ be a compact, connected Lie group, let $X$ be a pointed CW
complex, and let $P\to X$ be a principal $G$--bundle. The \emph{gauge
group} $\mathcal{G}(P)$ is the topological group of $G$--equivariant
bundle automorphisms of $P$ that cover the identity on $X$. When $X$
is finite, Crabb--Sutherland showed that there are only finitely many
homotopy types among the gauge groups $\mathcal{G}(P)$ as $P$ ranges
over all principal $G$--bundles over $X$~\cite{CrabbSutherland}. In
the fundamental case $X=S^4$ and $G$ simply connected and simple,
isomorphism classes of $G$--bundles are classified by
$\pi_4(BG)\cong \mathbb{Z}$, and we write $\mathcal{G}_k(G)$ for the
gauge group of the bundle classified by $k\in \mathbb{Z}$.

Gottlieb identified the classifying space of $\mathcal{G}_k$ with a
component of a mapping space~\cite{GottliebGauge}: specifically,
$B\mathcal{G}_k$ is homotopy equivalent to $\mathrm{Map}_k(S^4,BG)$,
the connected component of the mapping space $\mathrm{Map}(S^4,BG)$
containing the classifying map $k\colon S^4\to BG$.
This identification yields a fundamental evaluation fibration
\[
\Omega^3_0 G \;\longrightarrow\; B\mathcal{G}_k
\xrightarrow{\;\mathrm{ev}\;} BG.
\]
For $G=G_2$, this expresses $B\mathcal{G}_k$ as a twisted
mapping space over the base $B\G$.

The mod~$2$ cohomology of $B\G$ is classically known: Borel showed that
\[
H^*(B\G;\mathbb{F}_2) \cong \mathbb{F}_2[x_4,x_6,x_7]
\]
is a polynomial algebra on generators in degrees $4,6,7$
\cite{BorelCohomology}. The action of the Steenrod algebra $\mathcal{A}$ on
$H^*(B\G)$ can be described explicitly via Wu formulas, and from the
viewpoint of the hit problem one is interested in minimal
generating sets for the unstable $\mathcal{A}$--module $\mathbb{F}_2[x_4,x_6,x_7]$.

We work $2$--locally throughout and write $H^*(-)$ for
$H^*(-;\mathbb{F}_2)$. Consider the Serre spectral sequence of the
evaluation fibration
\begin{equation}\label{eq:Serre-SS}
E_2^{s,t}(k)\;\cong\; H^s(B\G;H^t(\Omega^3_0G_2))
\;\Longrightarrow\; H^{s+t}(B\mathcal{G}_k).
\end{equation}
The $E_2$--term is a module over $H^*(B\G)$ and an unstable module
over the Steenrod algebra. The twisting of this spectral sequence
depends on $k$ through the connecting map
\[
\partial_k\colon G_2 \longrightarrow \Omega^3_0 G_2,
\]
which is the triple adjoint of the Samelson product $\langle
k\cdot i_3,1\rangle$ by work of
Kishimoto--Theriault--Tsutaya~\cite{KishimotoTheriaultTsutayaG2}.
Kameko's calculation of the $2$--primary order of $\langle i_3,1\rangle$
implies that, after localization at $2$, the map $\partial_k$ depends
only on $k\bmod 4$.

The cohomology of the fibre $\Omega^3_0 G$ can in principle be
computed by iterated Eilenberg--Moore spectral sequences applied to
the path-loop fibrations of $BG$ and $G$.
Choi and Yoon conjectured that, for any simply connected finite
$H$--space $X$, the Eilenberg--Moore spectral sequences converging
to the mod $p$ homology of $\Omega^2 X$ and $\Omega^3 X$ collapse at
$E_2$~\cite{ChoiYoonEMSS}. This conjecture was subsequently proved by
Lin~\cite{LinEMSS}; in particular, for any compact simple Lie group
$G$ and any prime $p$, the Eilenberg--Moore spectral sequences computing
$H_*(\Omega^2 G;\mathbb F_p)$ and $H_*(\Omega^3 G;\mathbb F_p)$
collapse at $E_2$. Combined with the classical description of
$H^*(B\G)$, this implies that $H^*(\Omega^3_0G_2)$ is of finite type in
each degree and that its generators occur in bounded degrees. For our
purposes, however, we only need to know the very first nontrivial
cohomology group of $\Omega^3_0G_2$.

Let
\[
M \;=\; H^*(\Omega^3_0G_2;\mathbb{F}_2)
\]
and write $M^{\le N}=\bigoplus_{j\le N}M^j$ for its truncation in
degrees $\le N$. Our main theorem describes the Serre spectral
sequence \eqref{eq:Serre-SS} in total degree $\le 10$ and singles out
a distinguished scalar $\varepsilon(k)\in\mathbb{F}_2$ which already
captures the effect of $k$ on the first nontrivial fibre class.

\begin{equation}\label{eq:Serre-SS}
E_2^{s,t}(k)\;\cong\; H^s(\BG;H^t(\Omega^3_0\G))
\;\Longrightarrow\; H^{s+t}(B\mathcal{G}_k).
\end{equation}

In this paper we begin a complementary study of the mod~$2$
cohomology of the classifying spaces $B\mathcal{G}_k(G_2)$. Our goal
is to understand the structure of $H^*(B\mathcal{G}_k;\Ftwo)$ as an
unstable module over the mod~$2$ Steenrod algebra $\A$ in a low range
of degrees. Using the evaluation fibration
\[
\Omega_0^3 G_2 \longrightarrow B\mathcal{G}_k \xrightarrow{\;\mathrm{ev}\;} BG_2
\]
together with Serre and Eilenberg--Moore spectral sequences, we
study the Serre spectral sequence
\[
H^s(BG_2;H^t(\Omega^3_0G_2)) \Longrightarrow H^{s+t}(B\mathcal{G}_k)
\]
in total degree $\le 10$. A careful analysis of the homotopy groups
of $G_2$ shows that
\[
H^j(\Omega^3_0G_2;\Ftwo)=0\quad\text{for }1\le j\le 4,
\qquad
H^5(\Omega^3_0G_2;\Ftwo)\neq 0,
\]
so the first positive-degree generator of the fibre cohomology occurs
in degree $5$. As a consequence, there is a distinguished class
\[
u_5\in H^5(\Omega^3_0G_2;\Ftwo)
\]
whose only possible Serre differential in total degree $\le 10$ is a
$d_6$--differential
\[
d_6(u_5) = \varepsilon(k)\,x_6
\]
from $u_5$ to the degree-$6$ generator $x_6\in H^6(BG_2;\Ftwo)$, for a
certain scalar $\varepsilon(k)\in\Ftwo$ depending on the bundle
class~$k$.
\end{abstract}

We work $2$--locally throughout and write $H^*(-)$ for
$H^*(-;\Ftwo)$. Consider the Serre spectral sequence of the
evaluation fibration
\begin{equation}\label{eq:Serre-SS}
E_2^{s,t}(k)\;\cong\; H^s(\BG;H^t(\Omega^3_0\G))
\;\Longrightarrow\; H^{s+t}(B\mathcal{G}_k).
\end{equation}
The $E_2$--term is a module over $H^*(\BG)$ and an unstable module
over the Steenrod algebra. The twisting of this spectral sequence
depends on $k$ through the connecting map
\[
\partial_k\colon \G \longrightarrow \Omega^3_0 \G,
\]
which is the triple adjoint of the Samelson product $\langle
k\cdot i_3,1\rangle$ by work of
Kishimoto--Theriault--Tsutaya~\cite{KishimotoTheriaultTsutayaG2}.
Kameko's calculation of the $2$--primary order of $\langle i_3,1\rangle$
implies that, after localization at $2$, the map $\partial_k$ depends
only on $k\bmod 4$.

Let
\[
M \;=\; H^*(\Omega^3_0\G;\Ftwo)
\]
and write $M^{\le N}=\bigoplus_{j\le N}M^j$ for its truncation in
degrees $\le N$. Our main theorem describes the Serre spectral
sequence \eqref{eq:Serre-SS} in total degree $\le 10$ and singles out
a distinguished scalar $\varepsilon(k)\in\Ftwo$ which already
captures the effect of $k$ on the first nontrivial fibre class.

\item In total degree $j\le 10$ the differentials
 \[
 d_r\colon E_r^{0,t}(k)\longrightarrow E_r^{r,t-r+1}(k)
 \]
 with $2\le r\le 5$ vanish on all classes in bidegree $(0,t)$, and
 the only page on which a differential with source in $E_r^{0,5}(k)$
 can be nonzero is $r=6$. After choosing a generator
 $x_6\in H^6(\BG)\cong E_2^{6,0}(k)$ and $u_5\in M^5\cong E_2^{0,5}(k)$
 there is a uniquely determined scalar $\varepsilon(k)\in\Ftwo$ such that
 \[
 d_6(u_5) = \varepsilon(k)\, x_6
 \]
 in $E_6^{6,0}(k)$.

\item Let $U\subset E_2^{*,*}(k)$ denote the $H^*(\BG)$--submodule
 generated by $u_5$. In total degree $j\le 10$ all differentials
 $d_r$ with $2\le r\le 5$ vanish on $U$, and every differential
 \[
 d_6\colon E_6^{s,5}(k)\longrightarrow E_6^{s+6,0}(k)
 \]
 with $s+5\le 10$ satisfies
 \[
 d_6(x\cdot u_5) = x\cdot d_6(u_5)
 \qquad\text{for }x\in H^s(\BG).
 \]
 In particular, within $U$ the entire effect of the Serre
 differentials in total degree $\le 10$ is determined by the single
 scalar $\varepsilon(k)\in\Ftwo$.

\begin{remark}
Theorem~\ref{thm:main} should be viewed as a structural reduction. It
does not claim an explicit formula for $\varepsilon(k)$, nor does it
assert that $\varepsilon(k)$ is nontrivial. Kameko's result shows
that the $2$--local homotopy type of $\mathcal{G}_k(\G)$ depends on
$k\bmod 4$ through the $2$--primary order of the Samelson product
$\langle i_3,1\rangle$ \cite{KishimotoTheriaultTsutayaG2,
KamekoG2gauge}. It is natural to expect that $\varepsilon(k)$ is a
function of this $2$--primary data; however, determining
$\varepsilon(k)$ explicitly requires detailed knowledge of
$H^*(\Omega^3_0\G)$ and of the connecting map $\partial_k$ on mod~$2$
cohomology, and we leave this as an open problem.
\end{remark}

\item[(ii)] Lemma~\ref{lem:full-bidegree-vanish} only analyzes the differentials
with source in the $H^*(BG_2)$--submodule
\[
U \;=\; H^*(BG_2)\cdot u_5 \;\subset\; E_2^{*,*}(k).
\]
We do not claim that $M^j=H^j(\Omega^3_0G_2;\F_2)$ vanishes for $j\ge 6$,
and additional fibre generators in these degrees could in principle
support nontrivial differentials in the Serre spectral sequence.
Such differentials, however, do not change the description of the
possible $d_6$--differentials on $U$ in total degree $\le 10$, which is
all that is needed for Theorem~\ref{thm:main}. A determination of $M$
in higher degrees would require an explicit computation via
Eilenberg--Moore spectral sequences, and lies beyond the scope of the
present paper.
\end{itemize}
\end{remark}

\begin{note}\label{rem:epsilon-zero}
The case $k=0$ admits a particularly simple description. The class
$0\in\pi_4(BG_2)$ corresponds to the trivial principal
$G_2$--bundle $S^4\times G_2\to S^4$, and by Gottlieb's theorem \cite{GottliebGauge}, we have
\[
B\mathcal{G}_0 \;\simeq\; \mathrm{Map}_0(S^4,BG_2),
\]
where $\mathrm{Map}_0(S^4,BG_2)$ denotes the component containing the
constant map. The evaluation map $\mathrm{ev}\colon \mathrm{Map}_0(S^4,BG_2)\to BG_2$
admits a section $s\colon BG_2\to \mathrm{Map}_0(S^4,BG_2)$ defined by sending
$y\in BG_2$ to the constant map at $y$. The existence of this section
implies that the homomorphism
\[
\mathrm{ev}^*\colon H^*(BG_2)\longrightarrow H^*(B\mathcal{G}_0)
\]
is injective. In the Serre spectral sequence for the evaluation fibration
with $k=0$, the class $x_6\in H^6(BG_2)\cong E_2^{6,0}$ must therefore
survive to $E_\infty$ and cannot lie in the image of any differential.
In particular, $x_6$ cannot be hit by $d_6(u_5)$, forcing $d_6(u_5)=0$.
Thus, in our notation, $\varepsilon(0)=0$.
\end{note}

\begin{remark}
Theorem~\ref{thm:main} should be viewed as a structural reduction.  It
does not claim an explicit formula for $\varepsilon(k)$, nor does it
assert that $\varepsilon(k)$ is nontrivial.  Kameko's result shows
that the $2$--local homotopy type of $\mathcal{G}_k(G_2)$ depends on
$k\bmod 4$ through the $2$--primary order of the Samelson product
$\langle i_3,1\rangle$ \cite{KishimotoTheriaultTsutayaG2,
KamekoG2gauge}.  It is natural to expect that $\varepsilon(k)$ is a
function of this $2$--primary data; however, determining
$\varepsilon(k)$ explicitly requires detailed knowledge of
$H^*(\Omega^3_0G_2)$ and of the connecting map $\partial_k$ on mod~$2$
cohomology, and we leave this as an open problem.
\end{remark}

\medskip

The paper is organised as follows.  In Section~\ref{sec:prelim} we
review the necessary background on gauge groups, Samelson products,
the mod~$2$ cohomology of $B\G$ and the Steenrod algebra.  In
Section~\ref{sec:SS-setup} we describe the evaluation fibration and
its Serre spectral sequence, and we recall the relationship between
the connecting map and the Samelson product. Finally, Section~\ref{sec:low}
contains the proof of Theorem~\ref{thm:main}, including a detailed
analysis of the bidegrees for $j\le 10$ and the vanishing of
$H^j(\Omega^3_0G_2)$ for $1\le j\le 4$.

The post-theorem introduction does not give a step-by-step proof sketch, but it does indicate where the proof is carried out and what it uses: Section~\ref{sec:SS-setup} “describe[s] the evaluation fibration and its Serre spectral sequence,” and “recall[s] the relationship between the connecting map and the Samelson product.” Then Section~\ref{sec:low} “contains the proof of Theorem~\ref{thm:main}, including a detailed analysis of the bidegrees for $j\le 10$ and the vanishing of $H^j(\Omega^3_0G_2)$ for $1\le j\le 4$.”

The post-theorem introduction does not give a step-by-step proof sketch, but it does indicate where the proof is carried out and what it uses: Next the authors describe the evaluation fibration and its Serre spectral sequence, and recall the relationship between the connecting map and the Samelson product. Then later they establish the main theorem, including a detailed analysis of the bidegrees for $j\le 10$ and the vanishing of $H^j(\Omega^3_0G_2)$ for $1\le j\le 4$.

[Low-degree structure of $H^*(B\mathcal{G}_k)$]\label{thm:main}
Work $2$--locally and fix an integer $k$.  Consider the Serre spectral
sequence
\begin{equation}\label{eq:Serre-SS}
E_2^{s,t}(k)\;\cong\; H^s(\BG;H^t(\Omega^3_0\G))
\;\Longrightarrow\; H^{s+t}(B\mathcal{G}_k).
\end{equation}
for the fibration
$\Omega^3_0G_2\to B\mathcal{G}_k\to B\G$, and let
$M=H^*(\Omega^3_0G_2;\mathbb{F}_2)$.
\begin{enumerate}[label=\textup{(\roman*)}]
  \item In total degree $j=s+t\le 10$ the $E_2$--term is naturally
  isomorphic, as a bigraded $H^*(B\G)$--module, to
  \[
  E_2^{*,*}(k)^{\le 10} \;\cong\;
  H^*(B\G)\otimes M^{\le 10},
  \]
  with $E_2^{s,t}=H^s(B\G)\otimes M^t$.

  \item Using the low-degree homotopy groups of $G_2$ and the Hurewicz
  theorem, one has
  \[
  M^j = H^j(\Omega^3_0G_2;\mathbb{F}_2)=0\quad\text{for }1\le j\le 4,
  \qquad
  M^5\neq 0.
  \]
  In particular, there exists a nonzero class
  \[
  u_5 \in M^5\cong E_2^{0,5}(k)
  \]
  and $u_5$ is the first positive-degree generator of
  $H^*(\Omega^3_0G_2)$.

  \item In total degree $j\le 10$ the differentials
  \[
  d_r\colon E_r^{0,t}(k)\longrightarrow E_r^{r,t-r+1}(k)
  \]
  with $2\le r\le 5$ vanish on all classes in bidegree $(0,t)$, and
  the only page on which a differential with source in $E_r^{0,5}(k)$
  can be nonzero is $r=6$.  After choosing a generator
  $x_6\in H^6(B\G)\cong E_2^{6,0}(k)$ and $u_5\in M^5\cong E_2^{0,5}(k)$
  there is a uniquely determined scalar $\varepsilon(k)\in\mathbb{F}_2$ such that
  \[
  d_6(u_5) = \varepsilon(k)\, x_6
  \]
  in $E_6^{6,0}(k)$.

  \item Let $U\subset E_2^{*,*}(k)$ denote the $H^*(B\G)$--submodule
  generated by $u_5$.  In total degree $j\le 10$ all differentials
  $d_r$ with $2\le r\le 5$ vanish on $U$, and every differential
  \[
  d_6\colon E_6^{s,5}(k)\longrightarrow E_6^{s+6,0}(k)
  \]
  with $s+5\le 10$ satisfies
  \[
  d_6(x\cdot u_5) = x\cdot d_6(u_5)
  \qquad\text{for }x\in H^s(B\G).
  \]
  In particular, within $U$ the entire effect of the Serre
  differentials in total degree $\le 10$ is determined by the single
  scalar $\varepsilon(k)\in\mathbb{F}_2$.

  \item For each $j\le 10$ the spectral sequence induces a finite
  filtration of $H^j(B\mathcal{G}_k)$ whose associated graded
  contains, as a direct summand, the image of $U$ in
  $E_\infty^{*,*}(k)^j$.  This summand is explicitly computable from
  $H^*(B\G)\otimes M^{\le 10}$ together with $\varepsilon(k)$, while
  the remaining extension term depends only on $M^{\le j}$ and the
  multiplicative structure of the Serre spectral sequence.
\end{enumerate}
In particular, in total degree $j\le 10$ the dependence of
$H^j(B\mathcal{G}_k;\mathbb{F}_2)$ on the bundle class $k$ that is visible
through the first nontrivial fibre class $u_5$ is completely encoded
in the single Serre differential $d_6(u_5)=\varepsilon(k)x_6$.

Question:
Working 2-locally and with mod-2 cohomology, let \(k\in\mathbb Z\), let \(\mathcal G_k=\mathcal G_k(G_2)\) denote the gauge group of the principal \(G_2\)-bundle over \(S^4\) classified by \(k\in \pi_4(BG_2)\cong \mathbb Z\), and consider the evaluation fibration \(\Omega_0^3G_2\to B\mathcal G_k\to BG_2\). Let
\[
E_2^{s,t}(k)\cong H^s\!\left(BG_2;H^t(\Omega_0^3G_2;\mathbb F_2)\right)\Longrightarrow H^{s+t}(B\mathcal G_k;\mathbb F_2)
\]
be its Serre spectral sequence, write \(M=H^*(\Omega_0^3G_2;\mathbb F_2)\), and for \(N\ge 0\) write \(M^{\le N}=\bigoplus_{j\le N}M^j\). Which statement holds for every such integer \(k\)?

In total degree \(j=s+t\le 10\), the \(E_2\)-term is naturally isomorphic as a bigraded \(H^*(BG_2;\mathbb F_2)\)-module to \(H^*(BG_2;\mathbb F_2)\otimes M^{\le 10}\), with \(E_2^{s,t}(k)=H^s(BG_2;\mathbb F_2)\otimes M^t\). Moreover, \(M^j=0\) for \(1\le j\le 4\) and \(M^5\neq 0\), so there exists a nonzero class \(u_5\in M^5\cong E_2^{0,5}(k)\), and \(u_5\) is the first positive-degree generator of \(H^*(\Omega_0^3G_2;\mathbb F_2)\). In total degree \(\le 10\), all differentials \(d_r:E_r^{0,t}(k)\to E_r^{r,t-r+1}(k)\) with \(2\le r\le 5\) vanish on classes in bidegree \((0,t)\), and the only page on which a differential with source in \(E_r^{0,5}(k)\) can be nonzero is \(r=6\). After choosing generators \(x_6\in H^6(BG_2;\mathbb F_2)\cong E_2^{6,0}(k)\) and \(u_5\in M^5\cong E_2^{0,5}(k)\), there is a unique scalar \(\varepsilon(k)\in \mathbb F_2\) such that \(d_6(u_5)=\varepsilon(k)x_6\) in \(E_6^{6,0}(k)\). If \(U\subset E_2^{*,*}(k)\) is the \(H^*(BG_2;\mathbb F_2)\)-submodule generated by \(u_5\), then in total degree \(\le 10\) all \(d_r\) with \(2\le r\le 5\) vanish on \(U\), and every differential \(d_6:E_6^{s,5}(k)\to E_6^{s+6,0}(k)\) with \(s+5\le 10\) satisfies \(d_6(x\cdot u_5)=x\cdot d_6(u_5)\) for \(x\in H^s(BG_2;\mathbb F_2)\). Hence, within \(U\), the entire effect of the Serre differentials in total degree \(\le 10\) is determined by the single scalar \(\varepsilon(k)\). Finally, for each \(j\le 10\), the spectral sequence induces a finite filtration of \(H^j(B\mathcal G_k;\mathbb F_2)\) whose associated graded contains, as a direct summand, the image of \(U\) in the total-degree-\(j\) part of \(E_\infty\); this summand is explicitly computable from \(H^*(BG_2;\mathbb F_2)\otimes M^{\le 10}\) together with \(\varepsilon(k)\), while the remaining extension term depends only on \(M^{\le j}\) and the multiplicative structure of the Serre spectral sequence. Equivalently, in total degree \(j\le 10\), the dependence of \(H^j(B\mathcal G_k;\mathbb F_2)\) on the bundle class \(k\) that is visible through the first nontrivial fibre class \(u_5\) is completely encoded by the single differential \(d_6(u_5)=\varepsilon(k)x_6\).

Choices:
B. In total degree \(j=s+t\le 10\), the \(E_2\)-term is naturally isomorphic as a bigraded \(H^*(BG_2;\mathbb F_2)\)-module to \(H^*(BG_2;\mathbb F_2)\otimes M^{\le 10}\), with \(E_2^{s,t}(k)=H^s(BG_2;\mathbb F_2)\otimes M^t\). Moreover, \(M^j=0\) for \(1\le j\le 5\) and \(M^6\neq 0\), so there exists a nonzero class \(u_6\in M^6\cong E_2^{0,6}(k)\), and \(u_6\) is the first positive-degree generator of \(H^*(\Omega_0^3G_2;\mathbb F_2)\). In total degree \(\le 10\), all differentials \(d_r:E_r^{0,t}(k)\to E_r^{r,t-r+1}(k)\) with \(2\le r\le 5\) vanish on classes in bidegree \((0,t)\), and the only page on which a differential with source in \(E_r^{0,6}(k)\) can be nonzero is \(r=7\). After choosing generators \(x_7\in H^7(BG_2;\mathbb F_2)\cong E_2^{7,0}(k)\) and \(u_6\in M^6\cong E_2^{0,6}(k)\), there is a unique scalar \(\varepsilon(k)\in \mathbb F_2\) such that \(d_7(u_6)=\varepsilon(k)x_7\) in \(E_7^{7,0}(k)\). If \(U\subset E_2^{*,*}(k)\) is the \(H^*(BG_2;\mathbb F_2)\)-submodule generated by \(u_6\), then in total degree \(\le 10\) all \(d_r\) with \(2\le r\le 6\) vanish on \(U\), and every differential \(d_7:E_7^{s,6}(k)\to E_7^{s+7,0}(k)\) with \(s+6\le 10\) satisfies \(d_7(x\cdot u_6)=x\cdot d_7(u_6)\) for \(x\in H^s(BG_2;\mathbb F_2)\). Hence, within \(U\), the entire effect of the Serre differentials in total degree \(\le 10\) is determined by the single scalar \(\varepsilon(k)\).
C. In total degree \(j=s+t\le 10\), the \(E_2\)-term is naturally isomorphic as a bigraded \(H^*(BG_2;\mathbb F_2)\)-module to \(H^*(BG_2;\mathbb F_2)\otimes M^{\le 10}\), with \(E_2^{s,t}(k)=H^s(BG_2;\mathbb F_2)\otimes M^t\). Moreover, \(M^j=0\) for \(1\le j\le 4\) and \(M^5\neq 0\), so there exists a nonzero class \(u_5\in M^5\cong E_2^{0,5}(k)\). In total degree \(\le 10\), all differentials \(d_r:E_r^{0,t}(k)\to E_r^{r,t-r+1}(k)\) with \(2\le r\le 5\) vanish on classes in bidegree \((0,t)\), and the only page on which a differential with source in \(E_r^{0,5}(k)\) can be nonzero is \(r=6\). After choosing generators \(x_6\in H^6(BG_2;\mathbb F_2)\cong E_2^{6,0}(k)\) and \(u_5\in M^5\cong E_2^{0,5}(k)\), there is a unique scalar \(\varepsilon(k)\in \mathbb F_2\) such that \(d_6(u_5)=\varepsilon(k)x_6\) in \(E_6^{6,0}(k)\).
D. In total degree \(j=s+t\le 10\), the \(E_2\)-term is naturally isomorphic as a bigraded \(H^*(BG_2;\mathbb F_2)\)-module to \(H^*(BG_2;\mathbb F_2)\otimes M^{\le 10}\), with \(E_2^{s,t}(k)=H^s(BG_2;\mathbb F_2)\otimes M^t\). Moreover, \(M^j=0\) for \(1\le j\le 4\) and \(M^5\neq 0\), so there exists a nonzero class \(u_5\in M^5\cong E_2^{0,5}(k)\), and \(u_5\) is the first positive-degree generator of \(H^*(\Omega_0^3G_2;\mathbb F_2)\). In total degree \(\le 10\), all differentials \(d_r:E_r^{0,t}(k)\to E_r^{r,t-r+1}(k)\) with \(2\le r\le 5\) vanish on classes in bidegree \((0,t)\), and the only page on which a differential with source in \(E_r^{0,5}(k)\) can be nonzero is \(r=6\). After choosing generators \(x_6\in H^6(BG_2;\mathbb F_2)\cong E_2^{6,0}(k)\) and \(u_5\in M^5\cong E_2^{0,5}(k)\), there is a unique scalar \(\varepsilon\in \mathbb F_2\), independent of \(k\), such that \(d_6(u_5)=\varepsilon x_6\) in \(E_6^{6,0}(k)\) for every integer \(k\). If \(U\subset E_2^{*,*}(k)\) is the \(H^*(BG_2;\mathbb F_2)\)-submodule generated by \(u_5\), then in total degree \(\le 10\) all \(d_r\) with \(2\le r\le 5\) vanish on \(U\), and every differential \(d_6:E_6^{s,5}(k)\to E_6^{s+6,0}(k)\) with \(s+5\le 10\) satisfies \(d_6(x\cdot u_5)=x\cdot d_6(u_5)\) for \(x\in H^s(BG_2;\mathbb F_2)\). Hence, within \(U\), the entire effect of the Serre differentials in total degree \(\le 10\) is determined by the single scalar \(\varepsilon\), and therefore is independent of the bundle class \(k\).
E. In total degree \(j=s+t\le 10\), the \(E_2\)-term is naturally isomorphic as a bigraded \(H^*(BG_2;\mathbb F_2)\)-module to \(H^*(BG_2;\mathbb F_2)\otimes M^{\le 10}\), with \(E_2^{s,t}(k)=H^s(BG_2;\mathbb F_2)\otimes M^t\). Moreover, \(M^j=0\) for \(1\le j\le 4\) and \(M^5\neq 0\), so there exists a nonzero class \(u_5\in M^5\cong E_2^{0,5}(k)\), and \(u_5\) is the first positive-degree generator of \(H^*(\Omega_0^3G_2;\mathbb F_2)\). In total degree \(\le 10\), all differentials \(d_r:E_r^{0,t}(k)\to E_r^{r,t-r+1}(k)\) with \(2\le r\le 5\) vanish on classes in bidegree \((0,t)\), and the only page on which a differential with source in \(E_r^{0,5}(k)\) can be nonzero is \(r=6\). After choosing generators \(x_6\in H^6(BG_2;\mathbb F_2)\cong E_2^{6,0}(k)\) and \(u_5\in M^5\cong E_2^{0,5}(k)\), there is a uniquely determined scalar \(\varepsilon(k)\in\mathbb F_2\) such that \(d_6(u_5)=\varepsilon(k)x_6\) in \(E_6^{6,0}(k)\). If \(U\subset E_2^{*,*}(k)\) is the \(H^*(BG_2;\mathbb F_2)\)-submodule generated by \(u_5\), then in total degree \(\le 10\) all \(d_r\) with \(2\le r\le 5\) vanish on \(U\), and every differential \(d_6:E_6^{s,5}(k)\to E_6^{s+6,0}(k)\) with \(s+5\le 10\) satisfies \(d_6(x\cdot u_5)=x\cdot d_6(u_5)\) for \(x\in H^s(BG_2;\mathbb F_2)\). Finally, for each \(j\le 10\), the spectral sequence induces a finite filtration of \(H^j(B\mathcal G_k;\mathbb F_2)\) whose associated graded is completely and explicitly determined by \(H^*(BG_2;\mathbb F_2)\otimes M^{\le 10}\) together with \(\varepsilon(k)\); in particular, there are no additional extension terms depending on higher fibre classes or on the multiplicative structure of the Serre spectral sequence.

drops the claims about the submodule U and the associated-graded direct-summand/extension-term conclusion

The stem gives setup and notation but does not state the correct conclusion or uniquely signal choice A. There is no explicit answer leakage, though the structure suggests a theorem-selection task.

The item is not a direct restatement of the stem, but it largely asks the examinee to recognize the strongest theorem-like formulation among close variants. It is more theorem recall/identification than an independently generated conclusion.

Some reasoning is needed to compare subtle claims about the first possible differential, dependence on k, and extension control. However, the task is mostly discriminating among prewritten statements rather than producing or deriving the conclusion from scratch.

Several distractors are plausible and target real mathematical failure modes: wrong differential page (B), wrong parameter dependence (D), and overclaiming control of extensions (E). But choice C appears to be a weaker true statement, creating ambiguity and weakening distractor quality; also the correct choice is unusually comprehensive/long.

A technically sophisticated MCQ with little answer leakage and reasonably plausible distractors, but it is weakened by ambiguity because a weaker true option appears among the distractors, and the task leans more toward theorem recognition than genuinely generative reasoning.


Question:
Working 2-locally and with mod-2 cohomology, let \(k\in\mathbb Z\), let \(\mathcal G_k=\mathcal G_k(G_2)\) denote the gauge group of the principal \(G_2\)-bundle over \(S^4\) classified by \(k\in \pi_4(BG_2)\cong \mathbb Z\), and consider the evaluation fibration \(\Omega_0^3G_2\to B\mathcal G_k\to BG_2\). Let
\[
E_2^{s,t}(k)\cong H^s\!\left(BG_2;H^t(\Omega_0^3G_2;\mathbb F_2)\right)\Longrightarrow H^{s+t}(B\mathcal G_k;\mathbb F_2)
\]
be its Serre spectral sequence, write \(M=H^*(\Omega_0^3G_2;\mathbb F_2)\), and for \(N\ge 0\) write \(M^{\le N}=\bigoplus_{j\le N}M^j\). Which statement holds for every such integer \(k\)?

Choices:
B. In total degree \(j=s+t\le 10\), the \(E_2\)-term is naturally isomorphic as a bigraded \(H^*(BG_2;\mathbb F_2)\)-module to \(H^*(BG_2;\mathbb F_2)\otimes M^{\le 10}\), with \(E_2^{s,t}(k)=H^s(BG_2;\mathbb F_2)\otimes M^t\). Moreover, \(M^j=0\) for \(1\le j\le 5\) and \(M^6\neq 0\), so there exists a nonzero class \(u_6\in M^6\cong E_2^{0,6}(k)\), and \(u_6\) is the first positive-degree generator of \(H^*(\Omega_0^3G_2;\mathbb F_2)\). In total degree \(\le 10\), all differentials \(d_r:E_r^{0,t}(k)\to E_r^{r,t-r+1}(k)\) with \(2\le r\le 5\) vanish on classes in bidegree \((0,t)\), and the only page on which a differential with source in \(E_r^{0,6}(k)\) can be nonzero is \(r=7\). After choosing generators \(x_7\in H^7(BG_2;\mathbb F_2)\cong E_2^{7,0}(k)\) and \(u_6\in M^6\cong E_2^{0,6}(k)\), there is a unique scalar \(\varepsilon(k)\in \mathbb F_2\) such that \(d_7(u_6)=\varepsilon(k)x_7\) in \(E_7^{7,0}(k)\). If \(U\subset E_2^{*,*}(k)\) is the \(H^*(BG_2;\mathbb F_2)\)-submodule generated by \(u_6\), then in total degree \(\le 10\) all \(d_r\) with \(2\le r\le 6\) vanish on \(U\), and every differential \(d_7:E_7^{s,6}(k)\to E_7^{s+7,0}(k)\) with \(s+6\le 10\) satisfies \(d_7(x\cdot u_6)=x\cdot d_7(u_6)\) for \(x\in H^s(BG_2;\mathbb F_2)\). Hence, within \(U\), the entire effect of the Serre differentials in total degree \(\le 10\) is determined by the single scalar \(\varepsilon(k)\).
C. In total degree \(j=s+t\le 10\), the \(E_2\)-term is naturally isomorphic as a bigraded \(H^*(BG_2;\mathbb F_2)\)-module to \(H^*(BG_2;\mathbb F_2)\otimes M^{\le 10}\), with \(E_2^{s,t}(k)=H^s(BG_2;\mathbb F_2)\otimes M^t\). Moreover, \(M^j=0\) for \(1\le j\le 4\) and \(M^5\neq 0\), so there exists a nonzero class \(u_5\in M^5\cong E_2^{0,5}(k)\). In total degree \(\le 10\), all differentials \(d_r:E_r^{0,t}(k)\to E_r^{r,t-r+1}(k)\) with \(2\le r\le 5\) vanish on classes in bidegree \((0,t)\), and the only page on which a differential with source in \(E_r^{0,5}(k)\) can be nonzero is \(r=6\). After choosing generators \(x_6\in H^6(BG_2;\mathbb F_2)\cong E_2^{6,0}(k)\) and \(u_5\in M^5\cong E_2^{0,5}(k)\), there is a unique scalar \(\varepsilon(k)\in \mathbb F_2\) such that \(d_6(u_5)=\varepsilon(k)x_6\) in \(E_6^{6,0}(k)\).
D. In total degree \(j=s+t\le 10\), the \(E_2\)-term is naturally isomorphic as a bigraded \(H^*(BG_2;\mathbb F_2)\)-module to \(H^*(BG_2;\mathbb F_2)\otimes M^{\le 10}\), with \(E_2^{s,t}(k)=H^s(BG_2;\mathbb F_2)\otimes M^t\). Moreover, \(M^j=0\) for \(1\le j\le 4\) and \(M^5\neq 0\), so there exists a nonzero class \(u_5\in M^5\cong E_2^{0,5}(k)\), and \(u_5\) is the first positive-degree generator of \(H^*(\Omega_0^3G_2;\mathbb F_2)\). In total degree \(\le 10\), all differentials \(d_r:E_r^{0,t}(k)\to E_r^{r,t-r+1}(k)\) with \(2\le r\le 5\) vanish on classes in bidegree \((0,t)\), and the only page on which a differential with source in \(E_r^{0,5}(k)\) can be nonzero is \(r=6\). After choosing generators \(x_6\in H^6(BG_2;\mathbb F_2)\cong E_2^{6,0}(k)\) and \(u_5\in M^5\cong E_2^{0,5}(k)\), there is a unique scalar \(\varepsilon\in \mathbb F_2\), independent of \(k\), such that \(d_6(u_5)=\varepsilon x_6\) in \(E_6^{6,0}(k)\) for every integer \(k\). If \(U\subset E_2^{*,*}(k)\) is the \(H^*(BG_2;\mathbb F_2)\)-submodule generated by \(u_5\), then in total degree \(\le 10\) all \(d_r\) with \(2\le r\le 5\) vanish on \(U\), and every differential \(d_6:E_6^{s,5}(k)\to E_6^{s+6,0}(k)\) with \(s+5\le 10\) satisfies \(d_6(x\cdot u_5)=x\cdot d_6(u_5)\) for \(x\in H^s(BG_2;\mathbb F_2)\). Hence, within \(U\), the entire effect of the Serre differentials in total degree \(\le 10\) is determined by the single scalar \(\varepsilon\), and therefore is independent of the bundle class \(k\).
E. In total degree \(j=s+t\le 10\), the \(E_2\)-term is naturally isomorphic as a bigraded \(H^*(BG_2;\mathbb F_2)\)-module to \(H^*(BG_2;\mathbb F_2)\otimes M^{\le 10}\), with \(E_2^{s,t}(k)=H^s(BG_2;\mathbb F_2)\otimes M^t\). Moreover, \(M^j=0\) for \(1\le j\le 4\) and \(M^5\neq 0\), so there exists a nonzero class \(u_5\in M^5\cong E_2^{0,5}(k)\), and \(u_5\) is the first positive-degree generator of \(H^*(\Omega_0^3G_2;\mathbb F_2)\). In total degree \(\le 10\), all differentials \(d_r:E_r^{0,t}(k)\to E_r^{r,t-r+1}(k)\) with \(2\le r\le 5\) vanish on classes in bidegree \((0,t)\), and the only page on which a differential with source in \(E_r^{0,5}(k)\) can be nonzero is \(r=6\). After choosing generators \(x_6\in H^6(BG_2;\mathbb F_2)\cong E_2^{6,0}(k)\) and \(u_5\in M^5\cong E_2^{0,5}(k)\), there is a uniquely determined scalar \(\varepsilon(k)\in\mathbb F_2\) such that \(d_6(u_5)=\varepsilon(k)x_6\) in \(E_6^{6,0}(k)\). If \(U\subset E_2^{*,*}(k)\) is the \(H^*(BG_2;\mathbb F_2)\)-submodule generated by \(u_5\), then in total degree \(\le 10\) all \(d_r\) with \(2\le r\le 5\) vanish on \(U\), and every differential \(d_6:E_6^{s,5}(k)\to E_6^{s+6,0}(k)\) with \(s+5\le 10\) satisfies \(d_6(x\cdot u_5)=x\cdot d_6(u_5)\) for \(x\in H^s(BG_2;\mathbb F_2)\). Finally, for each \(j\le 10\), the spectral sequence induces a finite filtration of \(H^j(B\mathcal G_k;\mathbb F_2)\) whose associated graded is completely and explicitly determined by \(H^*(BG_2;\mathbb F_2)\otimes M^{\le 10}\) together with \(\varepsilon(k)\); in particular, there are no additional extension terms depending on higher fibre classes or on the multiplicative structure of the Serre spectral sequence.

The stem sets up notation and the spectral sequence but does not explicitly or implicitly reveal the correct choice. The key distinctions among the options are not leaked in the prompt.

The item is not a direct restatement of the stem, but it mainly tests recognition of a precise theorem-level conclusion about the spectral sequence rather than deriving a new consequence from simpler premises.

There is some reasoning pressure because the choices differ in subtle ways (first nonzero degree, page of the differential, dependence on k, extension claims). However, the pressure is weakened because choice C is a weaker true statement contained in A, while the stem asks only which statement 'holds,' not which strongest statement holds.

Several distractors are mathematically plausible and target realistic errors: wrong degree/page shift (B), incorrect k-independence (D), and overclaiming collapse/extension control (E). But C is a poor distractor because it appears to be genuinely true, making the item ambiguous rather than cleanly single-answer.

Technically sophisticated and free of answer leakage, but flawed as an MCQ because it appears to have more than one true option unless the prompt explicitly asks for the strongest conclusion. This reduces both discriminative power and generative reasoning value.


\begin{thm}\label{thm:Kitaev.vanishing}
If $U,V\in\mathcal{L}^\times$ satisfy
\begin{equation}\label{eqn:Kitaev.class}
(U-\mathbf{1})(V-\mathbf{1})\in\mathcal{L}^1\quad\mathrm{and}\quad (V-\mathbf{1})(U-\mathbf{1})\quad \in\mathcal{L}^1,
\end{equation}
then $\det(UVU^{-1}V^{-1})=1$.
\end{thm}

\label{sec:introduction}
We write $\mathcal{L}\equiv \mathcal{L}(\mathcal{H})$ for the ring of bounded linear operators on a Hilbert space $\mathcal{H}$, and $\mathcal{L}^1\equiv \mathcal{L}^1(\mathcal{H})$ for the ideal of \emph{trace class} operators. The group of invertibles in $\mathcal{L}$ is denoted $\mathcal{L}^\times$, and $\mathbf{1}$ denotes the identity operator on $\mathcal{H}$.

The following Theorem is our main result.

\section{Introduction}\label{sec:introduction}
We write $\mathcal{L}\equiv \mathcal{L}(\mathcal{H})$ for the ring of bounded linear operators on a Hilbert space $\mathcal{H}$, and $\mathcal{L}^1\equiv \mathcal{L}^1(\mathcal{H})$ for the ideal of \emph{trace class} operators. The group of invertibles in $\mathcal{L}$ is denoted $\mathcal{L}^\times$, and $\mathbf{1}$ denotes the identity operator on $\mathcal{H}$.

The Fredholm determinant is conjugation invariant, and is a group homomorphism from the invertible determinant class operators to the non-zero complex numbers. From these algebraic properties alone, it is straightforward to show that $\det(UVU^{-1}V^{-1})=1$ whenever any of $U,V$ or $UV$ is determinant class. Nevertheless, in general, $\det(UVU^{-1}V^{-1})$ can take \emph{any} non-zero value \cite{AV,EF}, and it is closely related to the rich subject of traces of commutators \cite{HH}, as well as $K$-theory \cite{Brown}.
Now, Kitaev's condition \eqref{eqn:Kitaev.class} implies
\begin{equation}\label{eqn:Kitaev.implies.det.class}
\begin{aligned}
UVU^{-1}V^{-1}-\mathbf{1}&=(UV-VU)U^{-1}V^{-1}\\
&=\big((U-\mathbf{1})(V-\mathbf{1})-(V-\mathbf{1})(U-\mathbf{1})\big)U^{-1}V^{-1}\in\mathcal{L}^1,
\end{aligned}
\end{equation}
so that $\det(UVU^{-1}V^{-1})$ is automatically well-defined, but it is not at all obvious that it must be trivial. Furthermore, because Kitaev's condition occurs naturally in geometric and physics problems (see Section \ref{sec:quantized.traces}), the veracity of Theorem \ref{thm:Kitaev.vanishing} is of considerable general interest for applications.

\section{Proof of determinant trivialization theorem}\label{sec:proof}
For each $n\geq 1$, we write $M_n(-)$ for the $n\times n$ matrix ring with entries in $(-)$, and $\GL_n(\mathcal{L})$ for the group of invertibles in $M_n(\mathcal{L})$. We may identify $M_n(\mathcal{L})$ with $\mathcal{L}(\mathcal{H}^{\oplus n})$, and $M_n(\mathcal{L}^1)$ with the ideal $\mathcal{L}^1(\mathcal{H}^{\oplus n})$. Thus, there are Fredholm determinant homomorphisms
\[
{\det}^{(n)}:\{U\in \GL_n(\mathcal{L})\,:\,U-\mathbf{1}^{\oplus n}\in M_n(\mathcal{L}^1)\}\to\mathbb{C}^\times,\qquad n\geq 1,
\]
and they are compatible with $\det=\det^{(1)}$ in the sense that
\begin{equation}\label{eqn.det.stable}
{\det}^{(n)}\begin{pmatrix} U & 0 \\ 0 & \mathbf{1}^{\oplus (n-1)}\end{pmatrix} = {\det}^{(1)}\, U,\qquad n\geq 1.
\end{equation}

By combining the Steinberg relations with the Kitaev condition \eqref{eqn:Kitaev.class}, and exploiting the redundancy Lemma \ref{lem:redundant.elementary}, we can rewrite $\det(UVU^{-1}V^{-1})$ as the determinant of a \emph{single} commutator of \emph{elementary matrices}:
\begin{prop}\label{prop:algebraic.step}
Let $U,V\in\mathcal{L}^\times$ satisfy the Kitaev condition . Then
\begin{equation}\label{eqn:algebraic.simplification}
\det(UVU^{-1}V^{-1})={\det}^{(2)}\big(e_{12}(U-\mathbf{1})e_{21}(V-\mathbf{1})e_{12}(\mathbf{1}-U)e_{21}(\mathbf{1}-V)\big).
\end{equation}
\end{prop}
\begin{proof}
According to the identities \eqref{eqn:mult.commutator.stabalized}--\eqref{eqn:Whitehead.big.UV},
\begin{equation}\label{eqn:det.alg.calc.first.factorization}
\det(UVU^{-1}V^{-1})={\det}^{(3)} \big(d_{12}(U)d_{13}(V)d_{12}(U)^{-1}d_{13}(V)^{-1}\big),
\end{equation}
where
\begin{equation}\label{eqn:for.shuffling}
\begin{aligned}
d_{12}(U) & = e_{21}(U^{-1})e_{12}(\mathbf{1}-U)e_{21}(-\mathbf{1})e_{12}(\mathbf{1}-U^{-1})\\
d_{13}(V) & = e_{31}(V^{-1})e_{13}(\mathbf{1}-V)e_{31}(-\mathbf{1})e_{13}(\mathbf{1}-V^{-1}).
\end{aligned}
\end{equation}
We would like to swap the positions of $d_{12}(U)$ and $d_{13}(V)$ in \eqref{eqn:det.alg.calc.first.factorization}, in order to cancel $d_{12}(U)^{-1}d_{13}(V)^{-1}$. This entails moving the elementary matrix factors of $d_{12}(U)$ past those of $d_{13}(V)$. Because of \eqref{eqn:Steinberg.commutator.123}, we will pick up some commutator terms of the form $e_{ik}(ST)$ or $e_{ki}(-TS)$ along the way.
In general, a complicated collection of commutator terms will be acquired.

Fortunately, for the purposes of computing the determinant \eqref{eqn:det.alg.calc.first.factorization}, Lemma \ref{lem:redundant.elementary} tells us that we can discard any $e_{ik}(ST)$ or $e_{ki}(-TS)$ term with $ST\in\mathcal{L}^1$ or $TS\in \mathcal{L}^1$, since this kind of elementary matrix has trivial determinant. The Kitaev condition \eqref{eqn:Kitaev.class} enters here: Because $(U-\mathbf{1})(V-\mathbf{1})$ is trace class, for $i\neq k$,
\begin{align*}
&{\det}^{(3)}\big(W e_{ij}(U-\mathbf{1})e_{jk}(V-\mathbf{1}) W^\prime\big)\\
&={\det}^{(3)}\big(W e_{ik}((U-\mathbf{1})(V-\mathbf{1}))e_{jk}(V-\mathbf{1})e_{ij}(U-\mathbf{1}) W^\prime\big) & (\mathrm{Eq.}\,\eqref{eqn:Steinberg.commutator.123})\\
&={\det}^{(3)}\big(W e_{jk}(V-\mathbf{1})e_{ij}(U-\mathbf{1}) W^\prime\big) & (\mathrm{Lemma}\;\ref{lem:redundant.elementary})
\end{align*}
holds whenever $W,W^\prime$ are invertible operators such that the above determinants are defined. Similarly with $U$ and $V$ swapped. The above property is conveniently expressed as an extra ``commutativity'' relation,
\[
e_{ij}(U-\mathbf{1})e_{jk}(V-\mathbf{1})\simeq e_{jk}(V-\mathbf{1})e_{ij}(U-\mathbf{1}),\qquad i\neq k,
\]
\emph{where ``$\simeq$'' indicates equality after taking a determinant}.
Kitaev's condition \eqref{eqn:Kitaev.class} also implies
\[
(U-\mathbf{1})V=U-\mathbf{1}=V(U-\mathbf{1}),\qquad
(V-\mathbf{1})U=V-\mathbf{1}=U(V-\mathbf{1})\qquad \mathrm{mod}\;\mathcal{L}^1,
\]
The extra relations made available by Kitaev's condition \eqref{eqn:Kitaev.class} are summarized below:
\begin{equation}\label{eqn:Steinberg.Kitaev.extra.identity.2}
\begin{aligned}
e_{ij}(U-\mathbf{1})e_{jk}(V-\mathbf{1}) &\simeq e_{jk}(V-\mathbf{1})e_{ij}(U-\mathbf{1}), & i\neq k,\\
e_{ij}(V-\mathbf{1})e_{jk}(U-\mathbf{1}) &\simeq e_{jk}(U-\mathbf{1})e_{ij}(V-\mathbf{1}), & i\neq k;\\
e_{ij}((U-\mathbf{1})V)&\simeq e_{ij}(U-\mathbf{1})\simeq e_{ij}(V(U-\mathbf{1})),\\
e_{ij}((V-\mathbf{1})U)&\simeq e_{ij}(V-\mathbf{1})\simeq e_{ij}(U(V-\mathbf{1})).
\end{aligned}
\end{equation}
As Kitaev's condition \eqref{eqn:Kitaev.class} holds with $U$ replaced by $U^{-1}$ and/or $V$ replaced by $V^{-1}$, the same replacements can be made in \eqref{eqn:Steinberg.Kitaev.extra.identity.2}.

Now specialize to the case $R=\mathcal{L}$ and $I=\mathcal{L}^1$. For $U,V\in \mathcal{L}^\times$ with determinant class $UVU^{-1}V^{-1}$, the above general definitions give
\begin{equation}\label{eqn:Brown.observation}
\begin{aligned}
\check{\det}\circ\partial\{u,v\}
&=\check{\det}\big[d_{12}(U)d_{13}(V)d_{12}(U)^{-1}d_{13}(V)^{-1}\big]_{K_1(\mathcal{L},\mathcal{L}^1)}\\
&=\check{\det}\left[\begin{pmatrix}UVU^{-1}V^{-1} & 0 & 0 \\ 0 & \mathbf{1} & 0 \\ 0 & 0 & \mathbf{1}\end{pmatrix}\right]_{K_1(\mathcal{L},\mathcal{L}^1)}\\
&=\det(UVU^{-1}V^{-1}).
\end{aligned}
\end{equation}
This $K$-theoretic nature of $\det(UVU^{-1}V^{-1})$ was first observed by Brown \cite{Brown}.

\subsection{Application to quantized traces}\label{sec:quantized.traces}\label{sec:application.quantized.traces}
Let $A,B\in\mathcal{L}$, and suppose
\begin{equation}\label{eqn:quantized.trace.hypothesis}
[A,B]\in \mathcal{L}^1\;\;\mathrm{and}\;\;(e^A-\mathbf{1})(e^B-\mathbf{1})\in\mathcal{L}^1.
\end{equation}
Then, as in \eqref{eqn:Kitaev.implies.det.class}, $(e^B-\mathbf{1})(e^A-\mathbf{1})$ is trace class as well. Thus, $e^A,e^B$ satisfy Kitaev's condition \eqref{eqn:Kitaev.class}, so Theorem \ref{thm:Kitaev.vanishing} guarantees that $\det(e^Ae^Be^{-A}e^{-B})=1$. By the Pincus--Helton--Howe formula \eqref{eqn:HH.formula}, we learn that the condition \eqref{eqn:quantized.trace.hypothesis} implies the trace quantization,
\[
\frac{1}{2\pi i}\mathrm{Tr}[A,B]\in\mathbb{Z}.
\]

This result was first suggested by A.~Kitaev in his influential paper \cite[pp.~56]{Kitaev}, for the purpose of demonstrating the integrality of certain trace formulae appearing in modern quantum physics (see Section \ref{sec:quantized.traces}). We shall therefore refer to \eqref{eqn:Kitaev.class} as the \emph{Kitaev condition} on an invertible pair $U,V\in \mathcal{L}^\times$.

\medskip
Recall that an operator $T\in\mathcal{L}$ is \emph{trace class} if for an(y) orthonormal basis $\mathcal{B}$, the sum $\sum_{e\in \mathcal{B}}\big\langle e\big||T|e\big\rangle$ is finite. In this case, its \emph{trace}
\[
\mathrm{Tr}(T)=\sum_{e\in \mathcal{B}}\big\langle e\big|Te\big\rangle \in\mathbb{C}
\]
is well-defined, independently of the choice of $\mathcal{B}$. An operator $U\in\mathcal{L}$ is \emph{determinant class} if $U-\mathbf{1}\in \mathcal{L}^1$, in which case, its \emph{(Fredholm) determinant} $\det(U)\in\mathbb{C}$ is well-defined. See \cite[Chapter 3]{Simon} for a detailed treatment. The identity
\[
\det(\exp(T))=\exp(\mathrm{Tr}(T)),\qquad T\in\mathcal{L}^1,
\]
shows that a trivial determinant is intimately related to integrality of a corresponding trace (up to a $2\pi i$ factor).

The Fredholm determinant is conjugation invariant, and is a group homomorphism from the invertible determinant class operators to the non-zero complex numbers. From these algebraic properties alone, it is straightforward to show that $\det(UVU^{-1}V^{-1})=1$ whenever any of $U,V$ or $UV$ is determinant class. Nevertheless, in general, $\det(UVU^{-1}V^{-1})$ can take \emph{any} non-zero value \cite{AV,EF}, and it is closely related to the rich subject of traces of commutators \cite{HH}, as well as $K$-theory \cite{Brown}.
Now, Kitaev's condition \eqref{eqn:Kitaev.class} implies
\begin{equation}\label{eqn:Kitaev.implies.det.class}
\begin{aligned}
UVU^{-1}V^{-1}-\mathbf{1}&=(UV-VU)U^{-1}V^{-1}\\
&=\big((U-\mathbf{1})(V-\mathbf{1})-(V-\mathbf{1})(U-\mathbf{1})\big)U^{-1}V^{-1}\in\mathcal{L}^1,
\end{aligned}
\end{equation}
so that $\det(UVU^{-1}V^{-1})$ is automatically well-defined, but it is not at all obvious that it must be trivial. Furthermore, because Kitaev's condition occurs naturally in geometric and physics problems (see Section \ref{sec:quantized.traces}), the veracity of Theorem \ref{thm:Kitaev.vanishing} is of considerable general interest for applications.

More than fifteen years after Kitaev's paper \cite{Kitaev} appeared, some progress was made in \cite{EF}, where Theorem \ref{thm:Kitaev.vanishing} was proved under the extra assumption that $(U^*-\mathbf{1})(V-\mathbf{1})$ and $(V-\mathbf{1})(U^*-\mathbf{1})$ are also trace class. Borel functional calculus played a key role in \cite{EF}, and it is unclear whether the techniques there can be improved to address the full \emph{adjoint-free} Theorem \ref{thm:Kitaev.vanishing}. Nevertheless, the result of \cite{EF} shows that there are no easy counterexamples to Theorem \ref{thm:Kitaev.vanishing}.

Our approach to Theorem \ref{thm:Kitaev.vanishing} is completely different, and we believe that it gets to the heart of the problem. Our proof is presented in Section \ref{sec:proof}, and comprises two main steps, both of which use the Kitaev condition \eqref{eqn:Kitaev.class} in crucial ways.
\begin{enumerate}
\item An \emph{algebraic} step, Prop.~\ref{prop:algebraic.step}, simplifies $\det(UVU^{-1}V^{-1})$ to $\det(E_1E_2E_1^{-1}E_2^{-1})$, where $E_1,E_2$ are so-called elementary matrices. This is the most novel part of the proof. Experts in algebraic $K$-theory may notice that the abstract concept of Steinberg symbols in $K_2$-theory is guiding our manipulations. We choose to keep the presentation concrete, and defer the $K$-theoretic interpretation to Section \ref{sec:K.perspective}.
\item An \emph{analytic} step computes $\det(E_1E_2E_1^{-1}E_2^{-1})=1$ using a formula of Pincus--Helton--Howe (Eq.~\eqref{eqn:HH.formula}), and Lidskii's trace theorem \cite{Lidskii}.
\end{enumerate}
Finally, the physical context for Theorem \ref{thm:Kitaev.vanishing} is discussed in Section \ref{sec:quantized.traces}.

The proof of Theorem~\ref{thm:Kitaev.vanishing} (given in Section~\ref{sec:proof}) has two main steps, both using the Kitaev condition \eqref{eqn:Kitaev.class} in crucial ways:
\begin{enumerate}
\item An \emph{algebraic} step (Prop.~\ref{prop:algebraic.step}) that “simplifies $\det(UVU^{-1}V^{-1})$ to $\det(E_1E_2E_1^{-1}E_2^{-1})$, where $E_1,E_2$ are so-called elementary matrices.” The text notes that “Steinberg symbols in $K_2$-theory [are] guiding our manipulations,” though the presentation is kept concrete.
\item An \emph{analytic} step that “computes $\det(E_1E_2E_1^{-1}E_2^{-1})=1$ using a formula of Pincus--Helton--Howe (Eq.~\eqref{eqn:HH.formula}), and Lidskii's trace theorem.”
\end{enumerate}
Additionally, the introduction observes that Kitaev's condition implies the commutator is determinant-class via \eqref{eqn:Kitaev.implies.det.class}, so $\det(UVU^{-1}V^{-1})$ is well-defined, though “it is not at all obvious that it must be trivial.”

The proof of \ref{thm:Kitaev.vanishing} (given later) has two main steps, both using the Kitaev condition
\begin{equation}\label{eqn:Kitaev.class}
(U-\mathbf{1})(V-\mathbf{1})\in\mathcal{L}^1\quad\mathrm{and}\quad (V-\mathbf{1})(U-\mathbf{1})\quad \in\mathcal{L}^1,
\end{equation}
in crucial ways:
\begin{enumerate}
\item An \emph{algebraic} step. We first prove the following proposition.
\begin{prop}\label{prop:algebraic.step}
Let $U,V\in\mathcal{L}^\times$ satisfy the Kitaev condition . Then
\begin{equation}\label{eqn:algebraic.simplification}
\det(UVU^{-1}V^{-1})={\det}^{(2)}\big(e_{12}(U-\mathbf{1})e_{21}(V-\mathbf{1})e_{12}(\mathbf{1}-U)e_{21}(\mathbf{1}-V)\big).
\end{equation}
\end{prop}
This “simplifies $\det(UVU^{-1}V^{-1})$ to $\det(E_1E_2E_1^{-1}E_2^{-1})$, where $E_1,E_2$ are so-called elementary matrices.” The text notes that “Steinberg symbols in $K_2$-theory [are] guiding our manipulations,” though the presentation is kept concrete.
\item An \emph{analytic} step that “computes $\det(E_1E_2E_1^{-1}E_2^{-1})=1$ using a formula of Pincus--Helton--Howe,
\begin{equation}\label{eqn:HH.formula}
\det(e^Ae^Be^{-A}e^{-B})=\exp(\mathrm{Tr}[A,B]).
\end{equation}
and Lidskii's trace theorem.”
\end{enumerate}
Additionally, the introduction observes that Kitaev's condition implies the commutator is determinant-class via \eqref{eqn:Kitaev.implies.det.class}, so $\det(UVU^{-1}V^{-1})$ is well-defined, though “it is not at all obvious that it must be trivial.”

Question:
Let mathcal{H} be a Hilbert space, let \(\mathcal{L}(\mathcal{H})\) be the algebra of bounded linear operators on \(\mathcal{H}\), let \(\mathcal{L}^1(\mathcal{H})\) be the ideal of trace-class operators, let \(\mathcal{L}(\mathcal{H})^\times\) denote the invertible bounded operators, and let \(\mathbf{1}\) be the identity operator. Suppose \(U,V\in \mathcal{L}(\mathcal{H})^\times\) satisfy
\[
(U-\mathbf{1})(V-\mathbf{1})\in \mathcal{L}^1(\mathcal{H})
\quad\text{and}\quad
(V-\mathbf{1})(U-\mathbf{1})\in \mathcal{L}^1(\mathcal{H}).
\]
For an invertible operator of the form \(\mathbf{1}+K\) with \(K\in \mathcal{L}^1(\mathcal{H})\), write \(\det\) for its Fredholm determinant. Under these hypotheses, which statement about the multiplicative commutator \(UVU^{-1}V^{-1}\) holds?

The Fredholm determinant of the commutator is trivial: \(\det(UVU^{-1}V^{-1})=1\).

Choices:
B.
The multiplicative commutator is always determinant class under these hypotheses, and its Fredholm determinant is given by the trace formula
\[
\det(UVU^{-1}V^{-1})=\exp\!\big(\operatorname{Tr}((U-\mathbf{1})(V-\mathbf{1})-(V-\mathbf{1})(U-\mathbf{1}))\big).
\]
C.
The multiplicative commutator is determinant class, i.e.
\[
UVU^{-1}V^{-1}-\mathbf{1}\in \mathcal{L}^1(\mathcal{H}).
\]
D.
If \((U-\mathbf{1})(V-\mathbf{1})\in \mathcal{L}^1(\mathcal{H})\), then automatically
\[
\det(UVU^{-1}V^{-1})=1,
\]
without any need to assume \((V-\mathbf{1})(U-\mathbf{1})\in \mathcal{L}^1(\mathcal{H})\).
E.
Under these hypotheses the commutator itself is trace class relative to the identity in the stronger sense that
\[
UVU^{-1}V^{-1}=\mathbf{1},
\]
so in particular its Fredholm determinant equals \(1\).

analytic-step trace formula applied directly to U,V rather than logarithmic elementary-matrix commutator

drop the final trivialization conclusion and keep only determinant-class/well-definedness

two-sided Kitaev condition reduced to one-sided trace-class product hypothesis

replace determinant triviality of the commutator by actual operator triviality

The stem states the hypotheses and asks for the resulting conclusion, but it does not explicitly reveal that the determinant equals 1 or strongly telegraph choice A over the others.

This is close to a direct theorem-recall item: the hypotheses are presented in essentially theorem form and the correct option states the standard conclusion. However, the presence of nearby alternatives (weaker true statement, overstrong statement, and altered hypotheses) prevents it from being a pure tautological restatement.

The item requires some reasoning to distinguish the exact conclusion from a merely weaker consequence (C), an unjustified stronger claim (E), and a one-sided hypothesis variant (D). Still, for a student who recognizes the theorem, the answer is fairly immediate rather than deeply generative.

The distractors are mathematically plausible and target realistic failure modes: confusing determinant triviality with trace-class membership (C), overextending to a one-sided condition (D), inserting an appealing but unjustified trace formula (B), or strengthening determinant triviality to operator equality (E).

A solid, high-quality theorem-based MCQ with strong distractors and no answer leakage, but it remains somewhat recall-oriented and only moderately non-tautological.