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\title{A tensor-opposite proof of the anyon-proliferation identity}
\author{FPRD Lab\\Agentic AI under the direction of Joshua Gay}
\date{7 September 2026; public edition 8 September 2026}

\begin{document}
\maketitle

\begin{center}
\small\itshape Candidate proof. Internally audited; no external specialist or peer review.
\end{center}

\begin{abstract}
Let $\cT$ be a non-degenerate braided fusion category and let
$\cB\subset\cT$ be a full braided fusion subcategory.  We prove that
condensing the canonical diagonal algebra
\[
 A_{\cB}=\bigoplus_{x\in\Irr(\cB)}x\boxt i_-(x^*)
 \quad\text{in}\quad \cT\boxt\Z(\cB)
\]
recovers $\cT$ as the category of local modules.  Here
$i_-(z)=(z,c_{-,z}^{-1})$ is the negative central embedding.  The
point is to run the usual Morita-center/alpha-induction argument over the
\emph{tensor-opposite} category $\cB^{\mathrm{mop}}$.  Transporting its center
back to $\Z(\cB)$ converts the evaluation half-braiding into exactly $i_-$.
This removes the symmetry hypothesis in the familiar symmetric-case proof.
\end{abstract}

\section{Statement and conventions}

Cheng's proposed anyon-proliferation identity, equations (II.1)--(II.2)
of \href{https://arxiv.org/html/2608.20604v1}{[1]}, asks whether adjoining
$\Z(\cB)$ and condensing the diagonal algebra returns the original
category $\cT$. The question includes braided subcategories $\cB$ that
are neither non-degenerate nor symmetric. The argument below is a
candidate proof of that general categorical identity. It does not derive
a microscopic Hamiltonian or the dynamics of a phase transition.


We work over an algebraically closed field of characteristic zero.  A half-braiding
on $z$ is written
\[
 \sigma_y:z\otimes y\longrightarrow y\otimes z.
\]
For a braided fusion category $\cB$, let
\[
 i_+(z)=(z,c_{z,-}),\qquad i_-(z)=(z,c_{-,z}^{-1})
\]
denote the positive and negative embeddings into $\Z(\cB)$.  Thus $i_-$ is
braided when its source is $\cB^{\mathrm{rev}}$, the category with reversed
braiding.  We reserve $\cB^{\mathrm{mop}}$ for the category with the same
underlying category and reversed tensor product,
\[
 x\odot y:=y\otimes x.
\]
These two reversals must not be conflated.

The object $A_{\cB}$ carries the standard diagonal connected etale algebra
structure: equivalently, it is the coend/evaluation algebra associated with the
tensor functor $\cB\boxt\cB^{\mathrm{rev}}\to\cB$.  This is the algebra intended
in Cheng's equation~(II.2).

\begin{theorem}\label{thm:main}
Let $\cT$ be a non-degenerate braided fusion category and let
$\cB\subset\cT$ be a full braided fusion subcategory.  Then there is a braided
equivalence
\[
 \boxed{\bigl(\cT\boxt\Z(\cB)\bigr)^0_{A_{\cB}}\simeq\cT.}
\]
For unitary categories and unitary inclusions, the equivalence may be taken
unitary.
\end{theorem}

\section{The tensor-opposite chirality calculation}

The braiding defines a strong monoidal equivalence
\[
 J:\cB^{\mathrm{mop}}\longrightarrow\cB,
 \qquad J(x)=x,
 \qquad J_2(x,y)=c_{x,y}:x\otimes y\longrightarrow y\otimes x.
\]
The monoidal coherence for $J$ is exactly the braid relation.  It induces a
braided equivalence $\Z(J):\Z(\cB^{\mathrm{mop}})\to\Z(\cB)$.

If $(z,\gamma)\in\Z(\cB^{\mathrm{mop}})$, then, when read in $\cB$,
\[
 \gamma_y:y\otimes z\longrightarrow z\otimes y.
\]
Transport through $J$ gives the half-braiding
\begin{equation}\label{eq:centertransport}
 \widehat\gamma_y
 =c_{y,z}^{-1}\,\gamma_y\,c_{z,y}:
 z\otimes y\longrightarrow y\otimes z.
\end{equation}
In particular, the half-braiding
\begin{equation}\label{eq:evaluationhb}
 \gamma_y=c_{z,y}^{-1}:y\otimes z\longrightarrow z\otimes y
\end{equation}
is carried to
\[
 \widehat\gamma_y=c_{y,z}^{-1}.
\]
Thus the center object selected by right evaluation in the tensor-opposite
module construction becomes the negative embedding $i_-(z)$ in $\Z(\cB)$.
Notice that this is not the generally invalid operation
$(z,\sigma)\mapsto(z,\sigma^{-1})$ inside one fixed center.

\section{Morita construction and the induced algebra}

Set
\[
 \cN:=\cT\boxt\cB^{\mathrm{mop}},\qquad \cM:=\cT.
\]
Make $\cM$ a left $\cN$-module category by
\begin{equation}\label{eq:action}
 (t\boxt b)\triangleright m:=t\otimes m\otimes b.
\end{equation}
The reversal in $\cB^{\mathrm{mop}}$ makes the associativity immediate:
successive right factors occur as $b'\otimes b$, exactly as in the tensor
product of $\cN$.  The module category is exact and indecomposable.  Let
\[
 \cD:=\Fun_{\cN}(\cM,\cM)
\]
be the Morita-dual fusion category, with tensor product given by composition.
Morita invariance supplies a braided equivalence
\begin{equation}\label{eq:schauenburg}
 \Sigma:\Z(\cN)\xrightarrow{\sim}\Z(\cD),
 \qquad \FPdim(\cD)=\FPdim(\cN)=\FPdim(\cT)\FPdim(\cB).
\end{equation}
We choose the Schauenburg equivalence characterized on underlying objects by
sending a central object $n\in\Z(\cN)$ to the $\cN$-module endofunctor
$n\triangleright-$.  This convention fixes the possible opposite-category
ambiguity in what follows.

Put $\cC:=\cT\boxt\Z(\cB)$.  Using the positive embedding of $\cT$ and
the center transport above, define a fully faithful braided functor
\[
 s:=i_+^{\cT}\boxt\Z(J)^{-1}:\cC
 \longrightarrow
 \Z(\cT)\boxt\Z(\cB^{\mathrm{mop}})
 \simeq\Z(\cN).
\]
Let
\begin{equation}\label{eq:alpha}
 \alpha:=U_{\cD}\circ\Sigma\circ s:\cC\longrightarrow\cD,
\end{equation}
where $U_{\cD}$ forgets a half-braiding.  This is a central tensor functor.

We now identify its induced algebra.  Write
$\Z(J)^{-1}(Z)=(b,\gamma)$ for $Z\in\Z(\cB)$.  On the module category
$\cM$, the object $\alpha(t\boxt Z)$ is the endofunctor
\begin{equation}\label{eq:endofunctor}
 m\longmapsto t\otimes m\otimes b.
\end{equation}
For $a\in\cT$ and $y\in\cB$, its $\cN$-module constraint is the composite
\[
 t\otimes a\otimes m\otimes y\otimes b
 \xrightarrow{\ c_{t,a}\ }
 a\otimes t\otimes m\otimes y\otimes b
 \xrightarrow{\ \gamma_y\ }
 a\otimes t\otimes m\otimes b\otimes y.
\]

\begin{lemma}\label{lem:hom}
For $t\in\cT$ and $Z\in\Z(\cB)$ there is a natural isomorphism
\begin{equation}\label{eq:homformula}
 \operatorname{Hom}_{\cD}\bigl(\alpha(t\boxt Z),\Id_{\cM}\bigr)
 \cong
 \bigoplus_{x\in\Irr(\cB)}
 \operatorname{Hom}_{\cT}(t,x)\otimes
 \operatorname{Hom}_{\Z(\cB)}\bigl(Z,i_-(x^*)\bigr).
\end{equation}
Under these isomorphisms, composition of module-natural evaluation maps is
the standard coend multiplication of the diagonal algebra.
\end{lemma}

\begin{proof}
First take $t$ and the underlying object $b$ of $Z$ simple.  A
$\cN$-module-natural map from the endofunctor in~\eqref{eq:endofunctor} to
the identity is determined by its value at the tensor unit.  It can be nonzero
only when $t\simeq b^*$.  Up to scalar its components are
\[
 t\otimes m\otimes b
 \xrightarrow{c_{t,m}\otimes\mathrm{id}_b}
 m\otimes t\otimes b
 \xrightarrow{\mathrm{id}_m\otimes\mathrm{ev}}
 m.
\]
Compatibility with the right $\cB^{\mathrm{mop}}$-action says, for every
$y\in\cB$, that
\[
 (\mathrm{ev}\otimes\mathrm{id}_y)(\mathrm{id}_t\otimes\gamma_y)
 =
 (\mathrm{id}_y\otimes\mathrm{ev})(c_{t,y}\otimes\mathrm{id}_b).
\]
Rigidity identifies this condition with
$\gamma_y=c_{b,y}^{-1}$.  By~\eqref{eq:centertransport}, this is equivalent
to $Z\simeq i_-(b)$.  Additivity and semisimplicity give
\eqref{eq:homformula}.

For the final assertion, restrict $\alpha$ to
$\cB\boxt i_-(\cB^{\mathrm{rev}})$.  Evaluation at the tensor unit identifies
the preceding module-natural maps with the evaluation pairing for the tensor
functor $\cB\boxt\cB^{\mathrm{rev}}\to\cB$.  The monoidal constraint is the
same crossing used in~\eqref{eq:centertransport}.  Therefore the lax-monoidal
product on the right adjoint is the coend/evaluation product, not merely an
algebra structure on the same underlying object.  Equivalently, on simple
summands the product is obtained by resolving
$x\otimes y$ into simples and composing the dual evaluation maps; this is the
standard diagonal algebra multiplication.  This also proves the assertion
without choosing bases, by the dinatural universal property of the coend.
This is also exactly the ``as algebras'' step in the proof of
Lan--Kong--Wen, Lemma~4.18, after replacing the symmetric category there by
$\cB^{\mathrm{mop}}$: the alpha-induction square and unit-intersection argument
are unchanged, while tensor reversal supplies the coherent module action.
\end{proof}

Let $\alpha^R$ be the right adjoint of $\alpha$.  By
\cref{lem:hom} and semisimple Yoneda,
\begin{equation}\label{eq:inducedalgebra}
 \alpha^R(\mathbf1_{\cD})\cong
 \bigoplus_{x\in\Irr(\cB)}x\boxt i_-(x^*)=A_{\cB}
\end{equation}
as connected etale algebras, not only as objects.

The central-functor theorem identifies $\cC_{A_{\cB}}$ monoidally with the
fusion subcategory $\operatorname{im}(\alpha)\subset\cD$.  But
\[
 \FPdim(\cC_{A_{\cB}})
 =\frac{\FPdim(\cT)\FPdim(\cB)^2}{\FPdim(\cB)}
 =\FPdim(\cD).
\]
Hence $\alpha$ is dominant and
\begin{equation}\label{eq:moduleequivalence}
 \cC_{A_{\cB}}\simeq\cD.
\end{equation}
Under this equivalence, the free-module central lift of $\cC$ agrees with
$\Sigma s$.

\section{Extracting the local sector}

Because $\cC$ is non-degenerate, the standard center theorem for local modules
gives a braided equivalence
\begin{equation}\label{eq:dmno}
 \Z(\cC_{A_{\cB}})\simeq
 \cC\boxt\bigl(\cC^0_{A_{\cB}}\bigr)^{\mathrm{rev}},
\end{equation}
and identifies the first factor with the central lift of the free-module
functor.  Combining~\eqref{eq:schauenburg}, \eqref{eq:moduleequivalence}, and
the factor tracking just noted, the centralizer of
$\bigl(\cC^0_{A_{\cB}}\bigr)^{\mathrm{rev}}$ is transported to $s(\cC)$ in
$\Z(\cN)$.

Now
\[
 \Z(\cN)
 \simeq \Z(\cT)\boxt\Z(\cB^{\mathrm{mop}})
 \simeq \cT_+\boxt\cT_-\boxt\Z(\cB^{\mathrm{mop}}),
\]
where $\cT_+\simeq\cT$ and $\cT_-\simeq\cT^{\mathrm{rev}}$ are the two
standard factors in $\Z(\cT)$.  By construction,
\[
 s(\cC)=\cT_+\boxt\Z(\cB^{\mathrm{mop}}).
\]
All three displayed factors are non-degenerate, so the Mueger centralizer of
$s(\cC)$ is exactly $\cT_-$.  Comparing with~\eqref{eq:dmno} yields
\[
 \bigl(\cC^0_{A_{\cB}}\bigr)^{\mathrm{rev}}\simeq\cT^{\mathrm{rev}}.
\]
Reversing both braidings proves \cref{thm:main}.

In the unitary setting, every functor used above is built from unitary
braidings, adjunctions, and Morita equivalences.  The resulting braided
equivalence can therefore be unitarized; it then preserves the canonical
unitary ribbon structures.

\section{Audit notes}

\begin{enumerate}[leftmargin=2em]
\item If $\cB=\mathrm{Vec}$, then $A_{\cB}=\mathbf1$ and the construction
reduces tautologically to $\cT$.
\item If $\cB$ is non-degenerate, the proof reduces to cancelling the
Lagrangian diagonal algebra between one $\cB$ factor in $\cT$ and the
$\cB^{\mathrm{rev}}$ factor of $\Z(\cB)$.
\item If $\cB$ is symmetric, $\cB^{\mathrm{mop}}\simeq\cB$ through the
symmetry and the argument specializes to Lan--Kong--Wen, Lemma~4.18.
\item A same-chirality attempt with the ordinary tensor category $\cB$ selects
$i_+$ rather than $i_-$.  The tensor-opposite step is therefore essential.
\item Equality of Frobenius--Perron dimensions and Witt classes is not used as
a substitute for the equivalence.  Dimension enters only after the induced
algebra and central functor have been identified.
\end{enumerate}

\section*{References}
\small
\begin{enumerate}[label={[\arabic*]},leftmargin=2.4em]
\item M. Cheng, \emph{Note on proliferation transitions of non-Abelian
anyons}, \href{https://arxiv.org/abs/2608.20604v1}{arXiv:2608.20604v1} (2026), especially equations (II.1)--(II.2).
\item T. Lan, L. Kong, and X.-G. Wen, \emph{Modular extensions of unitary
braided fusion categories and $2+1$D topological/SPT orders with symmetries},
Commun. Math. Phys. 351 (2017), 709--739; \href{https://arxiv.org/abs/1602.05936}{arXiv:1602.05936}, especially Lemma~4.18.
\item A. Davydov, M. Mueger, D. Nikshych, and V. Ostrik, \emph{The Witt group
of non-degenerate braided fusion categories}, J. Reine Angew. Math. 677
(2013), 135--177; \href{https://arxiv.org/abs/1009.2117}{arXiv:1009.2117}, especially Lemma~3.5 and
Corollary~3.30.
\item P. Schauenburg, \emph{The monoidal center construction and bimodules},
J. Pure Appl. Algebra 158 (2001), 325--346.
\item P. Etingof, D. Nikshych, and V. Ostrik, \emph{Fusion categories and
homotopy theory}, Quantum Topol. 1 (2010), 209--273.
\item D. Reutter, \emph{Uniqueness of unitary structure for unitarizable
fusion categories}, Commun. Math. Phys. 397 (2023), 37--52;
\href{https://arxiv.org/abs/1906.09710}{arXiv:1906.09710}, especially Theorem~5.
\end{enumerate}

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