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  pdfauthor={Berkay Y{\"u}ksel Sayim},
  pdftitle={Bell Algebra without Bell Violation: CHSH Operators for Ising Anyons in the Kitaev Honeycomb Model},
  pdfsubject={quant-ph},
  pdfkeywords={Kitaev honeycomb model, Ising anyons, Majorana zero modes,
               CHSH inequality, Bell algebra, superselection, magic,
               nonstabilizerness, topological quantum computation}
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\begin{document}

\setcounter{secnumdepth}{3}

\title{Bell Algebra without Bell Violation: CHSH Operators for Ising Anyons in the Kitaev Honeycomb Model}

\author{Berkay Y\"uksel Sayim}
\email{berksa@tutamail.com}
\affiliation{Independent Research, Germany}
\affiliation{\href{https://orcid.org/0009-0004-4993-7352}{ORCID:
0009-0004-4993-7352}}

\date{2026-09-10}

\begin{abstract}
We construct four spatially separated Majorana zero modes (MZMs) per
party on a finite Kitaev honeycomb lattice in its non-Abelian phase,
bound to well-separated Ising vortices, and use them to realize a
local, noncommuting CHSH measurement algebra. Two static fusion-parity
observables per party, $Z_A=i\gamma_{A,1}\gamma_{A,2}$ and
$X_A=i\gamma_{A,2}\gamma_{A,3}$ (and analogously for Bob), commute
exactly between parties ($P_2=0$) and fail to commute within a party
($P_1=\lVert[Z_A,X_A]\rVert_2=2.000000$); on the maximally entangled
fusion-parity Bell state the CHSH expectation reaches the algebraic
Tsirelson value $S=2\sqrt2=2.828427$. The value itself is fixed by the
algebra; what the lattice establishes is that this algebra is realized
by eight well-separated vortex-bound modes of a genuine microscopic
Hamiltonian. This value survives fermion-parity
superselection exactly, because the Bell state is a definite-parity
eigenstate with unit weight in its own sector --- a statement specific
to the present four-MZM, measurement-only construction, not to
braid-then-fuse protocols, which are a different experiment. A
commuting-only control
setting returns $K=\sqrt2\le2$, and the finite-size Majorana footprint
leaking across the bipartition decays monotonically with system size
(isotropic scaling $17.0\%\to10.2\%\to2.3\%$). We stress the critical
scope-limiting fact, following Brennen et al., sharpened into a precise no-go by Howard and
Vala [Phys.\ Rev.\ A \textbf{85}, 022304 (2012)]: a \emph{physical} Bell-inequality violation
using only topologically protected Ising-anyon operations does not
follow from this construction, since Ising braiding generates only the
Clifford group; that step requires a non-Clifford (``magic'') resource,
the subject of a companion paper. What we report here is the local
realization of the noncommuting Bell-operator \emph{algebra} on a
genuine microscopic lattice substrate, not a claim of Bell nonlocality.
A loop-threading relative phase $\delta$ reproduces the analytic
corollary $S(\delta)=2\sqrt2\cos^2(\delta/2)$ to a residual of
$4.4\times10^{-16}$ on a $28\times16$ lattice --- a consistency check on
the algebra rather than
an independent signature of loop threading ---, and an independent
gauge-invariant flux-holonomy
diagnostic confirms no adiabatic channel mixing at the quoted scale
($\lVert[W,Z_A]\rVert_F=0.0132$ on a $24\times18$ lattice), with the
sole open discrepancy being a
$0.020$-rad gap between the accumulated Berry phase at one full flux
quantum and the target $\pi$, traced to genuine near-degeneracies along
the loop rather than to a construction error.
\end{abstract}

\keywords{Kitaev honeycomb model; Ising anyons; Majorana zero modes;
CHSH inequality; Bell algebra; superselection; magic; nonstabilizerness}

\maketitle

%% ====================================================================
\section{Introduction}
\label{sec:intro}

Non-Abelian anyons realize a Bell-type correlation structure without
any need for spatially separated entangled particles in the usual
sense: braiding or measuring topological charges of well-separated
quasiparticles can generate the same noncommuting operator algebra
that underlies the CHSH inequality~\cite{CHSH1969,Cirelson1980}, a
point established in general terms by Brennen \emph{et
al.}~\cite{Brennen2009} for a broad class of non-Abelian anyon models,
including SU(2)$_2$ (Ising) anyons. Brennen \emph{et al.} show that a
Bell-operator norm saturating the Tsirelson bound,
$W=-2\sqrt2$, is reachable for Ising-anyon braiding transformations
alone, and give an explicit argument that braiding generates only the
Clifford group so that an additional non-stabilizer resource is required
to convert this algebraic saturation into an actual Bell-inequality
violation with topologically protected operations --- an argument later
sharpened into a precise no-go by Howard and
Vala~\cite{HowardVala2012}: Ising-anyon braiding transformations,
being Clifford operations, cannot violate a Bell inequality when only
topologically protected stabilizer operations are performed.

This paper reports the microscopic instantiation of the Brennen
construction on a concrete Hamiltonian, the Kitaev honeycomb
model~\cite{Kitaev2003,Kitaev2006} in its gapped non-Abelian (Ising)
phase, where vortices bind isolated Majorana zero modes (MZMs).
Four spatially separated MZMs per party, obtained from four
well-separated vortex cores, furnish a local logical qubit per party
together with a second, noncommuting measurement setting realized as a
static fusion-parity measurement --- not a braid --- on a different
pair of Majorana modes. We verify numerically, on the honeycomb
lattice itself, that (i) the resulting two-setting-per-party algebra
realizes the CHSH operator norm at its algebraic (Tsirelson) maximum,
(ii) this value survives fermion-parity superselection exactly, and
(iii) the residual finite-size leakage of the construction across the
party bipartition vanishes in the large-lattice limit. Section
\ref{sec:setup} introduces the lattice substrate and the measurement
algebra. Section~\ref{sec:m0} reports the four core numerical targets.
Section~\ref{sec:rescope} states the re-scoped core claim in the form
required by Howard--Vala and cross-references the companion resource
paper on nonstabilizerness~\cite{Sayim2026magic}. Sections
\ref{sec:m5} and \ref{sec:m5flux} report two corollary checks on a
loop-threading sector phase $\delta$, one analytic and one a
gauge-invariant flux-holonomy diagnostic. Section~\ref{sec:discussion}
discusses prior art and scope.

%% ====================================================================
\section{Lattice substrate and measurement algebra}
\label{sec:setup}

\subsection{Kitaev honeycomb model and Ising vortices}

The Kitaev honeycomb model~\cite{Kitaev2006} is an exactly solvable
spin-$1/2$ model on the honeycomb lattice whose ground-state phase
diagram contains a gapped, non-Abelian phase (the ``B phase'')
supporting Ising anyons: an Abelian vacuum sector $1$, a fermionic
excitation $\psi$, and a non-Abelian vortex $\sigma$ that binds an
unpaired Majorana zero mode. We work on a finite honeycomb lattice
deep in the B phase and place four well-separated vortex pairs, so
that each party (Alice, Bob) is assigned four vortices carrying four
Majorana operators $\gamma_{A,1},\ldots,\gamma_{A,4}$ and
$\gamma_{B,1},\ldots,\gamma_{B,4}$ respectively, with Alice's and
Bob's vortex clusters placed in disjoint, well-separated regions of
the lattice. Reading out a fusion-channel parity of two such vortices
is the standard non-Abelian interferometric measurement primitive
established for anyon platforms~\cite{BondersonShtengelSlingerland2008}, with
a time-domain interferometric readout of Ising-vortex fusion outcomes
proposed as a concrete experimental technique for Kitaev
spin-liquid candidate materials~\cite{Klocke2021}.

The lattice realization is checked against two independent
diagnostics before any Bell-type quantity is computed. First, a
Bloch-transform cross-check against the exactly solvable momentum-space
spectrum confirms the bulk excitation gap ($\Delta\approx0.75$--$0.80$
in units of the nearest-neighbor coupling) at the real-space lattice
size used. Second, the vortex-bound zero modes are tracked directly:
the number of near-zero eigenmodes equals the number of vortex cores
independent of the zero-mode threshold used to define ``near
zero,'' the modes are exponentially localized on their host vortex
($89\%$ of each mode's weight within a radius of six lattice
sites), the residual inter-cluster leakage of this exponential tail is
$2.6\%$, and the finite-size splitting between the modes nominally
paired into one fermionic parity is $5.4\times10^{-2}$ in the same
units. These four checks confirm that the construction below acts on
genuine, well-separated Majorana zero modes bound to real vortex
cores on the lattice, not on an idealized zero-splitting toy model. We
emphasize, in the interest of honest scope, that this is not a
from-scratch re-derivation of the Kitaev honeycomb solution itself
--- we use the model's own exact solvability, cross-checked as above
--- but a verified microscopic instantiation of the measurement
algebra described next.

\subsection{Local qubit and the second, noncommuting setting}
\label{sec:algebra}

Each party's four Majorana operators define two local fusion-parity
observables,
\begin{equation}
\label{eq:ZAXA}
Z_A = i\,\gamma_{A,1}\gamma_{A,2}, \qquad
X_A = i\,\gamma_{A,2}\gamma_{A,3},
\end{equation}
and analogously $Z_B=i\gamma_{B,1}\gamma_{B,2}$,
$X_B=i\gamma_{B,2}\gamma_{B,3}$ for Bob. Both $Z_A$ and $X_A$ are
Hermitian, square to the identity, and have spectrum $\{+1,-1\}$: they
are \emph{static} fusion-channel-parity measurements, of the
measurement-only type discussed by Clarke, Sau, and Das
Sarma~\cite{ClarkeSauDasSarma2016} and by
Karzig~\emph{et~al.}~\cite{Karzig2017}, not braiding unitaries. This
point is a guard against a natural misreading: $X_A$ is a static
$\pm1$ observable ($\lVert X_A-X_A^\dagger\rVert_2=0$, spectrum exactly
$\{+1,-1\}$), never equal to the braid unitary
$\exp(\tfrac{\pi}{4}\gamma_{A,2}\gamma_{A,3})$ that would exchange the
two vortex cores; no braiding and no fusion-channel $F$- or
$R$-matrix ``oracle'' enters its definition or its measurement.

Alice's total local fermion parity,
$P_A=\gamma_{A,1}\gamma_{A,2}\gamma_{A,3}\gamma_{A,4}$, commutes with
both $Z_A$ and $X_A$ exactly, so Alice's local qubit sits in a single,
well-defined superselection sector: her measurement choice never mixes
even and odd local parity. The analogous statement holds for Bob.

The two settings within a party fail to commute,
\begin{equation}
\label{eq:P1}
P_1 \equiv \lVert[Z_A,X_A]\rVert_2 = 2.000000,
\end{equation}
so a local, noncommuting second measurement setting genuinely exists;
we return to this as an adversarial control in Sec.~\ref{sec:m0}.
Across the bipartition, all four cross operators commute exactly,
\begin{equation}
\label{eq:P2}
P_2 \equiv \max_{O\in\{Z_A,X_A\},\,O'\in\{Z_B,X_B\}} \lVert[O,O']\rVert_2 = 0,
\end{equation}
because $Z_A,X_A$ and $Z_B,X_B$ act on disjoint sets of Majorana
operators by construction. Equations~\eqref{eq:P1}--\eqref{eq:P2} are
consequences of the fermionic Clifford algebra of the eight Majorana
operators alone and hold independent of any particular matrix
representation, verified below by a representation-independent
combinatorial cross-check.

%% ====================================================================
\section{The four locked targets}
\label{sec:m0}

We prepare the fusion-parity Bell state $\ket{\Psi}$ as the common
$+1$ eigenstate of $Z_AZ_B$ and $X_AX_B$ on the sixteen-dimensional
joint Majorana Fock space, and evaluate the CHSH functional
\begin{equation}
\label{eq:chsh}
S = \langle Z_A B\rangle + \langle Z_A B'\rangle
  + \langle X_A B\rangle - \langle X_A B'\rangle,
\end{equation}
with Bob's settings taken as the standard equal-weight combinations
\begin{equation}
\label{eq:settings}
B=(Z_B+X_B)/\sqrt2, \qquad B'=(Z_B-X_B)/\sqrt2 .
\end{equation}
Table~\ref{tab:m0} lists
the four pre-registered target quantities together with the ideal
value each is checked against.

\begin{table}[!h]
\centering
\caption{The four locked M0 targets, evaluated on the Kitaev-honeycomb
lattice construction of Sec.~\ref{sec:setup}. All values reproduced
independently from an unrelated, mirrored Jordan--Wigner ordering and
by a representation-independent combinatorial set-overlap rule (see
text); reproduction scripts deposited (Data and Code Availability).}
\label{tab:m0}
\begin{tabular}{lcc}
\toprule
Quantity & Value & Ideal \\
\midrule
$P_1=\lVert[Z_A,X_A]\rVert_2$        & $2.000000$ & $2$ \\
$P_2$ (cross-party commutator)       & $0$ (exact) & $0$ \\
$S$ (CHSH, ideal Bell state)         & $2.828427$ & $2\sqrt2$ \\
$K$ (commuting-only control)         & $1.414214$ & $\sqrt2\le2$ \\
\bottomrule
\end{tabular}
\end{table}

The commuting-only control $K$ replaces the second setting by an
operator built exclusively from the pair $\{\gamma_{A,1},\gamma_{A,2}\}$
(i.e., omitting the third Majorana mode entirely), which by
construction commutes with $Z_A$; the resulting CHSH expectation is
capped at $K\le2$, the classical bound. This is the falsifiability
check for the whole construction: without a genuine third, local
Majorana mode furnishing a noncommuting second setting, the
construction collapses to the same one-dimensional trap familiar from
braiding-only constructions~\cite{Brennen2009}, and $S$ can never
exceed $2$. That the full four-Majorana-per-party construction instead
reaches $S=2.828427$ shows the noncommuting second setting is doing
genuine work.

Two further consistency checks accompany Table~\ref{tab:m0}. First,
the operator norm of the full Bell operator itself saturates at
$\lVert\text{CHSH-operator}\rVert_2=2.828427$, so $S$ cannot exceed
the Tsirelson value on any state; the value in Table~\ref{tab:m0} is a
genuine quantum maximum, not an artifact of state tuning. Second, the
Bell state $\ket{\Psi}$ is maximally entangled across the Alice/Bob
cut: its Schmidt coefficients are $(1/\sqrt2,1/\sqrt2,0,0)$ and its
entanglement entropy is $1.000$ bit, the maximum for a two-qubit pair
--- the CHSH value in Table~\ref{tab:m0} reflects genuine bipartite
entanglement of the underlying fusion-parity qubits, not a
classically-correlated artifact of the construction.

All four targets, together with the representation-independence check
and the adversarial controls above, were reproduced by an entirely
independent implementation: a mirrored Jordan--Wigner string
convention (opposite ordering of the eight Majorana operators), plus a
representation-independent combinatorial rule for operator
commutation --- for two products of Majorana operators indexed by
finite sets $S,T$, $\prod_{i\in S}\gamma_i$ and $\prod_{j\in T}\gamma_j$
commute if and only if $|S||T|-|S\cap T|$ is even --- checked against
the explicit matrix representation on all sampled operator pairs, with
full agreement. Because this rule depends only on set overlap and not
on any specific representation, its agreement with the matrix
computation rules out a representation-specific artifact.

Finally, we quantify the residual finite-size leakage of the
construction: at a fixed lattice size, the mean cross-party
support weight of Alice's modes on Bob's region is $8.3\%$, with the
Majorana-splitting/gap ratio at $7.3\%$; scanning isotropically over
increasing lattice size (on $24\times12$, $32\times16$ and
$44\times22$ lattices, with inter-cluster separations $d=18$, $24$
and $33$), this support weight decreases monotonically as
$17.0\%\to10.2\%\to2.3\%$. These are the two finite-size quantities
that would lift the commutator $P_2$ away from zero at finite size; in
the construction reported above $P_2$ is exactly zero, and both
quantities vanish as $L\to\infty$. We report the point value and the finite-size series
separately to avoid conflating a single lattice-size measurement with
the scaling trend.

%% ====================================================================
\section{Re-scoped core claim}
\label{sec:rescope}

The numbers of Sec.~\ref{sec:m0} establish that the Kitaev honeycomb
model, restricted to measurement-only operations on well-separated
Ising vortices, realizes \emph{locally} the noncommuting CHSH
Bell-operator \emph{algebra}: two noncommuting settings per party
($P_1=2$), exact cross-party commutativity ($P_2=0$), and a CHSH
expectation value on the corresponding maximally entangled fusion-parity
state that reaches the algebraic Tsirelson value
$S=2\sqrt2=2.828427$, surviving fermion-parity superselection exactly
because the Bell state is a definite-parity eigenstate
($\langle P_{\mathrm{tot}}\rangle=+1$) with unit weight in its own
sector, so that the physical parity projection acts as the identity on
it --- the same superselection-survival mechanism identified for
Majorana-based nonlocal entanglement by Romito and
Gefen~\cite{RomitoGefen2017}.

The lattice modes enter the algebra already orthonormal: they are real
Schur vectors of the real antisymmetric $\hat A$, so
$G^\mathsf{T}G = I$ is an identity of the construction rather than a
numerical finding, holding to the accuracy of the Schur decomposition
($2.3\times10^{-14}$) before any orthonormalization step. The Tsirelson
value is accordingly fixed by the algebra, as stated above; what the
lattice supplies is that this algebra is realized by eight
well-separated vortex-bound modes, with the finite quantities --- the
Majorana-splitting/gap ratio and the mean cross-party support weight
--- reported separately and entering no Bell value.

What does \emph{not} follow from this, and is the central scope
restriction of this paper, is a \emph{physical} Bell-inequality
violation obtained using only topologically protected Ising-anyon
operations. Howard and Vala~\cite{HowardVala2012} show that Ising-anyon
braiding transformations generate only the Clifford group, and that
Clifford operations alone --- however they realize the CHSH
\emph{algebra} --- cannot produce a genuine Bell-inequality violation
without an additional non-Clifford (``magic'') resource; this
sharpens the earlier, more general Clifford-braiding argument of
Brennen \emph{et al.}~\cite{Brennen2009} (Sec.~\ref{sec:discussion}).
The construction of this paper uses only Clifford combinations of
static fusion-parity observables on topologically protected Majorana
zero modes: Alice measures the parities $Z_A$ and $X_A$ directly, while
the CHSH settings on Bob's side are the $45^\circ$-rotated combinations
$B,B'$ of Eq.~\eqref{eq:settings}. No non-Clifford gate is introduced
anywhere in Secs.~\ref{sec:setup}--\ref{sec:m0}, and the rotations are
Clifford; the Howard--Vala condition additionally requires the
measurements to be stabilizer operations --- so it is precisely a
realization of
the Bell \emph{algebra} and not of a Bell \emph{violation} in the
physical, topologically-protected sense. We treat $B,B'$ at the level
of the ideal logical Majorana algebra; their realization as individual
topologically protected measurements is not constructed here. The
parity observables are bilinears in the matter Majoranas at spatially
separated vortex cores and are therefore not by themselves invariant
under the $\mathbb{Z}_2$ gauge generators
$D_i = b^x_i b^y_i b^z_i c_i$; the gauge-invariant observable carries a
Wilson line of the link variables between the two
cores~\cite{Petrova2014}. Within a fixed flux sector --- the setting
used throughout --- that Wilson line is a number $\pm1$ fixed by the
enclosed flux, so the dressed and undressed operators act identically
and every value reported here is unchanged. Whether and how a
non-stabilizer resource can be added on top of the present construction
to convert the algebraic saturation of Table~\ref{tab:m0} into an
actual physical violation is the subject of a companion paper
establishing where standard correlation witnesses are, and are not,
sensitive to nonstabilizerness (``magic'') in the related family of
$\mathrm{SU}(2)_k$ anyon models~\cite{Sayim2026magic}; we cross-reference
that paper's results bidirectionally and make no independent claim
about magic resources here.

%% ====================================================================
\section{Loop-threading sector phase: analytic corollary}
\label{sec:m5}

As a corollary check that the construction of Sec.~\ref{sec:setup}
behaves as expected under a controlled physical deformation, we
introduce a loop-threading relative phase $\delta$ on the maximally
entangled fusion-parity Bell state,
$\ket{\Psi(\delta)}=(\ket{00}+e^{i\delta}\ket{11})/\sqrt2$ in the
$\{Z_A,Z_B\}$ eigenbasis, and evaluate the CHSH expectation at Alice's
settings held fixed at their $\delta=0$-optimal values. The resulting
curve is the analytic corollary of the Bell-state algebra of
Sec.~\ref{sec:m0},
\begin{equation}
\label{eq:Sdelta}
S(\delta) = 2\sqrt2\,\cos^2(\delta/2),
\end{equation}
which we verify at eight pre-registered values of $\delta$ between $0$
and $\pi$, finding a maximum residual
$\max_\delta\lvert S(\delta)-2\sqrt2\cos^2(\delta/2)\rvert=4.4\times10^{-16}$
against a pre-registered tolerance of $8.9\times10^{-16}$, and confirm
by direct fine scan that $S(\delta)$ decreases monotonically over
$\delta\in[0,\pi]$ and never exceeds $2\sqrt2$. The nonlocality
threshold, where $S(\delta)$ crosses the classical bound $S=2$, is at
\begin{equation}
\delta^\star = 2\arccos\bigl(2^{-1/4}\bigr) = 1.143718\ \mathrm{rad} = 65.53^\circ.
\end{equation}

We flag explicitly, following the self-testing literature on
local-unitary invariance of the Tsirelson
value~\cite{SupicBowles2020}, that Eq.~\eqref{eq:Sdelta} by itself is
\emph{not} an independent microscopic signature of loop threading: by
a Heisenberg-picture equivalence, the same curve is produced by a
purely kinematic detector-misalignment scan of a fixed Bell state
against a rotated measurement axis, with zero residual difference from
Eq.~\eqref{eq:Sdelta} to numerical precision. What is genuinely ownable
about the sector-phase construction is the underlying algebra of
Sec.~\ref{sec:setup} together with the two topologically protected
($\mathbb{Z}_2$-native) endpoints $\delta=0$ (Tsirelson saturation) and
$\delta=\pi$ (classical baseline $S=0$); we treat $S(\delta)$ itself as
a corollary consistency check on the algebra, not as a free-standing
result.

%% ====================================================================
\section{Continuous-flux holonomy: an honest partial}
\label{sec:m5flux}

A stronger, and physically distinct, question is whether the phase
$\delta$ can be generated \emph{dynamically}, by adiabatically
threading a physical flux quantum through a loop enclosing one of
Alice's vortex pairs, rather than imposed by hand on the reference
state. We evaluate this using a gauge-invariant diagnostic: the
discrete non-Abelian Wilson-loop holonomy $W$ obtained by parallel
transport of the near-zero-mode subspace around a closed flux loop
$\Phi:0\to2\pi$, together with the basis-covariant localization
operator $Z_A$ constructed from the enclosed-vortex projector at
$\Phi=0$. Both $W$ and $\lVert[W,Z_A]\rVert_F$ are, by construction,
invariant under any change of basis of the near-zero-mode subspace
(the non-Abelian Wilczek--Zee gauge freedom of the degenerate
subspace~\cite{WilczekZee1984}, not the $\mathbb{Z}_2$ gauge
redundancy generated by the $D_i$); we
have verified this numerically to machine precision
($\Delta$(eigenphases)$\,{=}\,7.6\times10^{-16}$ under an explicit
random basis rotation) and across three random-number seeds
(spread $2.0\times10^{-15}$), and confirmed lattice convergence between
grid resolutions $M=64$ and $M=128$ (relative change $\sim0.3\%$).

Three of five pre-registered criteria for this diagnostic pass
outright: the holonomy $W$ is near-trivial, with eigenphases bounded
by $6.2\times10^{-3}$ over the full loop; the commutator
$\lVert[W,Z_A]\rVert_F=0.0132$ shows no channel mixing between the
localized near-zero modes; and the Berry/Wilson integral is computed
directly from the loop, not fitted to any target curve. The fourth
criterion is an honest labeling requirement (that the continuous-flux
construction is presented as an effective description, not claimed to
be natively realized on the exactly solvable model) and is satisfied
by construction.

The fifth criterion, and the sole open discrepancy, is that the
accumulated single-mode Berry phase at one full flux quantum does not
close exactly on the target value $\pi$:
$\delta_{\mathrm{full}}\bmod2\pi=3.1212$ against the target
$\pi=3.1416$, a gap of $0.020$~rad. We traced this discrepancy to its
source rather than treating it as an unresolved anomaly: tracking the
four near-zero eigenvalues along the flux loop reveals genuine,
near-exact level degeneracies at two points on the loop (minimum gaps
of $2.2\times10^{-18}$ and $8.7\times10^{-19}$, at the floating-point
floor), at which single-mode adiabatic following is not well defined.
The maximum-overlap tracking used to extract a single-mode phase
necessarily follows a \emph{diabatic} branch through such a crossing,
which can accumulate an extra phase inconsistent with the adiabatic
value, while the four-dimensional subspace holonomy $W$ --- immune to
internal relabeling at a crossing within the tracked subspace ---
remains near-trivial throughout. The two facts, single-mode
$\delta\approx\pi$-with-a-gap and subspace $W\approx\mathbb{1}$, are
consistent answers to two different questions, not a contradiction.

We therefore report this diagnostic as an honest \textbf{partial}: the
gauge-invariant, basis-independent quantities
($W$, $\lVert[W,Z_A]\rVert_F=0.0132$) show no adiabatic channel
mixing, and the continuous-$\delta$ construction remains valid as an
\emph{effective} description of the Ising topological quantum field
theory monodromy; but the single-mode Berry-phase closure at the full
flux quantum is not established to the pre-registered tolerance, and
we do not claim it. What survives unconditionally from this section is
the pair of $\mathbb{Z}_2$-native endpoints $\delta=0,\pi$ and the
parity-selective fusion-qubit algebra of Sec.~\ref{sec:setup}, both
already established without reference to the continuous flux
construction. We note, for the record, that an earlier per-mode
diagnostic of this same construction used an unseeded eigensolver
pipeline and reported a basis-dependent commutator norm that varied
by an order of magnitude across repeated runs (range
$0.18$--$3.3$); that diagnostic was not gauge-invariant and has been
superseded by the seed-stable, basis-independent construction
reported in this section. Only the values given above --- $W$,
$\lVert[W,Z_A]\rVert_F=0.0132$, and the criteria of this section ---
are cited anywhere in this paper.

%% ====================================================================
\section{Discussion}
\label{sec:discussion}

\paragraph{Relation to Brennen \emph{et al.}} Brennen, Iblisdir,
Pachos, and Slingerland~\cite{Brennen2009} give the general argument,
for a broad class of non-Abelian anyon models including SU(2)$_2$
(Ising anyons), that a Bell-operator norm can saturate the Tsirelson
bound ($W=-2\sqrt2$ for the Ising case) using topologically protected
braiding transformations alone, and observe explicitly that these
transformations generate only the Clifford group, so an additional
non-Clifford gate is required to turn this algebraic saturation into
an actual Bell violation --- three years before the sharper Howard--Vala
no-go~\cite{HowardVala2012}. Ising anyons are topologically classical
in exactly this sense: the algebraic saturation of the Tsirelson bound
is available from braiding alone, but converting it into a genuine
violation is not, precisely because braiding alone cannot supply the
required non-Clifford resource. We make no priority claim on
Ising-anyon CHSH saturation, on the value $2\sqrt2$ itself, or on the
Clifford-braiding-cannot argument: all three are established by
Brennen \emph{et al.} and sharpened by Howard and Vala. What this paper
contributes beyond that prior art is the explicit microscopic
instantiation of the algebra on the Kitaev honeycomb Hamiltonian
itself (Sec.~\ref{sec:setup}), the sector-phase axis $\delta$ and its
loop-threading corollaries (Secs.~\ref{sec:m5}--\ref{sec:m5flux}), and
the explicit verification that superselection does not reduce $S$
(Sec.~\ref{sec:m0}) --- none of which appear in Brennen \emph{et al.},
whose continuous family is parametrized by a state amplitude rather
than by a sector/loop-threading phase. A recent hardware demonstration
realizes universal gates on the smallest non-Abelian quantum double
($S_3$, 54 trapped-ion qubits) precisely by promoting anyon
\emph{fusion} to an active computational primitive --- braiding alone
remains nonuniversal there --- an experimental instance of the same
resource dichotomy that scopes our witness statement~\cite{Lo2026}.

\paragraph{Relation to Michalski \emph{et al.}} Michalski, Konderak,
Bruzda, and Augusiak~\cite{Michalski2025} construct a
device-independent self-testing equivalence for an abstract class of
Jordan--Wigner Clifford generators motivated by Majorana-fermion
certification, at the level of the abstract operator algebra, with no
microscopic lattice, honeycomb, or vortex construction. Their result
and the present one are complementary rather than competing: theirs is
platform-independent self-testing of an abstract generator algebra,
ours is a microscopic, sector-resolved lattice realization of a
closely related algebra on a specific Hamiltonian.

\paragraph{Relation to Hu, Deng, and Chen.} Hu, Deng, and
Chen~\cite{HuDengChen2009} propose an all-versus-nothing (Hardy-type)
violation of local realism on the same Kitaev honeycomb lattice,
using the Abelian toric-code (Z$_2$ gauge theory) sector of the model.
Their construction and ours share a Hamiltonian family but occupy
different phases and different correlation structures: theirs is an
Abelian all-versus-nothing/GHZ-type violation in the toric-code
sector; ours is a non-Abelian CHSH \emph{algebra}, not a violation, in
the Ising (non-Abelian) sector. We do not claim any all-versus-nothing
result, and their construction does not bear on the non-Clifford
requirement that is the central scope restriction of Sec.~\ref{sec:rescope}.

\paragraph{Relation to Takahashi \emph{et al.}} Takahashi, Yamada,
Udagawa, Mizushima, and Fujimoto~\cite{Takahashi2022} study the same
platform --- Ising vortices trapped at lattice vacancies of the
Kitaev honeycomb spin liquid --- and propose a nonlocal spin (and
associated conductance) correlation as an experimental signature of
the trapped anyons. Their construction and ours share the underlying
lattice and the non-Abelian vortex substrate, but not the observable:
theirs is a nonlocal spin-correlation signature aimed at
distinguishing genuine Ising-anyon physics from trivial local
artifacts, with no CHSH functional, Bell operator, or measurement
algebra constructed; ours is the Bell-operator algebra of
Sec.~\ref{sec:setup} evaluated on fusion-parity observables. We flag
the overlap in platform explicitly so the two are not conflated.

\paragraph{Why Ising and not Fibonacci.} A natural question is whether
the sector-phase construction of Sec.~\ref{sec:m5} generalizes to
Fibonacci anyons, whose braiding is dense in $\mathrm{SU}(2)$ and
therefore does supply a non-Clifford resource by braiding alone. We
record a negative finding here, qualitatively: the naive analog of the
sector-phase deformation, obtained by replacing a Fibonacci braid
generator $R_\tau$ by $R_\tau\,e^{i\delta}$, breaks the Yang--Baxter
consistency of the braid representation, so it does not correspond to
any physically consistent Fibonacci braiding transformation; we are
not aware of a physically consistent Fibonacci-sector analog of
Eq.~\eqref{eq:Sdelta}. This is a documented negative observation, not
a numerical result, and we do not pursue it further here.

\paragraph{Scope.} This paper establishes a local realization of the
CHSH Bell-operator algebra on a microscopic, spatially separated
Majorana construction, together with a controlled sector-phase
corollary and an honestly partial continuous-flux diagnostic. It does
not establish a physical Bell-inequality violation using only
topologically protected operations (Sec.~\ref{sec:rescope}); it makes
no claim about the non-stabilizer resource required for such a
violation, which is addressed for a related anyon family in a
companion paper~\cite{Sayim2026magic}; and the Majorana zero modes used
here are a theoretical construction on an idealized lattice
Hamiltonian, with no claim made about, or connection asserted to, any
specific solid-state or hybrid platform experiment.

%% ====================================================================
\section*{Data and Code Availability}
The reproduction scripts and raw JSON outputs supporting the values
reported in this paper are deposited at
\href{https://doi.org/10.5281/zenodo.21134322}{doi:10.5281/zenodo.21134322}.
Paper, figures, and data are released under the Creative Commons
Attribution 4.0 International (CC~BY~4.0) license; the deposited source
code is released under the Apache License, Version 2.0
(see \texttt{LICENSE} and \texttt{LICENSE-CODE}).
The deposit contains the four locked targets of Table~\ref{tab:m0}, the
sector-phase curve, the flux-holonomy diagnostic and the finite-size
scan, together with the scripts that reproduce them on the
Kitaev-honeycomb lattice construction of Sec.~\ref{sec:setup}.

\section*{Use of AI tools}

In the research, processing, and writing of this paper and its results I worked together with generative AI tools, in practice a system of multiple coordinated AI instances that I set up and orchestrate (large language models, mainly Claude, by Anthropic, inside Claude Code). At their current context sizes I found it far more effective to work with several specialized instances, each with its own role and its own harness of rules and parameters that I designed and refined through feedback, than to load a single instance with all of the material; for my workflow that would have been inefficient, though this depends on the individual implementation. I lead this collaboration: I choose the research directions, set the goals, and make the final decisions in open exchange with the AI, learning actively as the work proceeds. The AI carries out the drafting, including the mathematical and technical parts, the numerical computation, and the literature search, under my direction. The AI works autonomously only task by task, within the structure I develop through feedback: it completes a task, and at open questions that need me it stops until the point is settled before the next step. Along the way I witness and take many of the decisions that shape the path, and it is common for me to spot things that need improvement. The work spans many separate runs, and a single simulation or build task alone can take up to an hour, so it could not happen all together in one autonomous run; and had I let the AI do all of it together alone, even if it is possible, it would no longer be my work but the AI's.

I run multiple verifications at the different stages of the work and one before release, including cross-checks with an unrelated AI model from a different company, and all references are checked against the original sources. In the end what matters are human eyes, a principle that is itself written into the parameters of my system: I reach out to experts after publishing for review and feedback, so I learn what is solid and what must be corrected or falsified. My scripts for reproduction and review are released with this record. These tools are not authors; I am the author, and I take full responsibility for all scientific content and decisions leading to these results and their publication.

\section*{Acknowledgments}
No external funding was received. The author declares no competing
interests.

%% ====================================================================
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\end{document}