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\begin{document}

\title[Exact rank thresholds for state-independent contextuality]{Exact projector-rank thresholds\\ for state-independent contextuality}
\author{Sunay Venkat}
\date{11 November 2026}

\begin{abstract}
For every integer $r\ge1$ we construct a finite exclusivity graph $F_r$ with the following two
properties. First, $F_r$ has a projective representation by pairwise distinct projectors of rank $r$
that exhibits state-independent contextuality. Second, no projective representation of $F_r$ by
projectors of any common rank $s<r$, in any finite dimension, exhibits state-independent
contextuality; in every such representation the maximally mixed state is noncontextual. This
establishes, for every $r$, the ``rank efficient'' scenarios sought by H\"ohn, Xu and Kleinmann, whose Section~7 reports numerical evidence for specific examples at $r=2,3$. The graph is the disjunctive product of an
odd cycle $C_{2r+1}$ with the orthogonality structure of a rank-one contextuality set whose
fractional chromatic number exceeds its dimension by an arbitrarily small amount. Such sets are
obtained from the infinite family of Kochen--Specker parity proofs in $\R^4$ of Elford and
Lison\v{e}k, together with a coning construction that raises the dimension by one.
\end{abstract}

\maketitle

\section{Introduction}

A finite family of projectors exhibits \emph{state-independent contextuality} (SIC) if, for every
quantum state, the outcome probabilities are incompatible with every noncontextual hidden-variable
model \cite{YuOh,BudroniRMP}. In the graph-theoretic approach \cite{CSW,RH,CKB} the relevant datum is
the exclusivity graph, which records which projectors are orthogonal. Most known SIC sets consist of
rank-one projectors. Xu, Yu and Kleinmann \cite{XYK} showed that there are exclusivity graphs for
which SIC is possible with projectors of rank two but impossible with rank one.

H\"ohn, Xu and Kleinmann \cite{HXK} gave a systematic construction, based on the disjunctive graph
product with odd cycles, of graphs that admit SIC at some rank but at no rank up to a prescribed $r$,
with explicit examples for $r\le5$, and conjectured that such examples exist for every $r$. This
conjecture was recently proved by Westberg \cite{Westberg}, using the arc graphs from Kohlmeyer and
Kruer's solution of Erd\H{o}s problem 108 \cite{KK,NW}. The prescribed-exact-rank question is separate from this arbitrary-threshold result.
In \cite[Section~7]{HXK} the authors seek graphs that are \emph{rank efficient}, ``in the sense that
rank $r$ is sufficient for SIC, while for $r-1$ no representation with SIC can be found'', and report
numerical evidence for $r=2$ and $r=3$. Westberg states that his theorem does not ``determine a
prescribed exact minimum rank'' \cite[Section~6]{Westberg}. The arbitrary-threshold constructions do not identify a prescribed first successful rank.
Westberg also excludes unequal ranks bounded by the threshold and supplies a faithful successful
representation; our exact-rank result below concerns uniform ranks and does not assert faithfulness.

We settle the question for every $r$.

\begin{theorem}\label{thm:main}
For every integer $r\ge1$ there is a finite graph $F_r$ such that:
\begin{enumerate}
\item[(i)] $F_r$ has a projective representation $(\Pi_v)_v$ on some $\C^{p}$ by pairwise distinct
projectors of rank $r$, and weights $W_v\ge0$, with
\[\sum_v W_v\Pi_v=\lambda\,\one \qquad\text{and}\qquad \alpha(F_r,W)<\lambda .\]
In particular this representation exhibits state-independent contextuality.
\item[(ii)] For every $1\le s<r$ and every projective representation of $F_r$ by projectors of rank
$s$ on a finite-dimensional Hilbert space, the probability vector of the maximally mixed state lies
in $\STAB(F_r)$. In particular no such representation exhibits state-independent contextuality.
\end{enumerate}
\end{theorem}

For $r\ge2$ one may take $F_r=B\vee C_{2r+1}$, where $B$ is described in \cref{sec:proof}; then
$p=D(2r+1)$, where $D\ge4$ is the least integer coprime to $r$, and the SIC ratio of the
representation in (i) is at least $1+\frac{4}{D(N-1)}$ for a parameter $N$ of order $r^2$.

Westberg excludes low ranks through a large chromatic number, but his result does not establish
prescribed exact thresholds. The present proof does not use the Kohlmeyer--Kruer theorem. We use the
rigidity of projective representations of odd cycles and the integrality of the dimension. The idea
is as follows. If $B$ contains a clique of size $D$, any rank-$s$ representation of $B\vee C_{2r+1}$
contains a rank-$sD$ representation of $C_{2r+1}$, which forces the dimension to be at least
$2sD+\lceil sD/r\rceil$. If $\gcd(D,r)=1$ and $s<r$ the rounding gains at least $1/r$. If moreover the
fractional chromatic number of $B$ exceeds $D$ by a sufficiently small amount, this gain pushes the
dimension to at least $s\,\chif(B\vee C_{2r+1})$, and then the maximally mixed state is noncontextual. At
rank $r$ there is no rounding and a tensor-product representation is contextual. The same template
underlies \cite[Theorem~3]{HXK}; the new ingredients are the use of the rounding term to pin the
threshold at $r$ itself, and the supply of rank-one SIC sets $B$ with $\chif(B)-D$ arbitrarily small
in every dimension $D\ge4$.

\section{Definitions}

All graphs are finite and simple; all Hilbert spaces are finite-dimensional and complex. We follow
\cite{HXK}.

A \emph{projective representation} (PR) of a graph $G$ on $\C^d$ assigns to each vertex $v$ a
nonzero orthogonal projector $\Pi_v$ such that $\Pi_u\Pi_v=0$ for every edge $uv$. It has
\emph{rank $s$} if $\rank\Pi_v=s$ for all $v$. Let $\STAB(G)\subset\R^{V(G)}$ be the convex hull of
the indicator vectors of the independent sets of $G$, and for $w\colon V(G)\to\R_{\ge0}$ let
$\alpha(G,w)$ be the maximum of $\sum_{v\in S}w_v$ over independent sets $S$. A PR \emph{exhibits
SIC} if for no density matrix $\rho$ the vector $(\tr\rho\Pi_v)_v$ lies in $\STAB(G)$. If
$\sum_vW_v\Pi_v\succeq\lambda\one$ with $\lambda>\alpha(G,W)$, then every state violates the
inequality $\sum_vW_vp_v\le\alpha(G,W)$, which is valid on $\STAB(G)$, so the PR exhibits SIC.

The fractional chromatic number $\chif(G)$ is the least total weight of a fractional colouring, that
is, of a function $y\ge0$ on independent sets with $\sum_{S\ni v}y_S\ge1$ for every $v$. The
\emph{disjunctive product} $G\vee H$ has vertex set $V(G)\times V(H)$, with $(u,v)\sim(u',v')$ if
$u\sim u'$ or $v\sim v'$. The cycle $C_n$ has vertex set $\Z_n$ and edges $\{j,j+1\}$.

\section{Lemmas}

\begin{lemma}[Odd cycles]\label[lemma]{lem:cycle}
Let $(Q_j)_{j\in\Z_{2r+1}}$ be projectors of rank $m$ on $\C^d$ with $Q_jQ_{j+1}=0$ for all $j$.
Then $d\ge 2m+\lceil m/r\rceil$.
\end{lemma}

\begin{proof}
Let $V_j$ be the range of $Q_j$ and $T_j=V_0\cap V_2\cap\dots\cap V_{2j}$ for $0\le j\le r$. Both
$T_j\subseteq V_{2j}$ and $V_{2j+2}$ are contained in $V_{2j+1}^{\perp}$, which has dimension $d-m$.
Hence $\dim T_{j+1}\ge\dim T_j+m-(d-m)$, and so $\dim T_r\ge m-r(d-2m)$. Since $2r$ and $0$ are
adjacent, $T_r\subseteq V_0\cap V_{2r}=\{0\}$. Thus $d\ge2m+m/r$, and $d$ is an integer.
\end{proof}

This is the dimension count behind the value $2+1/r$ of the projective rank of $C_{2r+1}$
\cite[Theorem~6.15.3, p.~140]{Roberson}, credited there to joint work with Laura Man\v{c}inska; it sharpens \cite[Lemma~4]{HXK}.

\begin{lemma}[{\cite[Lemma~2.2]{Westberg}}]\label[lemma]{lem:cube}
If $0\le t\le1/\chif(G)$, then $t\,\one\in\STAB(G)$.
\end{lemma}

\begin{proof}
Normalising an optimal fractional colouring gives a random independent set containing each vertex
$v$ with probability $m_v\ge1/\chif(G)$. Retain each of its vertices independently with probability
$t/m_v$. The result is a random independent set that contains each vertex with probability $t$.
\end{proof}

\begin{lemma}[Products, cf.\ {\cite[Theorem~1]{HXK}}]\label[lemma]{lem:prod}
\textup{(a)} $\alpha(G\vee H,w\otimes a)\le\alpha(G,w)\,\alpha(H,a)$ for weights $w,a\ge0$.
\textup{(b)} $\chif(G\vee H)\le\chif(G)\,\chif(H)$.
\end{lemma}

\begin{proof}
(a) The coordinate projections $S_G,S_H$ of an independent set $S$ of $G\vee H$ are independent and
$S\subseteq S_G\times S_H$. (b) If $I,J$ are independent then so is $I\times J$; take the product of
two fractional colourings.
\end{proof}

\begin{lemma}[The cycle factor, {\cite[Section~4]{HXK}}]\label[lemma]{lem:E}
Index the standard basis of $\C^{2r+1}$ by $\Z_{2r+1}$ and let $E_j$ be the projector onto the
coordinates $jr,jr+1,\dots,jr+r-1$. Then $(E_j)$ is a PR of $C_{2r+1}$ of rank $r$ by pairwise
distinct projectors, $\sum_jE_j=r\,\one$, $\alpha(C_{2r+1})=r$ and $\chif(C_{2r+1})\le(2r+1)/r$.
\end{lemma}

\begin{proof}
The coordinate blocks of $E_j$ and $E_{j+1}$ are consecutive of total length $2r<2r+1$, hence
disjoint. As $\gcd(r,2r+1)=1$, the blocks are pairwise distinct and each coordinate $c$ lies in
exactly $r$ of them. The set $I_c$ of those $j$ is independent, because adjacent blocks are disjoint.
The $2r+1$ sets $I_c$ cover each vertex $r$ times, so $\chif\le(2r+1)/r$. Finally
$\alpha(C_{2r+1})=r$.
\end{proof}

\begin{definition}\label[definition]{def:datum}
A \emph{datum of dimension $D$ with parameters $(\lambda,\alpha_0,T)$} consists of a graph $B$, unit
vectors $x_b\in\C^D$ $(b\in V(B))$ spanning pairwise distinct rays, with $P_b=|x_b\rangle\langle x_b|$,
and weights $w_b\ge0$, such that
\begin{enumerate}
\item[(D1)] $x_b\perp x_{b'}$ for every edge $bb'$ of $B$;
\item[(D2)] $\sum_bw_bP_b=\lambda\,\one$;
\item[(D3)] $\alpha(B,w)\le\alpha_0<\lambda$;
\item[(D4)] $B$ contains a clique of size $D$;
\item[(D5)] $\chif(B)\le T$.
\end{enumerate}
\end{definition}

By (D1)--(D3) a datum is a rank-one SIC set. By (D4), $\chif(B)\ge D$; we need data with $T-D$ small.

\begin{lemma}[Base data]\label[lemma]{lem:base}
Let $p,q\ge3$ be odd and coprime and $N=pq$. There is a datum of dimension $4$ with parameters
$\lambda=N/2$, $\alpha_0=(N-1)/2$ and $T=4+\frac4{N-1}$.
\end{lemma}

\begin{proof}
By Elford and Lison\v{e}k \cite[Theorem~1 and Proposition~1]{EL} there are an orthogonal matrix $M$
of order $N$ on $\R^4$, vectors $a,b\in\R^4$, and an integer $\rho$ with $\rho\equiv1\pmod p$ and
$\rho\equiv-1\pmod q$, such that for every $i\in\Z_N$
\[\beta_i=\{M^ia,\;M^{i-1}a,\;M^ib,\;M^{i-\rho}b\}\]
is an orthogonal basis of $\R^4$, and the $2N$ vectors $M^ia$, $M^ib$ $(i\in\Z_N)$ are pairwise
linearly independent. Thus $M^ia$ lies in exactly the bases $\beta_i,\beta_{i+1}$ and $M^ib$ in
exactly $\beta_i,\beta_{i+\rho}$. Let $\Gamma$ be the graph on $\Z_N$ with edges
$\{i,i+1\}$ and $\{i,i+\rho\}$; it is simple and $4$-regular because $\rho\not\equiv\pm1\pmod N$ and
$N$ is odd. Identify the edge $\{i,i+1\}$ with the ray of $M^ia$ and $\{i,i+\rho\}$ with the ray of
$M^ib$; then the four edges at the vertex $i$ are the four rays of $\beta_i$. Let $B=L(\Gamma)$ be
the line graph, $x_e$ the normalised vector of the edge $e$, and $w\equiv1$.

(D1) Adjacent edges lie in a common basis. (D2) Each ray lies in exactly two bases, so
$\sum_eP_e=\frac12\sum_{i}\sum_{e\ni i}P_e=\frac N2\one$. (D3) An independent set of $L(\Gamma)$ is a
matching of $\Gamma$, which has at most $(N-1)/2$ edges since $N$ is odd. (D4) The four edges at a
vertex. (D5) Since $\gcd(\rho,N)=1$, each of the two edge classes $\{i,i+1\}$ and $\{i,i+\rho\}$ is a
Hamiltonian cycle of odd length $N$. A cycle of length $N$ has a matching with $(N-1)/2$ edges, and
its $N$ rotations cover every edge of the cycle exactly $(N-1)/2$ times. Giving weight $2/(N-1)$ to
each of these $2N$ matchings yields a fractional colouring of $B$ of total weight $4N/(N-1)$.
\end{proof}

\begin{remark}
The 18-ray set of Cabello, Estebaranz and Garc\'ia-Alcaine \cite{CEG} has $N=9$ bases and the same
parameters; its basis graph is $K_3\,\square\,K_3$. Pulling its rays back along the covering maps
$C_{3a}\,\square\,C_{3a}\to C_3\,\square\,C_3$ ($a$ odd) gives an alternative proof of
\cref{thm:main} without the distinctness of the projectors. The family of \cite{EL} gives distinct
rays.
\end{remark}

\begin{lemma}[Coning]\label[lemma]{lem:cone}
From a datum of dimension $D$ with parameters $(\lambda,\alpha_0,T)$ one obtains a datum of dimension
$D+1$ with parameters $(D\lambda+\alpha_0,\,(D+1)\alpha_0,\,T+1)$.
\end{lemma}

Note that $(D\lambda+\alpha_0)-(D+1)\alpha_0=D(\lambda-\alpha_0)>0$ and that $T-D$ is unchanged.

\begin{proof}
Let $e_0,\dots,e_D$ be the standard basis of $\C^{D+1}$ and $J_i\colon\C^D\to e_i^\perp$ the
coordinate embedding. Since unitary images of a datum are data, we may choose unitaries
$U_0,\dots,U_D$ of $\C^D$ such that no vector $U_ix_b$ has a zero coordinate; the unitaries violating
this for a given $i$, $b$ and coordinate form a nowhere dense closed subset of the unitary group, and
there are finitely many conditions.

The graph $B'$ consists of $D+1$ disjoint copies of $B$, with vertices $(i,b)$, and further vertices
$a_0,\dots,a_D$. Its edges are the copies of the edges of $B$, the pairs $a_i\sim(i,b)$ for all $b$,
and the pairs $a_i\sim a_j$ for $i\ne j$. Put $x'_{(i,b)}=J_iU_ix_b$, $x'_{a_i}=e_i$,
$w'_{(i,b)}=w_b$ and $w'_{a_i}=\alpha_0$.

\emph{Distinct rays.} The vector $x'_{(i,b)}$ vanishes in coordinate $i$ and in no other
coordinate. Hence rays from different copies differ, rays in the same copy differ because $J_iU_i$ is
injective, and no $x'_{(i,b)}$ is a multiple of some $e_j$ because $D+1\ge3$.

(D1) $J_iU_i$ is an isometry with range $e_i^\perp$. (D2)
$\sum_i\bigl[\lambda(\one-|e_i\rangle\langle e_i|)+\alpha_0|e_i\rangle\langle e_i|\bigr]
=(D\lambda+\alpha_0)\one$. (D3) An independent set of $B'$ contains at most one of the $a_i$. If it
contains none, its weight is at most $(D+1)\alpha_0$. If it contains $a_i$, it contains no vertex of
copy $i$, so its weight is at most $\alpha_0+D\alpha_0$. (D4) A $D$-clique of copy $i$ together with
$a_i$.

(D5) We have $T\ge\chif(B)\ge D$. Regard a fractional colouring of $B$ of total weight $T$ as a
piecewise constant map from $[0,T)$ to independent sets under which every vertex is present for
total time at least $1$. Divide $[0,T+1)$ into unit intervals $I_0,\dots,I_D$ and a final interval
of length $T-D$. During $I_i$ colour $a_i$. Copy $j$ runs its schedule during the $D$ intervals
$I_i$, $i\ne j$, and during the final interval, which have total length $T$. At every time the
coloured set is a union of independent sets of distinct copies together with at most one vertex
$a_i$, whose copy $i$ is then idle; it is therefore independent. Every vertex is coloured for time
at least $1$, so $\chif(B')\le T+1$.
\end{proof}

\section{Proof of Theorem~\ref{thm:main}}\label{sec:proof}

For $r=1$ any rank-one SIC set, for instance that of \cref{lem:base}, satisfies (i), and (ii) is
vacuous. Let $r\ge2$. Let $D\ge4$ be the least integer with $\gcd(D,r)=1$, and choose odd coprime
$p,q\ge3$ with
\[N=pq\ge4(r-1)(2r+1)+1.\]
Apply \cref{lem:base}, and then \cref{lem:cone} $D-4$ times. This gives a datum $B$ of dimension $D$
with parameters $(\lambda,\alpha_0,T)$, where $\lambda>\alpha_0$ and
\begin{equation}\label{eq:small}
T-D=\frac4{N-1}\le\frac1{(r-1)(2r+1)} .
\end{equation}
Put $F_r=B\vee C_{2r+1}$.

\subsection*{Proof of (i)}
On $\C^D\otimes\C^{2r+1}$ let $\Pi_{(b,j)}=P_b\otimes E_j$, with $E_j$ as in \cref{lem:E}, and
$W_{(b,j)}=w_b/r$. These projectors have rank $r$ and are pairwise distinct, because the $P_b$ are
pairwise distinct and so are the $E_j$. If $(b,j)\sim(b',j')$ then $P_bP_{b'}=0$ or $E_jE_{j'}=0$, so
$(\Pi_v)$ is a PR of $F_r$. By (D2) and \cref{lem:E},
\[\sum_vW_v\Pi_v=\Bigl(\sum_bw_bP_b\Bigr)\otimes\Bigl(\frac1r\sum_jE_j\Bigr)=\lambda\,\one .\]
By \cref{lem:prod}(a), (D3) and $\alpha(C_{2r+1})=r$ we get $\alpha(F_r,W)\le\alpha_0<\lambda$.

\subsection*{Proof of (ii)}
Let $1\le s<r$ and let $(\Pi_v)$ be a PR of $F_r$ of rank $s$ on $\C^d$. Let $K$ be a clique of size
$D$ in $B$ and put $Q_j=\sum_{b\in K}\Pi_{(b,j)}$ for $j\in\Z_{2r+1}$. For fixed $j$ the vertices
$(b,j)$, $b\in K$, are pairwise adjacent, so $Q_j$ is a projector of rank $sD$. If $j\sim j'$ then
every $(b,j)$ is adjacent to every $(b',j')$, so $Q_jQ_{j'}=0$. By \cref{lem:cycle},
\[d\ge2sD+\Bigl\lceil\frac{sD}{r}\Bigr\rceil .\]
Since $\gcd(D,r)=1$ and $0<s<r$, the integer $r$ does not divide $sD$, hence
$\lceil sD/r\rceil\ge sD/r+1/r$ and
\[d\ge\frac{sD(2r+1)}{r}+\frac1r .\]
On the other hand, by \cref{lem:prod}(b), \cref{lem:E} and (D5), $\chif(F_r)\le T(2r+1)/r$, and by
\eqref{eq:small}
\[s\,\chif(F_r)\le\frac{sD(2r+1)}{r}+\frac{s(T-D)(2r+1)}{r}\le\frac{sD(2r+1)}{r}+\frac1r\le d .\]
For the maximally mixed state every vertex has probability $s/d\le1/\chif(F_r)$. By \cref{lem:cube}
the probability vector lies in $\STAB(F_r)$. \qed

\begin{remark}[Coprimality]
If $\gcd(D,r)=g>1$ then for $s=r/g<r$ the rounding term vanishes and the argument for (ii) fails.
Parity proofs exist only in even dimension \cite{EL}, so an odd $D$, needed for even $r$, is not
available from \cref{lem:base} alone. This is the purpose of \cref{lem:cone}.
\end{remark}

\begin{remark}[Size]
With $p=3$ the smallest admissible parameters give, for $r=2,3,4,5,6$: $N=21,57,111,177,267$;
$|V(F_r)|=1075,798,10035,3894,34775$; Hilbert space dimension $25,28,45,44,65$; and SIC ratio at
least $26/25$, $57/56$, $277/275$, $177/176$, $667/665$. No attempt was made to optimise these.
\end{remark}

\section{Scope and open points}

\begin{itemize}
\item \emph{Uniform rank.} Part (ii) concerns representations in which all projectors have the same
rank $s<r$, the setting of \cite{HXK}. Whether a representation of $F_r$ with unequal ranks, all
smaller than $r$, can exhibit SIC is not decided here.
\item \emph{Faithfulness.} The projectors in (i) are pairwise distinct, but the representation is not
claimed to be faithful: non-adjacent vertices of $F_r$ may carry orthogonal projectors.
\item \emph{Ranks above $r$.} Nothing is claimed for ranks $s>r$ that are not multiples of $r$.
\item \emph{Optimality.} The number of projectors and the size of the violation are far from optimal.
\end{itemize}

\section*{Verification}

A separate Python verification script was run for $r=2,\dots,6$ and for the base pairs
$(p,q)=(3,5),(3,7),(5,7),(5,9),(7,11)$. It checks edge orthogonality with absolute tolerance
$10^{-9}$, the weighted matrix identity (D2) with absolute tolerance $10^{-7}$, and distinctness
by requiring every off-diagonal absolute inner product of normalised rays to be below $1-10^{-9}$.
These are floating-point consistency checks, not exact certificates of the vector identities.
In exact integer or rational arithmetic it checks (D4), the independence and covering property
of every colour class in the explicit fractional colouring, and
$2sD+\lceil sD/r\rceil\ge sT(2r+1)/r$ for every $s<r$ in those instances.
All checks passed. The script is retained separately from the Hexagon deposit.

The mathematical proof does not depend on these computations. In particular, condition (D3),
the general lemmas and the all-ranks conclusion are justified by the proofs above, not by a
finite test. This paper has not been formalised in a proof assistant or peer reviewed.

\section*{AI disclosure}

The author reports that the argument was generated entirely by Claude Fable~5.2 (Anthropic),
including the construction and its mathematical justification. The choice of problem, the
replacement of the original base construction by the family of Elford and Lison\v{e}k, and the
adversarial checks of novelty and correctness were subsequently developed through further
interaction with the model. The model also produced the verification code, the exposition and the
\LaTeX{} manuscript. This disclosure describes the provenance of the argument; it does not
supersede the attribution of previously published mathematical methods cited in the paper or
constitute independent verification of the proofs. The human author retains responsibility for the
correctness of the mathematical claims, the accuracy of this disclosure, and the decision to submit
or disseminate the work. OpenAI Codex subsequently assisted with source verification,
editorial corrections, rerunning the numerical and combinatorial checks, compilation, and
preparation of this submission package on 11 November 2026. This further AI-assisted review is
not independent human verification.

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