Two-basepoint Terwilliger algebras and the quantum symmetry of prime-order circulants

Abstract

Which vertex-transitive graphs of prime order have quantum symmetry? The question of Banica, Bichon and Chenevier is open in the dense regime of Paley graphs, where coherent-algebra methods give no information. To each such graph we attach a two-basepoint Terwilliger algebra of its cyclotomic scheme and study the module it generates from the basepoints: fullness forces the quantum permutation algebra to be commutative, and the module admits no intermediate state, containing either exactly two point masses or all $p$ of them. One point mass, captured at any depth, therefore suffices, and Chassaniol's orbital criterion is the depth-one case. Three consequences follow. A sharp counting argument replaces the Banica--Bichon--Chenevier threshold $p>6^{\varphi(k)}$ by the quadratic bound $p>(k-1)(k-2)+2$, where $k$ is the type. Four certificates, each a short list of additions modulo $p$, settle $C_{31}(2,4,8,15)$ and $C_{41}(4,10,16,18)$, the two graphs left open by Chassaniol, and complete the classification for type at most $10$ without machine assistance. An exact computation extends the dichotomy ``quantum symmetry if and only if complete or empty'' to all prime orders $p\le250$, settling the Paley graphs $P_{p}$ with $p\le241$, the first beyond $P_{17}$. What remains is the capture of a single explicit vector: the midpoint $2^{-1}$ of the two basepoints.