SQAN: Simplicial Q-Attention on the BERT Hypersphere — A Neuro-Symbolic Architecture with Angular Logic Tensor Grounding

We propose SQAN, a neuro-symbolic architectural framework that treats the BERT embedding space as a curved semantic manifold — the unit hypersphere S^767 — and builds its entire reasoning apparatus from a single geometric primitive: the angular distance d_angular(u,v) = arccos(<u,v>)/pi. The central observation is that Logic Tensor Networks already compute truth values angularly (for unit vectors, the grounding is a sigmoid of a cosine), while standard transformer attention uses Euclidean dot products normalised by sqrt(d) and passed through softmax — operations that flatten the angular structure. SQAN grounds both the topological weighting scheme and the logical inference layer in the same angular geometry: a Vietoris-Rips complex built with geodesic distance, Q-coherence weights providing a topologically grounded measure of simplicial tightness, and a spherical grounding function shown to be a monotone affine transformation of the LTN dot-product grounding. A necessity corollary establishes that any scheme sharing a scalar variable with LTN grounding on S^767 must be angular. We illustrate the framework on a Romanian sentence processed through bert-base-multilingual-cased: the angular filtration recovers three semantically grounded intra-group bonds, and the spherical grounding separates intra-group from inter-group pairs by a factor reflecting the cosine structure of LTN grounding. We interpret G = 1/2 at orthogonality as Lukasiewicz indecidability, connecting the construction to the non-bivalent logic tradition from Aristotle's De Interpretatione onward — through Lukasiewicz's three-valued logic (1920), through t-norm fuzzy logics, to contemporary neuro-symbolic systems. This paper is an architectural proposal with a single worked example; systematic empirical validation is deferred to future work. The contribution lies in the unified geometric derivation and in making the angular structure of LTN grounding operational at the topological layer.