Gleason-Type Uniqueness of Born's Rule for Spin-$\tfrac{1}{2}$ in the PDL Framework

We subsequently reanalyse the mixed-triangle measurement scheme previously formulated within PDL, in which a minimal $(4,6)$ closure encodes a spin doublet and a Stern–Gerlach-type device is represented by a finite active surface composed of mixed triangles. We prove that, under mild symmetry constraints on the apparatus, the effective probabilities generated by this scheme satisfy our operational axioms and thus fall within the scope of the uniqueness theorem. It follows that Born’s rule for spin-$\tfrac12$ measurements is not merely implementable in PDL, but is uniquely enforced by the stipulated operational and combinatorial conditions. This result provides an initial link between Gleason-type derivations and the discrete relational substrate underlying PDL, and suggests the feasibility of a more general uniqueness programme for quantum probabilities within this framework.