Sharp Observability Horizons and Witness Amplification in Dihedral Subfactor Quadrilaterals

Let (D_n= y,z y^2=z^2=1,(yz)^n=1) be the dihedral group of order (2n), acting outerly on a type (II_1) factor (M). Put (N=MD_n), (P=Mʸ), and (Q=Mᶻ), and let (p=e_P), (q=e_Q) be the corresponding intermediate Jones projections in the basic construction for (N⊂ M). We identify exactly the finite-dimensional von Neumann algebra generated by (p) and (q): it is the copy of ( C[D_n]) in (M D_n), and therefore has dimension (2n=[M:N]). We then establish a sharp tracial observability horizon. Every joint tracial polynomial in (p,q) of total degree below (2n) has exactly the same value as for two free Bernoulli-(1/2) projections, while the first possible separation occurs at degree (2n), where the finite dihedral closure relation produces a unit trace gap. For the positive angle compression (A_n=qpq), we derive all moments explicitly and prove that the first closure-sensitive moment occurs only at order (n), with gap (4⁻ⁿ). At the same tracial word degree, an involutive reflection witness has gap (1), yielding an exact witness-amplification factor (4^n). The observability threshold is exactly reciprocal to the smallest Sano–Watatani angle,[ddetθₘᵢₙ=2π.] Consequences are also obtained for alternating conditional expectations and relative-entropy capacity. The results exhibit a precise distinction between relational depth and the signal quality of a chosen representation of the same finite closure relation.