Sharp Observability Horizons and Witness Amplification in Dihedral Subfactor Quadrilaterals

Let (D_n=\langle y,z\mid y^2=z^2=1,(yz)^n=1\rangle) be the dihedral group of order (2n), acting outerly on a type (II_1) factor (M). Put (N=M^{D_n}), (P=M^{\langle y\rangle}), and (Q=M^{\langle z\rangle}), and let (p=e_P), (q=e_Q) be the corresponding intermediate Jones projections in the basic construction for (N\subset M). We identify exactly the finite-dimensional von Neumann algebra generated by (p) and (q): it is the copy of (\mathbb C[D_n]) in (M\rtimes 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,[d_{\mathrm{det}}\theta_{\min}=2\pi.] 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.