Direct von Neumann Entropy of the Angle Operator for Dihedral Subfactor Quadrilaterals

Bakshi, Guin and Pal asked for computable formulas for the direct von Neumann entropy H(|\Theta|^2) of the angle operator between intermediate subalgebras, beyond the formulas available for its Fourier dual. We answer this problem for the full dihedral fixed-point family. Let D_n be the dihedral group of order 2n acting outerly on a type II_1 factor M, and considerN=M^{D_n}\subset P=M^{\langle y\rangle},\quad Q=M^{\langle z\rangle}\subset M.If \Theta_n=e_Pe_Q, then with respect to the normalized Markov trace the spectral measure of |\Theta_n|^2 is\mu_n=\frac12\delta_0+\frac1{2n}\sum_{j=0}^{n-1}\delta_{\cos^2(\pi j/n)}. This yields an explicit direct entropy formula for every n\ge2. We determine its limit, derive an exact Fourier-aliasing expansion, prove a parity law, and obtain the first finite-size corrections: even n approach the limit from below with leading term -\zeta(3)/(2n^3), whereas odd n approach from above with leading term 3\zeta(3)/(8n^3). The spectral measures converge to one half an atom at zero plus one half the arcsine law. Although the Jones index [M:N]=2n diverges, the direct angle entropy converges to\frac12\log2-\frac14.Thus this scalar entropy is not a coercive proxy for common-core index in the dihedral family, even though the full angle spectrum determines n exactly. The formulas agree with the low-index case treated by Bakshi–Guin–Pal and provide an explicit infinite-family partial answer to their concluding problem.