Statistical Pharmacology via Kakutani Dichotomy II: The Correlated Kakutani Index, the 99.95% Covariance Dominance Theorem, and Spectral Fingerprints of Cryptic Allostery

This is the second paper of the series "Statistical Pharmacology via Kakutani Dichotomy". It answers Open Problem (1) of Paper I (DOI 10.5281/zenodo.23005921): the independent Bernoulli switches of Paper I are replaced by a joint Gaussian conformational ensemble with off-diagonal couplings, P_A = N(mu_A, C_A) and P_B = N(mu_B, C_B) on R^N, and the correlated Kakutani index is defined as the Bhattacharyya distance CKI = D_B = D_mean + D_cov, whose exponential exp(-D_B) is the second-order Kakutani affinity. The conformational pharmacological spectral fingerprint F_L = (Delta lambda_i, theta_i) records, mode by mode, the amplitude log-ratio and the rotation angle of the collective motions. Three theorems are proved. (i) Commuting-rotation decomposition (Theorem 2.3): D_cov = D_cov^comm + D_cov^rot with D_cov^comm = (1/2) sum_i ln cosh(Delta lambda_i / 2) and D_cov^rot >= 0, the inequality following from the Ky Fan-Lidskii majorization of the eigenvalues of C_A + C_B and the strict convexity of -ln, with equality if and only if C_A and C_B share an ordered common eigenbasis. (ii) The 99.95% covariance dominance theorem (Theorem 3.2): for Kac-Murdock-Szego covariance networks C(i,j) = rho^|i-j| with rho_A = 0.35, rho_B = 0.55 and a mean shift Delta mu_i = 0.12 i^(-0.75) in the subcritical regime of Paper I, D_mean(N) increases monotonically to the finite limit 0.0024990 with an N^(-1/2) tail whose prefactor c_mu^2 / (8 g(0)) is explicit, whereas D_cov(N) = s N + c_0 + O(q^N) with the Szego rate s = 0.0094968218 and the strong-Szego constant c_0 = -0.0093715854 obtained in closed form by Jensen's formula on the unit circle; the two constants reproduce the measured D_cov(N) to 1e-13 for every N >= 50, the infinite ensembles are mutually singular by the Feldman-Hajek theorem, and the covariance share reaches 99.949% at N = 501 (D_cov = 4.74854, D_mean = 0.0024408, D_B = 4.75098). (iii) The Gram-matrix rewiring theorem (Theorem 4.1): a Type II ligand pair that opens a distal allosteric channel through the convex mixture C_B = (1/2) C_A + (1/2) M, with M = E^T C_A E a Gram matrix of unit diagonal, has RMSD = 0, Delta RMSF = 0, marginal DKI K_N = 0 and D_mean = 0 identically, yet D_cov(501) = 3.97, of which 71.6% is the non-commuting rotation part, with lambda_min(C_B) >= (1/2) lambda_min(C_A) > 0.24074 (measured 0.24079). Every constant quoted is verified against the numerical experiment to the precision of double arithmetic, and the paper closes with an assessment and with the question for Paper III: the spectral Kakutani criticality of covariance perturbations that decay along the mode index. This record contains the paper (PDF and LaTeX source), the Python code that generates all data, figures and analytic constants (code/), the raw numerical data (data/), the four-panel figure in PDF and PNG (figures/), and the derived tables (results/). All results reproduce in seconds with NumPy, SciPy and Matplotlib; see README.md. Source repository: https://github.com/Ruqing1963/kakutani-statistical-pharmacology-II