Spectral Kakutani–Feldman–Hájek Criticality of Correlated Conformational Modes: Mode Amplitude Perturbation, Iso-Spectral Givens Rotation, and Tail-Increment Finite-Size Elimination

This is the third paper of the series "Statistical Pharmacology via Kakutani Dichotomy". Paper II (DOI 10.5281/zenodo.23012217) decomposed the covariance part of the correlated Kakutani index of two Gaussian conformational ensembles into a commuting amplitude part and a non-commuting rotation part, D_cov = D_comm + D_rot, and asked what happens when a ligand's effect on the collective modes decays along the mode index. This paper answers that question. Two spectral perturbation mechanisms are defined on the cone of positive definite matrices. Mechanism A (mode amplitude perturbation): lambda_i^B = lambda_i^A (1 + c i^(-alpha_lambda)) in a common eigenbasis, with the exact per-mode contribution (1/2) ln[(1 + delta/2)/sqrt(1 + delta)] = sum_k a_k delta^k, a_k = (-1)^k (1 - 2^(1-k))/(4k). Mechanism B (iso-spectral Givens rotation): all eigenvalues unchanged, the k-th adjacent pair of modes of anisotropy ratio r rotated by theta_k = c_theta k^(-alpha_theta), with the exact per-pair contribution (1/2) ln[1 + kappa(r) sin^2 theta_k], kappa(r) = (r-1)^2/(4r). Both are normalised to the leading coefficient A = 0.0100 of Paper I. Main results. (1) Unified three-regime Euler–Maclaurin asymptotics of D_cov(N) for both mechanisms with all limit constants in closed form, and the Feldman–Hájek dichotomy at alpha_c = 1/2 for mode amplitudes and for mode orientations separately; the predictions agree with the numerical experiment to 1e-11 or better for N >= 100 (Theorem 3.1). (2) The doubling tail increment D_tail(N) = D_cov(2N) - D_cov(N) annihilates the Euler–Maclaurin constant A zeta(2 alpha), so its regression exponent equals 1 - 2 alpha on the whole range 0 < alpha < 1, including the negative subcritical values; the bias of the naive exponent at alpha = 0.45 falls from +0.073 to +0.003 (A) and from +0.062 to +0.0001 (B) (Theorem 4.1). (3) A refined finite-size formula that reproduces the naive exponent to four decimals, 0.1729 at alpha = 0.45 where the two-term formula of Paper I gives 0.1601 (Theorem 4.2). (4) The geodesic parity theorem: in the geodesic coordinate eta = ln(lambda^B/lambda^A) of the SPD cone the per-mode divergence (1/2) ln cosh(eta/2) is even, the odd terms of Mechanism A are the curvature of the Euclidean coordinate delta = e^eta - 1, and the resulting shift Delta gamma = alpha c (1-2alpha)/(1-3alpha) (2^(1-3alpha)-1)/(2^(1-2alpha)-1) N^(-alpha) is verified; the geodesic variant lambda^B = lambda^A exp(c i^(-alpha)) and the Givens mechanism, both even, give gamma_tail(0.45) = +0.1001 (Theorem 4.3). (5) The joint (alpha_lambda, alpha_theta) phase diagram: the infinite ensembles are equivalent iff min(alpha_lambda, alpha_theta) > 1/2, with effective exponent max(1 - 2 min(alpha_lambda, alpha_theta), 0), confirmed on a 19 x 19 grid with no misclassified point (Theorem 5.1); the finite-N two-power-law cross-over formula reproduces all 361 measured grid exponents to 4e-4 and explains every characteristic entry of the phase table in closed form (Corollary 5.3). The rotation-only singular phase is the iso-spectral signature of cryptic allostery: every eigenvalue, every residue fluctuation amplitude and every spectral functional unchanged, yet the ensembles mutually singular. This record contains the paper (PDF and LaTeX source), the Python code that generates all data, the figure and every analytic constant (code/), the raw numerical data (data/), the four-panel figure in PDF and PNG (figures/), and the derived tables with a text verification summary (results/). All results reproduce in about a minute with NumPy, Matplotlib and mpmath; see README.md. Source repository: https://github.com/Ruqing1963/kakutani-statistical-pharmacology-III. Version 1.1 (2026-09-28). Equation (3.1) now carries the correct negative sign of the N^{−s−1} Euler–Maclaurin term, H_N(s) = N^{1−s}/(1−s) + ζ(s) + N^{−s}/2 − s N^{−s−1}/12 + O(N^{−s−3}); version 1.0 printed it with a plus sign and the companion script evaluated the Euler–Maclaurin prediction columns of Tables 1 and 2 with that sign. Only the N = 10 and N = 100 rows of those columns changed (deviations now at most 8 × 10^{−9}); Table 4 and every theorem, constant and exponent are unchanged. Version 1.0 remains available under its original DOI.