Statistical Pharmacology via Kakutani Dichotomy I: The Marginal Drug Kakutani Index and the α_c = 1/2 Criticality Theorem in High-Dimensional Conformational Ensembles

This is the first paper of a series that transports the central mechanism of "Localized Bost–Connes systems" (R. Chen, 2026, DOI 10.5281/zenodo.22883179) — a phase transition that is a tail event of an infinite product, invisible to every finite factor — from arithmetic thermodynamics to the statistical geometry of protein conformational ensembles. Two ligands with indistinguishable binding free energies and residue-by-residue almost identical conformational footprints can nevertheless drive a protein into different, even disjoint, downstream functional states. We model the conformational space of a protein with N two-state local switches as {0,1}^N and two ligands as product measures P_A = ⊗ Bernoulli(p_i), P_B = ⊗ Bernoulli(q_i). We define the marginal drug Kakutani index (DKI) K_N = Σ h_i, the sum of the local squared Hellinger distances, the Kakutani affinity Π_N = Π (1 − h_i/2), and the doubling tail increment D_tail(N) = K_{2N} − K_N. By Kakutani's theorem the infinite-dimensional ensembles are either equivalent or mutually singular according as Σ h_i converges or diverges. Main results. (1) An exact quartic Taylor expansion of the local Hellinger distance in the shift δ_i = q_i − p_i, whose cubic term vanishes identically at the maximal-entropy point p = 1/2, where h(δ) = 2 − √(1+2δ) − √(1−2δ) = δ² + (5/4)δ⁴ + (21/8)δ⁶ + … exactly. (2) For the power-law allosteric decay model q_i = p + c·i^(−α) (p = 1/2, c = 0.1, A = c²/(4p(1−p)) = 0.01), the marginal drug Kakutani phase trichotomy: K_N ~ A·N^(1−2α)/(1−2α) for 0 < α < 1/2 (singular), K_N = A(ln N + γ) + C_∞(1/2) + O(1/N) at α = 1/2 (singular, Π_N ~ N^(−A/2)), and K_N → A·ζ(2α) + C_∞(α) < ∞ for α > 1/2 (equivalent); α_c = 1/2 is the unique critical exponent. (3) At criticality the doubling increment converges to the universal constant A·ln 2. (4) An Euler–Maclaurin finite-size scaling theory: the exponent read from a finite regression window is γ_eff(N; α) = (1 − 2α) / (1 + (1 − 2α) ζ(2α) N^(2α−1)), so the systematic upward bias near α_c is the simple pole of ζ at 2α = 1 and not a numerical error, and the doubling-tail estimator cancels it identically. Numerical verification. At N = 10^5 the measured values K_N = 6.311486, 0.1211103, 0.0262136, 0.01658727 for α = 0.25, 0.5, 0.75, 1 agree with the analytic predictions to relative accuracy 10^(−10); the measured D_tail(10^5) at α = 1/2 agrees with A·ln 2 to 3.5 × 10^(−6); the anomalous regression exponent γ_num(0.45) = 0.1610 is predicted analytically as 0.1601 at the window centre and 0.1613 on the window. The paper closes with an assessment and with the problem left to Paper II: whether the off-diagonal covariance network of a correlated ensemble can drive the Kakutani transition on its own when every marginal is unchanged. This record contains the paper (PDF and LaTeX source), the Python code that generates all data and figures (code/), the raw numerical data (data/), the four-panel figure in PDF and PNG (figures/), and the derived tables with every analytic constant used in the text (results/). All results reproduce in seconds with NumPy, Matplotlib and mpmath; see README.md. Source repository: https://github.com/Ruqing1963/kakutani-statistical-pharmacology-I