Statistical Pharmacology via Kakutani Dichotomy IV: KMS Inverse-Temperature Criticality β_c = 1, Pocket-to-Distal Tail Allostery, and the Four-Quadrant Ligand Classification

This is the fourth paper of the series Statistical Pharmacology via Kakutani Dichotomy, which transports the tail-event phase transition of Localized Bost–Connes systems (R. Chen, DOI 10.5281/zenodo.22883179) to the statistical geometry of protein conformational ensembles. Papers I–III established that the Kakutani–Feldman–Hájek dichotomy between two ligand-induced ensembles is decided by the decay exponent of the perturbation along the ordered modes, with the critical exponent α_c = 1/2, and that the doubling tail-increment estimator reads this exponent without the pole bias of naive regression. Paper IV places the spectral perturbation model of Paper III in a statistical-mechanical setting. An inverse temperature β, interpreted as the stiffness of the allosteric coupling network, controls the decay exponent of the distal-mode perturbation through α(β) = 1/(2β), so that the Kakutani transition at α_c = 1/2 becomes a KMS-type phase transition at β_c = 1. The conformational space is divided into a binding pocket (20 modes, docking-visible), an allosteric core (50 modes, finite elastic channel) and a distal tail of up to 10^5 loop modes. The main results are: closed-form saturation of the pocket and core divergences; the KMS trichotomy in the inverse temperature, with equivalent ensembles for β < 1, logarithmic critical growth D = A ln N + C at β = 1 and singular power-law growth with exponent 1 − 1/β for β > 1; a refined naive-slope identity; a sub-leading cross-over theorem for the doubling-tail exponent that combines the quartic geodesic term of order N^{−1/β} with the Euler–Maclaurin lattice term of order N^{−1} and reproduces the measured error at all 18 temperatures to 2 × 10^{−8}; a tail-allostery amplification theorem, in which the pocket contribution stays at 10^{−3} while the distal share reaches 99.9 % and the tail-allostery ratio reaches 5953; and a four-quadrant classifier of ligands in the plane of docking score change and cumulative Kakutani index. On a synthetic library of 60 ligands the classifier is correct in all cases, whereas a docking-only activity call is correct in half of them and misses every cryptic tail-allosteric ligand. The record contains the numerical experiment (Python, NumPy, Matplotlib, SciPy), the three CSV data files (temperature scan, spatial profile, ligand screen), the four-panel figure in PDF and PNG, the analytic companion script that derives every table from closed forms, the derived table files, and the 16-page paper in LaTeX and PDF with full proofs. All results are reproducible from the repository in under one minute. The Euler–Maclaurin expansion used here carries the correct sign of the N^{−s−1} term; the first versions of Papers I and III printed it with the opposite sign, which affected only the small-N rows of their prediction columns, and their repositories have been corrected. Authors: Zhengyi Chen (Guangxi Key Laboratory of Drug Discovery and Optimization, School of Pharmacy, Guilin Medical University, Guilin 541199, P. R. China) and Ruqing Chen (GUT Geoservice Inc., Montreal, Québec, Canada). Funding: National Natural Science Foundation of China (No. 22464010), Guangxi Natural Science Foundation of China (No. 2025GXNSFAA069294), and the 2025 Bagui Youth Top Talent Project. Code under the MIT License; paper, figures and data under CC BY 4.0. Source repository: https://github.com/Ruqing1963/kakutani-statistical-pharmacology-IV.