Kakutani–KMS Theory of Ligand-Induced Conformational Phase Transitions

Measure-Class Obstructions, Covariance Criticality, and Tail Allostery in Computational Pharmacology

Zhengyi Chen¹, Ruqing Chen²

¹ Guangxi Key Laboratory of Drug Discovery and Optimization, School of Pharmacy, Guilin Medical University, Guilin 541199, P. R. China — chenzhengyi@glmc.edu.cn ² GUT Geoservice Inc., Montreal, Québec, Canada — ruqing@hotmail.com

This manuscript integrates the series Statistical Pharmacology via Kakutani Dichotomy, Papers I–VII, into one article (paper/kakutani_pharmacology_paper.tex, 16 pages).

1. Introduction. The blindness of local and tensorized invariants. Theorem 1.1 is proved in full: identical pocket laws, identical per-residue marginals and equal binding free energies are compatible with mutually singular global ensembles, and singularity sets in exactly at decay exponent 1/2. Proposition 1.2 shows that pocket-local energies can never produce singularity.
2. First-order index. The marginal index DKI and the critical exponent α_c = 1/2 (Paper I).
3. Correlated index. CKI, the exact three-way decomposition, and 99.95 % covariance dominance, with the Szegő rate recomputed independently (Paper II).
4. Collective modes. Spectral Kakutani–Feldman–Hájek criticality of amplitudes and rotations (Paper III).
5. KMS scaling and broken phases. KMS scaling β_c = 1, tail allostery, and the chemical obstruction group (Papers IV and V).
6. Transport cost. The cone-geodesic Wasserstein cost W_tail and alchemical path topology (Paper VI).
7. Pipeline and representation. The MD estimation pipeline kakutani_pharma (Paper VII), and the tail-augmented representation z_local ⊕ z_symmetry ⊕ z_tail with a proposition on its information content.
8. Discussion. What is established, what is not, and what follows.

All numerical evidence comes from analytic models and synthetic trajectories. Validation on all-atom protein–ligand molecular dynamics has not been done.

code/verify_synthesis_theorems.py checks the statements that this article proves or recomputes itself, and writes results/synthesis_verification.txt:

• the expansion of the local index to eighth order (the remainder over δ⁸ tends to 429/64);
• the pair divergence of Theorem 1.1(2) against a direct Bhattacharyya computation, and its partial sums for α = 1/4, 1/2 and 3/4;
• Proposition 1.2 on a Gaussian example;
• the Szegő rate s = 0.0094968218, the constant c₀ = −0.0093715854 for N = 50 to 1000, and D_mean, D_cov and CKI at N = 501.

All checks pass. The theorems of Papers I–VII are verified by the scripts archived with each paper.

.
├── paper/kakutani_pharmacology_paper.{tex,pdf}
├── figures/                              # figures of Papers IV-VII
├── code/verify_synthesis_theorems.py
├── results/synthesis_verification.txt
├── CITATION.cff, LICENSE, .gitignore
└── README.md

pip install numpy scipy mpmath
python code/verify_synthesis_theorems.py     # prints OVERALL: PASS
cd paper
pdflatex kakutani_pharmacology_paper.tex && pdflatex kakutani_pharmacology_paper.tex && pdflatex kakutani_pharmacology_paper.tex

The figures in figures/ are copied from Papers IV–VII.

https://github.com/Ruqing1963/kakutani-statistical-pharmacology-synthesis

Paper I 10.5281/zenodo.23005921 · II 10.5281/zenodo.23012217 · III 10.5281/zenodo.23015709 · IV 10.5281/zenodo.23018236 · V 10.5281/zenodo.23027704 · VI 10.5281/zenodo.23029360 · VII 10.5281/zenodo.23039869 · monograph 10.5281/zenodo.22883179.

@article{ChenChen2026KakutaniKMS,
  author  = {Chen, Zhengyi and Chen, Ruqing},
  title   = {Kakutani--KMS Theory of Ligand-Induced Conformational Phase Transitions: Measure-Class Obstructions,
             Covariance Criticality, and Tail Allostery in Computational Pharmacology},
  year    = {2026},
  doi     = {10.5281/zenodo.23042828},
  url     = {https://doi.org/10.5281/zenodo.23042828}
}

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; text and figures under CC BY 4.0 (see LICENSE).