Objective  To construct a formalized, structured, and equivalent mathematical representation framework for the basic theories of traditional Chinese medicine (TCM), and to illustrate the mathematical expression of the Bianzheng Lunzhi (辨证论治, syndrome differentiation and treatment) process using the Guizhi Tang (桂枝汤) syndrome as an example.

Methods  Bianzheng (辨证, syndrome differentiation) attributes and the properties of Chinese materia medica (CMM) were selected as the major cognitive elements and were symbolized and quantified. The quantified Bianzheng attributes were used to construct a state space, while the effects of CMMs were abstracted as state-correction operators. Within an n-dimensional Euclidean space, state attributes, ordering relations, a personalized constitutional baseline, state-transition expressions, and convergence expressions were constructed, and a schematic calculation based on Guizhi Tang syndrome was performed.

Results  A mathematical representation framework comprising an n-dimensional state space S, a personalized constitutional baseline vector B, a state-correction operator O, a harmonization coefficient α, and compatibility coefficients λ_rs was established. A first-order affine state-transition equation and a generalized state-transition equation incorporating nonlinear compatibility terms were formulated. Under the schematic parameter settings, the initial state of the Guizhi Tang syndrome changed from S₁ = (2, 2, − 2, − 2)^T to S₂ = (0.8, 0.4, − 0.4, − 0.8)^T, while the health index V decreased from 4.00 to 1.26, indicating that the state approached the predefined healthy equilibrium.

Conclusion  The framework developed in this study can provide a unified, describable, and computable mathematical representation of syndrome states, formula-related therapeutic actions, and state changes in the Bianzheng Lunzhi process. It may provide a foundation for the formal study of the basic theories of TCM and for subsequent calibration using clinical data.