ABSTRACT

Theory of both Zangxiang (藏象) and Jingluo (经络) is useful in understanding disease. By using mathematical reasoning based on Yin Yang Wu Xing Theory in Traditional Chinese Medicine (TCM), this paper demonstrates the treatment principle: “Searching for a root cause of disease in cure, treatment of both the root-cause and symptoms at the same time” (治病求本, 标本兼治). It means that for a human body, there is the mathematical structure of both Zangxiang and Jingluo as her/his second physiological system. It can be used to determine both the root-cause and symptoms of the sick organ by using both Zangxiang and Jingluo. In general, for the human blood pH value, the normal range of theory is [7.34539, 7.45461] nearly to [7.35, 7.45], and the center is 7.4. The first or second transfer law of a human body’s energies changes according to the different blood pH values whether in the normal range or not. Human disease treatment should protect and maintain the balance of two incompatibility relations: the loving relationship and the killing relationship. As an application, acupuncture is used to treat limb-girdle muscular dystrophy.

Keywords:

Traditional Chinese Medicine (TCM), Yin Yang Wu Xing Theory, Steady Multilateral Systems, Incompatibility Relations, Side Effects, Medical and Drug Resistance Problem

1. Introduction

Theory of both Zangxiang (藏象) and Jingluo (经络) is useful in understanding disease. Zangxiang and Jingluo are two general mathematical structures. Human body blood pH value of both Zangxiang and Jingluo is a general parameter linking together the complexity of relations between organ pairs, the organ itself, the capabilities for intervention reaction and self-protection of the body and mind as a whole, related to the environment, food, health and personal history, air, water, earth, climate, season, etc. Human body blood pH value is as useful in understanding disease as the average is in statistics, or as the expected value is in probability calculation.

The studies of Prof. Macro Pianesi, President of the International Association UN PUNTO MACROBIOTICO, began as a pioneer work continuing the work of Japanese Nyoiti Sakurazawa, known in the West as Georges Ohsawa, the founding father of Macrobiotics. For example, thanks to these studies and macrobiotic cooking, over 25,000 disease patients have been cured in the last 8 years in the state of Africa alone, even with understandable and interesting economic implications.

Macrobiotic diet therapy trials have started in other parts of the world as well, including China: Prof. Wang Bin (Director of the Department of Medical Matters, Tibetan and Ethnic Medicine Hospital, University of Traditional Chinese Medicine, Beijing), Dr. Weiguo Ma and Dr. Bing Xu (also from the University of Traditional Chinese Medicine, Beijing), and Dr. Yujing Quian (Doctor in Traditional Chinese Medicine at Capital Normal University), who reported their experiences in a conference held in Italy on October 18, 2010.

Mario Pianesi never ceases to thank the Ancient Chinese for the depth of their thinking, and for the value they give to food in their reflections and observations. Among others, Prof. Mario Pianesi identifies an important indicator for human health: the value of blood pH, which, under normal conditions, ranges from 7.35 to 7.45, and the center is 7.4. Outside this range (acid: Yin condition; alkaline: Yang condition), disease appears. Almost always, when there is a disease, the condition of blood is a Yin condition, little is a Yang condition.

There are a lot of evidences (e.g., experimental identification for probability and real applications) to support this viewpoint, such as, Shirakabe et al. [1] , Kaur et al. [2] , Aly et al. [3] , Intven et al. [4] , Patel et al. [5] , Handman et al. [6] , Natalia et al. [7] , Barfod et al. [8] , Zhu et al. [9] , Zhu et al. [10] , Robert et al. [11] , Mona et al. [12] , Saritas et al. [13] , Stevens et al. [14] , and so on.

The body begins to activate the necessary mechanisms to restore this parameter to its appropriate range. If the body is unable to restore optimal pH levels, the disease may become chronic and lead to dire consequences.

Zhang et al. [15] - [23] have started a great interest and admired works for Traditional Chinese Medicine (TCM), where, through mathematical reasoning, they demonstrate the presence of incompatibility relations, which are predominant in daily life, yet absent in traditional Aristotelian Western logic.

Many people as Western person are beyond all doubt, the Yin Yang Wu Xing theory is superior to the traditional true-false logic, which does not contemplate incompatibility relations, which Zhang [19] has expertly explained from a mathematical standpoint.

The work Zhang [15] [16] has started, allows many people like Western person to think of a true re-foundation of mathematical language, to make it a better suited tool for the needs of mankind and the environment. Although so doing, Zhang [19] also brings to light the difficulty of establishing the values of both the intervention reaction coefficients ${\rho }_1,{\rho }_2$ and the self-protection coefficient ${\rho }_3$ as parameters with due accuracy.

In this paper, the introduction of a parameter such as a blood pH value will be suggested, in order to facilitate the understanding and the calculation of these values. This paper ventures to suggest this with all due to respect, because it be believed that the path Zhang [15] [16] has started, in such an understandable way from the mathematical point of view, will be very useful for all mankind searching for tools to understand the mechanisms of life.

The article proceeds as follows. Section 2 contains a parameter model and basic theorems, in order to explain the intervention reaction coefficients ${\rho }_1,{\rho }_2$ and the self-protection coefficient ${\rho }_3$ through the introduction of a parameter model to study the normal range of a human blood pH value, while the first or second transfer law of a human body’s energies is demonstrated in Section 3, proved through the concept of both relation costs and the relationship analysis of steady multilateral systems. Furthermore, if the range of a human blood pH value is divided into four parts from small to large, for the body in every part, the prevention or treatment method of a human body’s diseases as the treatment principle of TCM is given respectively in Section 4. As an application, acupuncture is used to treat limb-girdle muscular dystrophy in Section 5 and conclusions are drawn in Section 6.

2. Parameter Model and Basic Theorems

The concepts and notations in Zhang [18] are start and still used.

Let $\phi =\left(\sqrt{5}-1\right)/2=0.61803399$ be the gold number. Denoted ${\rho }_0=0.5897545123$, namely healthy number. It is because the healthy number ${\rho }_0$ can make the healthy balance conditions ${\rho }_1={\rho }_3,{\rho }_2={\rho }_1{\rho }_3$ and $1-{\rho }_2{\rho }_3={\rho }_1+{\rho }_2{\rho }_3$ hold if ${\rho }_1={\rho }_0,{\rho }_2={\rho }_0^2$ and ${\rho }_3={\rho }_0$. Assuming ${{\rho }^{\prime }}_0=0.68232780$, namely unhealthy number. It is because under a poor self-protection ability, the unhealthy number ${{\rho }^{\prime }}_0$ can make the following unhealthy balance conditions hold:

$\begin{array}{l}{\rho }_1-{\rho }_3={\rho }_3={{\rho }^{\prime }}_0/2=0.34116390,\\ {\rho }_2-{\rho }_1{\rho }_3={\rho }_1{\rho }_3={\left({{\rho }^{\prime }}_0\right)}^2/2=0.23278561\\ 1-{\rho }_2{\rho }_3={\rho }_1+{\rho }_2{\rho }_3\end{array}$

if ${\rho }_1={{\rho }^{\prime }}_0,{\rho }_2={\left({{\rho }^{\prime }}_0\right)}^2=0.46557123$ and ${\rho }_3={{\rho }^{\prime }}_0/2$. Thus ${\rho }_0<\phi <{{\rho }^{\prime }}_0$.

A parameter model of a human blood pH value in a mathematical sense based on Yin Yang Wu Xing Theory of TCM is reintroduced by using the functions $\lambda \left(x\right)$ and $\rho \left(x\right)$ of the human blood pH value x described as follows.

Let $x\in \left(7,7.8\right)$ be a human blood pH value, where the values 7 and 7.8 are the minimum and maximum acceptable the blood pH value. Denoted the center value 7.4 is the target as the expectation of the human blood pH value. Define a function $\lambda \left(x\right)$ of the blood pH value x in below:

$\begin{array}{c}\lambda \left(x\right)=\frac{\left|x-7.4\right|}{\left(7.8-x\right)\left(x-7\right)},\forall x\in \left(7,7.8\right)\\ =\{\begin{array}{cc}\frac{x-7.4}{\left(7.8-x\right)\left(x-7\right)},& 7.8>x\ge 7.4;\\ \frac{7.4-x}{\left(7.8-x\right)\left(x-7\right)},& 7<x<\mathrm{7.4.}\end{array}\end{array}$ (1)

A parameter model is considered as

$\rho \left(x\right)=\frac{1/2}{\lambda \left(x\right)+1/2},\forall x\in \left(7,7.8\right).$ (2)

Theorem 2.1 Under model (2), the following statements hold.

1) The one that $0<\rho \left(x\right)=\frac{1/2}{\lambda \left(x\right)+1/2}\le 1$ is equivalent to the other that $0\le \lambda \left(x\right)=\frac{1-\rho \left(x\right)}{2\rho \left(x\right)}<+\infty$, where $\lambda \left(x\right)$ is a monotone decreasing function of x if $x\in \left(7,7.4\right]$ or a monotone increasing function of x if $x\in \left[7.4,7.8\right)$; and $\rho \left(x\right)$ is a monotone decreasing function of $\lambda \left(x\right)$ if $\lambda \left(x\right)\in \left[0,+\infty \right)$; and $\lambda \left(x\right)$ is a monotone decreasing function of $\rho \left(x\right)$ if $\rho \left(x\right)\in \left(0,1\right]$.

(2) If $1\ge \rho \left(x\right)\ge {\rho }_0$, then

$\lambda \left(x\right)=\frac{1-\rho \left(x\right)}{2\rho \left(x\right)}\le \frac{1-{\rho }_0}{2{\rho }_0}={\rho }_0^2\le \rho {\left(x\right)}^2\le 1;$ $\frac{\lambda \left(x\right)}{\rho \left(x\right)}=\frac{1-\rho \left(x\right)}{2\rho {\left(x\right)}^2}\le \frac{1-{\rho }_0}{2{\rho }_0^2}={\rho }_0\le \rho \left(x\right)\le 1;$ and $\frac{\lambda \left(x\right)}{\rho {\left(x\right)}^2}=\frac{1-\rho \left(x\right)}{2\rho {\left(x\right)}^3}\le \frac{1-{\rho }_0}{2{\rho }_0^3}=1.$

3) If $0<\rho \left(x\right)<{\rho }_0$, then

$\lambda \left(x\right)=\frac{1-\rho \left(x\right)}{2\rho \left(x\right)}>\frac{1-{\rho }_0}{2{\rho }_0}={\rho }_0^2>\rho {\left(x\right)}^2>0;$

$\frac{\lambda \left(x\right)}{\rho \left(x\right)}=\frac{1-\rho \left(x\right)}{2\rho {\left(x\right)}^2}>\frac{1-{\rho }_0}{2{\rho }_0^2}={\rho }_0>\rho \left(x\right)>0;$ and $\frac{\lambda \left(x\right)}{\rho {\left(x\right)}^2}=\frac{1-\rho \left(x\right)}{2\rho {\left(x\right)}^3}>\frac{1-{\rho }_0}{2{\rho }_0^3}=1.$

4) Taking $0<{\rho }_1=\rho \left(x\right)<{\rho }_0,{\rho }_2=\rho {\left(x\right)}^2$ and ${\rho }_3=c\rho \left(x\right)$ where $0\le c\le 1$, there are

${\rho }_1-{\rho }_3=\rho \left(x\right)\left(1-c\right)\ge 0,{\rho }_2-{\rho }_1{\rho }_3=\rho {\left(x\right)}^2\left(1-c\right)\ge 0,$

and $\left({\rho }_1+{\rho }_2{\rho }_3\right)=\rho \left(x\right)+c\rho {\left(x\right)}^3<1-{\rho }_2{\rho }_3=1-c\rho {\left(x\right)}^3,$

where

$\left|\left({\rho }_1+{\rho }_2{\rho }_3\right)-\left(1-{\rho }_2{\rho }_3\right)\right|>2\left(1-c\right){\rho }_0^3=\left(1-c\right)\mathrm{0.41024.}$

5) Taking $1\ge {\rho }_1=\rho \left(x\right)\ge {\rho }_0,{\rho }_2=\rho {\left(x\right)}^2$ and ${\rho }_3=c\rho \left(x\right)$ where $0\le c\le 1$, there are

firstly, ${\rho }_1-{\rho }_3=\rho \left(x\right)\left(1-c\right)\ge 0,{\rho }_2-{\rho }_1{\rho }_3=\rho {\left(x\right)}^2\left(1-c\right)\ge 0$ and $\left({\rho }_1+{\rho }_2{\rho }_3\right)=\rho \left(x\right)+c\rho {\left(x\right)}^3\ge 1-{\rho }_2{\rho }_3=1-c\rho {\left(x\right)}^3$ if $1\ge c\ge \frac{1-\rho \left(x\right)}{2\rho {\left(x\right)}^3}=\frac{\lambda \left(x\right)}{\rho {\left(x\right)}^2}\ge 0$;

secondly, ${\rho }_1-{\rho }_3=\rho \left(x\right)\left(1-c\right)>\rho \left(x\right)/2,{\rho }_2-{\rho }_1{\rho }_3=\rho {\left(x\right)}^2\left(1-c\right)>\rho {\left(x\right)}^2/2$ and $\left({\rho }_1+{\rho }_2{\rho }_3\right)=\rho \left(x\right)+c\rho {\left(x\right)}^3<1-{\rho }_2{\rho }_3=1-c\rho {\left(x\right)}^3$ where $\left|\left({\rho }_1+{\rho }_2{\rho }_3\right)-\left(1-{\rho }_2{\rho }_3\right)\right|\le {\left({{\rho }^{\prime }}_0\right)}^3=0.31767$ if $0\le c<\frac{1-\rho \left(x\right)}{2\rho {\left(x\right)}^3}=\frac{\lambda \left(x\right)}{\rho {\left(x\right)}^2}\le \frac{1}{2}$ in which $1>\rho \left(x\right)\ge {{\rho }^{\prime }}_0$;

thirdly, $\begin{array}{l}{\rho }_1-{\rho }_3=\rho \left(x\right)\left(1-c\right)\ge \rho \left(x\right)/2,\\ {\rho }_2-{\rho }_1{\rho }_3=\rho {\left(x\right)}^2\left(1-c\right)\ge \rho {\left(x\right)}^2/2\end{array}$ and $\left({\rho }_1+{\rho }_2{\rho }_3\right)=\rho \left(x\right)+c\rho {\left(x\right)}^3<1-{\rho }_2{\rho }_3=1-c\rho {\left(x\right)}^3$ where $\left|\left({\rho }_1+{\rho }_2{\rho }_3\right)-\left(1-{\rho }_2{\rho }_3\right)\right|\le 2{\rho }_0^3=0.41024$ if $0\le c\le \frac{1}{2}<\frac{1-\rho \left(x\right)}{2\rho {\left(x\right)}^3}=\frac{\lambda \left(x\right)}{\rho {\left(x\right)}^2}\le 1$ in which ${\rho }_0\le \rho \left(x\right)<{{\rho }^{\prime }}_0$;

finally, ${\rho }_1-{\rho }_3=\rho \left(x\right)\left(1-c\right)<\rho \left(x\right)/2,{\rho }_2-{\rho }_1{\rho }_3=\rho {\left(x\right)}^2\left(1-c\right)<\rho {\left(x\right)}^2/2$ and $\left({\rho }_1+{\rho }_2{\rho }_3\right)=\rho \left(x\right)+c\rho {\left(x\right)}^3<1-{\rho }_2{\rho }_3=1-c\rho {\left(x\right)}^3$ where $\left|\left({\rho }_1+{\rho }_2{\rho }_3\right)-\left(1-{\rho }_2{\rho }_3\right)\right|<{\left({{\rho }^{\prime }}_0\right)}^3=0.31767$

if $\frac{1}{2}<c<\frac{1-\rho \left(x\right)}{2\rho {\left(x\right)}^3}=\frac{\lambda \left(x\right)}{\rho {\left(x\right)}^2}\le 1$ in which ${\rho }_0\le \rho \left(x\right)<{{\rho }^{\prime }}_0$.

In particular, when c is nearly to 1/2, there are

${\rho }_1-{\rho }_3=\rho \left(x\right)\left(1-c\right)\to \rho \left(x\right)/2,{\rho }_2-{\rho }_1{\rho }_3=\rho {\left(x\right)}^2\left(1-c\right)\to \rho {\left(x\right)}^2/2$

and the following statements hold.

a) The absolute value $\left|\left({\rho }_1+{\rho }_2{\rho }_3\right)-\left(1-{\rho }_2{\rho }_3\right)\right|$ is nearly to 0 if $0<c<\frac{1-\rho \left(x\right)}{2\rho {\left(x\right)}^3}=\frac{\lambda \left(x\right)}{\rho {\left(x\right)}^2}\le \frac{1}{2}$ in which $1>\rho \left(x\right)\ge {{\rho }^{\prime }}_0$.

b) The value $\left[\left({\rho }_1+{\rho }_2{\rho }_3\right)-\left(1-{\rho }_2{\rho }_3\right)\right]$ is included in the interval $\left[-{\rho }_0^3=-0.20512,0\right)$ respectively if $0<c\le \frac{1}{2}<\frac{1-\rho \left(x\right)}{2\rho {\left(x\right)}^3}=\frac{\lambda \left(x\right)}{\rho {\left(x\right)}^2}\le 1$ in which ${\rho }_0\le \rho \left(x\right)<{{\rho }^{\prime }}_0$.

c) The value $\left[\left({\rho }_1+{\rho }_2{\rho }_3\right)-\left(1-{\rho }_2{\rho }_3\right)\right]$ is included in the interval $\left[-{\rho }_0^3=-0.20512,0\right)$ respectively if $\frac{1}{2}<c<\frac{1-\rho \left(x\right)}{2\rho {\left(x\right)}^3}=\frac{\lambda \left(x\right)}{\rho {\left(x\right)}^2}\le 1$ in which ${\rho }_0\le \rho \left(x\right)<{{\rho }^{\prime }}_0$. #

Corollary 2.1 Under model (2), the following statements hold.

1) For any $0<d<1$, there is a unique solution $u\in \left(7,7.4\right)$ and there is also a unique solution $v\in \left(7.4,7.8\right)$, such that

$\lambda \left(7.4\right)=0\le \lambda \left(x\right)=\frac{1-\rho \left(x\right)}{2\rho \left(x\right)}\le \lambda \left(u\right)=\lambda \left(v\right)=\left(1-d\right)/\left(2d\right),$

$\rho \left(u\right)=\rho \left(v\right)=d\le \rho \left(x\right)=\frac{1/2}{\lambda \left(x\right)+1/2}\le 1=\rho \left(7.4\right).$

2) The condition $x\in \left[\text{7}\text{.35,7}\text{.45}\right]$ is equivalent to each of the following conditions:

$\lambda \left(7.4\right)=0\le \lambda \left(x\right)=\frac{1-\rho \left(x\right)}{2\rho \left(x\right)}\le \lambda \left(7.35\right)=\lambda \left(7.45\right)=0.31743,$

$\rho \left(7.35\right)=\rho \left(7.45\right)=0.61167\le \rho \left(x\right)=\frac{1/2}{\lambda \left(x\right)+1/2}\le 1=\rho \left(7.4\right).$

3) The condition $x\in \left[7.35129,7.44871\right]$ is equivalent to each of the following conditions:

$\lambda \left(7.4\right)=0\le \lambda \left(x\right)=\frac{1-\rho \left(x\right)}{2\rho \left(x\right)}\le \lambda \left(7.35129\right)=\lambda \left(7.44871\right)=0.30902,$

$\rho \left(7.35129\right)=\rho \left(7.44871\right)=\phi \le \rho \left(x\right)=\frac{1/2}{\lambda \left(x\right)+1/2}\le 1=\rho \left(7.4\right).$

4) The condition $x\in \left[7.34539,7.45461\right]$ is equivalent to each of the following conditions:

$\lambda \left(7.4\right)=0\le \lambda \left(x\right)\le \lambda \left(7.34539\right)=\lambda \left(7.45461\right)={\rho }_0^2=0.34781,$

$\rho \left(7.34539\right)=\rho \left(7.45461\right)={\rho }_0\le \rho \left(x\right)=\frac{1/2}{\lambda \left(x\right)+1/2}\le 1=\rho \left(7.4\right).$

5) The condition $x\in \left[7.36307,7.43693\right]$ is equivalent to each of the following conditions:

$\lambda \left(7.4\right)=0\le \lambda \left(x\right)\le \lambda \left(\text{7}.\text{363}0\text{7}\right)=\lambda \left(\text{7}.\text{43693}\right)={\left({{\rho }^{\prime }}_0\right)}^2/2=\text{0}\text{.23279,}$

$\rho \left(7.36307\right)=\rho \left(7.43693\right)={{\rho }^{\prime }}_0\le \rho \left(x\right)=\frac{1/2}{\lambda \left(x\right)+1/2}\le 1=\rho \left(7.4\right).\#$

Theorem 2.1 and Corollary 2.1 can be found in Zhang [23] .

Remark 1. In west, through experiment or through practice observation, many researchers [1] - [14] have obtained the normal range of human blood pH value as $x\in \left[7.35,7.45\right]$. But in TCM, from Yin Yang Wu Xing Theory, Zhang [17] has already determined: ${\rho }_0\le {\rho }_1\le 1$ for the normal range of a healthy body. Taking ${\rho }_1=\rho \left(x\right),{\rho }_2=\rho {\left(x\right)}^2$ and ${\rho }_3=c\rho \left(x\right)$ where $0\le c\le 1$ for a human body which has the capabilities of both intervention reaction and self-protection. From Corollary 2.1, the condition ${\rho }_0\le {\rho }_1=\rho \left(x\right)\le 1$ is equivalent to $x\in \left[7.34539,7.45461\right]$. In other words, in Theory of TCM, the normal range of human blood pH value is considered as $x\in \left[7.34539,7.45461\right]$, nearly to $x\in \left[7.35,7.45\right]$. Of course, little difference of the two intervals which makes the diagnosis of disease as a result, there may be no much difference as a suspect. In fact, TCM uses the rule ${\rho }_0\le {\rho }_1\le 1$ from the Yin Yang Wu Xing Theory instead of the normal range of human blood pH value. The equivalence of Corollary 2.1 shows that TCM is The scientific.

Zhang [17] has already determined: a body is said healthy when the intervention reaction coefficient ${\rho }_1$ satisfies $1\ge {\rho }_1\ge {\rho }_0$. In logic and practice, it’s reasonable that ${\rho }_1+{\rho }_2$ is near to 1 if the input and output in a human body are balanced, since an output organ is absolutely necessary other organs of all consumption. In case: ${\rho }_1+{\rho }_2=1$, all the energy for intervening organ can transmit to other organs which have neighboring relations or alternate relations with the intervening organ. The condition $1\ge {\rho }_1\ge {\rho }_0$ can be satisfied when ${\rho }_2={\rho }_1{\rho }_3$ and ${\rho }_3={\rho }_1$ for an organ since ${\rho }_1+{\rho }_2=1$ implies ${\rho }_1=\phi \approx 0.61803\ge {\rho }_0$. In this case, ${\rho }_2={\phi }^2\approx 0.38197$. If this assumption is set up, then the intervening principle: “Real disease with a healthy body is to rush down its son and virtual disease with a healthy body is to fill its mother” based on the Yin Yang Wu Xing theory in image mathematics by Zhang et al. [20] , is quite reasonable. But, in general, the ability of self-protection is often insufficient for a usual human body, i.e., ${\rho }_3$ is small. A common standard is ${\rho }_3=\left(1-{\rho }_1\right)/\left(2{\rho }_2\right)\approx 1/2$ which comes from the balance condition $\left(1-{\rho }_2{\rho }_3\right)=\left({\rho }_1+{\rho }_2{\rho }_3\right)$ of the loving relationship if ${\rho }_1+{\rho }_2\approx 1$. In other words, there is a principle which all losses are bear in a human body. Thus the general condition is often ${\rho }_1\approx 0.61803\ge {\rho }_3\approx 0.5\ge {\rho }_2\approx 0.38197$. Interestingly, they are all near to the golden numbers. It is the idea to consider the unhealthy number ${{\rho }^{\prime }}_0=0.68232780$ since the poor condition of self-protection ability ${\rho }_3={{\rho }^{\prime }}_0/2=0.34116390$ can make the unhealthy balance conditions hold

$\begin{array}{l}{\rho }_1-{\rho }_3={\rho }_3={{\rho }^{\prime }}_0/2=0.34116390,\\ {\rho }_2-{\rho }_1{\rho }_3={\rho }_1{\rho }_3={\left({{\rho }^{\prime }}_0\right)}^2/2=0.23278561\\ 1-{\rho }_2{\rho }_3={\rho }_1+{\rho }_2{\rho }_3\end{array}$

if ${\rho }_1={{\rho }^{\prime }}_0$ and ${\rho }_2={\left({{\rho }^{\prime }}_0\right)}^2=0.46557123$.

By Theorem 2.1 and Corollary 2.1, $x\in \left[7.35,7.45\right]$ implies $1\ge {\rho }_1=\rho \left(x\right)\ge 0.61167=\rho \left(7.35\right)=\rho \left(7.45\right)$.

And $x\in \left[7.35129,7.44871\right]$ implies

$1\ge {\rho }_1=\rho \left(x\right)\ge \phi =\rho \left(7.35129\right)=\rho \left(7.44871\right)$.

And $x\in \left[7.34539,7.45461\right]$ implies

$1\ge {\rho }_1=\rho \left(x\right)\ge {\rho }_0=\rho \left(7.34539\right)=\rho \left(7.45461\right),$

where $\lambda \left(7.34539\right)=\lambda \left(7.45461\right)=\frac{1-{\rho }_0}{2{\rho }_0}={\rho }_0^2$

since $\left(1-{\rho }_0^3\right)=\left({\rho }_0+{\rho }_0^3\right)$.

And $x\in \left[7.36307,7.43693\right]$ implies

$1\ge {\rho }_1=\rho \left(x\right)\ge {{\rho }^{\prime }}_0=\rho \left(7.36307\right)=\rho \left(7.43693\right),$

where $\lambda \left(7.36307\right)=\lambda \left(7.43693\right)=\frac{1-{{\rho }^{\prime }}_0}{2{{\rho }^{\prime }}_0}=\frac{{\left({{\rho }^{\prime }}_0\right)}^2}{2}$

since $\left(1-{{\rho }^{\prime }}_0\right)={\left({{\rho }^{\prime }}_0\right)}^3$.

The last one is the healthy interval in a person’s self-protection ability poor conditions. The interval range than the normal human body’s health requirements too strict, only the first three interval ranges are considered as a normal human body’s health. If keep two decimal places, then the first three intervals are the same as $x\in \left[7.35,7.45\right]$. This shows that range $x\in \left[7.35,7.45\right]$ is stable. The interval as the normal range of human blood pH value may be also appropriate. To conservative estimates, the interval of the largest length of the first three range intervals is used, i.e., $x\in \left[7.34539,7.45461\right]$, as the theoretical analysis of the normal range. In fact, the range $x\in \left[7.34539,7.45461\right]$ is better than the range $x\in \left[7.35,7.45\right]$. It is because the range $x\in \left[7.34539,7.45461\right]$ satisfies the healthy balance conditions ${\rho }_1={\rho }_3,{\rho }_2={\rho }_1{\rho }_3$ and $\left(1-{\rho }_2{\rho }_3\right)\le \left({\rho }_1+{\rho }_2{\rho }_3\right)$ if ${\rho }_1=\rho \left(x\right),{\rho }_2=\rho {\left(x\right)}^2$ and ${\rho }_3=c\rho \left(x\right)\to {\rho }_1$. In other words, the parameter ${\rho }_1=\rho \left(x\right)\ge {\rho }_0$ or the range $x\in \left[7.34539,7.45461\right]$ is the healthy running condition of both the killing relationship and the loving relation at the same time. But neither are the others. The human blood pH value must be precise calculation to keep at least 5 decimal places can ensure correct because of its sensitivity to the diagnosis of disease.#

Remark 2. Western Medicine is different from TCM because the TCM has a concept of Chi or Qi (气) as a form of energy. From the energy concept, that one organ or subsystem of the human body is not running properly (or disease, abnormal), is that the energy deviation from the average of the organ is too large, the high (real disease) or the low (virtual disease). But there do not exist these concepts of both real diseases and virtual diseases in Western medicine. For the normal range of blood pH value of some human body as $x\in \left[7.34539,7.45461\right]$, in TCM, if $x>7.45461$, the body is considered as a real disease since the human blood pH value is too high; if $x<7.34539$, the body is considered as a virtual disease since the human blood pH value is too low. Thus TCM identifies an important indicator for a human body’s health: the human blood pH value, which, under normal conditions, ranges from 7.34539 to 7.45461. Outside this range (acid: Yin condition; alkaline: Yang condition), disease appears. Almost always absolutely, when there is virtual disease, the condition of human blood pH value is a Yin condition; when there is real disease, the condition of human blood pH value is a Yang condition.#

Remark 3. Obviously, when applying the hypothesis of Theorem 2.1 and Corollary 2.1 to other fields than a human body’s health, it is necessary to identify a global parameter in each field that is able to yield a general Yin or Yang condition in relation to the average behavior of the studied phenomenon, and that maintains the equations at a sufficiently simple level of writing and application. In fact, let $x\in \left(\min ,\max \right)$ where the values min and max are the minimum and maximum acceptable the index x. Denoted the value $t_0$ is the target as the expectation of the index x such that $\rho \left(t_0\right)=1$. In Equations (1) and (2), replace $\text{7},\text{7}.\text{8},\text{7}.\text{4}$ by $\min ,\max ,t_0$, respectively. The equivalent condition of a healthy body ${\rho }_0\le {\rho }_1=\rho \left(x\right)\le 1$ is considered as $x\in \left[u,v\right],\min <u<t_0<v<\max$, where

$\rho \left(u\right)=\rho \left(v\right)={\rho }_0\le {\rho }_1=\rho \left(x\right)=\left(1/2\right)/\left[\lambda \left(x\right)+\left(1/2\right)\right]\le 1=\rho \left(t_0\right),$

$\begin{array}{l}\lambda \left(t_0\right)=0\le \lambda \left(x\right)=\left(1-\rho \left(x\right)\right)/\left(2\rho \left(x\right)\right)\le {\rho }_0^2=\lambda \left(u\right)=\lambda \left(v\right)\\ =\rho {\left(u\right)}^2=\rho {\left(v\right)}^2\le {\rho }_2=\rho {\left(x\right)}^2\le {\rho }_1=\rho \left(x\right)\le 1.\#\end{array}$

3. Relationship of the Steady Multilateral Systems

3.1. Energy Changes of a Steady Multilateral System

In order to apply the reasoning to other fields rather than a human body’s health, Zhang [17] has started a steady multilateral system imitating a human body. A most basic steady multilateral system is as follows.

Theorem 3.1 [20] For each element x in a steady multilateral system V with two incompatibility relations, there exist five equivalence classes below:

$X=\left\{y\in V|y~x\right\},X_S=\left\{y\in V|x\to y\right\},X_K=\left\{y\in V|x\Rightarrow y\right\},$

$K_X=\left\{y\in V|y\Rightarrow x\right\},S_X=\left\{y\in V|y\to x\right\},$

which the five equivalence classes have relations in Figure 1.#

The Yin Yang Wu Xing model can be written as follows. Define

$V_0^5=X,V_1^5=X_S,V_2^5=X_K,V_3^5=K_X,V_4^5=S_X$

corresponding to wood, fire, earth, metal, water, respectively, and assume $V^5=V_0^5+V_1^5+V_2^5+V_3^5+V_4^5$ where $V_i^5\cap V_j^5=\varnothing ,\forall i\ne j$ (hereinafter the same).

And take ${\Re }^5=\left\{R_0^5,R_1^5,R_2^5,R_3^5,R_4^5\right\}$ satisfying

$R_r^5={\displaystyle \underset{i=0}{\overset{4}{\sum }}{V}_i^5}\times V_{\mathrm{mod}\left(i+r,5\right)}^5,\forall r\in \left\{0,1,2,3,4\right\},R_i^5*R_j^5=R_{\mathrm{mod}\left(i+r,5\right)}^5,$

where $V_i^5\times V_j^5=\left\{\left(x,y\right):x\in V_i^5,y\in V_j^5\right\}$ is the Descartes product in set theory and

$R_i^5*R_j^5=\left\{\left(x,y\right):\exists u\in V\text{suchthat}\left(x,u\right)\in R_i^5,\left(u,y\right)\in R_j^5\right\}$ is the multiplication relation operation. The relation multiplication of $*$ is isomorphic to the addition of module 5. Then $\left(V^5,{\Re }^5\right)$ is a steady multilateral system with one equivalent relation $R_0^5$ and two incompatibility relations $R_1^5={\left(R_4^5\right)}^{-1}$ and $R_2^5={\left(R_3^5\right)}^{-1}$ where ${\left(R_i^5\right)}^{-1}=\left\{\left(x,y\right):\left(y,x\right)\in R_i^5\right\}$ is the inverse relation operation. The Yin and Yang mean the two incompatibility relations and the Wu Xing means the collection of five disjoint classifications of $V^5=V_0^5+V_1^5+V_2^5+V_3^5+V_4^5$. The model is called Yin Yang Wu Xing model, denoted simply by $V^5=\left\{0,1,2,3,4\right\}$.

It can be proved by Theorem 3.2 in Zhang [23] that the steady multilateral system in Theorem 3.1 is the reasoning model of Yin Yang Wu Xing in TCM if there is an energy function $\phi (*)$ satisfying

$\begin{array}{l}\frac{\Delta \phi \left(X\right)}{\Delta }\to \frac{\text{d}\phi \left(X\right)}{\text{d}X}=\left(1-{\rho }_2{\rho }_3\right)=\left(1-c\rho {\left(x\right)}^2\right)>0;\\ \frac{\Delta \phi \left(X_S\right)}{\Delta }\to \frac{\text{d}\phi \left(X_S\right)}{\text{d}X}=({\rho }_1+{\rho }_2{\rho }_3)=\rho \left(x\right)\left(1+c\rho {\left(x\right)}^2\right)>0;\\ \frac{\Delta \phi \left(X_K\right)}{\Delta }\to \frac{\text{d}\phi \left(X_K\right)}{\text{d}X}=-\left({\rho }_1-{\rho }_3\right)=-\rho \left(x\right)\left(1-c\right)<0;\end{array}$

$\begin{array}{l}\frac{\Delta \phi \left(K_X\right)}{\Delta }\to \frac{\text{d}\phi \left(K_X\right)}{\text{d}X}=-\left({\rho }_2-{\rho }_1{\rho }_3\right)=-\rho {\left(x\right)}^2\left(1-c\right)<0;\\ \frac{\Delta \phi \left(S_X\right)}{\Delta }\to \frac{\text{d}\phi \left(S_X\right)}{\text{d}X}=\left({\rho }_2-{\rho }_1{\rho }_3\right)=\rho {\left(x\right)}^2\left(1-c\right)>0,\\ \text{if}\text{incraese}\text{the}\text{energy}\text{of}X\left(\forall \Delta \phi \left(X\right)=\Delta >0\right).\end{array}$

The parameter ${\rho }_v={\rho }_1+{\rho }_2{\rho }_3$ is called the coefficient of the vital or righteousness energy. The parameter ${\rho }_e=1-{\rho }_2{\rho }_3$ is called the coefficient of the evil energy. A human body is called healthy if the vital or righteousness coefficient ${\rho }_v={\rho }_1+{\rho }_2{\rho }_3$ is greater than or equal to the evil coefficient ${\rho }_e=1-{\rho }_2{\rho }_3$. Otherwise, the human body is called unhealthy. For a healthy human body, the transfer law of the Yang vital or righteousness energy in the Yin Yang Wu Xing Model is

$\begin{array}{l}\text{Wood}\left(X\right)\to \text{Fire}\left(X_S\right)\to \text{Earth}\left(X_K\right)\to \text{Metal}\left(K_X\right)\\ \to \text{Water}\left(S_X\right)\to \text{Wood}\left(X\right).\end{array}$

Figure 1 in Theorem 3.1 is the figure of Yin Yang Wu Xing theory in Ancient China. The steady multilateral system V with two incompatibility relations is equivalent to the logic architecture of reasoning model of Yin Yang Wu Xing theory in Ancient China. What describes the general method of human bodies can be used in the human body.

By non-authigenic logic of TCM, i.e., a logic which is similar to a group has nothing to do with the research object [20] , in order to ensure the reproducibility such that the analysis conclusion can be applicable to any a human body, a logical analysis model can be chosen which has nothing to do with the object of study. The Tao model of Yin and Yang is a generalized one which means that two is basic. But the Tao model of Yin Yang is simple in which there is not incompatibility relation. The analysis conclusion of Tao model of Yin Yang cannot be applied to an incompatibility relation model. Thus the Yin Yang Wu Xing model with two incompatibility relations of Theorem 3.1 will be selected as the logic analysis model in this paper.

On the other hand, the steady multilateral system $\left(V^2,{\Re }^2\right)=\left(V_0^2+V_1^2,\left\{R_0^2,R_1^2\right\}\right)$ is called the Tao model, denoted simply by $V^2=\left\{0,1\right\}$, if it satisfies the following conditions:

$\begin{array}{l}{R}_r^2={\displaystyle \underset{i=0}{\overset{1}{\sum }}{V}_i^2}\times V_{\mathrm{mod}\left(i+r,5\right)}^2,r\in \left\{0,1\right\},R_i^2*R_j^2=R_{\mathrm{mod}\left(i+r,2\right)}^2,\\ R_0^2=\left\{\left(0,0\right),\left(1,1\right)\right\},R_1^2=\left\{\left(0,1\right),\left(1,0\right)\right\}.\end{array}$

The relation multiplication of $*$ is isomorphic to the addition of module 2. The element 1, or 0 is called a Yang force or a Yin force respectively. For a healthy human body, the transfer law of the Tao force in the Tao model is from Yang to Yin.

In TCM, any material can be found, not Yang is Yin. No matter of Yin and Yang are unable to see, known as dark matter, or nonphysical. Therefore, the Tao force often exists in the physical world. Any steady multilateral system only force under the action of the Tao, may be to perceive.

Furthermore, the steady multilateral system

$\left(V^6,{\Re }^6\right)=\left(V_1^6+\cdots +V_6^6,\left\{R_1^6,\cdots ,R_6^6\right\}\right)$

is called the Telluric effluvium model, denoted simply by $V^6=\left\{e,\left(12\right),\left(13\right),\left(23\right),\left(123\right),\left(132\right)\right\}$, if it satisfies the following conditions:

$\begin{array}{l}{R}_r^6={\displaystyle \underset{i=1}{\overset{6}{\sum }}{V}_i^6}\times V_{i*r}^6,\forall r\in \left\{1,2,\cdots ,6\right\},\\ R_i^2*R_j^2=R_{i*r}^2,\end{array}$

$\begin{array}{ccccccc}i*r& 1=e& 2=(12)& 3=(13)& 4=(23)& 5=(123)& 6=(132)\\ 1=e& 1=e& 2=(12)& 3=(13)& 4=(23)& 5=(123)& 6=(132)\\ 2=(12)& 2=(12)& 1=e& 5=(123)& 6=(132)& 3=(13)& 4=(23)\\ 3=(13)& 3=(13)& 6=(132)& 1=e& 5=(123)& 4=(23)& 2=(12)\\ 4=(23)& 4=(23)& 5=(123)& 6=(132)& 1=e& 2=(12)& 3=(13)\\ 5=(123)& 5=(123)& 4=(23)& 2=(12)& 3=(13)& 6=(132)& 1=e\\ 6=(132)& 6=(132)& 3=(13)& 4=(23)& 2=(12)& 1=e& 5=(123)\end{array}$

The number 1, or 2, or 3 is called the tengen (天元), the earth material (地元), the people ability (人元), respectively. The set of $\left\{1,2,3\right\}$ is called three types of talent or material. It is with elements, $e,\left(12\right),\left(13\right),\left(23\right),\left(123\right),\left(132\right)$. The each of elements, $e,\left(12\right),\left(13\right),\left(23\right),\left(123\right),\left(132\right)$, is called the primordial energy (元气), essence derived from food (谷气), defensive energy (卫气), essential substance circulating in the channels and blood vessels (营气), genuine energy (真气), pectoral energy (宗气), respectively. Another name is respectively

$\begin{array}{l}\text{shaoyang(e)(}少阳\text{),yangming((12))(}阳明\text{),taiyang((13))(}太阳\text{),}\\ \text{jueyin((23))(}厥阴\text{),shaoyin((123))(}少阴\text{),taiyin((132))(}太阴\text{)}\text{.}\end{array}$

Generally positive or Yang material, they are able to be perceived, but few can see the material itself, can only use signs. Therefore, the Yang energy symptoms of the set $M_1=\left\{e,\left(12\right),\left(13\right)\right\}$ calls the marrow energy (髓); The Yin energy of the set $M_2=\left\{\left(123\right)\right\}$ calls the blood energy (血); The Yin energy of the set $M_3=\left\{\left(132\right)\right\}$ calls the saliva energy (津); The Yin energy of the set $M_4=\left\{\left(23\right)\right\}$ calls the essence of water and grain (水谷精微).

Growth and conveyance in the six energies $e,\left(12\right),\left(13\right),\left(23\right),\left(123\right),\left(132\right)$ known as the six roots (根); As the fruit of these six energies $e,\left(12\right),\left(13\right),\left(23\right),\left(123\right),\left(132\right)$ known as the six fruits (结); Storage of these four energies $M_1,M_2,M_3,M_4$ known as the four seas(四海); Energy exchange of the four kinds of $M_1,M_2,M_3,M_4$ known as the four streets(四街). Of course, for a healthy human body, the transfer law of each of the six energies $e,\left(12\right),\left(13\right),\left(23\right),\left(123\right),\left(132\right)$ is from its root (root-causes) (根) to its fruit (symptoms) (结).

Western Medicine is different from TCM because the TCM has a concept of Chi or Qi (气) as a form of energy of steady multilateral systems. It is believed that this energy exists in all things of steady multilateral systems (living and non-living) including air, water, food and sunlight. Chi is said to be the unseen vital force that nourishes the human body and sustains the human body’s life. It is also believed that an individual is born with an original amount of Chi at the beginning of human body’s life and as a human body grows and lives, the human body acquires or attains Chi or energy from “eating” and “drinking”, from “breathing” the surrounding “air” and also from living in its environment. The steady multilateral system having an energy is called the anatomy system or the first physiological system. And the first physiological system also affords Chi or energy for the steady multilateral system’s meridian system (Zang Xiang (藏象) and Jing-Luo (经络)) which forms a parasitic system of the steady multilateral system, called the second physiological system of the steady multilateral system. The second physiological system of the steady multilateral system controls the first physiological system of the steady multilateral system. A steady multilateral system would become ill or dies if the Chi or energy in the steady multilateral system is imbalanced or exhausted, which means that ${\rho }_1=\rho \left(x\right)\to 0,{\rho }_2=\rho {\left(x\right)}^2\to 0$ and ${\rho }_3=c\rho \left(x\right)\to 0$.

For example, in TCM, a human body as the first physiological system following the Yin Yang Wu Xing theory was classified into five equivalence classes as follows:

wood (X) = {liver, bravery, soul, ribs, sour, east, spring, birth};

xiang-fire ( $X_S^x$ ) = {pericardium, the triple energizer, nerve, the blood, bitter taste, the south, summer, growth};

earth( $X_K$ ) = {spleen, stomach, willing, meat, sweetness, center, long summer, combined};

metal ( $K_X$ ) = {lung, large intestine, boldness, fur, spicy, west, autumn, accept};

water ( $S_X$ ) = {kidney, bladder, ambition, bone, salty, the north, winter, hiding};

jun-fire ( $X_S^j$ ) = {heart, small intestine, nerve, the blood, bitter taste, whole body, whole direction, throughout the year, overall growth}.

Fire ( $X_S$ ) = xiang-fire ( $X_S^x$ ) $\cup$ jun-fire( $X_S^j$ ).

There is only one of both loving and killing relations between every two classes. General close is loving, alternate is killing.

In every category of internal, think that they are with an equivalent relationship, between each two of their elements there is a force of similar material accumulation of each other. It is because their pursuit of the goal is the same, i.e., follows the same “Axiom system”. It can increase the energy of the class at low cost near to zero if they accumulate together. Any nature material activity follows the principle of maximizing so energy or minimizing so cost. In general, the size of the force of similar material accumulation of each other is smaller than the size of the loving force or the killing force in a stable human body. The stability of any human body first needs to maintain the equilibrium of the killing force and the loving force. The key is the killing force. For a stable human body, if the killing force is large, i.e., ${\rho }_3=c\rho \left(x\right)$ becomes larger by Theorems 3.4 and 3.5 below, which needs positive exercise, then the loving force is also large such that the force of similar material accumulation of each other is also large. They can make the human body more stable. If the killing force is small, i.e., ${\rho }_3=c\rho \left(x\right)$ becomes smaller by Theorems 3.4 and 3.5 below, which means little exercise, then the loving force is also small such that the force of similar material accumulation of each other is also small. They can make the human body becoming unstable. The Chi or energy is also called the food hereafter for simply. In order to get the food, by Attaining Rule in Zhang [23] , the second physiological system must make the first physiological system intervened, namely exercise. It is because only by intervention on the first physiological system, the second physiological system can be to get food.

The second physiological system of the steady multilateral system controls the first physiological system of the steady multilateral system, abiding by the following rules.

Definition 3.1 (Zangxiang (藏象)) Assume the Yin Yang Wu Xing model $V^5$ is implemented by the Tao force of the Tao model $V^2$. Then the steady multilateral system $V^2\times V^5=\left\{\left(i,j\right)|i\in V^2,j\in V^5\right\}$ is called the Zangxiang or the Five Zang-organs model of the steady multilateral system.

The Zangxiang model satisfies as follows:

$\begin{array}{l}{R}_{\left(r,r^{\prime }\right)}^{\left(2,5\right)}={\displaystyle \underset{\left(i,i^{\prime }\right)=\left(0,0\right)}{\overset{\left(1,4\right)}{\sum }}{V}_{\left(i,i^{\prime }\right)}^{\left(2,5\right)}}\times V_{\left(\mathrm{mod}\left(i+r,2\right),\mathrm{mod}\left(i^{\prime }+r^{\prime },5\right)\right)}^{\left(2,5\right)},\forall \left(r,r^{\prime }\right)\in V^2\times V^5,\\ R_{\left(i,i^{\prime }\right)}^{\left(2,5\right)}*R_{\left(j,j^{\prime }\right)}^{\left(2,5\right)}=R_{\left(\mathrm{mod}\left(i+j,2\right),\mathrm{mod}\left(i^{\prime }+j^{\prime },5\right)\right)}^{\left(2,5\right)}.\end{array}$

Each of the elements,

$\begin{array}{l}(1,0),(0,0),(1,1),(0,1),(1,2),\\ (0,2),(1,3),(0,3),(1,4),(0,4),\end{array}$

is called respectively

$\begin{array}{l}Jia(1,0),Yi(0,0),Bing(1,1),Ding(0,1),Wu(1,2),\\ Ji(0,2),Geng(1,3),Xin(0,3),\mathrm{Re}n(1,4),Gui(0,4),\end{array}$

corresponding to the Chinese words respectively:

$\begin{array}{l}甲(1,0),乙(0,0),丙(1,1),丁(0,1),戊(1,2),\\ 己(0,2),庚(1,3),辛(0,3),壬(1,4),癸(0,4),\end{array}$

also corresponding to the notations in Theorem 3.1 respectively:

$\begin{array}{l}{X}^{+}(1,0),X^{-}(0,0),X_S^{+}(1,1),X_S^{-}(0,1),X_K^{+}(1,2),\\ X_K^{-}(0,2),K_X^{+}(1,3),K_X^{-}(0,3),S_X^{+}(1,4),S_X^{-}(0,4).\end{array}$

Here, each of the elements: $X,X_S,X_K,K_X,S_X$ is called wood, fire, earth, metal, water, respectively, and this is ${*}^{+}$ Yang, ${*}^{-}$ is Yin.

In TCM, another name of the Zangxiang system is the ten Heavenly Stems (天干). It means in the changes of energy transmission, plays a major role, is under the power of positive (Yang), although it is invisible, the reason of behind the sign.

In the ten heavenly stems, Yang is respectively:

$\begin{array}{l}\text{Jia}{X}^{+}\text{(1,0)likingYangwood,Bing}{X}_S^{+}\text{(1,1)likingYnagfire},\\ \text{Wu}{X}_K^{+}\text{(1,2)likingYangearth,Geng}{K}_X^{+}\text{(1,3)likingYangmetal,}\\ \text{Ren}{S}_X^{+}\text{(1,4)likingYangwater;}\end{array}$

Yin is respectively:

$\begin{array}{l}\text{Yi}{X}^{-}\text{(0,0)likingYinwood}，\text{Ding}{X}_S^{-}\text{(0,1)likingYinfire}，\\ \text{Ji}{X}_K^{-}\text{(0,2)likingYinearth}，\text{Xin}{K}_X^{-}\text{(0,3)likingYinmetal}，\\ \text{Gui}{S}_X^{-}\text{(0,4)likingYinwater}\text{.}\end{array}$

Yin is in the inside (里), Yang is in the outside (表). The relationship between the inside and the outside is the liking relationship.

The laws of the Zangxiang system or the ten Heavenly Stems model are summarized in Figure 2.#

Definition 3.2 (Jingluo) Assume the Telluric effluvium model $V^6$ is implemented by the Tao force of the Tao model $V^2$. Then the steady multilateral system $V^2\times V^6=\left\{\left(i,j\right)|i\in V^2,j\in V^6\right\}$ is called the Jingluo or the Six Fu-organs model of the steady multilateral system.

The Jingluo model satisfies as follows:

$\begin{array}{l}{R}_{\left(r,r^{\prime }\right)}^{\left(2,6\right)}={\displaystyle \underset{\left(i,i^{\prime }\right)=\left(0,1\right)}{\overset{\left(1,6\right)}{\sum }}{V}_{\left(i,i^{\prime }\right)}^{\left(2,6\right)}}\times V_{\left(\mathrm{mod}\left(i+r,2\right),i^{\prime }*r^{\prime }\right)}^{\left(2,6\right)},\forall \left(r,r^{\prime }\right)\in V^2\times V^6,\\ R_{\left(i,i^{\prime }\right)}^{\left(2,6\right)}*R_{\left(j,j^{\prime }\right)}^{\left(2,6\right)}=R_{\left(\mathrm{mod}\left(i+j,2\right),i^{\prime }*j^{\prime }\right)}^{\left(2,6\right)}.\end{array}$

Each of the elements,

$\begin{array}{cccccc}(0,e)& (1,e)& (0,(12))& (1,(12))& (0,(13))& (1,(13))\\ (0,(23))& (1,(23))& (0,(132))& (1,(132))& (0,(123))& (1,(123))\end{array}$

is called respectively

$\begin{array}{cccccc}\text{Zi(0,e)}& \text{Hai(1,e)}& \text{Chen(0,(12))}& \text{Mao(1,(12))}& \text{Shen(0,(13))}& \text{Wei(1,(13))}\\ \text{Chou(0,(23))}& \text{Xu(1,(23))}& \text{Si(0,(132))}& \text{Yin(1,(132))}& \text{You(0,(123))}& \text{Wu(1,(123));}\end{array}$

Corresponding to the Chinese words respectively:

$\begin{array}{cccccc}子\text{(0,e)}& 亥\text{(1,e)}& 辰\text{(0,(12))}& 卯\text{(1,(12))}& 申\text{(0,(13))}& 未\text{(1,(13))}\\ 丑\text{(0,(23))}& 戌\text{(1,(23))}& 巳\text{(0,(132))}& 寅\text{(1,(132))}& 酉\text{(0,(123))}& 午\text{(1,(123));}\end{array}$

Also corresponding to the notations in Theorem 3.1 as the five viscera and six entrails respectively:

$\begin{array}{cccccc}{X}^{+}(0,e)& X_S^{x+}(1,e)& X_K^{+}(0,(12))& K_X^{+}(1,(12))& S_X^{+}(0,(13))& X_S^{j+}(1,(13))\\ X^{-}(0,(23))& X_S^{x-}(1,(23))& X_K^{-}(0,(132))& K_X^{-}(1,(132))& S_X^{-}(0,(123))& X_S^{j-}(1,(123)).\end{array}$

Here, each of the elements: $X,X_S^x,X_K,K_X,S_X,X_S^j$ is called wood, xiang-fire (相火), earth, metal, water, jun-fire (君火) respectively, and this is ${*}^{+}$ Yang, ${*}^{-}$ is Yin.