c. Mathematical Model and Implications of the 1st Law

c. Mathematical Model and Implications of the 1st Law

In this mathematical model, the definition of political capacity  implies that political phenomena are explained based on Samjae capacities^[Ch.3.101]. The persuasiveness of this position will be recognized when the scope of the mathematical model to explain political phenomena is broad.

The formula^[Fmla.3.1.2] that the strength(\( F_{RC} \) ) of each political force and the scope(\( R_{NG} \)) of its effect are inversely proportional can be combined with the inequality that expresses the strength of each Samjae capacity to infer the spatiotemporal range^[Ch.3.103] in which each Samjae capacity acts. As mentioned earlier, armed capacity is powerful in a small area but only acts temporarily, economic capacity is visible in a moderate range and acts gradually, and ideological capacity is weak in a wide area but acts very persistently.

Furthermore, the order of interaction among Samjae capacities^[Ch.3.105] can be derived from the strength of each Samjae capacity and [Fmla.3.1.3] or [Fmla.3.1.11]. For example, in the case of [Fmla.3.1.11], which determines the size of a political actor's economic capacity over time (\( E(t) \) ), the size of armed capacity ( \( A \)) acts as an exponent in [Fmla.3.1.11]. Therefore, the larger the armed capacity, the more geometrically the political actor's economic capacity increases over time. The same calculation applies to economic capacity and value capacity.

If there are three political actors [A], [B], and [C], and the value of [A]'s armed capacity is 1, while [B] and [C]'s armed capacity are 1.2 and 1.5, respectively, then the change in the size of the economic capacity of the three political actors over time(t) is shown in the following graph.

 

[Diag.3.A.9] Change in economic capacity according to difference in Armed capacity

In this graph, until the time point , the economic capacity of each political actor does not show a significant difference. This indicates that the size of the economic capacity up to this point depends more heavily on the initial value of economic capacity ( \( E_{0} \)). Each political actor's economic capacity is initially more influenced by their own capital or land, among other things. However, over time, it is increasingly influenced by the size of armed capacity.

From the late 14th to the 16th century, the Ottoman Empire provides an example of the 'timar' system. The Ottoman Empire distributed land called 'timar,' which included the right to collect taxes, to its soldiers. The greater the political power of an individual, the larger the timar and tax collection rights they held. In other words, if their armed capacity was significant as a warrior at that time, this would correspond to the case where the value of M in [Diag.3.A.9] is large, and accordingly, the value of \( E(t) \) also increased. Another example would be the 'Danegeld' received by the Vikings who invaded England at the end of the 10th century in exchange for peace. The amount of Danegeld increased from 22,000 pounds in 991 to 16,000 pounds in 994, 24,000 pounds in 1002, 30,000 pounds in 1007, and 48,000 pounds in 1012.

On the other hand, if the initial value  \( E_{0} \)  of [B] is twice the initial value  \( E_{0} \)   of [다], the above [Diag.3.A.9] changes as follows.

[Diag.3.A.10] Change in economic capacity according to difference in Armed capacity 2

As seen in the Diagram, until time  \( t_{2} \)  , the economic capacity of [나], who has a larger initial value, would be greater, but after that, the economic capacity of [다], who has a larger armed capacity, becomes larger. This fact demonstrates that the difference in armed capacity strongly influences the changes in economic capacity, which is also evidenced by the fact that those who currently control power also control enormous wealth. For example, since the Han Dynasty (B.C. 202- A.D. 220), Chinese bureaucrats monopolized the distribution of iron and salt, raising national revenue easily by selling them at arbitrarily high prices. Therefore, it is an inevitable result for Putin and Xi Jinping, who wield powerful power, to accumulate enormous wealth.

On the other hand, if the value of armed capacity ( \( A \) ) becomes 0, the sizes of economic capacity and value capacity both become 1, and they remain unchanged over time. Meanwhile, when the value of each capacity reaches 1, it means that the initial values of  \( E_{0} \)   and  \( | \vec{I_{0}} | \)   have both disappeared, and as a result, all power of each actor is taken away, leaving only the physical body of the actor.

To summarize the above content, when considering the continuity of survival, failure in a strong military competition would result in the end of survival. Therefore, armed capacity has a strong political impact. When the armed capacity reaches a certain balance, economic capacity is then affected on that balance, and then ideological capacity may increase or decrease on the economic capacity balance.