New Method for Solving the Collatz Conjecture 7 with High-Prize Money
Chapter 6: Propositions on CS-Space Equivalent to the Collatz Conjecture (article 7)
Please read "Introduction to a New Method in Chapter 1 (article 1)"
before reading Chapter 2 (article 2).
Chapter 1 also posts links to each chapter.
When translating from Japanese to English, subtle nuances of the original articles may not be fully conveyed. If you would like to view the original Japanese articles, please see Chapter 1 of Japanese Article 1.
Summary Video of the New Method in English
6.1 Proposition on CS-Space Equivalent to the Collatz Conjecture (Weak Proposition)
Chapter 2 through Chapter 5 provide all the necessary foundation for understanding the proposition on the CS-space equivalent to the Collatz conjecture. This chapter mainly explains the figures, graphs, and formulas presented so far, adding additional graphs as needed for explanation, and finally introduces the equivalent proposition (weak proposition).
The Lorenz plot introduced in Chapter 5 is a technique for finding patterns in seemingly random signals. It is quite powerful.
6.1.1 Comparison of two types of Lorenz plots
Figure 6-1 shows the step diagram (top) and the progress graph (bottom) displaying only the odd numbers until the initial value of 27 reaches 1, as described in Chapter 2. It consists of 42 points.
The graph in Figure 6-1 does not show any obvious pattern. However, let's try applying the Lorenz plot to it. The result is shown in the left figure of Figure 6-2. There are 41 plot points, which are given by the sequence:
$${(27,41), (41,31), (31,47), \cdots ,(35,53), (53,5), (5,1)}$$
Indeed, some patterns are now visible. There are two or three plot points that seem to be approximable by a straight line. The right figure shows the Lorenz plot of all plot points divided by the maximum value of 3077 to fit between 0 and 1. It is, of course, similar to the left figure.
Indeed, some patterns are observed when the Lorenz plot is applied without using the C-transformation. The Lorenz plot is a simple yet powerful tool. It might be interesting to create Lorenz plots for many more initial values, normalized by their respective maximum values, and combine them into a graph like the one on the right. The left figure should also be created simultaneously.
Now, let's get started on to the main topic.
Figure 6-3 shows the CS-vibration diagram, which is a graph using
the C-transformed values, $${c(No)}$$ of the 42 odd numbers $${No}$$ in the upper figure of Figure 6-1. Compared to the lower figure of Figure 6-1 before the Lorenz plot, it is obvious that there is much more periodicity and regularity, so it is natural to say that the Lorenz plot after C-transformation will show more regularity than the right figure of Figure 6-2.
The No values are transformed into values between 0 and 1 by the
C-transformation, but the maximum values of the intermediate steps are not used. In other words, the No values are intrinsically transformed to a range between 0 and 1 by the C-transformation.
The Lorenz plot after C-transformation is shown in Figure 6-4. A graph including the CS-line has already been introduced in several previous chapters. Specifically, it is a graph that plots 41 values:
$${(c(27),c(41))}$$,$${((c(41),c(31))}$$, $${\cdots}$$, $${(c(53),c(5))}$$, $${(c(5),c(1))}$$
6.1.2 How to read a CS-plot cobweb diagram
Figure 6-5 shows the cobweb graph with the CS-lines, $${ f_{c}(x) , f_{s}(x)}$$ (solid lines), added to Figure 6-4.
$$
\begin{array}{}
&(6.1)& & f_{c}(x)=\cfrac{3}{2}x+\cfrac{1}{2}& &(0\le x <1/3)& \\\\\
&(6.2)& & f_{s}(x)=\cfrac{3}{4}x-\cfrac{1}{4}& &(1/3\le x <1)
\end{array}
$$
The cobweb graph with only the first and last four points is shown in Figure 6-6 because the original graph with too many horizontal and vertical grid lines makes it difficult to see the step-by-step progression.
and the Last Four Points (Right) of Figure 6-5
The z-values of Figure 6-6 are as follows:
$${z_0=(c(27),c(41))}$$
$${z_1=(c(41),c(31))}$$
$${z_2=(c(31),c(47))}$$
$${z_3=(c(47),c(71))}$$
$${z_{38}=(c(23),c(35))}$$
$${z_{39}=(c(35),c(53))}$$
$${z_{40}=(c(53),c(5))}$$
$${z_{41}=(c(5),c(1))}$$
In Chapter 3, a sequence of odd numbers that reaches an odd number $${m_o}$$ in one step is defined as $${cs(m_o)}$$.
$${cs(1)}$$, which reaches $${1}$$ in one step, is $${cs(1)=\{1,\ 5, \ 21, \ 85, \cdots\}}$$ and has an infinite number of elements.
Also, $${c(1)=0}$$. Additionally, there is no other odd number $${No}$$ such that $${c(No)=0}$$ besides $${1}$$.
Therefore, in the CS (Lorenz) plot, the final point is always on the x-axis.
All other points are above the x-axis. The number $${1}$$ is the only periodic number with a period of 1 that reaches 1 in one step. It has only one final point,$${(c(1),c(1))=(0,0)}$$.
Furthermore, once $${1}$$ is reached, The sequence continues $${1 \to 1\to 1 \to \cdots}$$ in the general Collatz space. Therefore, all final points become the origin $${\bm{(0,0)}}$$ (ultimate point) in the extended space where all points after reaching 1 are mapped to 1.
Here, the odd number elements in $${\bm{cs(1)}}$$ are denoted specifically by $${\bm{O}}$$.
In this section, $${O}$$ is used to define a weak proposition.
6.1.3 Proposition on CS-Space Equivalent to the Collatz Conjecture (Weak Proposition)
Based on the above, the Collatz conjecture on the general Collatz space that states "For any odd number, triple it and add 1. For any even number, divide it by 2. Repeat this process and any positive integer will eventually reach 1 in a finite number of steps" is equivalent to the following proposition on the CS space (weak proposition) :
＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊
Under the C-transformations, $${\bm{c(O)}}$$ and $${\bm{c(No)}}$$ for positive odd integers, $${\bm{O}}$$ and $${\bm{No}}$$,
any $${\bm{c(No)}}$$ in the CS-space reaches a point on the x-axis, $${\bm{(c(O),0)}}$$, in a finite number of steps, and the ultimate point is the origin $${(0,0)}$$.
＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊＊
The CS-plot points are not always on the CS-line (slightly deviated and close to it). (With strong propositions, they do lie on the CS line!)
In the case of even numbers, it is not necessary to consider them in the general Collatz space because they always reach an odd number.
However, in the CS space, even numbers can be inevitably ignored for the following reasons, as explained in Section 2.3 of Chapter 2.
For example, in the sequence $${13 \to 40 \to 20 \to 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1}$$, $${c(40)=c(20)=c(10)=c(5)}$$ and $${c(16)=c(8)=c(4)=c(2)=c(1)}$$ are inevitably true.
If you find the content of the article or the proof to be "I see", please consider evaluating it. This will encourage further analysis.
Variables, Definitions, and Functions up to Section 6.1 Chapter 6:
$${N}$$, $${No}$$,
total number of steps, step number, CS-vibration, C-transformation,
CS-plot, CS-linear-equations, 3C+1 transformation, 3S+1 transformation,
(general) Collatz space, CS-line, S-transformation, CS-space, t-tree,
Lorenz plot, period orbit, periodic point, eventually periodic point,
cobweb diagram, fixed point, final point, ultimate point,
type $${6t+5}$$, $${6t+1}$$, $${4t+3}$$, $${8t+1}$$, etc.
$${c(No)}$$, $${z_{s}}$$, $${f_{c}(x)}$$, $${f_{s}(x)}$$, distance$${D}$$,
$${Nc}$$, $${Ns}$$, $${m_o}$$, $${m_{c}}$$, $${m_{s}}$$, $${t_{c}}$$, $${t_{s}}$$, $${t}$$, $${cs(m_o)}$$, $${Mo}$$, $${s(Mo)}$$, $${O}$$,
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