Proof of the Collatz hypothesis

I will continue the topic begun by the previous post with the proof of the Collatz hypothesis .

What is this hypothesis? Take any natural number $$$n$. If it is even, then divide it by $$$2$, and if it is odd, then multiply by $$$3$and add $$$1$(we get $$$3n + 1$). On the resulting number, perform the same actions, and so on. Collatz's hypothesis is that whatever initial number $$$n$we neither took, sooner or later we will get a unit.

Proof of the Collatz hypothesis

We reformulate the hypothesis as follows: take any natural number $$$n$. If it is even, then divide it by $$$2$until it loses the parity property, and then we transfer it to the base number system $$$\ dfrac {1} {3}$and add $$$1$until the base of the number system becomes the opposite $$$n$. Collatz’s hypothesis in such a formulation is that whatever initial number $$$n$we did not take it, sooner or later it will happen, that is, the equation

$$

$$\ dfrac {1} {pn} = \ dfrac {1} {3 ^ m + m}$$

Where $$$m$- the number of odd steps, $$$p$- some rational number with unknown properties, has a solution for any natural $$$n$which is obvious so.

Proven .
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But nothing is clear, especially why I reformulated the hypothesis in this way.

Some basic concepts of the theory of entropy

A system is an indefinable concept on the basis of which unprovable definitions are built:
Entropy - the information capacity of the system.
Extropy is the opposite of entropy.
Phase transition - a decrease in entropy by one order of magnitude, occurring with the accumulation of sufficient and necessary amount of knowledge.

Evolution of small n

Consider a specific example: take from the formulation of the hypothesis n and turn it into $$$n$. How to achieve this? Reducing ignorance, that is, asking questions that are possible only a definite answer "yes" or "no." Go:
1. n - the plane? Not. This answer reduced our ignorance about n to the knowledge of “n is not a plane”, but did not tell us anything about the properties of n, that is, it reduced the entropy, but did not increase extropy.
2. n - a mathematical object? Yes. This answer has increased extropy, now we know that n is a mathematical object, therefore it has all the properties of a mathematical object, in particular, it is a variable or a constant.
3. n - constant? Not. This answer again reduced the entropy and made a phase transition. The amount of accumulated information "n is a mathematical object" and "n is not a constant" turned into its quality - output " $$$n$- the variable "and now allows us to reduce the entropy of the data above definitions.

Entropy (in mathematics) is an inherent property of a mathematical object, a measure of our ignorance about it as a system, a value measured in bits of entropy. Informally: the answer is "no" to the question.

Extropy (in mathematics) is the opposite of entropy (in mathematics), a measure of our knowledge of a mathematical object as a system, a value measured in extropy bits. Informally: the answer is "yes" to the question.

Phase transition (in mathematics) - a decrease in entropy (in mathematics) by one bit of entropy, which occurs when a sufficient and necessary amount of knowledge is accumulated.

A little magic theory of entropy

So how did I get my equivalent of the Collatz conjecture? Assume that initially $$$n$had a look $$$8m$that is, contained $$$3$bits of extropy: “yes” answers to the questions “does $$$n$on $$$2,4,8$respectively? ”and a certain number of bits of entropy defined by a variable $$$m$. Division operation by $$$2$we lowered extropy three times, in the end it became equal to zero, the entropy remained unchanged and equal $$$2 ^ r$where $$$r$- the number of digits in the binary number $$$m$. By transferring to a number system with a fractional basis, each time we made a phase transition (in mathematics), because we received knowledge " $$$m$divided by $$$3$", that is, to put it in plain language programmers, made a bit shift to the left and replaced zero in the rightmost digit by one. As a result, for a finite number of shifts, we got a number from a completely undefined number, all the digits in which one, that is, a number completely defined whose entropy is equal to zero, which is written in decimal notation $$$1$. Q.E.D.