---

—  Collatz Conjecture Solution   —
EXPLAINED
—  Collatz Conjecture Explained   —

Collatz Conjecture Solved by
Collatz Conjecture Murgu Arrow
Guarderd By
8 New Murgu Formulas as Key, see Murgu Formulas
Perfect Grid To Infinity ,
Collatz Conjecture Murgu Inverse Method,
Conventional named odds in Perfect Grid With appropriate Proprieties,
and for a easy Collatz Mind Image,
3D Logical Apple Tree Collatz Conjecture Representation
Collatz Conjecture Murgu Table2To3 Wich lead to
Collatz Conjecture Murgu Arrow Dual Linear Noncompeting Functions.

With patience , after making a mind image about: Perfect Grid To Infinity, 3D Apple Tree Collatz Conjecture Logical Representation, Logical Eternal Triads Nodes(LETN) & Logical Dead Nodes(LDN)(even if do not accept how are named!) , small understunding of New Collatz Conjecture Murgu Formulas, Murgu Inverse Method, and, Collatz Conjecture Murgu Arrow Beauty_Logical you will can Build in mind all Collatz Conjecture Patterns as a Tree.
Only Human Can Image Collatz Conjecture.
     In Reality Collatz Conjecture Murgu Arrow, or Murgu Arrow, or Murgu_Table2To3 Contain all Collatz Conjecture , when every Node (x) of every 2 Linear Functions can be dual connected to Even's with Formula
(2^k)(x)       k>1 to infinity.
that do not modify Infinity_Per_6 Collatz Conjecture paterns number
only will send start Nodes Pont to Infinity
Do not let your mind eyes to be cheated by
(2^k)(x)       k>1 to infinity.
and special by
(2^k)(1)       k>1 to infinity 1=Unity.
Also, do not forgot Logical Trunk, which include those(last)!.

In this material are maybe more repetition and will keep those for a time , as to help
Collatz Conjecture Human Mind Image
As Example:
(2^2k)(1)      create ((2^2k)(1)-1)/3      
metaphorical are like The Bow for Murgu Arrow.
Collatz Conjecture Murgu Arrow beauty logical consists in the fact :
Murgu_Table2To3 is a Collatz Conjecture complete map if associate to it
((2^2k)(1)-1)/3      and
(2^k)(x)       k>1 to infinity. x any odd.
(2^k)(1)       k>1 to infinity 1=Unity.
This material remain as working material for any times, then can't to exclude yrt any small errors but neither one is essence anyway!

With Your Accept
Collatz Conjecture Murgu Arrow
will be included in Book Of Millennium 3
REMARK:
Collatz Conjecture can't have complete solution via Proofs, but only by demonstratig
Infinity Pay respect for. That can do only Collatz Conjecture Murgu Arrow which via Collatz Conjecture Murgu_Table2To3 demonstrated Collatz Conjecture is backed by 2 linear Functions non concurent on Positive Integers ❲1, ∞❫.
If accept or not Change The Name gave to Perfect Grid To Infinity Components for Analytic remained components after excluding Evens : 2 Kind of Logical Eternal Triads and a Logical Dead Node or Logical Left Closure, demonstrated LDN exclusion for Collatz Conjecture problems creators, all remain back on LET(Logical Eternal Triad's ).
Murgu_Table2To3
come with 2 Linear Noncompeting Functions , one starting from Unity (1) , and second from 3.
Knowing now all (Even Numbers ang LDN's ) are dependent connected to LETN and for every its first Node Connection are increasing with constant values demonstrated:

Infinity Pay Respect for Collatz Conjecture
With your accept YET will keep parts from old materials and forgive my if can be a bit disipated ! Also for maybe obsesive repetitions!

A Remark: as will see I excluded for analyze Even Positive Integers as those have a unique role in Collatz Conjecture of trespassing to an ODD. That do not affect our solution. The single thing affected will be:
Infinity_Per_6 became Infinity_Per_3 if keep Murgu Inverse Method and Collatz Procedures
But in Reality Remain Infinity_Per_6 in both of them as only every Path Point Of Start will be Sent to Infinity(heavy to follow). As all Start Nodes before were LDN now will be
(2^k(LDN) ) where k = infinity
Then if Follow Collatz Conjecture or Murgu Inverse Method, All Collatz Conjecture paths which are different:
Infinity_Per_6,
But first come with a beauty - Odds Numbers weared By Evens as Flowers.
Second have also a surprise. Never will meet , in a pattern, an even divisible with 3 if don't start from one. Also never inside of any pattern will not meet any even 3 divisible. Even Numbers Divided by 3 became closure and every LDN=d will have infinity obtained by formula (2^kd) placed in line starting from (2d). Then if start the pattern from (2ⁱd), never will meet the rest to (2^kd).
Infinity_Per_6 because ELDN (Even LDN) will be (2^kd) with k= ∞
Murgu Inverse Method have a litle clarity over it. Make all more easy.
Starting from Unity can make a software what to generate (until at a big odd ) a MAP Collatz Conjecture patterns.
Are importants Those Reference to Infinity, as we can refer any to Infinity.
Also we will work with two kind of INFINITY - Infinity and infinity as part of.
But Now in a 3d Apple Tree Even Numbers Come as Flowers of Odds x with Petals sonding Infinity and infinity as Number.
That include Logical Dead Nodes.

    Collatz Conjecture Murgu Arrow(Murgu_Table2To3 ), Solved Collatz Conjecture
demonstrated that, Infinity Pay respect for Collatz Conjecture as every Logical Eternal Triad is connected linear to another or to a Logical Dead Node
and eliminate the danger coming for aparent Collatz Conjecture divergence wich can create a Leaf Effect as can See at Murgu Conjecture Vicious Redundanci (3x-1).
Collatz Conjecture - Murgu_Table2To3 - for sure will get a role in Science.

Collatz Conjecture Solved Principal Key
Collatz Conjecture Murgu Arrow
or 3xPlus1_Murgu_Table2To3

In reality, to understand Collatz Conjecture maybe will need to see all material published anterior .
But if any Math skill, for sure down explanation will be sufficient.
To solve Collatz Conjecture, needed :
8 new formulas , named Collatz Conjecture Murgu Formulas
To divide Positive Numbers in 6 components groups - Perfect Grid to Infinity. (in negative Integers Collatz Conjecture is Murgu Vicious Redundancy Conjecture).
A virtual representation as Collatz Conjecture 3D Apple Tree Logical Representation, as to help to image it.
A conventional numing of every grid odd numbers (3 as numbar) as 2 kind of Logical Triads AND Logical Dead Nodes or Logical Closures.
Collatz Conjecture Solved solution is easy now , but to understand is heavy as need to make in mind more logical inversions. Keep in mind we excluded , not for Collatz Conjecture, but but for its logic, and using Math correct rules Even Numbers and as will see Logical Dead Nodes (all odds divided one or more times by 3).
Then we remained for Collatz Conjecture Analitic with 2 Infinity Per 6 , odds Logical Eternal Triads.
All this maked possible by Murgu Inverse Method and any Murgu Formulas.
Maybe I am repeating, but for helping to understand , I will Explain it More Times.

For Clearity we separed Even by odds , as odds number Active and even Passive .
As will see we in any senses excluded Even Positive Integers From Demonstrating Collatz Conjecture , as are only Passing Nodes to Active Nodes, knowing every odd x have infinity pasive nodes via (x(2^k)). Then , helped by Murgu Inverse Method Formulas we can follow only Collatz Conjecture Active Nodes.
If You Consider those Legal Nodes (active nodes) , Then
Infinity_PER_6 is reference to all Collatz Patterns.

Collatz Conjecture Solution
mean to demonstrate 2 Issues:
Infinity Pay Respect For Collatz Conjecture. Every Odd To Infinity via Collatz Conjecture will End in Unity.
to exclude via demonstration --- Not any Leaf Effect --- see Murgu Conjecture Vicious Redundancy. This include even reverse to Infinity.

Reminder:
Collatz Conjecture Solutions can't be afforded via PROOFS.
A proof can't certify for Infinity.
Now, via New Murgu Concepts even Binary Proof for Collatz Conjecture can make a step more , but Infinity don't let it to be more then a Proof.
Then Solution claim accurate DEMONSTRATION and The Problem is . What to demonstrate for Collatz Conjecture?
Murgu Conjecture Vicious Redundancy show clear one.
DANGER OF LEAF EFFECTS
and then also a possible
ROOT REVERSE TO INFINITY
To demonstrate it need to demonstrate every dual logical Collatz connection possible is a unique duality for Collatz Procedure and UNITY is Father for all as this dual connection is leading Up To Down.
Then , To organise the work by simplify. For Demonstration will exclude even numbers as all of them are already connected to Odds via division with 2 via infinity connections via formula (x(2^k)).
A remark is needed, infinity is part of Infinity.
By "excluding" even numbers will remain with the half for which to demonstrate pay respect for Collatz Conjecture.(Isn't a real exclusion because if demonstrate for odds automatic we demonstrate for even numbers as every even by division with 2 fall in odd).
Now remain for Analytic the half of positive Integers (Infinity/2 or ∞/2).

By here need to try to understand Collatz Conjecture Murgu 3D Apple Tree Logical Representation and Perfect Grid To Infinity. As Example can image how all odds wear infinity evens as a flower.
     I every grid of 6 components remained 3 for which to demonstrate that pay respect for Collatz Conjecture.
For it, Murgu Inverse Method (using 2 formulas ) will help more then a lot via 3 absolute truths non observable direct via Collatz Conjecture procedures.
(x(2^k) - 1)/3 = q    where q is top(up) Collatz Conjecture x connection.
Any of them (Logical Eternal Triad's have infinity) but it don't mean yet a Leaf Effect, instead only a danger. 1. Formulas are coming from Collatz Conjecture Procedure extension and may be used , but new is Inverse Method.
2. Detected (and then demonstrate Demonstration ) all odds divisible one or more times by 3 do not have up or left or top connections but a Flower with infinity petals of even numbers.
3. Every Grid have 2 kind of Logical Eternal Triad with infinity Top Connections. Function Of Grid Positin (first or second) those are offered by 2 Murgu Formulas, which are down.
Every also , as Logical Dead nodes , have a Flower offered by infinity petals conncetions to dependent even numbers. Those only give beauty to Collatz Conjecture, then we can real problem of Collatz Conjecture only via odds.
4. The demonstrated fact that Logical Dead Nodes do not have Left and Top Odd Connections exclude Logical Dead Nodes from our possible problems abd tath because a LDN have only Right Connection via Collatz Procedure 3x+1 WHICH THEN CAN'T BE another LDN but ONLY LETN (logical Eternal Triad).
5. That pass the problem to Logical Eternal Triad's.
Logical Eternal Triad's became a kind of Key for Collatz Conjecture Solution.

By here all 8 Murgu Collatz Conjecture Formulas see Murgu Formulas play any roles in understanding. Murgu Inverse Method Became a key and because is provided even by Collatz Conjecture procedure (3x + 1 ) do not need demonstration.
Using those will get an incredible result connecting Collatz Conjecture to 2 Linear Functions nonConcurent as starting one from 3 and one from 5 both connecting to unity as first grid assymetry exception.
In fact Unity Eternal Triad Father and via Murgu Inverse Method all Collatz Conjecture Patterns Generator.
REMARK:
All Even Numbers have a Unique Connection to a single Odd Number when an Odd Number have infinity bunch Connections of Even Numbers via Formula
q = o(2^k) where q even o odd
like a Flower Petals.
Picking the flowers will remain with Collatz Conjecture Tree but good for ANALYTIC.
infinity but unicity?

For future Math From Collatz Conjecture maybe all wich will survive etern will be
3xPlus1_Murgu_Table2To3 or 3xPlus1_Murgu_Arrow or Murgu Arrow
alreaddy Know as :
Collatz Conjecture Murgu Arrow Collatz Conjecture Murgu Arrow , maybe Mathematics correct name Collatz Conjecture Murgu Table2To3, used Murgu Formulas for every Eternal Triad's for First Top Connection , which we named Left Connection and GOT INCREDIBLE RESULT.
Maybe, nice Idea was to compare those via a Table2to3 , which instead leaded to 2 linear Functions which can have now a complete demonstration for Existence and caracteristics.

Now as know via Murgu formulas every Eternal Triad have infinity Top Connections, but every one is independent , coming from another pattern and are using inverse method, then this assure unicity for Top to Right or Down.

Collatz Conjecture Murgu Arrow, via Murgu Table2To3 detected liniarity and will see , demonstrated Those Pay Respect To Infinity. That mean Infinity Pay Respect for Collatz conjecture.
A words joke wearing truth. Take the time and will understood and even Image Collatz Conjecture which came also with any Beauty:
1. more then sure Murgu Table2To3 will find its usefulness in Science and not only.
2. if you accept Collatz Conjecture patterns as I described , We Get All Collatz Conjecture Patterns which at the last have one component different. 3. in a 3D Representation even odd (including LDN) appear to have in Top a Flower of infinity even numbers with petals sonding infinity. 4. can be morem even logic contained is a beauty logical.
To hope it will help more for understanding! Also will keep a part from old explanation.

Collatz Conjecture Explained will be motive of change time by time and add until all with any Math skill will understood it .
But Collatz Conjecture is solved by 2023 by 3xPlus1_Murgu_Table2To3(3xPlus1_Murgu_Arrow) as Collatz Conjecture Solution.
Collatz Conjecture Solution wear a lot of Logical Beauty which maybe are heavy to explain with clearity and have 3 important results :
1. Only Human Mind can Image Collatz Conjecture
2. Infinity_Per_6, Collatz Complete Paths (Patterns wich have at The Last a different Logical Node)
3. Infinity_Per_18 Logical Trunk Direct Connections(not finished calculation) etc..
This will not affect
Collatz Conjecture Solved
Collatz Conjecture Murgu Arrow as Principal Key Of who brought New Logical Eternal Triads Formulas but also
Collatz Conjecture Murgu Arrow Dual Linear Noncompeting Functions. Arrowhead is Unity.

But if need to analyze any different capitols, down are any links for

For, Collatz Conjecture Murgu Arrow
Beauty Logical

For, Collatz Conjecture Murgu Formulas
Murgu Formulas

First, Is clear understood for sure
Even Positive Integers Exclusion , as even will return procedural to odd, because Every Even Positive Integer by one or multiple division with 2 fall in a odd. This will make a little easy the work in using Murgu Inverse Method via adequate formulas.
Now
To understand Collatz Conjecture even

Collatz Conjecture 3D (or 2D) Apple Tree Corner Web Logical Representation
Is Important and brought any in, as:
possibility to make distinct more kind of Logical Nodes Connections which also pay respect for Collatz Procedures. This was possible, or revealed also By Murgu Inverse Method and for it need to deffine function of any logical caracteritics Odd Integers in 3 category:
1. Logical Dead Nodes or Logical Left Side Closures,
All Odd Positive Integers which are divided one or more times by 3.
2. LFETN_1i , Logical First Eternal Triad Node (first as position in Grid)
3. LSETN_2i , Logical Second Eternal Triad Node (second as position in Grid)
both last 2 and 3 are Logical Eternal Triads wich share common properties as:
Logical Left Connection(which can be another LETN or a LDN )
Logical Top Connections which are anothers LETN , are Parts of Infinity as infinity.
Logical Right or Down Connection ofered by (3x+1) itself. Understanding this logical representation help Collatz Conjecture understanding but also a happy fact
ONLY HUMAN MIND CAN IMAGE - Collatz Conjecture.
Down is a try to see it for a small amount and without check of any error yet, and using Murgu Inverse Method.

This PNG reveal partially our Inverse Method refering only to odd positive Integers. For Murgu Inverse Method Only Infinity is the barrier in revealing Collatz Conjecture all paths, all connections and all beauty.

All Collatz Conjecture Murgu New Concepts are Logical Connected, but to be explained need to be separed.
Maybe next as importance to be explained is
Perfect Grid To Infinity
For Collatz Conjecture and Murgu Conjecture Vicious Redundancy (mirror image of Collatz Conjecture in Negative) , ZERO do not exist. Then first tentation is to divide Positive Integer in Grids of 3 components ((1,2,3),(4,5,6), (7,8,9),(10,11,12), ..), but as we saw via Murgu Inverse Method, Even Numbers and Odds in Grids present small different proprieties. Then it lead us to grids of 6 components as will see those will share common proprieties for Collatz Procedures .
((1,2,3,4,5,6), (7,8,9,10,11,12), (13,14,15,16,17,18),..)
As know Murgu Inverse Method and simple observation Even Numbers fall in a odd via Collatz procedures, let us to work only with Odds, as to easy The Work.
((1,3,5),(7,9,11), (13,15,17),(19,21,23), ..)
As can see every grid contain :
Logical First Eternal Triad Node, LFETN_1i
Logical Dead Node or Logical Left Closure LDN_i . Logical Left Closure(LDN) is every Odd divided one or more times by 3. As will see, a LDN do not have Left or Right Connection but only Down-Right via (3x+1) and this connection will be to a LETN or directly to Trunk , which mean directly to LETN UNITY.
Logical Second Eternal Triad Node, LSETN_2i

Demonstration,
all LDN don't generate Left or Top Connection

Analytic for Perfect Grid To Infinity reveal a lot as :
Every LDN not Left and Top Connection is verry important because having only Right-Down connection. Then Reverse Logic lead it, automatic is an LETN . That is more then important, because demonstrate
every LDN have a Unique Connection and is to a LETN.
Only Collatz Procedures starting from LDN can be Collatz Complete Paths.
A Collatz Procedure Starting from a LDN, in its Path, never will pass(or meet) via another LDN.
A Collatz Procedure Starting from a LETN, in its Path, never will pass(or meet) via a LDN.
This need demonstration and can be right here

A Left Side Connection is revealed By 2 possible Murgu Formulas:
(4d_i -1 = 3 u_i)     d is a LDN in grid i
OR
(2d_i -1 = 3 u_i)       d is a LDN in grid i
we know d is d_i = 3 v ehere v an odd
then
(12v -1 = 3 u_i)    
OR
(6v-1 = 3 u_i)      
If will note (4v = t ) and (2v = g)
(3t -1 = 3 u_i)    
OR
(3g-1 = 3 u_i)      
Impossible
Is Clear this exclude also Top Connections, for any(all) LDN, and Murgu Inverse Method and Reverse Logic helped a lot.
THIS EXCLUDE LDN FROM DANGERING Collatz Conjecture but via a
MATH OVERPASS THE PROBLEM TO LOGICAL ETERNAL TRIADS
Collatz Conjecture Murgu Arrow will solve it.
A Logical Beauty coming for it
Collatz Conjecture Complete Paths
Infinity_PER_6 .

Murgu Inverse Method and Reverse Logic.
Murgu Inverse Method is based on a first place Reverse Logic of Collatz Conjecture Procedure (3x+1).
As we excluded virtual Even Numbers, then we can redirect this procedure to Odds via
((3x+1)=(2^k)o) where x and o Odds k Positive Integer
((((2^k)o)-1)=3x) base formula for Murgu Inverse Method

As will see Murgu Inverse Method is leaded by 4 Formulas and a applied only to Logical Eternal Triad including Unity . Now can make Javascript , Perl, C++, C, C#, Rust , etc., which to reveal Collatz Conjecture Paths, but limit remain Infinity also for every programing language.
Infinity_PER_6
then, prov
ONLY HUMAN MIND CAN IMAGE COLLATZ CONJECTURE and USA_AEPDF

Now a simple observation, we have in every grid 2 kind of Logical Eternal Triads,
Logical First Eternal Triad Node, LFETN_1i
and
Logical Second Eternal Triad Node, LSETN_2i
with simple Math motivated conditions will separe it in 2. To note LFETN_1i with W_1i and LSETN_2i with W_2i , those will be :
((((2^2k)o)-1)=3W_1i)    k >0
((((2^2k+1)o)-1)=3W_2i)    k >=0
Via those Equations an using added skill about Perfect Grids To Infinity, now can make programs in Javascript, Perl, C++ which to generate all Collatz Conjecture Paths, but
Infinity is a big issue even for Computers
Sory to repeat, but to remind to everyone, we can be proud, Anyway PROV
ONLY HUMAN MIND CAN IMAGE COLLATZ CONJECTURE

From those 2, we extract also 2 to make distinct Left Side Connection node from Top's.
((4o)-1=3W_1i)   
((((2)o)-1)=3W_2i)   

THOSE 2 EQUATION HAVE BIG IMPORTANCE FOR
Collatz Conjecture Solutions

As saw before excluded LDN as "PROBLEM" from, BY Passing it To Logical Eternal Triads
and as will see are base for solving it there via Collatz Conjecture Murgu Arrow.
Here is a Link for:

For, Collatz Conjecture Murgu Arrow Beauty Logical

Maybe also Logical Eternal Triads need a small Explanation,

First, I insisted for The Name "Logical Eternal Triad" as express perfect those entities.
Those refer 3 Connections, Left, Top and Right-Down - then Triad. Top Connections are infinity, generating eternal connections to Infinity.
As already know a grid contain 3 Odds Numbers. One is a Logical Dead Node and guarded in the front and back by 2 Logical Eternal Triads which virtual , using Murgu Inverse Method every one "generate" infinity(Part of Infinity ) Logical Nodes any times are LDN .
As know LDN leave all the weight on Eternal Triad - and it - lead to
Table 2 To 3
which to reveal if possibly all Logical Eternal Triads Left Side Connections.
Possisibly mean - leaded via Functional - then to determine, in our case, 2 function which to certify Unicity for this Dualism.
this mean, if exist, all Essence Connections (Left Connections ) assure for Collatz Procedure (Reverse Logic aspect- left with right) unicity for way Down to UNITY.
And will see
Collatz Conjecture Murgu Arrow
did it. There we initialy did , fastening, a small confusion or indices , but for a Math skilled is nothing.
As we will expose next Collatz Conjecture Murgu Formulas, down is a link if wish to pass

For, Collatz Conjecture Murgu Arrow Beauty Logical

Collatz Conjecture Murgu Formulas.
Collatz Conjecture Solved, seem easy now, but heavy to explain.
And it because have its Logical Beauty which imply a number of new concepts and new formulas.
I will Say 8, even if 2 are coming from (3x+1). Can be 12, but Anyway 8, can't be negate.

Collatz Conjecture Solutions.
was brought by
Collatz Conjecture Murgu Arrow
short
Murgu Arrow
which wear it in new dress.
Collatz Conjecture New Name -Infinity_Per_6
A Poetic Truth.
Collatz Conjecture Murgu Arrow
new 8 Collatz Murgu Formulas
Perfect Grid To Infinity
Eternal Triad and LLDN
brought for, or wear
Collatz Conjecture
in Beauty Logical and
first EXACT INFINITY REFERENCE
Infinity_PER_6
Every Murgu New Collatz Conjecture Concepts have importance on, even ,
3D Collatz Conjecture Logical Apple Tree Representation
and it because only Human can Mind Image it, and is important.
Also , Murgu Collatz Conjecture String Locators Formulas, for locating : grids, 2 for Logical Eternal Triads and one for Logical Dead Nodes.
Keep in mind , Murgu Arrow show only Eternal Triads Left Connection Node , when for a Mind Image need to not forgot Top which are infinity for every Eternal Triad , where infinity is a part of Infinity and can't be defined exactly. Instead
Infinity_PER_6 can
Complete Paths Number
and also Number of Perfect Grids. Then
Collatz Conjecture Murgu Arrow is principal key of
Collatz Conjecture Solved,
BUT Also Collatz Conjecture Murgu Formulas play an essence roll in.

As key of Collatz Conjecture Solved, I will keep Collatz Conjecture Murgu Arrow Link.

For, Collatz Conjecture Murgu Arrow
Beauty Logical

    MURGU FORMULAS (IN LOGICAL MIRROR)ASSURE LOGICAL CONTINUITY AND UNICITY for A DUAL LOGICAL CONNECTION FOR ALL ODDS TO INFINITY FOR COLLATZ PROCEDURES
THEN FOR ALL POSITIVE INTEGER.

    MURGU FORMULAS can be 12 , but important are 8. That because can exclude Murgu Lemas(2) , and unity 4 for LET in 2. To understand theirs role, need to see any new concepts in, as:
6 components Grids to Infinity
Collatz Procedures used via logical reverse Murgu Inverse Method
Observation, logical can exclude Positive Even Integers to easy content
then , a Grid will contain 2 Logical Eternal Triad and a Logical Dead Nodes

MURGU FORMULAS

   4 Collatz Conjecture Murgu String Locators Formulas
Grid Locator    G_i = 1 + 6i
First Logical Eternal Triad Locator    (LET1)_i = 1+6i
Second Logical Eternal Triad Locator    (LET1)_i = 5+6i
Logical Dead Node Locator    (LDN)_i = 0+3i    i>0 and i odd only

Now, will folow 6 new Formulas which are essence of solution for Collatz Conjecture. 2 of them are used for a long time , but without positive results.
2 of them are like 4, obtained in exercise to rebuild Collatz Conjecture starting from Unity,
Collatz Conjecture Murgu Inverse Method.
Those refer only Logical Eternal Triad , but Logical Keft Dead Nodes are not uncovered. Those don't have Left And Top Logical Connections , then only one which is Logical Right And Down. Then for moment can be out of our imediat scope and return separately to it.
Those are:
(2^(2k) v_1i -1 = 3 u_i)    for   LET₁

(2^(2k+1)v_2i -1 = 3 u_i)     for   LET₂
where v_1i is a ET , first in Grid and v_2i is a ET , second in Grid i
Those are Logical Top Commections, are infinity but as will see every one assure duality and unicity for Collatz Procedure. Logical Top Commections do not affect Murgu Arrow , because every one is coming as Right-Down from anoter LETN wich is (3x+1) formula.
Discovering of infinity as Infinity Parts fact prove
ONLY HUMAN CAN IMAGE COLLATZ CONJECTURE
Image here play a role of verb , action.
Murgu Inverse Method Separed from Those Formulas 2 as Left Side Logical Connections:
(4v_1i -1 = 3 u_i)    for   LLETN₁

(2v_2i -1 = 3 u_i)     for   LLETN₂
where v_1i is a ET , first in Grid and v_2i is a ET , second in Grid i
Those last 2 have importance in demonstrating unicity of logical connections and even in geting Collatz Conjecture Murgu Arrow Formulas.
Collatz Conjecture Murgu Arrow , became essence key for Collatz Conjecture Solutiomn.
Maybe Logical forte point of Collatz Conjecture Murgu Arrow is
TABLE 2 TO 3
WHICH is possible because in Collatz Conjecture Perfect Grids To Infinity in everi Grid , for logical analytic, remain 3 Components LETN_1i, LDN_i and LETN_2i (where i is grid indice and 1 or 2 ET position in grid) .
Because LDN don't have Left Connection we can make this table which reveal 2 New Collatz Conjecture Murgu Formulas which can offer closure for .

Errata
In first published Murgu Arrow material, I Don't make special important specification :
Murgu Arrow formulas are offering every Left Logical Side ET Node Connection which can to be any times a LDN(it was seid) , and in reality for every kind of ET at every third time. Then Formulas will be:
LFETNC_1(i+1) = LFETNC_1(i) + 8 for   i grid indice,1 position
LSETNC_2(i+1) = LSETNC_2(i) + 4 for   i grid indice,2 position
This do not affect demonstration , are only conventional remark with role of distinction.

Those Formulas offer to
Collatz Conjecture to be exactly to Collatz Procedures, Liniarity, Unicity and important , not Concurency , as between W_1i and W_2i stand up 5 as not 2 divisible. As known, Logical Left Side Dead Nodes or Logical Left Side Closures do not present a Possible Danger on
Logical Leaf Effect (Logical Circle Rotation via Collatz Procedures)
or, Root Reversion to Infinity.

Anyway , If we Want to make more study for and on
Left Side Dead Nodes or Logical Left Side Closures
Murgu 3XPlus3 Conjecture and (3X-3) may be a help.
Now, via new concepts exposed For Mathematics , Collatz Conjecture present 2 dangers:
Leaf Effect as Murgu Conjecture Vicious Redundancy which create multiple roots.
reverse root to Infinity which mean to exist any positive Integer for which via Collatz Procedures never to fall in Unity.

To eliminate all Doubts I will keep also old Collatz Conjecture Murgu Arrow Demonstration which in essence is correct
For, Collatz Conjecture Murgu Arrow Beauty Logical

Collatz Conjecture Murgu Arrow

If You remind , we got, in a 3D APLLE Tree Logical Representation Analyse, Positive Integers Separated -in 6 Components Perfect Grids To Infinity. Every One(grid) Contain a Logical Left Side Dead Node, and 2 Logical Eternal Triad Generators Which Differ by Murgu Collatz Conjecture Formulas(4^kW or 2^kW)
First (LETGN1): (4W-1)/3 - Second(LETGN2) in Grid: (2W-1)/3
generate Left Side new connection, LETGN or LLDN and also To TOP Will Be generayed infinity(FOR LETG only).
For This Scope we Generated A Table where First Column folow Grids Eternal Triads Only abd Top Row Grids Eternal Triads and Dead Nodes(results of Murgu Formulas for Left connections).
The Table is Down Table 2To3
The Table we get can be considered Consistent Proof, but we Demonstrated which those undreds proofs show.
But Table reveal a Small Math Beauty
Collatz Conjecture Murgu Arrow
and
If whish to extend The Table became easy task.
As see a edge of arrow evolve by +4 to last, and another edge by +8 to the last
AND Every Edge is provided by first ET in the grid and second edge by sec. in the grid.
The table Show for Every Eternal Triad we have a Left Side Connection and Its Lefta Side Connction via Collatz Procedure Fall In It.

Apparent we are losing any Dead Nodes or Left side Closures
This is hapening because Table is 2To3 and important is Logical Eternal Triads can't be.
Those are going direct into Logical Trunk then Unity. Example 21 - (21x3 +1)=64.
Table 2To3 became Principal Key Of Collatz Conjecture Solution, because if can be demonstrate those dual dependencies , then clear wil guarde Collatz Conjecture with 2 linear functionality.

DEMONSTRATION

Like said before, LDN passing all weight on Logical Eternal Triads lead us to a
Table 2To3 ANALYTIC
Eternal Triads - Table 2To3
and did it using Murgu Inverse Method 2 Murgu Formulas for geting Logical Left nodes connection. As we have 2 kind of Eternal Triads , will note those as
LLFNC_1i, where 1 - first in grid i, LLSNC_2i where 2 - second in grid i .
To substitute those for easy with
T_1i, where 1 - first in grid i, T_2i where 2 - second in grid i .
Building the tabel for, can't to escape from observation a fact revealed down
T_1(i+1) = T_1i + 8
T_2(i+1) = T_2i + 4

But, even if we have a lot of proofs for , we can legalize those linear functions but only after demonstration. We can do it via 2 Methods.
By geting T_1(i+1) and T_2(i+1) , then T_1(i) and T_2(i) from anything else and compare.(Method 1)
Or, using Add Absurd.(Method 2)

For it need to connect
T_1(i+1) and T_2(i+1), T_1(i) and T_2(i)
to theirs Logical Eternal Triads via Murgu formulas.
(3(T_1(i+1))) = (4(W_1(i+1))-1)   E.1
(3(T_1(i))) = (4(W_1(i))-1)   E.2
we know also (W_1(i+1)) = ((W_1(i))+6) and then E.1 can be write
(3(T_1(i+1))) = (4(W_1(i))-1+24)   E.1
or (3(T_1(i+1))) = ((3(T_1(i)))+24) and dividing by 3

T_1(i+1) = T_1i + 8

for Logical Eternal Triad Second position in grid

(3(T_2(i+1))) = (2(W_2(i+1))-1)   F.1
(3(T_2(i))) = (2(W_2(i))-1)   F.2
we know also (W_2(i+1)) = ((W_2(i))+6) and then F.1 can be write
(3(T_2(i+1))) = (4(W_2(i))-1+12)   F.1
or (3(T_2(i+1))) = ((3(T_2(i)))+12) and dividing by 3

T_2(i+1) = T_2i + 4

For Method 2 can consider old one EXPOSED IN OLD MATERIALS
THIS DEMONSTRATION CERTIFY 2 NEW FUNCTIONS FOR COLLATZ CONJECTURE.
THOSE SHOW LINEARITY, AND THEN, NOT CONCURENCY FOR -
EXCLUDE LEAF EFFECT AND MAKE THE SENSE INFINITY TO UNITY (BY REVERSE LOGIC) UNIQUE.

Those can be connected to grid position into defined Perfect Grid to Infinity , where first grid is 0.
For First Logical Eternal Triad in Grid i, its Logical Left Connection wiil be
LLETC_1i = 1 + 8i where i is Grid i LETN position 1. FC.1
For Second Logical Eternal Triad in Grid i, its Logical Left Connection wiil be
LLETC_2i = 3 + 4i where i is Grid i LETB position 2. FC. 2
To hope not anoter notations mistakev ! Anyway FC.1 and FC.2 can be verified as True.

—  Collatz Conjecture SOLVED    —
2 Solutions for Collatz Conjecture

WE DEMONSTRATED(and brought) 2 IMPORTANT FUNCTIONS FOR Collatz Conjecture via
Collatz Conjecture Murgu Arrow
that closed Collatz Conjecture whith A Beauty Logical SOLUTION
Apparent any is loss there , but is nothing more then a confusion made by Reverse Logical understanding .
For it we make an another table for the same Logical Eternal Triads (the same Grids) but Logical Right-Down Connections, as to see how Collatz Procedure create apparent Logical Divergence

First 6 grids Logical Eternal Triads Collatz Procedure Connections.

Any LDN do not appear into table 2To3 because are Top Connections of UNITY as generators and fall direct into Trunk.
Example 21 is a LDN and determine a Collatz Path with 21 - Unity via (3x+1)
(3(21) + 1) = (63 + 1) =64
Also 85 , BUT AS LETGN

Those can be refered to Infinity like Infinity_PER_6 and if not mistake are
Infinity_PER_18 as fasting calculus.
now can say
Collatz Conjecture SOLVED
Collatz Conjecture HAVE SOLUTION
Murgu Inverse Method Excluded EVEN Positive Integers from
I will say via right motivation
Every Even is connected to a odd(x) via (2^k (x))
As Logical Falling Nodes. AND to introduce them as Collatz Nodes change only Collatz Conjecture Complete Paths , for
Infinity_PER_6 .

Table 2TO3 Show Unicity for every Eternal Triad Collatz Procedure related.
For 1/3 (Odds Numbers part)Logical Dead Nodes Uncovered BUT
That is because are Direct Logical Dead Nodes to Unity via Trunk.
REMINDER: Every Logical Dead Nodes have unique connection right-down to a LETN including Unity
, then is covered also here as those are n's connection of a Dead Nodes to a LETN. For our final demonstration
Infinity Pay Respect for Collatz Procedure
This is not a problem
Above I gave an example 21, and another can be 45 Wich is second connection of 17 as first top one. Left Side connection for 17 is 11, when Top first one is 45.
Collatz Conjecture Murgu Arrow , became essence key for Collatz Conjecture Solutiomn.
Maybe Logical forte point of Collatz Conjecture Murgu Arrow is
TABLE 2 TO 3
WHICH is possible because in Collatz Conjecture Perfect Grids To Infinity in everi Grid , for logical analytic, remain 3 Components LETN_1i, LDN_i and LETN_2i (where i is grid indice and 1 or 2 ET position in grid) .
Because LDN son't have Left Connection we can make this table which reveal 2 New Collatz Conjecture Murgu Formulas which can offer closure for .

Collatz Conjecture Murgu Arrow Logical Eternal Triads Formulas as
Murgu Dual Linear Noncompeting Functions
Demonstrated
Every Logical Eternal Triad LeftSide Connection is Unique Bigest Then Logical Eternal Value, AND not Concurency as Not 2 of them can have the same.

LLETC_1i = 1 + 8i where i is Grid i LETN position 1. FC.1
LLETC_2i = 3 + 4i where i is Grid i LETB position 2. FC. 2
Eliminate possibility Of Leaf Effects and Reorder Collatz Conjecture Apparent Divergence to Unique Evolving Direction Infinity To UNITY . NOT Reverse ROOT

Collatz Conjecture Solved

Infinity Pay Respect for Collatz Procedure

To understand Math Logical Play Divergence to Functional Convergence Compare
First 6 grids Logical Eternal Triads Collatz Procedure Connections.
WITH
Table 2To3
Nothing Math Logical can't contest here, reverse logical Left Connections Formulas because are coming from "mirroring" Collatz Procedure itself excluding maybe small nottations errors .

Down is Table 2To3, also old Collatz Conjecture Murgu Arrow . but also which may present interest. Any Example which can help understanding at link:
For, helping example

	1	3	5	7	9	11	13	15	17	19	21	23	25	27	29	31	33	35	37	39	41	43	45	47	49	51	53	55	57	59	61	63	65	67	69	71	73	75	77	79	81	83	85	87	89	91	93	95	97	99	101	103	105	107	109	111	113	115	117	119	121	123	125	127	129	131	133	135	137	139	141	143	145	147	149	151	153	155	157	159	161	163	165	167	169	171	173	175	177	179	181	183	185	187	189	191	193	195	197	199	201
1	1
5		3
7					9
11				7
13									17
17						11
19													25
23								15
25																	33
29										19
31																					41
35												23
37																									49
41														27
43																													57
47																31
49																																	65
53																		35
55																																					73
59																				39
61																																									81
65																						43
67																																													89
71																								47
73																																																	97
77																										51
79																																																					105
83																												55
85																																																									113
89																														59
91																																																													121
95																																63
97																																																																	129
101																																		67
103																																																																					137
107																																				71
109																																																																									145
113																																						75
115																																																																													153
119																																								79
121																																																																																	161
125																																										83
127																																																																																					169
131																																												87
133																																																																																									177
137																																														91
139																																																																																													185
143																																																95
145																																																																																																	193
149																																																		99
151																																																																																																					201
155																																																				103

This Table 2To3 is possible because:
1. Logical Dead Nodes don't have Logical Left and Top Connections.
2. Murgu Inverse Method to reveal Logical Left Eternal Triads Connection and Now show Top Connections even are infinity , are dual and independent.

	1		5	7		11	13		17	19		23	25		29	31		35	37		41	43		47	49		53	55		59	61		65	67		71	73		77	79		83	85		89	91		95	97		101	103		107	109		113	115		119	121		125	127		131	133		137	139		143	145		149	151		155	157		161	163		167	169		173	175		179	181		185	187		191	193		197	199
	1		1	11		17	5		13	29		35	19		11	47		53	7		31	65		71	37		5	83		89	23		49	101		107	55		29	119		125	1		67	137		143	73		19	155		161	41		85	173		179	91		47	191		197	25		103	209		215	109		7	227		233	59		121	245		251	127		65	263		269	17		139	281		287	145		37	299

Table 2To2 which show for first 34 Grids Logical Eternal Triads Collatz Procedure Connections.
Instead Of Table 2To3 Reverse Logic, here became clear divergence on.

In Collatz Conjecture
Infinity darts Unity Via
Collatz Conjecture Murgu Arrow.
Collatz Conjecture Murgu Arrow used Murgu Collatz Conjecture Inverse Method but only to reveal for every Logical Eternal Triad Generarators only Logical Left Side Connection.