Human Intel 2024 Logical Beauty
Beauty Logical
Only Human Can Image Collatz Conjecture
Poetic Pillar FOR
Human Intelligence Legacy


Collatz Conjecture Solution
& Logical Beauty
Collatz Conjecture END In
Murgu Leaf Effect Conjecture .

Under Humanity Natural Law Of Copyright

& USA ISBN-978-1-63972-042-2

Collatz Conjecture Solved and Brought any Beauty Logicals

Only Human Can Image Collatz Conjecture
Marker USA Murgu Arrow as Collatz Conjecture Pocket Map.

Murgu_Table2To3 is Collatz Conjecture Unique Solution



OBSERVATION:
This Material have small disorder in, but all demonstration are in , excluding a logical expresion
MURGU_CVR DO NOT HAVE AN ARROW HEAD AND THEM MURGU INVERSE METHOD CAN'T EVOLVE FOR IT A POCKET MAP STARTING FROM UNITY.
That explain leaf effect , but maybe never will can dem. , how many.
Anyway demonstrate :
Murgu Collatz Conjecture END
Collatz Conjecture Solved via Murgu_Table2To3 sustained by Marker USA Murgu Arrow , Collatz Conjecture Murgu Formulas , Murgu Perfect Grid to Infinity, Murgu Inverse Method and even Eternal Triads play a role in.
Normaly, this material for Collatz Conjecture is suficient, (do not need more losing time), but if time, will be ordered.
Also, any times will meet repetitions. Those are ok for first time reader , but for second time can seem dizzy, but, every time will be a first time reader.
In This material are used in any sense, but correct and demonstrated
Logical Inversion
& Logical Exclusions
Logical Exclusions are "For The Moment" , for easing demonstration, and will be returned in final to theirs place.
For that need to specify , virtual named LETN (2 in every Grid), have Intermediate infinity (not Infinity) connections which will be passed by in any Collatz Conjecture patterns , but 2 (Left and Right-Down ) present Logical Interest. The Top (infinity) are Intermediated by (4k) or by (22k+1) and are important as can image Collatz Conjecture Map.
Collatz Conjecture Logical Apple Tree Representation combined with Logical Grids To Infinity Representation will be util for.
Then, maybe is important to know:
LETN have Logical Left Connection, infinity Logical Top Connections - (Intermediated, but we speack about as to make a image for) - and Logical Right-Down one,
an EVEN have only 2, Logical Left_Top and Logical Right-Down . We named those Eternal Triads as stand as a Triad instead of
Logical Dead Nodes (all odds divided by 3 one ore more times) have a single connection Logical Right - Down (Intermediated via 2k are not important for demonstration but for Collatz Conjecture Map and Image).
    We used Logical Inversion 2 times , one time , via Murgu Inverse Method to gain any Collatz Conjecture Patterns , and second to get Murgu Table2To3 , which concluded(via demonstration) in 2 Linear Functions for every Logical Eternal Triads First Left Locical Connection Node which can be another Logical Eternal Triad or a Logical Dead Node.
Here, trhen, Logical Inversion seem a bit tricky as 2 infinity (not Infinity) to represent logic 3 infinity? But to not give up, we don't loss any. Apparent missed Logical connections are Contained in ((4k - 1)/3) which are going to Unity via divisions whith 2 after first (3x + 1) and
MAYBE a new Math Beauty

From Every infinity Odd Logical Eternal Triad we start Collatz Procedures , around remain also infinity open ways coming for others to Infinity, but for every one exist a Unique Logical Left Connection and also a Unique Logical Right-Down Connection Node which is 3XPlus1 divided one or more times by 2 until Odd.


Remaind -The Logic - via demonstrations we show all responsibility for
Infinity Pay Respect to Collatz Conjecture
Fall to Logical Eternal Triads
AND DEMONSTRATED THOSE HAVE A UNIQUE LEFT NODES CONNECTIONS
and infinity (the rest- Logical Top) are intermediated by multiple Collatz Procedures.
Sorry For Repetitions !, but being not understood for long time I considered to explain it coming from more directions.


Mathematical, Collatz Conjecture Solution is Murgu_Table2To3 , but also it reveal as Beauty Logical 2 Math Metaphors,
Marker USA Murgu Arrow &
Unity Arch To Infinity
Unity Arch To Infinity maybe forced poetic and can be only mind image. Is Trunk of Collatz Conjecture formed as odds Nodes
((4k -1)/3) , for k>1
and then additions of evens procedural connected
(2i(4k -1)/3) , for k>1 , i>0

Marker USA Murgu Arrow is a Metaphor but can be image from, Table 2To3 and as see are 2 Lines starting one from 3 one from 9 , having as head arrow 3,1,9 .
For making Mathematics accepted and loved I generated virtual it as a
Fine Art Essay
of Math in Metaphors.
Down are Collatz Conjecture Solution Explanations and Demonstrations for.

Forgive My, to begin with a new Explanation as I think Mathematicians don't understood yet for what Murgu_Table2To3 solved Collatz Conjecture and is Unique Solution !
As you can see Murgu Table2To3 brought 2 non concurent liniar functions describing and named, as math metaphor, Murgu Arrow (forgot The Trunk, for moment as excluded logical out of generatings problems ). Do not forgot, Murgu_Table2To3 is obtained via Collatz Conjecture Murgu Inverse Method and to connect in maind it to Collatz Conjecture procedures. Practic is simple, 2 Liniar Functions, Murgu Logical Triads String Locators as Functions, Generate theirs anothers connected Liniar Functions and bring also new 2 Murgu Formulas. The fact that all 4 are liniar for a Mathematicians say all.
I said Murgu Arrow is like a Collatz Conjecture Pocket Map, top which can connecting at the final The Trunk and for every LETGN1 , LETGN2 , and LDN theirs infinity connections(not Infinity), but those are connections endorsed by divisions with 2.
Collatz Conjecture Murgu_Table2To3 present an important property for every LETN (G mean generator)
Connection Unicity
for every LETN, and I said down in material.
As Logical exclusions let all part to LETN,
Unicity and Non Concurent Double Linearity
SOLVED COllatz Conjecture.

Via Murgu_Table2To3 Collatz Conjecture Reveal 4 or 5

Beauty Logical

TOP of Those
ONLY HUMAN CAN IMAGE Collatz Conjecture
an Happy Reminder Human Intelligence yet Top Intelligence
- gain a new stand for
Human Legacy on Earth

but also FIRST Math Metaphor
Collatz Conjecture Murgu Arrow.
and also big as poetic sense
UNITY Arch To Infinity
Collatz Conjecture have a UNIQUE Solution
Murgu_Table2To3
not as any proofs but DEMONSTRATION.
and for a Mathematician is enough to see it and Collatz Conjecture Murgu Formulas.


87 Years Old Mathematics Problem
Collatz Conjecture Solved
and by Method using All Strict Mathematics Rules.


Collatz Conjecture Solution
Collatz Conjecture Solved AS
Beauty Logical
AND FIRST Math Metaphor
Collatz Conjecture Murgu Arrow.
Collatz Conjecture have a UNIQUE Solution
Murgu_Table2To3
not as any proofs but DEMONSTRATION.


87 Years Old Mathematics Problem
Collatz Conjecture Solved
and by Method using All Strict Mathematics Rules.


This Material wasn't checked by any small errors, but for sure reach its Scope!
As Collatz Conjecture in all materials and false proofs show only Divergence. , following Collatz Conjecture Procedure
Here and or this is a Knowledge Labyrinth Extraction and heavy to explain.
Our Scope wasn't yet , Grammar, but, Collatz Conjecture ,then Is a hope in final and here , did it!
Large Example and tables will force us to work more to css styles , but yet are accepable in this page with your accept.



Logical Math is a heavy task as every where in Math is a danger of logical miracles which to lead to interpretation error.
For EVERYBODY, here , will meet 2 "Logical Exclusion" (For Evens and LDN) which placed all Collatz Conjecture responsibility to metaphoric Logical Eternal Triads Nodes (love it!) is out of error interpretation as Exclusion are demonstrated. Also all exclusions are reincluded, atached late to Murgu_Arrow which becamed Collatz Conjecture Pocket Map. Do not forgot , Murgu_Arrow as Math Metaphor and, as will see , Collatz Conjecture Doble Linear revelation.

For a MATHEMATICIAN, maybe aren't needed all repetitive EXPLANATIONS, and can easy go direct to
Murgu_Table2To3 (Solution)
after luck a bit to:
Formal needed - 3D Apple Tree Collatz Conjecture Logical Representation,
Perfect Grid To Infinity,
Collatz Conjecture Murgu Formulas,
Collatz Conjecture Murgu Inverse Method,
Logical Exclusion from Essence Of Problem.
Double Linearity - Murgu Arrow
BUT maybe Even They Need a Small Explanation
Apparence Of Divergence come from that - following Collatz Conjecture Procedures an Logical Eternal Triad have infinity (not Infinity) connections, when our procedure reveal one at the time.
Every Odd have Intermediate infinity connections to Evens
(2k(x)) via its intermediary (2(x))
but more important
Following , Murgu Inverse Method, which legaly work only whith ODDS, Only and only Logical Eternal Triads have infinity Up Connections
For First in Grid ((22k(x)-1)/3), k>1
For Second in Grid ((2(2k+1)(x)-1)/3), k>0
(for k=1 and k=0) are our first left connections part revealed in Murgu Arrow.
Don't forgot - Murgu_Table2To3 "demonstrated REVELATOR" of all
Logical Eternal Triad Nodes
as a
Collatz Conjecture POCKET MAP
which now can be EXTINDED as
ONLY HUMAN CAN IMAGE Collatz Conjecture
Do Not Forgot
MURGU_TABLE2TO3
brought 2 new demonstrated Formulas which build , Also DEMONSTRATED
2 Linear Functions Formulas
until there , and to understand need to became familiar whith Logical Inversion as those are mirroring it.




HERE, Collatz Conjecture Solved
via Demonstration and using drastic Math Rules avoiding proof as Math Method but only to reveal The Situation.

Collatz Conjecture Solved via Murgu Table2To3
Brought also any Logical Beauties to remark as part of them even may be util to strong Logic.
1. Infinity_Per_6_Plus_1 .
2. Math Metaphor - Murgu Arrow - with Headarrow 3-1-9 .
3. Collatz Conjecture Murgu Table2To3 as revealing how Collatz Conjecture is backed by 2 Linear Functions.
4. Only Human can Image Collatz Conjecture, making human proud as yet best intelligence.
5. Collatz Conjecture Murgu Inverse Method, as Collatz Conjecture Mapping.
Maybe best beauty logical got. We can now , starting from Unity to reveal Every Odd Integers Collatz Conjecture Connections a limit maybe only Infinity. Evens Integers are Contained by every odd X via
((2k)x ) then is a complete MAP.

As need to keep yet the material as was exposed and as working material need to go around even don't like to many repetitions, but, how soon is possible will be brought to a comercial form. For Now all which mater is what we get
Collatz Conjecture Solved.

Solution was brought by Math Name , Murgu_Table2To3, but can added also a Math Metaphor as neme
Murgu Arrow
As Mention, to Solution as will see will contribute also:
Murgu Lemas, Murgu Collatz Conjecture Formulas, Murgu Collatz Conjecture Inverse Method, important also Perfect Grid To Infinity and for "Only Human can Image Collatz Conjecture"
Murgu 3D Apple Tree Collatz Conjecture Logical Representation.
I am against of proof in Mathematics when is Involved Infinity, and for the Good Of Science , I hope , So Are You.
Now, reading any times all material will understood Logical Exclusions for Logical Dead Nodes (all odds Positive Integers divided one or more times by 3) , and all Evens Positive Integers .
Will see every Logical Dead Nodes Have infinity (not Infinity) connections but only one to a Odd, and this is by demonstration a Logical Eternal Triad.
All Evens Positive Integers will fall in a Logical Dead Node or Logical Eternal Triad by one ore more division with 2 . That let by Demonstration all back to Logical Dead Nodes .
Murgu_Table2To3 is a table wich via demonstration show Every Logical Dead Nodes key Connections are following 2 Linear Functions, and mean
Infinity Pay Respect For Collatz Conjecture.

A thing I let in suspense as hope will be solved by a high School Student.
But as maybe Internet is under
Hackerism Monoply
(see USA AEPDF by 9 years hide maybe by AI Internet Filters, But for sure handled by any hackers).
This is:
As we got 2 kind of Logical Eternal Triads , for what First into Grid Is generating Top Connections via ((22k - 1)/3) and second in Grid ((2(2k+1) - 1)/3). A position dependency.
This is simple to understand using Murgu Collatz Conjecture LDN Locators :
First LDN in Grid - (1 + 6 i)
Second LDN in Grid (5 + 6 i)
where i is Grid Number.
Then for first k = 0 to Infinity:
((2(2k+1) )(1 + 6 i)-1) never divided by 3
For Second k =1 to Infinity:
((2(2k) )(5+ 6 i)-1) never divided by 3
In any sense Murgu Lemas already did it.



    In Mathematics Proofs (as samples), in problems that have infinity as domain, Instead Of Demonstrations Can Induce Science Ambiguity.
In Mathematics Prooved must mean:
Demonstrated in general case, by using Strict Mathematics Rules.
Don't Kill My!, but sure, will be a time when to remark how Proof for Fermat's Last Theorem destroy any Science Beauty of Fermat's Last Theorem, first - Infinity Divergent Conjecture -
Infinity Divergent Conjecture is a heavy task but can be needed in Science , at the last for powers 4 and 5 as 2 and 3 solved by Fermat's Last Theorem VIA FMIE.
FMIE - Fermat Murgu Impossible Equations, Mathematics Method which used The Lost Fundamental Operator and SENT Fermat's Last Theorem in Fundamental Science as Second Fundamental (first is The Lost Fundamental) by Sending all Fermat's Last Theorem solutions for all powers in Irrational, including power 2.

AS example Fermat Murgu Table (don't cover all Integers but a large domain) show
X3 + Y3 - Z3
not equal with 5 as predicted in any Proof presentations
But also not equal with 3 , or 4 , or 13 ..
Maybe neither Fermat Murgu Impossible Equations can't demonstrate it, but CLEAR DEMONSTRATED

X3 + Y3 - Z3 ≠ 0     into Integers
AND it is Important as will see FOR ALL Science.

—  Collatz Conjecture Solution   —
EXPLAINED
—  Collatz Conjecture Explained   —



1. Murgu 3D Apple Tree Collatz Conjecture Logical Representation.
Helped to observe and remark any common propieties related to Collatz Conjecture of positive Integers and idea was part coming from Murgu Conjecture Vicious Redundancy.
From here separated Odd Integers in Logical Dead Nodes , Logical Eternal Triads . The names are relative, and can be changed.
A important remark is
The Trunk is formed by Evens of form
(2 4k )
and Odds of form
(((2 4k ) -1)/3 )

2. Perfect Grid To Infinity
Inspired by first one in any part, consists in , dividing Positive Integers [1, ∞] in groups of 6 Integers
((1,2,3,4,5,6), (7,8,9,10,11,12), (13,14,15,16,17,18),..)
This one, First one and Murgu Inverse Method helped us to make 2 observations important in DEMONSTRATION.
Logical Elimination. Logical Escape.
And then for Analytic, remain in every Grid only 3 Odds Integers. Now, Murgu 3D Apple Tree Collatz Conjecture Logical Representation and Perfect Grid To Infinity made a order by pointing to any facts via Murgu Inverse Method as observations.
Every Grid consists into 2 Logical Eternal Triad Nodes and a Logical Dead Node (Odds Divided one or more times by 3). Also as observations , but demonstrate into future, a Logical Dead Node don't generate Left and Tops Connections been connected to Tree only by one connection - Right-Down , which lead to Second Escape.

3. Logical Elimination.
is not a real Elimination but a reduction of moment.
In Fact, Collatz Conjecture Solution need to demonstrate
Infinity Pay Respect For Collatz Conjecture
and it mean an Integer posted to Infinity will fall into Unity via Collatz Procedures but also will not fall in a Leaf Effect (in a circle of multiple roots).

Logical Dead Node have a single connection to the trunk, and it can't be to another one , but to an Logical Eternal Triad Node
DEMONSTRATION
for every odd Integer Left and Tops Connection are obtained via 2 Murgu Formulas(notate by x any odd Integer)
((x(24k )-1)/3)  k=1 to ∞ or

((x(22k +1 )-1)/3)   k=1 to ∞
for a Logical Dead Node (x = 3b), then
(12b -1) or (6b - 1) never divided by 3
Need to remark as we use Murgu Inverse Method will refer any connection as Odd To Odd.


Then For Moment will eliminate all Logical Dead Node and all Evens Integers as every one via one or multiple divisions by 2 will fall in a Odd Integer. This do not need a demonstration.
Logical Dead Node not having Left and Tops connections automat can have also a connection to another Logical Dead Node, and "push" its Collatz Conjecture responibility to Logical Eternal Triad Node. Also need to know as util a Logical Eternal Triad Node have an Left Connection and infinity (Infinity part) Top Connections and a Right- Down one coming from its Generator. As example for 5 is 1.
Then in our attention for a time will remain into every Grid 2 Odds named by hazard and aspects Logical Eternal Triad Nodes, ex: Grid 0 [1,5], grid 1 [7 , 11] , grid 2 [13, 17], ...


All Responsability for Collatz Conjecture. Now, fall on
Logical Eternal Triad Node's.


To see if Logical Eternal Triad Node's will offer a ANSWER, need to see and Understand:
Murgu Lemas see Murgu Lemas but specially
Collatz Conjecture Murgu Formulas ,
Murgu Inverse Method ,


    If Understood:
- Perfect Grid to Infinity of 6 succesiv Pozitive Integers starting From UNITY (which is simple).
- Logical Exclusion for evens will let Logical Responsibility to ODDS, 3 of them in Each Grid.
- Logical Exclusion for Logical Dead Nodes - DEMONSTRATED - will let Logical Responsibility for 2 of them which formal were named Logical Eternal Triad and are different as position in Grid and then as mode how Generate its Left and Top Connections (Read your logic for Murgu Inverse Metyhod which will Map Collatz Conjecture, and after will can return to It).
-Follow and exercise with Murgu Inverse Method .
-Luck for first 6 Collatz Conjecture Murgu Formulas and specialy String Locators .
- By applying Murgu Inverse Method for every Logical Eternal Triad ( name is formal, but with your accept, describe theirs beauty) will get Murgu_Table2To3 which luck like a Math Metaphor unwaited Marker USA Murgu Arrow and 2 New Collatz Conjecture Murgu Formulas

THOSE WILL be DEMONSTRATED

and will be
Collatz Conjecture Solution Key

    Using Murgu Inverse Method to get Left Eternal Triad connections we get by Deduction Murgu_Table2To3 (part Of it) which Graphic inspire Math Metaphor Murgu Arrow.
This induce idea as proof of General Proprieties of Double Linearity, as every kind of Logical Eternal will Follow a Proper Linear Function for its Left Connection wich can be another Logical Eternal Triad or a Logical Dead Nodes .
HERE CAN BE 2 OR 3 LOGICAL INVERSIONS , but in time and exercise will be understood.
That let us to deduct 2 new Collatz Conjecture Murgu Formulas :
T1(i+1) = T1i + 8
T2(i+1) = T2i + 4
     Those are DEMONSTRATED down and can say
Collatz Conjecture - Problem Closed

    Now using Collatz Conjecture Murgu Formulas String Locators We will get 2 Linear Function which reveal for every kind of LET , Logical Left Connections follow a Linear Function Of Grid i .
LLET1i = (1 + 8 i) for first in Grid i, i>0
LLET2i = (3 + 4 i) for second in Grid i, i>1
That mean, every LETN have a UNIQUE Left Logical Connections
AND
as we said before, Collatz Conjecture PROBLEM fall in LETN and both Functions show unicity and also no Compet in any Points

Collatz Conjecture Solved
Collatz Conjecture Solution Murgu_Table2To3
and to give to Math rights to Essay via
Marker_USA_Murgu_Arrow

Observation:
The fact that Table is 2 To 3 is not a impediment to Solution. Do not forgot we make a Logic Inversion and apparent from every grid we have 3 odds , then luck like we are lossing one .
Not, the 3-th Component is hided in Trunk like a exponential "bow to Infinity"
((4k - 1)/3)   k>1
as a new Math Metaphor.
     Also, don't forgot we used Murgu Inverse Method which can make a Collatz Coinjecture Complete Map using Murgu Arrow as
Collatz Conjecture Pocket Map
completing it whith Logical Top Nodes . If Make a Mind Analyze after will see Collatz Procedures Will follow all those Patts.

     Murgu_Table2To3 reveal and certify by demonstration Collatz Conjecture as 2 Linear Functions which don't compet, offering for every Logfical Eternal Triad Unicity for its first Collatz Connection, and a Trunk as a exponential metaphoric "bow to Infinity".
    This DEMONSTRATED :
Infinity Pay Respect for Collatz Conjecture
and unicity of LETN first connection show
UNITY IS UNIQUE ROOT
But Murgu Inverse Method show how Unity Is Logical Eternal Generator Father for all.     Murgu Table2To3 brought 2 Math Metaphor's : Marker_USA_Murgu_Arrow and Trunk as "Arch To Infinity".
FORGIVE MY! for any small errors , but for sure not are about essence.

Murgu Lemas
1. All Positive Integers Of Form
((22k)-1 )
are divided one or more times by 3.
2. All Positive Integers Of Form
((22k+1)+1 )
are divided one or more times by 3.


   COLLATZ CONJECTURE 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. COLLATZ CONJECTURE MURGU FORMULAS are obtained directly from Collatz Procedures integrating Perfect Grid To Infinity and Aple Tree Logical Representation.

Then Murgu Formulas need no demonstrations excepting 4 of them which reveal
Logical Eternal Triads as Generators
used in Murgu Inverse Method and in revealing Math Metaphor Collatz Conjecture Murgu Arrow with a remark
2 of them reveal unique Logical Left Eternal Triad Connection and 2 of them Logical Tops infinity Logical Eternal Triad Connections which can be reunite in generalized 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 (every) Grid will contain 2 Logical Eternal Triad and a Logical Dead Node

MURGU FORMULAS

   4 Collatz Conjecture Murgu String Locators Formulas
Grid Locator    Gi = 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 = 3+6i   

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 as generators, but Logical Dead Nodes are not uncovered
as, appear as any Logical Eternal Triad Left connection .
Those don't have Left And Top Logical Connections , then only one which is Logical Right And Down to a Logical Eternal Triad . Then for moment can be out of our imediat scope and return separately to it.

Knowing it, can remark any Logical Laws about :
1. A LDN do not generate Left connection and Tops Connections. (This make The Differnce between LDN and LETN as all are Odds)
2. For all LDN all Right-Down Connections are unique and lead to a LETN.
3. A LDN do not generate Left connection exclude as absolute as two LDN to have direct connection.
Those laws refer LDN's.

Those 2 general which reveal
LDN as generators(in Murgu Inverse Method) are:
(2(2k) v1i -1 = 3 ui)    for   LETN1

(2(2k+1)v2i -1 = 3 ui)     for   LETN2
where v1i is a ET , first in Grid and v2i is a ET , second in Grid i
Those are Logical Top Commections and first one Left Connection, 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:
(4v1i -1 = 3 ui)    for   LLETN1

(2v2i -1 = 3 ui)     for   LLETN2
where v1i is a ET , first in Grid and v2i 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 and all 4 into Maping Collatz Conjecture using Murgu Inverse Method.
Collatz Conjecture Murgu Arrow is a Math Metaphor Name of
Collatz Conjecture Solution
Murgu Table2To3
and then those Equations Unicity need Demonstration as First and second Logical Eternal Triads Generators have its proper Function.



Demonstration
For First LETN1 in Grid, to demonstrate (2(2k+1) v1i -1 ) never divided by 3
(2(2k+1) v1i -1 = 3 ui)    for k=0 to ∞   LETN1
False
To write v1i = 1 + 6i and k= 0 then ((2 + 12i) -1 ) not divided by 3
for k =1 ((8 + 48i) -1 ) not divided by 3
((2(2k+1) +6i(2(2k+1))) - 1)/3 =
(( 2(2k+1) -1)/3) + (2i(2(2k+1)))
As already know from Murgu Lema ( 2(2k+1) +1) divided by 3 then
( 2(2k+1) -1) never divided by 3 .

For Second LETN1 in Grid to demonstrate (2(2k) v2i -1 ) never divided by 3
(2(2k) v2i -1 = 3 ui)    for k=2 to ∞   LETN2
False
To write v2i = 5 + 6i and k= 2 then ((20 + 24i) -1 ) not divided by 3
for k =3 ((320 + 384i) -1 ) not divided by 3
as 319 not divided by 3
(5(2(2k) +6i(2(2k))) - 1)/3 =
(((5 (2(2k) -1)/3)) + (2i(2(2k))) )
(2(2(2k) - (2(2k) + 1)/3 ) + (2i(2(2k))))
As already know from Murgu Lema ( 2(2k) - 1) divided by 3 then
( 2(2k) + 1) never divided by 3 .



, 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 LETN1i, LDNi and LETN2i (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:
LFETNC1(i+1) = LFETNC1(i) + 8 for   i grid indice,1 position
LSETNC2(i+1) = LSETNC2(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 W1i and W2i 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.





Remark : Apparent , via Murgu Inverse Method for every odd, we lost connections to evens of form
((3x + 1)2k)    k =0 to ∞
Right Connections in our method contained by LETN Generator
ex: For 5 By Unity
Anyway we also can add those to Pocket Map of Collatz Conjecture after.
For Collatz Conjecture important is to demonstrate : Infinity Pay Respect For and not Leaf Cicling effect
Murgu_Table2To3 did it.


Murgu Inverse Method for Collatz Conjecture is based on Collatz Conjecture Procedures reversion, Collatz Conjecture Murgu Formulas and demonstrated Murgu_Table2To3 logical results.
Collatz Conjecture Murgu Inverse Method can do a Map Of Collatz Conjecture Positive Integers. For Negatives, Collatz Conjecture have a Mirror transfered into Positive by
Murgu Conjecture Vicious Redundancy (3x - 1)

Murgu Inverse Method

Is using Murgu Formulas Grid Eternal Triads Nodes Locator (if is first or second in its Grid) , and then , starting from unity and using Collatz Conjecture Murgu Formulas to get Valoric Left Eternal Triad connection which can be another LETN or a LDN.
Like we did for Murgu_Tables2To3 which can be considered a Map for first 34 Grids, a
Pocket Map
, but not incomplete , because we know NOW
For an LDN x Meet on Pocket Map we need to add only and only Evens Connections
((2k)x)   k = 1 to ∞
BUT FOR a LETN x
((2k)x)   k = 1 to ∞
but also
if first LETN in grid (((22k) x -1 )\3)    k =2 to ∞

OR if second LETN in grid (((22k+1) x -1 )\3)    k =1 to ∞

This Method can be extended for more Grids using strong computer and software program in a language which support BigInt.

Our Pocket Map apparent 1/3 connections are missed , which isn't real. Missed connection are included in Trunk as LDN wich have Right-Down connection in trunk

((22k)1 =3x +1 )    k =2 to ∞




Now we are ready to get Collatz Conjecture Solution
Murgu Table2To3
Reveal hiden and incredible Collatz Conjecture Double Linearity



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
((2(2k + 1)W -1)/3) or ((22kW-1)/3)

First (LETGN1): ((22kW-1)/3) - Second(LETGN2) in Grid: ((2(2k + 1)W -1)/3)
generate Left Side new connection, LETGN or LLDN and also To TOP Will Be generated infinity(FOR LETG only).
For This Scope we Generated A Table where First Column folow Grids Eternal Triads Only fstarting with Grid one , and the row all Odds which wiil be Grids Eternal Triads and Dead Nodes. Using Murgu Inverse Method Murgu Formulas for Logical Eternal Triads (reminded above) to get for every Logical Triad Left Connection
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
LLFNC1i, where 1 - first in grid i, LLSNC2i where 2 - second in grid i .
To substitute those for easy with
T1i, where 1 - first in grid i, T2i where 2 - second in grid i .
Building the tabel for, can't to escape from observation a fact revealed down
T1(i+1) = T1i + 8
T2(i+1) = T2i + 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 T1(i+1) and T2(i+1) , then T1(i) and T2(i) from anything else and compare.(Method 1)
Or, using Add Absurd.(Method 2)

For it need to connect
T1(i+1) and T2(i+1), T1(i) and T2(i)
to theirs Logical Eternal Triads via Murgu formulas.
(3(T1(i+1))) = (4(W1(i+1))-1)   E.1
(3(T1(i))) = (4(W1(i))-1)   E.2
we know also (W1(i+1)) = ((W1(i))+6) and then E.1 can be write
(3(T1(i+1))) = (4(W1(i))-1+24)   E.1
or (3(T1(i+1))) = ((3(T1(i)))+24) and dividing by 3

T1(i+1) = T1i + 8

for Logical Eternal Triad Second position in grid

(3(T2(i+1))) = (2(W2(i+1))-1)   F.1
(3(T2(i))) = (2(W2(i))-1)   F.2
we know also (W2(i+1)) = ((W2(i))+6) and then F.1 can be write
(3(T2(i+1))) = (4(W2(i))-1+12)   F.1
or (3(T2(i+1))) = ((3(T2(i)))+12) and dividing by 3

T2(i+1) = T2i + 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
LLETC1i = 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
LLETC2i = 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 (2k (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 LETN1i, LDNi and LETN2i (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.

LLETC1i = 1 + 8i where i is Grid i LETN position 1. FC.1
LLETC2i = 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                                                         

Murgu_Table2To3 which show for first 34 Grids Logical Eternal Triads Collatz Procedure Connections.
Instead Of Table 2To3 Reverse Logic, here became clear divergence on.



COLLATZ CONJECTURE SOLVED

WE DEMONSTRATED LOGICAL DEAD NODES ARE LEFT SIDE CLOSURES
THOSE CAN'T EVOLVE LEFT SIDE CONNECTIONS.
A COLLATZ PROCEDURE STARTING FROM LOGICAL ETERNAL TRIAD , NEVER WILL MEET OR PASS ANY LDN , LLDN.
COLLATZ PROCEDURES STARTING FROM LLDN ALSO WILL NOT PASS OR MEET ANOTHER LLDN, AND VIRTUAL AS TO DEFINE ALL COLLATZ PATHS CA BE RELATED AS COMPLETE PATH FOR ITS DEAD NODE

ALL COLLATZ COMPLETE PATHS ARE INFINITY/6 (∞/6)

NOW
Collatz Conjecture Murgu Arrow (VIA New Formulas) Cover Totaly Collatz Conjecture and brought(implay also Murgu Inverse Method) for it Logical Continuity which
Math can to afford now for Collatz Conjecture.
Even Collatz Paths are covered there , with a mention
for Logical Left Dead Nodes or Logical Left Closures node Logical Right Eternal Triad Connection can be obtained using one time Collatz Procedure.



Now Perfect Grid To Infinity
became easy for using

REDUCTION AD ABSURD



6 Components Positive Integers began to number with 0 , because of 0 grid assymmetry.
Now, to remind any important things before start:
don't forgot Murgu Lemas, Murgu Collatz Conjecture Formulas
in a grid have 2 kind of Eternal Triad
Collatz Conjecture Murgu Arrow Formulas

Now to consider by absurd for grid K for both Eternal Triads is a Truth that will have a Left Connection another Eternal Triads or Dead Nodes .
Then If for Grid (K + 1) also this is a truth
For Collatz Conjecture is a truth
do not present Root Reversion or Leaf Effect.
I repeat for Dead nodes do not need to do it as those have only rigths side connections wich are to a eternal triad or direct onto Trunk to Unity.
To take a grid Nr. K and to analyze it for Grid (K + 1)
As known, First Eternal Triad grid K will be (6K + 1)
and in (K + 1), (6K + 7) Then: Via Murgu Formulas





COLLATZ CONJECTURE SOLVED


COLLATZ CONJECTURE SOLVED
Keys for: Perfect Grid To Infinity
Murgu String Locators Formulas
Reverse Logic
Murgu Inverse Method
Collatz Conjecture Murgu Arrow.
Logical Dead Nodes demonstrated Exclusion from Problem Creators.
Logical Eternal Triads 2To3 Table.
New Reverse Logic Collatz Conjecture Murgu Dual Linear Functions .
DEMONSTRATED via Math rules.
Logical Eternal Triads First Left Connection Murgu Formulas
Logical Eternal Triads Top Connection Murgu Formulas
Maybe even Murgu Lemas hided in LDN not Logical Left and Top Connections>
and 3D(2D) Logical APPLE Tree Corner Web Collatz Conjecture Representation .
and we get First Infinity Exact Reference
Infinity_PER_6.
Make 86 unsolved Problem

COLLATZ CONJECTURE SOLVED
as Brilliant
Earth Science AND Culture Event.
Human Intelligence Victory.



Down, is a try image of a inverse method starting from Unity incomplete and maybe any small errors , but can help for Understanding.



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


Eternal Triads Top Connections , infinity, appear via a Murgu Formulas, because Infinity connections can't imaged. Those can be Imaged only in Human mind . Need to remark, for Collatz Procedures , every one present Unicity.




All Collatz Conjecture Paterns and so to say any presented as proofs
show DIVERGENCE and Then can be considered
Knowledge Labyrinth Extraction
Collatz Conjecture Solution
Collatz Conjecture Murgu Arrow



Down I will keep yet any tables wich can be used in understanding and everyone can contribute in times.


Format HTML seem to be single wich can cover a Collatz Conjecture Book.


999 1499 2249 1687 2531 3797 89 67 101 19 29 11 17 13 5 16 1
3997 1499 2249 1687 2531 3797 89 67 101 19 29 11 17 13 5 16 1

This Table Show how Collatz Conjecture Murgu Inverse Method can Complete a Collatz Path.



- - Grid Nr 31
187 281 211 317 119 179 269 101 19 29 11 17 13 5 1
187 249 LLDN
 191   287   215   323   485   91   137   103   155   233   175   263   395   593   445   167   251   377   283   425   319   479  719   1079   1619   2429   911   1367   2051   3077   577   433   325   61   23   35   53   5   1 
191 127 169 225 LLDN
- - Grid Nr 32
193 145 109 41 31 47 71 107 161 121 91 137 103 155 233 175 263 395 593 445 167 251377 283 319 479719 10791619 2429  911   1367   2051   3077   577   433   325   61   23   35   53   5   1 
193 171 LLDN
197 37 7 11 17 13 5 1
197 131 87 LLDN
- - Grid Nr 33
199 299 449 337 253 95 143 215 323 485 91 137 103 155 233 175 263 395 593 445 167 251377 283 425 319 479 719 1079 1619 2429 911 1367 2051 3077 577 433 325 61 23 35  53   5   1 
199 353 235 313 417 LLDN
203 305 229 43 65 49 37 7 11 17 13 5 1
203 135 LLDN
- - Grid Nr 40
241 181 17 13 5 1
241 321 LLDN
245 23 35 53 5 1
245 163 217 289 385 513 LLDN
- - Grid Nr 45
271 407 611 917 43 65 49 37 7 11 17 13 5 1
271 361 481 641 427 569 379 505 673 897 LLDN
275 413 155 233 175 263 395 593 445 167 251 377 283 425 319 479 719  1079   1619   2429  911  1367   2051   3077  577 433 325 61 23 35  53   5   1 
275 183 LLDN
- - Grid Nr 73
439 659 989 371 557 209 157 59 89 67 101 19 29 11 17 13 5 1
439 585 LLDN
443 665 499 749 281 211 317 119 179 269 101 19 29 11 17 13 5 1
443 295 393 LLDN


Example of Generating Collatz Conjecture Complete Paths Starting from an Eternal Triad and using both Procedures.



 999   1499   2249   1687   2531   3797   89   67   101   19   29   11   17   13   5   1 
993 745 559 839 1259 1889  1417   1063   1595   2393   1795   2693   505   379   569   427   641   481   361   271   407   611  917   43  65 49 37 7 11 17 13 5 1
987 1481 1111 1667 2501 469 11 17 13 5 1
981 23 35 53 5 1
975 1463 2195 3293 1235 1853 695 1043 1565 587 881 661 31 47 71 107 161 121 91 137 103 155233  175   263   395   593   445   167   251   377   283   425   319   479   719   1079   1619   2429   911   1367   2051   3077   577   433   325   61   23   35   53   5   1 
969 727 1091 1637 307 461 173 65 49 37 7 11 17 13 5 1

Example: Logical Dead Nodes are Complete Collatz Conjecture Paths.


Collatz Conjecture Solved
Even if are Math Curiosity Problems:
Murgu Conjecture Vicious Redundancy
Murgu 3xPlus3 Conjecture
and 3X - 3 . come as Provocations.


For ,any small grammar , I hope, errors, I will return when time for.



1. All Collatz Conjecture Murgu New Concepts are needed.

for geting
Collatz Conjecture Solutions
and
Collatz Conjecture Solved


Down , any Examples of Collatz Conjecture Patterns wich as Proofs LEAD to
Divergence
Then how many proofs we will get, this divergence will let a shadow for:
Collatz Conjecture Solutions
VIA Proofs.


191 287 431 647 971 1457 1093 205 77 29 11 17 13 5 1
10111 15167 22751 34127 51191 76787 115181 43193 32395 48593 36445 13667 20501 961 721 541 203 305 229 43 65 49 37 7 11 17 13 5 1
987 1481 1111 1667 2501 469 11 17 13 5 1
975 1463 2195 3293 1235 1853 695 1043 1565 587 881 661 31 47 71 107 161 121 91 137 103 155 233  175   263   395   593   445   167   251   377   283   425   319   479   719   1079   1619   2429   911   1367   2051   3077   577   433   325   61   23   35   53   5   1 

' Collatz Conjecture patterns which show or intuitively lead to Divergence.
Then a proof or multiples can't lead to a sure solution.