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Collatz Conjecture Unique Solution
Murgu Table2To3
which been validate multiple times by:
Murgu - Collatz Conjecture as Pocket Map
Marker USA Murgu Arrow
Murgu Infinity Collatz Table2To3 Functions
AND MATHEMATIC LOGICAL BEAUTY LOGICAL
Infinity Marker USA Murgu Arrows
Collatz Conjecture One Dimension Array.
Maybe first Group Theory with a final Absolute Truth Validation.

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Murgu Table2To3 or Collatz Table2To3 are Mathematic 2 Collatz Conjecture Murgu New Formulas

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((2^(2k+2)[1+6i]) - 1) = 3 Q_i
((2^(2l+1)[5+6j]) - 1) = 3 Q_j

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for k=l=0 , and if wish to see it before exploring what is down, here is a link for.
Murgu_Table2To3 - Collatz Table3To3, Click here

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For Infinity Murgu Table2To3 (above functions for k=l=0 to Infinity), where we give an small example - See :
Infinity Murgu_Table2To3 - Collatz Table3To3 , Click here

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Grids separation and new Mathematics Logic implied may can include Collatz Table2To3 into Group Theory beauty.

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For Mathematicians
Collatz Table2To3 Solved Collatz Conjecture
even by declaring :
Murgu Collatz Conjecture Grids
(I will say as Group Theory Method)
Splitting Grids in Odds and separed by Evens.
(via pure and legal observation - divisions with 2 lead to an Odd)
and as will be explained.
and demonstrating all around

Collatz Conjecture Murgu Inverse Method legacy for
Collatz Conjecture Odds To Odds Formulas
Separation, for work only, of now Odd Integers in Grids in LET₁ⁱ , LDNⁱ and LET₂ⁱ where i is Grid Number.
New Murgu Formulas implied to solve it.
and Murgu Table2To3 as result.

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Collatz Solved even will need an effort to understood, all Math rigor in demonstrations.

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Murgu Table2To3 or Collatz Table2To3 Functional Beauty revealing Functional Divergence as Double Linearity.
Infinity Murgu Table2To3 a Mathematics New Art Concept.
Table2To3 are Up Functions as Table to visualize theirs proprieties.

OBSERVATION :
As I consider Proof in Math as forced and imposed by intel tiredness state, then a give-up. need to say:
Collatz Table2To3 not Proof but an absolute Math rigor SOLUTION.
Infinity Murgu2To3 come to validate it for not skilled in Math
Murgu Table2To3 or Collatz Table2To3 not an P_Vs_NP or Proof solution , instead
A Complete Mathematics All Rigor.
As after more then 1 year , Murgu Table2To3 wasn't understood I will rexplain it
for every noon very skilled in Math, and hope I have talent for!

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As Murgu Table2To seem not understood yet
I don't see motivation for but will try to Reexplain.
Murgu Table2To3 followed by Infinity Murgu Table2To3 are
H-Intel and Science Logical Beauty
and then merit attention for:
Marker USA Murgu Arrow
Infinity Per 6
Murgu Inverse Method
as Spider following Infinity Murgu Table2To3 To weave Collatz Conjecture Web NODES
and to reveal
Collatz Conjecture One-Dimensional Array

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Merit All Science Attention.

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Notes:
A important key is separations of positives Integers in GRIDS of 6 Integers, which for sure don't need explanations then that is utile fact for our scope.
All Formulas, Murgu Inverse Method , splitting grids in LET₁ , LDN, LET₂, splitting Collatz Conjecture Formula in 3 parts, LDN don't have another ODD Integer UP connection, LET₁ and LET₂ Formulas ,
are explained and demonstrated
in 2024 Murgu Table2To3

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Then 2nd key for our scope is a Math pure Observation
All Evens positive Integers via division with 2, fall in a odd, and don't think to need any explanation. That led to idea of separating Infinity 6 grids in grids of 3 odds Integers.

Now the problem is how to control those grids using Collatz Procedures ?
A strong - strong IDEA helped on
- Collatz Conjecture Murgu Inverse Method -

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which broght a big double advantage :
possibility of avoiding Division with 2.
possibility or building (weaving) Collatz web nodes (into odds) starting from Unity.
That need explanations
writing first Collatz Formula for a general case
((3x + 1) = 2^k O ) - where O is an Odd
akso as
(3x = (2^k O -1)) --- we get br (x = (2^k O -1)/3)
for unique k - x and O unique .
Murgu Inverse Method Formula is perfect valid, but, brought also any caracteristics for our odds into grids, as
For LDN not applicable
For LET₁ and For LET₂ wear different form.
Then all this will need explanations and when is the case DEMONSTRATION.
--- Infinity Murgu Table2To3 bring Double Infinity Linear Functions which Practic Reveal for every Odd Positive Integer its Ways to Unity and Infinity.
Where Murgu Collatz Inverse Method play important Role
The Spider which Weave Collatz Web Nodes following Murgu_Tables2To3 LOGIC.
Follow My to see it!
That is what will do into next passages.

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Heavy Part Explained

First to explain Murgu Inverse Methods Formulas for every category in Grid.
Anyway New Murgu Formulas:
1 + 6i --- LET_1i
5 + 6i --- LET_2i
3 + 6i --- LDN_i
Not need explanations as clear express positions on grid i, and to be easy will use a short forms :
L_1i = (1 + 6i) -- L_1i real value
L_2i = (5 + 6i) -- L_2i
D_i = (3 + 6i) -- D_i
also not need demonstration
D_i cover all LDN forms as (3^k), ((3^k)L_1i ) and ((3^k)L_2i ).
For L_1i :
Murgu Inverse Method in general Form is
(3x = (2^k O -1))    clear not all divided by 3-- will be an odd
That will force us to treat appart formula for LET_1i, LET_2i, and LDN_i . Then to change x notation with Q -- an O odd positive Integer with one of ours grid odd. (3 Q_i ) = (2^k (L_1i ) -1)
(3 Q_i ) = (2^k (1 + 6i) -1)
(3 Q_i - 6i (2^k))= (2^k -1)
3(Q_i - i (2^k+1)) = (2^k -1)
in old materils this appear as Murgu Lema
(2^k -1) to be divided by 3 need to be of form
(2^(2k+2) -1) k =0 to Infinity
That mean for LET_1i Murgu Inverse Formula will be
(2^(2k+2)L_1i - 1) = 3Q_i
or (2^(2k+2)(1 +6i) - 1) = 3Q_i

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Now to demonstrated Murgu Inverse Formula for LET_2i .
L_2i = (5 + 6i) -- L_2i in grid i.
We will pass direct to Murgu inverse Formula for:
((2ⁿ[5+6i]) - 1) = 3 Q_i

(2ⁿ[5+6i] - 1) = (3 Q_i )
((2ⁿ(5) - 1) + 6i(2ⁿ)]) = (3 Q)
(2ⁿ(5) - 1) divided by 3 only for (n) of form (2k+1)
This is also in a Lema , but may be demonstrated using our new grids.
Both formulas may be demonstrated via relation between grids of 10 and grids 6 .
We demonstrated first one, then to do it for second.
To note with l =(2ⁿ(5) - 1)
In grids of 6 this mean this l will be now in grid
(2ⁿ(5) - 1)/6
(2^(n-1)(5))/3
(2⁽ⁿ⁾) will send 5 in a grid of form
(2⁽ⁿ⁾(5))/6
(2⁽ⁿ⁾(5) - 1) = 3 u then
in grids of 10 (2⁽ⁿ⁾(5)) will be the last
can write it also as (10j)
(10j - 1) = 3 u
(2⁽ⁿ⁾(5) - 1) = (10j - 1)

(2⁽ⁿ⁾(5) ) = (10j)
(2^(n-1) = j)

if n = (2k+2) then (2^(2k+1)(5) - 1) = 3 u
if n = (2k+1) then (2^(2k)(5) - 1) = 3 u
which is a truth in part as one above exclude k = l+1 which is (2^(2k+2)(5) - 1) = 3 u
then for - L_2i = (5 + 6i) -- L_2i in grid i.
Murgu inverse Formula for:

((2^(2k+1))(L_2i) - 1) = 3 Q

To make distinct Q coming from L_2i or L_1i will write Murgu Inverse Methods Formulas as:

((2^(2k+2))(L_1i) - 1) = 3 Q_1i
((2^(2l+1))(L_2j) - 1) = 3 Q_{2 j}

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via k,l,i,j those formula wear their independence and show to cover diferent domain and
Very Important revealed achivment

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express Collatz Unicity, write as

((2^(2k+2))(1 + 6i) - 1) = 3 Q_1i
((2^(2l+1))(5 + 6j) - 1) = 3 Q_{2 j}

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Those are Collatz Conjecture Murgu Formulas

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and theirs legacy over Collatz Conjecture been demonstrated
As said, multyple times, LDN_i not accept a Murgu Inverse Formulas but only Collatz Procedure. Motivation for what, will see and demonstrated soon.

That mean , LDN_i are acting like "closures". This and Collatz Unicity are 2 big revelations discovered via Murgu Inverse Method and pillars on solving Collatz Conjecture.

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Collatz Unicity

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is contained in Collatz Conjecture Murgu Formulas, and mean

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For all k,l,i,j covering (0 to Infinity) there will not be an Q_1i equal as value with an Q_2j .

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Logical Dead Nodes Act In Collatz Conjecture as Closures.

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This is one of pillars of solution, and mean a LDN do not have an UP Collatz Connection.
Demonstration :
((2^k)(3 D) - 1) = 3 Q
3((2^k)D - Q) = 1
This is Impossible

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But here we re-meet Double false redundancy Of Truth .
As Collatz Conjecture Procedure over a LDN have value - important also .
Thath mean, Collatz Conjecture Procedure reveal a connection Down , when Murgu Formulas reveal connection UP . For that I separed Odds in LET's and LDN.
this caracteristic is more important as born a important Collatz Conjecture rules :

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As an LDN do not have UP Collatz Connection , then is Down Collatz Connection is only and only a LET.

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As an LDN do not have UP Collatz Connection, a Collatz Pattern starting from a LDN will not pass another LDN.

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Those Rules are also Important for Collatz Conjecture Solution.

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Now we have all to start Murgu Table2To3
which is (are) Murgu LET's Formulas (equations) for k=l=0
(named Table for visualization and to remark new referince system - important for -)
Murgu Table2To3 don't need LDN_i Formulas as all LDN will fall via Collatz Procedures Into a LET And then if can bring all LET's to an functional which to include Collatz Procedures divergence into ordonate functional will - Solve Collatz Conjecture. Murgu Inverse Methods Formulas became key for:

((2^(2k+2))(L_1i) - 1) = 3 Q_1i
((2^(2l+1))(L_2j) - 1) = 3 Q_2j

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Those now became linear functions , one starting for Unity , second from 5

Q_1i = ((2^(2k+2))(L_1i) - 1)/3
Q_2j = ((2^(2k+1))(L_2j) - 1)/3

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Mention : 5 is connected to Unity via a collatz pure procedure (3(3) +1) = 10.
For Those Double Linear Functions every LET to Q unicity is contained in theirs linear Property .
Q_1i = ([(2^(2k+2))(1+6i)] - 1)/3
Q_2j = ((2^([2k+1))(5+6j)] - 1)/3

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as Step are 8 and 4 the chance as any (Q_1i = Q_2j) null.

Collatz Conjecture SOLVED BY MURGU TABLE2TO3 -
Table2To3 come from the fact, our Functions axe Ox contain all Odds Positive Integers when axe Oy only LET'S (exclude LDN's). BUT, but, as Collatz Conjecture was provocative 83 years for all H-Intel and to confirm totally its beauty, down will Explain Infinity Murgu Table2To3 (come also with Infinity Double Linear Functions) wich will make for Collatz Conjecture

Murgu Inverse Method - Collatz Conjecture Poket Map
like
Spider Which weave all Collatz Conjecture Nodes using
Infinity Murgu Table2To3 as correct logical tool.

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Infinity Murgu Table2To3 or Infinity Collatz Table2To3 Validate
Murgu Table2To3 Or Collatz Table2To3
and brought via
Marker USA Murgu Arrows

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A Mathematics Logical Beauty of Collatz Conjecture
Collatz Conjecture As One-Dimensional Array.

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How Collatz Table2To3 Solved Collatz Conjecture For Mathematicians?
Murgu Collatz Table2To3 are 2 Linear Functions with diferent slopes , one starting form Unity and second from 5 covering all LET's .
Collatz Procedure for 5 unite those into an Arrow.

The rules of Closures , LDN's integrated with "Collatz Unicity " make logical
- ALL Integer Fall Back into Unity -

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If LDN's UP demonstration was relevant, can pas this one!
Logical Dead Nodes

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Logical Dead Nodes are cause of Collatz Conecture Functional Divergent behavior, but as we see now, divergence not absolute, instead, functional one.
That is what we will demonstrate here.
We defined Logical Dead Nodes as all Odds Positive Integer divided one or more times by 3.
Murgu Collatz Conjecture Grid reper those as
LDN_i = (3 + 6i)
and occupy 3-rd pozition in every grid - aS EVERY GRID I HAVE AN 2 LDN's one odd and second in position 6-th even.
We will work only with odds as evens fall via 2 division in odds.
then a LDN may be writed as L_i = ((3^k)D_i) , or simple
L_i = (3D_i)
if include D as all Odds Positive Integers.
LDN's don't accept Murgu Inverse Method direct applied, and it because Murgu Inverse Method reveal an up cnnection to another odd when a LDN is a finnall Node Odds Connections .
Demonstration
3 (L_j) +1 = 2^k(1+6i)
or
3 L_j +1 = 2^k(5+6i)
For first 3 (3 D) +1 = 2^k(1+6i)
9D +1 = 2^k(1+6i)
3(3D + (i)(2^k+1)) = 2^k

2^k never divided by 3.

For Second : - 3 (L_j) +1 = 2^k(5+6i)
9D +1 = 2^k(5+6i)
3(3D + (i)(2^k+1)) = 5(2^k)
5(2^k) never divided by 3.
That mean a LDN do not have an Up Connection then only Down and down value can be revealed only by a Collatz Procedure.

Now to demonstrate an LDN do not fall down to another LDN.

If (3 L_j +1) = 2^kL_i j>i
(9 D_j +1) = 3(2^kD_i)
3(2^kD_i -3D_j ) = 1
IMPOSSIBLE

The Unicity
Of every LDN Collatz procedures Reflected value , as see above by a pure Collatz Procedure is
L_i = (3D_i) expresion by nature and definition reflect Unicity.
Collatz Procedures Effect over evry LDN also wear Unicity
Do not exist 2 LDN's wich following Collatz Procedures to fall in the same D.
IF BY ABSURD Consider :
3 (L_j) +1 = 2^k(D)
AND
3 (L_i) +1 = 2^l(D)
(3 (L_j) +1 ) (2^l) = (L_i +1 )(2^k)
for l>k
((L_j) +1 ) (2^(l-k)) = (L_i +1 )
(L_i +1 ) = ((L_j) +1 ) (2^(l-k))
(3D_i +1 ) = ((3D_j) +1 ) (2^(l-k))
we know
(3D_i +1 ) = 2^uU
((3D_j) +1 ) = 2^vV
then
2^uU = 2^vV(2^(l-k)) where U and V odd Integers
U/V = (2^(u-v-l+k))
IMPOSSIBLE even if (u-v-l+k) < 0

We demonstrated every LDN fall via Collatz Procedures in an LET and this is unique for every LDN and aloso all above demonstrations.
We Deomstrate Murgu_Table2To3 Solved Collatz Conjecture
Infinity Murgu Table2To3 which bring Infinity Double Linear Function are maping Collatz Conjecture into Odds Integers.
We can't say confirm as in reality build Collatz Conjecture Web Nodes.

Infinity Murgu Tables2To3 bring an Math beauty which I hope will be accepted by all Mathematicians .

Infinity Per 6
If we accept Murgu Collatz Reflective Table which esclude procedural Evens as we know all evens fall via divisions with 2 in an odd.
And also know Every Odd number is connected to Infinity via web threads as links to Infinity.
(2 ^∞ Q) where Q an Odd Positive Integer
then keep Murgu Collatz TableTo3 as Collatz Conjecture representative
The Ways Of Odds Positive Integers coming to Unity can be considered as starting From LDN's .
Then we get a way to refer INFINITY.
Infinity Per 6 as all distinctives Ways of Odds to Unity via Collatz Procedures refering directly The Odds.
Or maybe more cleary as know Murgu Inverse Method are connections Odd To Odd
Infinty Per 6
The Number of Murgu Collatz Inverse Metods Closures.

All those demonstrations are also in early 2024 Murgu Table2To3
Murgu Table2To3 normally not need demonstrations as every Math Skilled may deduce those, but , as so long time was not understand , I been coming time by time with explanations. That at any time became a little dizzy even for My. I hope this last one will clear all.

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NOW
as I hope Murgu Table2To3 or Collatz Table2To3 is explained for everyone, to analyze a beauty coming from.

Infinity Murgu Table2To3

As can see Murgu Collatz Inverse Method Formulas :

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((2^(2k+2)[1+6i]) - 1) = 3 Q_i
((2^(2l+1)[5+6j]) - 1) = 3 Q_j

have also as variable k and l with values from 0 to Infinity
we make a graph only for k=l=0 and to cover all will be an impossible task
But Those present Mathematic any proprieties which reveal Murgu Collatz Web Nodes as
perfect functional woven links.

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This born Infinity Murgu Table2To3

Starting k=l=0
((2⁽²⁾[1+6i]) - 1) = 3 Q_i
((2⁽¹⁾[5+6j]) - 1) = 3 Q_j

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k=l=1
((2⁽²⁺²⁾[1+6i]) - 1) = 3 Q_i
((2⁽²⁺¹⁾[5+6j]) - 1) = 3 Q_j

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k=l=2
((2⁽⁴⁺²⁾[1+6i]) - 1) = 3 Q_i
((2⁽⁴⁺¹⁾[5+6j]) - 1) = 3 Q_j

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k=l=3
((2^(2k+2)[1+6i]) - 1) = 3 Q_i
((2^(2l+1)[5+6j]) - 1) = 3 Q_j

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.
.
k=l=n
((2⁽²ⁿ⁺²⁾[1+6i]) - 1) = 3 Q_i
((2⁽²ⁿ⁺¹⁾[5+6j]) - 1) = 3 Q_j

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Those Functions stand in double for k =0 to Infinity and bring a new beauty.

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Murgu Collatz Web Structure
EXPRESSED FUNCTIONAL
If can Image it , then can to see also
USA Marker Murgu Arrow
To see it will be enogh to build into same graph (Table) Double Linear Functions for k=l=1
We will do it into and also for 1 , 5, 7, 11 for k=l=2 .

Our capacity for imaging computational aren't so big , but will be enogh as using imagination to see
now can to see part of Murgu Collatz Web Structure and to image then this web as all.
Offer to my a chance to say
Now a Intelligent Spider can to Web Unity To Infinity and reverse.

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For non Mathematician , need to bring plus explantions.

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Collatz Table2To3 or Murgu Table2To3 Solved Collatz Conjecture by
DEMONSTRATE
Every Odds LET's Collatz Nodes
have Collatz Unicity
LDN's are Collatz Nodes which via Collatz Procedures create apparence of "divergence can lead to multiple roots" instead of only Unity.
But we Demonstrated : Every LDN via Collatz Procedures fall in an LET, and there do not exist 2 LDN's wich to fall into The Same LET.
Excluding Unity , which as root , have a important property - fall into itself via both methods - "Do not any LET's which to lead to same odd value via Collatz Procedures or Murgu Inverse Methods Formulas" .

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Infinity Murgu Tables2To3 Come to Certify Mrgu Table2To3

Infinity Collatz Table2To Demonstrate even by formulas only
Murgu Table2To3 Collatz Unicity
Can't execute phisic those in any graph , only small example
but Formulas Demonstrate In Full it.

Infinity Murgu Tables2To3 also represent Perfect
Murgu Collatz Web Structure
Yes we can now to determine more simple a Collatz Patern from Infinity Murgu Table2To3 Graph as Infinity Double Linear Functions Table .
Read on Oy an Odd, Go Down on Ox and see its Intersection , reper is position on Oy , Go down and see Ox Intersection and so on . For it our table may be extended
but for small numbers we can use it and I take an example 51.

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51 - 77 - 29 - 11 - 17 - 13 - 5 - 1

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We ,by fastening, or Interest named orizontal axe as Oy and Verical one as Ox - forgive My for it!

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To See How Infinity make 1000 times esiest on geting Collatz Patterns I am going down with anothers 3 ex. .
49 - 37 - 7 - 11 - 17 - 13 - 5 - 1
53 - 5 - 1
25 - 19 - 29 - 11 - 17 - 13 - 5 -1

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Early 2024 Murgu Table2To3 was right about #Marker_USA_Murgu_Arrow and Infinity Murgu Table2To3 confirm it.
But also was Right about Murgu Table2To3 via Murgu Collatz Inverse Method became an Collatz Pocket Map .

But Important is
Infinity Per 6
If we now know every odd positives are connected to Infinity via infinity
2^∞Q
for which is absurd to expose practic Collatz Patterns as all fall in Q
Also Now know , every even positive Integer fall in an Odd.
Know
Murgu Collatz Grids of only Odds now can control Totally Collatz Conjecture.
Logical Dead Nodes are Collatz Conjecture Closures .
THEN
Infinity Per 6

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is a beauty earning for Science
But Over It Brought a new Math Issue , which don't keep of Collatz Issues , but revealed by
Infinity Murgu Table2To3

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All Prime Numbers and all Prime Numbers Produces fall via Murgu Inverse Method In an LDN.

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Murgu_Table2To3 - Collatz Table3To3
grade 1,2 and 3
Click here

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Also if want to compare with Murgu Table2To3 wich contain More LET's, can Clink on Down link>

Murgu_Table2To3 - Collatz Table3To3, Click here

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