Collatz Conjecture Complte Solution
Collatz Conjecture Nodes Murgu Formulas

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but as those were revealed by Murgu Table2To3 Formulas wearing Infinity Murgu Table2To3
For it will need Heavy Part Explained

To understand It need understood important parts as
Collatz Conjecture Murgu New Structure and New Logic Structural aproach.

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Collatz Conjecture Murgu New Structure
is about separation of positive Integers in Grids of 6 Integers each.
Then , as known (for reading Murgu Inverse Method ), started by checking Grids Elements for any common proprieties. First was a simple observation
Every Even in Grids fall via Collatz Procedure division with 2 in an Odd
Then excluded from work.
Second observed our now Odds Elements in Grids function of Position in have any common proprieties.
Real, this observation was an absolute result of -
Collatz Conjecture Murgu Inverse Method
which will be presented after will introduce Grids New Structure.

Those Odds in Grid been named and identified function of position in grid as:
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) -- (MCF.01)
L_2i = (5 + 6i) -- (MCF.02)
D_i = (3 + 6i) -- (MCF.03)
also not need demonstration
D_i cover all LDN forms as (3^k), ((3^k)L_1i ) and ((3^k)L_2i ).

Collatz Conjecture Murgu Inverse Method

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started from Collatz Procedure for Odds
(3x+1)= 2^k Q
and applied to Grid's Odds Integers brought 4 revelation and brought also new Common proprieties for grid elements.
first need remark in a primary form Murgu Inverse Method Formula is
(2^k O -1) = 3 Q -- (MCF.04) -- important
REVELATIONS Values

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1. Murgu Inverse Method lead Direct as result ODD to Odd (avoid 2 divisions ).
2. Murgu Inverse Method reveal,
Logical Dead Nodes don't have UP Collatz Connections

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can be considered closures and bring an important Collatz Conjecture Property as rules.
(demonstrated in old 2024 material)
2.1. Every Logical Dead Node - via Collatz Procedure will Fall only and only into LET.
2.2. Every Collatz patterns starting from a Logical Dead Node will not pass another LDN.
2.3. Do not exist 2 LDN to fall into same LET.
3. First LET in Grid (L_1i) will reveal its multiple up Collatz Connections Nodes , Only and only Formula (demonstrated in old materials)
(2^(2k+2)(L_1i) - 1) = 3Qⁱ -- k =0 to ∞
as L_1i are identified AS ( 1 + 6i ) can write it as
(2^(2k+2)(1 + 6i) - 1) = 3Qⁱ -- (MCF.05)

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4. Last LET in Grid (L_2i) will reveal its multiple up Collatz Connections Nodes , Only and only Formula (demonstrated in old materials)
(2^(2k+1)(L_2j) - 1) = 3Q^j -- k =0 to ∞
as L_2i are identified AS (5 + 6i) can write it as
(2^(2k+1)(5 + 6i) - 1) = 3Qⁱ -- (MCF.06)

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(MCF.05) AND (MCF.06) in essence, Solved Collatz Conjecture via theirs form and as Infinity Double Linear Functions which assure Collatz Unicity.

Collatz Connections Unique (Unicity) is included in (MCF.05) AND (MCF.06) FORM -

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This can be understood maybe only By Mathematicians.
But mean:
Every LET can have Multiples Collatz Connections (infinity not Infinity ) but every one treated as functional connections between 2 values (points), have at start point the same value but at the end is different and never repeat.
For a Mathematician, implicit Collatz Murgu Formulas, (MCF.05) AND (MCF.06) Cover All Positive Integers Collatz Connections

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FOR A very skilled MATHEAMTICIAN
Colatz Conjecture End HERE

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But need first to demonstrate Collatz Conjecture Solution for everybody, and for it Collatz Murgu Inverse Method brought more Scientific tools.

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Collatz Conjecture Murgu New Structural Grids and Collatz Conjecture Murgu Inverse Method are basic keys for, but also new logical adapted may be a key.
2024 and then 2025(as needed to explain) validate multiple times
Collatz Conjecture Solved
and all are coming with any Math Beauty in .

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Down will expose those Validation Values

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First is Murgu Table2To3 or Collatz Table2To3
IMPORATANT as reveal Marker USA Murgu Arrow
Murgu_Table2To3 - Collatz Table3To3, Click here
Murgu Table2To3 or Collatz Table2To3 are (MCF.05) AND (MCF.06) for k=l = 0
and show now (MCF.05) AND (MCF.06) COVER all via Double Linear Functions falling in Unity like an Arrow with head in Unity.
Known as Marker USA Murgu Arrow

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Second is more Strong :
Now is clear (MCF.05) AND (MCF.06) for k=l=0 TO ∞ brought Infinity Double Linear Functions , evryone falling into Unity Like Arrows.
Infinity Murgu Table2To3 or Infinity Collatz Table2To3

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Known Now as Marker USA Murgu Arrows
Collatz Connections Unicity is included into Functions form as implicit demonstration.
We also come with a small example for :
Murgu_Table2To3 - Collatz Table3To3
grade 1,2 and 3
Click here
Third is a Math Beauty :
Collatz Conjecture One-Dimensional Array

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(MCF.05) AND (MCF.06) as Infinity Arrows
Infinity Marker USA Murgu Arrows
Reveal All Collatz Nodes as an One_Dimensional Array named un 2024
Collatz Conjecture Pocket Map

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To show it we make 2 Javascript programs : one to generate The Array and second to reveal Instant , by using generated array (functions of how many grids array cover) to reveal in part or totally Collatz Pattern for an Odd Integers. Javascript was split in 2 programs to make instant answer for Collatz Patterns.
For Collatz Conjecture One-Dimensional Array, Click here
or
For Collatz Conjecture One-Dimensional Array, Click here
Now those Third make Collatz Conjecture Solution Understable for evry Skilled In Math.

Collatz Conjecture Solved

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Via Multiple Solutions :
Collatz Conjecture Murgu Formulas (MCF.05) AND (MCF.06) for skilled Mathematicians
Collatz Conjecture Murgu Table2To3 or Collatz Table2To3 for skilled Mathematicians
Collatz Conjecture Infinity Murgu Table2To3 or Infinity Collatz Table2To3 for strong skilled in Math
Collatz Conjecture Marker USA Murgu Arrows for all skilled in Math

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IMPORTANT is :
Collatz Conjecture Solved

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and brought any Math Beauty
New structural
New Logical Methods
Marker USA Murgu Arrow as possible New Science Toll
Collatz Conjecture AS One-Dimensional Array
(as remark:
Collatz Conjecture AS One-Dimensional Array is a reverse of Dual-Dimensional Array Infinity Marker USA Arrows - )
Mathematics Beauty FOR
Earth Science And Culture Treasure .

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Collatz Conjecture Unique And Complete Solution
Brought also a lot of Math Beauty and Logical Beauty.

Murgu_Table2To3 1st, 2nd 3rd Grades parts to image Murgu Collatz Web Structure.
I keept part of Murgu Table2To3 First Grade as comparative image.