AS COLLATZ CONJECTURE Unique Solution Murgu Table2To3 wasn't Understood Yet

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This html have included more repetitions , but all are exposed different as hope all skilled in Mathematics will understand
The Solution
as is important for future projects coming from Collatz Conjecture revelations after solved.
1. Table2To3 Utility.
2. Functional Divergence Study.
3. Murgu 1D Array Index Infinity_Per_6 - Computational Evolution METER and Tester .

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Then will start with
Collatz Conjecture Murgu Method and Formulas Legacy

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which contain a use of Collatz Unicity , which may be explained also after.

I explained more times On Youtube . com and on climaticdisorder . com by 2024 Murgu Table2To3 with its formulas and analyzed into new Collatz Conjecture - Coordinate System Table2To3 Solved Collatz Conjecture FOR Mathematicians , but seem they didn't understood it as don't begin to introduce it in theirs courses.
Then in 2024 - 2025 and 2026 I returned with a lot of validation and re explanations but seem - The same Effect. Then above of last material -here - I will try to bring or Last Explanation at what I think wasn't understood.

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Collatz Conjecture Solution

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motivated issues not understood yet
Try to explain .
Special Collatz Unicity

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I am sure everybody understood Group Theory grouping Integers into Grids of 6 Integers , and importance of it as lead to Murgu Inverse Method .
Collatz Conjecture Murgu New Formulas revealed by this Group Theory Method are key of new Method as lead to Collatz Conjecture Solution. Nobody , but Math Logic , can't contest Murgu Formulas Legacy over Collatz Conjecture as have base
((3x + 1) = Q) Q Even or Odd
Between those 9 Formulas , 2 are the key and became Murgu Table2To3 Collatz Conjecture Solution
((2^(2k+2)[1+6i]) - 1) = 3 Q_i     (C.E.-1)
((2^(2l+1)[5+6j]) - 1) = 3 Q_j     (C.E.-2)

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for k=l=0 (Marker USA Murgu Arrow) , and if wish to see it before exploring what is down, down will be a link for.
This is Table Nr, 1 or 0 , doesn't Mater , important is to understand those are now
Formulas and Functions
and both are 2 times infinity via k, l and i,j into a
New Collatz Conjecture Coordinate System - Table2To3

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To stay away of confusion , luck at those as going in Groups of 2 via k = l .
To understand Collatz Unicity revealed by those functions need to write those Equations related to our LET_1i and LET_2j which will note as T_1i and T_2j , indices i,j needed as to remark Q_1i and Q_2j .
((2^(2k+2)T_1i) - 1) = 3 Q_i     (C.E.-1.1)
((2^(2l+1)T_2i) - 1) = 3 Q_j     (C.E.-2.1)

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Strong properties of (C.E.-1.1) and (C.E.-2.1) are, the fact - Those are Odds To Odds functions, then not only avoid Evens , but Define Collatz Conjecture
Odds To Odds Structure Legacy
and via i,j cover for all k=l= 0 to Infinity, all LET_1i and LET_2j .
Every Table2To3 (are Infinity) Cover all LET_1i and LET_2j domain.

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Then to understand Collatz Unicity need to remark difference by Unique.
Collatz Unicity mean as LET_1i and LET_2j show infinity Collatz Connections first connection, k=l=0 -Table2To3 Nr. 0, is Unique as reveal the way to unity for theirs
Q_1j AND Q_2j .

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Collatz Unicity is not only revelation Of
Murgu Table2To3

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But show it is a Unique Solution For.
Infinity Murgu Table2To3
evolved in Infinity Marker USA Murgu Arrows
Validate Collatz Conjecture Solution .

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and lead also
- even I consider Proof in Mathematics a Give-up -
Lead to Collatz Conjecture PROOF OF PROOFS
Murgu 1D Array Index Infinity_Per_6

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Poetic Infinity_Per_6 is a Theoretic Reality.

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For a Mathematician , Above 2 equations solved Collatz Conjecture - and remain only to demonstrate theirs legacy for Collatz Conjecture , which we did multiple times.
Those for k=l=0 to Infinity will go as -Infinity- 2 Double Linear functions one starting from 1 and second from 5. Those (for every k=l) are unite into UNITY by Collatz Procedure (3*5+1=16)
Collatz Conjecture Murgu Formulas Legacy :

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1. both come from Collatz Conjecture Procedure For Odds-
((3x+1)=2^k D) Where D Odd Integer
2. Maybe important to remark "Double False Redundancy Of Truth" Effect , how
((3x+1)=2^k D)
hide (C.E.-1) and (C.E.-2) which are 2 inversions for ((3x+1)=2^k D) . This born , I will say, a new Structural Group Theory ,
Collatz Conjecture Group Theory Solution - Murgu Inverse Method
But including all , will say
Collatz Conjecture Murgu 6 Grid Group Theory Solution.

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3. We get 2 instead of one as Logical Eternal Triads are 2 kind in every Grid but all (LET_1i=(1 + 6i)) related to Collatz procedure have same property and also (LET_2j= (5 + 6 j)) have another's but common.
4. The middle component for every grid - Logical Dead Nodes - do not have an UP Connection . Demonstrated by 2024 and DOWN in another paragraph.
This is a very important propriety revealed by Murgu Inverse Method - and mean also -
Logical Dead Nodes are Collatz Conjecture Closures.

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(LDN_i = (3 + 6i )) and then above propriety give a chance to refer all Collatz Patterns Possible Number as Infinity Per 6 . But more important is the fact - a Logic Explanation only - of Collatz Conjecture Divergence as Functional Divergence.
5. As (C.E.-1) and (C.E.-2), are Odds to Odds Formulas make those Functions.
6. (C.E.-1) and (C.E.-2) legacy are clear , but will be explained multiple times down.
7. Collatz Conjecture Murgu Formulas by refer Odds to Odds became
Collatz Conjecture Functions

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and born first
new Group Theory Coordinate System
Table2To3 Coordinate System

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(Axe (Ox) all Positives Integers Odds - Axe (Oy) all LEt's )
Axe (Ox) and (Oy) been chose as reveal clear functions linearity.
8. As said- EVENS been "excluded" only for work
Evens now are back as (2^k D ) , where D is Odd Positive Integer.
Table2To3 First Group Theory Coordinate System.

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brought
Collatz Table2To3 First Group Theory Coordinate System - Legacy
Murgu Table2To3 First Group Theory Coordinate System - Legacy

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9. Murgu Table2To3 or Collatz Table2To3 brought a Science New Concept - Table2To3 , here - with your accept - via Collatz Conjecture Murgu Inverse Method as an Group Theory Method , But for sure can be adapted to lots Science Issues like it. That mean we created a new coordinate system.
Collatz Conjecture Murgu Coordinate System - Table2To3

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Legacy For is demonstrated by results and validate by :

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PROOF of All Proofs - Murgu 1D Array

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Murgu One Dimension Array or Collatz 1D Array is REAL and obtained using - Collatz Conjecture Murgu Inverse Method and Collatz Conjecture Murgu Formulas - VIA a Software generator in Javascript, Perl, Python, PHP, etc. . As Poetic Reality Murgu 1D Array INDEX is Infinity Per 6 but, "our reality" , in reality to approach big values we will to use BigInt and then Multiple Murgu 1D Arrays handled by another Program HANDLER like in
Murgu_Table2To3 - Collatz Table2To3, Click here

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Trust Mathematics as Science as Collatz One Dimension is a beauty of.

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Murgu 1D Array is not a PROOF
Murgu 1D Array is an Collatz Conjecture Integrator.

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Join to USA site collatzconectureend . com project of future
Murgu 1D Array Index Infinity_Per_6 as Computational Evolution METER And TESTER.
Index Infinity_Per_6 seem poetic , but is an Theoretic Reality.
This is heavy to understand and I am trying by early 2024 to explain but maybe I am not so good for.

As Maybe said before Murgu Inverse Method using Collatz Conjecture Murgu Formulas REVEALED and TRUTH from Collatz Conjecture
Murgu Collatz Conjectures Closures

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Those are those Positive Integers which I named into Group Theory repartition as Logical Dead Nodes
LDN_i = (3 + 6*i)
and we demonstrated - do not have so named UP Collatz Connection .
Demonstration is Simple :
2^k(3 + 6*i) -1 NEVER EQUAL (9 Q _i )
for any (Q _i ) Integer
But this bring also a beauty logical which may be considered a Poetic Collatz Conjecture Solution.
Collatz Conjecture Unite Unity With Infinity via WIRE
2^∞(LDN_i) Only and Only

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That mean:
Collatz Conjecture New Name - Infinity Per 6

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Is an associated name brought by solution, and , Collatz Conjecture remain Collatz Conjecture.
But it now determine exact The Index of Murgu 1D Array as - Infinity Per 6.

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Any time I didn't use * for Multiply as for sure , to say 6i auto imply 6*i .

for k=l=0 , Those equations are old Marker USA Murgu Arrow, and if wish to see it before exploring what is down, here is a link for.
Marker USA Murgu Arrow

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Arrow - as Collatz Procedure (3*5 +1 = 16) Unite them like an Arrow.
Murgu_Table2To3 - Collatz Table3To3, Click here
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.
To understand, My pure Explanations an repetitions, May Be the best to see those before, as help on.

For Mathematicians
(if pay a small attention and make profound analytic)

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Collatz Table2To3 Solved Collatz Conjecture
even by declaring :
Murgu Collatz Conjecture Grids
(I will say as Group Theory Method)
Splitting Grids in Odds and separated 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 re explain it
for every noon very skilled in Math, and hope, I have talent for!

As Murgu Table2To3 seem not understood yet
I don't see motivation for, but will try recursive approach.
Murgu Table2To3 need understood as utility concept which .
Murgu Table2To3 followed by Infinity Murgu Table2To3 are may help revealing (C.E.-1) and (C.E.-2) as linear functions here and maybe in another issues.

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 reveal
Collatz Conjecture One-Dimensional Array

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Merit All Science Attention.
as for sure, its form will have utile usage in all Science, Then a gain for.

Notes:
A important key is separation of positives Integers in GRIDS of 6 Integers, which for sure don't need explanations then that, is utile fact for our scope, trying a approach to Group Theory.
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 brought 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
also as
(3x = (2^k O -1)) --- we get
(x = (2^k O -1)/3)
for unique k - x and O unique .
Murgu Inverse Method Formula is perfect valid, but, brought also any caracteristic 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 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.

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 apart 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 materials this appear as Murgu Lemas
(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 Lemas , 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 different domain and
Very Important as revealed achievement

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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, multiple 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.

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 .
That mean, Collatz Conjecture Procedure reveal a connection Down , when Murgu Formulas reveal connection UP . For that I separated Odds in LET's and LDN.
this characteristic 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.

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 coordinate 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 structured 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(5) +1) = 16.
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) which will make for Collatz Conjecture

Murgu Inverse Method - Collatz Conjecture Pocket 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 different 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 -

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 Conjecture 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 reveal 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 write 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 final 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) expression by nature and definition reflect Unicity.
Collatz Procedures Effect over every LDN also wear Unicity
Do not exist 2 LDN's which 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 Demonstrate Murgu_Table2To3 Solved Collatz Conjecture
Infinity Murgu Table2To3 which bring Infinity Double Linear Function are mapping 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 exclude 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 distinct's Ways of Odds to Unity via Collatz Procedures refer directly The Odds.
Or maybe more clear as know Murgu Inverse Method are connections Odd To Odd
Infinity Per 6
The Number of Murgu Collatz Inverse Methods 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.

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 enough 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 enough 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 explanations.

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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 apparent 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 Murgu Table2To3

Infinity Collatz Table2To Demonstrate even by formulas only
Murgu Table2To3 Collatz Unicity
Can't expose all 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 Pattern 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 horizontal axe as Oy and Vertical one as Ox - forgive My for it!

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To See How Infinity make 1000 times easiest on getting Collatz Patterns I am going down with another's 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_Arrows 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 practically Collatz Patterns as all fall in Q , but pass via Q.
Also Now, know , as 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.

Assuming The risk of Multiple repetitions :

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Infinity Murgu Table2To3 are for viewing Infinity Collatz Conjecture Murgu Formulas for LET's. To "see" clear how those are covering Infinity by being Double Linear Infinity functions.
Those cover all Collatz Conjecture Nodes Connections and separated those in 2 domains
One ruled by Unity (1) , second by 5
But Collatz Procedure for 5 , (5*3 +1 = 16) UNITE those into arrows heading Unity.

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Infinity Collatz Conjecture Murgu Formulas

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Cover all Collatz Conjecture Nodes Connections

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A Truth Validated by Equations Form
By LDN rules
By "Collatz Unicity" extension - For k=l and every k≠l will not be Q_1i = Q_2j.
for k=l as Marker USA Murgu Arrows are going hand in hand
as (2^{(2k + 2)}(1 + 6i))≠(2^{(2k + 1)}(5 + 6i))
(2(1 + 6i))≠((5 + 6i))
( 12i)≠((3 + 6i))
2 i≠1 as i is Integer.
for i≠j
(2(1 + 6i))≠((5 + 6j))
( 6i - 6j)≠(3)
( 2(i - j))≠(1) as i,j Integers

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FOR k≠l
the way we used up maybe heavy - but -
Infinity Collatz Conjecture Murgu Formulas

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every One have different slope.
This is an essence

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because if will have ONE Q_1i = Q_2j this will be a ROOT and mean

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Collatz Conjecture not rule Infinity

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We Demonstrated (I hope, I was clear! )

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Repeat as to Repeat
Infinity Collatz Conjecture Murgu Formulas reveal all LET's have UP infinity (multiple not Infinity ) Collatz nodes Connections (Q_1i or Q_2j or LDN's ) , when all those, (Q_1i or Q_2j or LDN's ) every one will have DOWN a single way connection node - theirs LET-generator ant this is the

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The Secrecy

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Infinity Murgu Table2To3 handle Infinity as Collatz Conjecture legacy.
Infinity Collatz Conjecture Murgu Formulas

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Cover all Collatz Conjecture Nodes Connections

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Collatz Conjecture Rule Infinity

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ALL ABOVE demonstrated - Infinity Pay Respect For Collatz Conjecture -
and revealed of lot of logical beauty coming from - but -
work to convince also PROOF lovers brought a miracle
Proof Validator

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as following
Marker USA Murgu Arrows - Validators -
got ever PROOF Beauty
Collatz Conjecture as One-Dimension Array

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for it , can visit :
Collatz Conjecture as One-Dimensional Array
or
Collatz Conjecture as One-Dimensional Array

Collatz Conjecture Divide Integers In 2 Parts

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Infinity Collatz Conjecture Murgu Formulas
or Infinity Collatz Conjecture Murgu Functions
are 2 times Infinity, as are double linear functions .
Following those from k = l [0 to Infinity], going together as
Marker USA Murgu Arrows
as Collatz Procedure to 5 Unite those like arrows in Unity -
is clear those intersect never as in Negative Collatz became something else.
Intersection in one is forced by Collatz Procedure - (3(5) + 1 =16 )

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But a small miracle , is small, come from Collatz Conjecture .
Table2To3 reveal
Collatz Conjecture Divide Positive Integers In Two Domains.

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and it is a functional separation solving functional divergence.
This mean we are of a Group Theory Functional Division , as those 2 Domains can be playable only Functional.

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Demonstration :
is simple if calculation the slope in general cases for
((2^(2k+2))(1 + 6i) - 1) = 3 Q_1i
((2^(2l+1))(5 + 6j) - 1) = 3 Q_{2 j}

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for LET1, in points (7, 1), for LET2, in points (11, 5) Will get :
(2^(2k+2)) never equal with (2^(2l+1)) doesn't matter k and l .

That mean
Infinity Collatz Conjecture Murgu Functions for LET₁
never Intersect
Infinity Collatz Conjecture Murgu Functions for LET₂
in short
Collatz Conjecture Divide Positive Integers in Two Domains
and abstract
Table2To3 Divide FunctionalPositive Integers in 2 Domains

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Infinity Collatz Conjecture Murgu Formulas

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Collatz Conjecture Murgu Inverse Method

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Extracted From Collatz Conjecture Multiple Beautis and one is refering Infinity and may be important by doing it.
Infinity_Per_6 - Infinity Per 6 .

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As Number Of Collatz Patterns which cover all Collatz Conjecture Nodes.
DEMONSTRATION: All Odds are formally tied to Infinity via 2^∞ O , Where O Is an ODD Positive Integer. But not for all this is an real tier but only for LDN's as those are known now as "closures". For LET's those remain active participants To UP to Infinity New Odds Connections generators. Then only for LDN's 2^∞ O are " Closures into Infinity " for Collatz Conjecture. Now is clear only Collatz Patterns which start from those all will cover all Collatz Conjecture Nodes.
The number of those closures are - Infinity Per 6 - ∞/6 -
Infinity Per 6 - is an absolute Truth.

Is Strange then strange Mathematicians don't understood yet
Collatz Conjecture Murgu Inverse Method via new Collatz Conjecture Murgu Formulas
Cover All Collatz Conjecture

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Can be 2 motivations :
1. Not yet Understood how Evens are related there which maybe nonsense . Every Even is in via
(2^k Q) where q is an odd , k ∈(0, ∞ ).
For LDN (Logical Dead Nodes) those are direct links to infinity and every (2^k Q) will fall in its LDN via Divisions with 2. Those not longer will participate to "generate" another's Collatz Odds Nodes. Poetic, those are "Closures In Infinity " , but , via Collatz Procedures fall back into its LDN and Then in Unity.

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2. Heavy part to understand, even if I explained multiple times , but don't have talent for - Collatz Conjecture Murgu Formulas for LET's -

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

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contain in theirs forms are in theirs Infinity Double Linear acting as arrows for
k=l and k = 0 to Infinity
note : (1 + 6i) = LET_1i and (5 + 6j) = LET_2j
((2^(2k+2))(LET_1i ) - 1) = 3 Q_1i (L1.1)
((2^(2k+1))(LET_2j) - 1) = 3 Q_{2 j} (L2.2)

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and do a simple analytic : for every k=0 to ∞ , slope for (L1.1) different of (L2.2) slope
That mean those divide Collatz Nodes in 2 domains .
but more important if follow now i and j, for any value (maybe heavy to observe) , i=j =10
can see every one will have multiple to infinity (not Infinity) Q_1i and Q_2j .
Important is : respective LET show every Q node which via Collatz Procedures will pass via LET as Collatz Node. As i and j === 0 to ∞ -- this fact cover this property for all Positive Integers.

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Now

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As also, k=l === 0 to Infinity
Mathematic

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It will Cover All Collatz Conjecture Nodes including for every one above proprieties.

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I hope , I was a good teacher at the last for skilled in Mathematic !

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Anyway , with all My teaching power I advise everybody! Visit Marker USA Murgu Arrows - Validators -
got ever PROOF Beauty
Collatz Conjecture as One-Dimension Array

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for it , can visit :
Collatz Conjecture as One-Dimensional Array
or
Collatz Conjecture as One-Dimensional Array

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If want to participate to The Future.

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