Murgu Table2To3 as Collatz Conjecture Unique Solution began in 2023 autumn and reach first form around of Jun 2024. But also was coming with a lot of validations in 2024, 2025 and 2026 - as wasn't understood - need more explanations.
Collatz Conjecture Murgu Method and New Formulas Legacy related to Collatz Conjecture
was heavy to accept,then, will try one more time here.

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This will be maybe out of its historic, but will try to explain Mathematical and Logic for what in June 2024 Murgu Table2To3 Solved Collatz Conjecture.

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Even if historic is not this right as will be here, real solution start via
Restructure of Collatz Conjecture Domain via an
Group Theory Method

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by grouping Positive Integers into Grids of 6 Elements (Integers) and then every positive Integer will be included as one of Formulas :
(1+6i) - (2+6i) - (3+6i) - (4+6i) - (5+6i) - (6+6i) -- G.0

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Next step was a logical analytic of those grids related to Collatz Conjecture Procedures , and first temptation was (3x+1) itself. This born small hopes for Murgu Inverse Method. Murgu Inverse Method Legacy

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for Collatz Conjecture is natural one as is (3x+1) inverse use and natural revelation offered by analytic testes formulas became Odds to Odds direct relation.
Then if in (3x+1) = ((2^k)L), note x as Q will get a primary Murgu Inverse Method Formula.
(((2^k)L)-1) = 3 Q
Primary as will see long analytic will come with 2 Murgu Inverse Method Formulas .
Turning back to Group Theory Contribution will observe any useful things as distinction between Evens and Odds became a small key for Solution , then as
each Odd number have Infinity Collatz Images in Evens via
((2^k)L)
This is a Legacy for Excluding for work only Evens and easing logical analytic, but also is revealing into Group Theory G.0 Formulas write as
(1+6i) - 2(1+3i) - (3+6i) - 2(2+3i) - (5+6i) - 2(3+3i) -- G.0.1

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G.0.1 - 2(1+3i) - 2(2+3i) - 2(3+3i) send My for a loss time to analyze (2+3i), (3+3i) in a Collatz Logic, but was a losing time related to Collatz Conjecture as we have accurate Method and Formulas to solve Collatz Conjecture.
Evens exclusion Legacy for Collatz Conjecture Analytic is natural and included in
((2^k)L)
Now our Analytic will resume to Group Theory Grids containing for only
(1+6i) - (3+6i) - (5+6i) -- G.0.0

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For a good Control of proprieties we named those formal:
(3+6i) Logical Dead Nodes as will see in logical sense those are Collatz Conjecture Closures REVEALED by Murgu Inverse Method.
(1+6i) and (5+6i) as Logical Eternal Triads Nodes as are Collatz Conjecture Generators in Murgu Inverse Method Perspective. Then even Simple need to remark those as Equations which been used into DEMONSTRATIONS.
LET_1i = (1 + 6*i) as LN.1
LET_2j = (5 + 6*j) as LN.2
LDN_k = (3 + 6*k) as LN.3
G.0.0 mean also first step to Table2To3 as Collatz Conjecture New Coordinate System.

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Next step is Logical Analytic of G.0.0 members related to Murgu Inverse Method and then legacy Collatz Conjecture Solution.

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Following a Group Theory Model, and then using those Equations and Murgu Inverse Method into a formal new environment for (Table2To3 as Collatz Conjecture new coordinate system) make possible an absolute logical structural analyze as Function's , New Murgu Formulas as reverse mirroring of (3x + 1).
That mean we get new qualitative strong Mathematics Pillars for.

Logical Analytic of G.0.0 members related to Murgu Inverse Method

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Golden KEY of Collatz Conjecture Solution

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it started with attempts of using Primary Murgu Inverse Method for each G.0.0 members general Form and after work we get 3 important rules.
1. Murgu Lema for (1+6i) which brought
((2^(2k+2)[1+6i]) - 1) = 3 Q_i     (C.E.-1)
A New Collatz Conjecture Formula which will see have propriety of function and will return for.
Legacy of this Formula-Function related to Collatz Conjecture Solution was in all materials demonstrated and is simple and can be demonstrated now even by a High School Student Math Skilled.

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2. Murgu Lema for (5+6i) which brought
((2^(2k+1)[5+6j]) - 1) = 3 Q_j     (C.E.-2)
A New Collatz Conjecture Formula which will see have propriety of function and will return for.
Legacy of this Formula-Function related to Collatz Conjecture Solution was in all materials demonstrated and is simple and can be demonstrated now even by a High School Student Math Skilled.

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3. A base key for Collatz Conjecture Solution
Collatz Conjecture Logical Closures (3 + 6i)

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Revealed By Murgu Inverse Method and contained hide in Collatz Procedures.
Demonstration is simple as :

(((2^k)(3L))-1) ≠ 3 Q
but easy demonstration is
(((2^k)(L))-1) ≠ 9 Q

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I included it in a simple demonstrative formulas as (3L) include all Odds divided by 3. This wear now Collatz Conjecture Solution Legacy AND brought 3 Important Collatz Conjecture Rules.
I. All (3+6i) or (3L) , all Odds Divided one or more times by 3 do not have UP a Collatz Connection.
II. A Collatz Conjecture Pattern started from an (3 + 6i) never will meet second (3 + 6k).
III. A Collatz Conjecture Pattern started from an (1 + 6i) or from an (5 + 6i) never will meet an (3 + 6k).
Those 3 rules are important for Collatz Conjecture Solution.
Logical Dead Nodes Murgu Rules .

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Next Step will make a return to Group Theory Method as we get new Structural Logical rules.

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As those new rules legacy for Collatz Conjecture Solution been demonstrated became clear a new structure related to Collatz Conjecture Solution and first is Table2To3 as Collatz Conjecture new Logical Coordinate System.

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For it , following new rules we renamed
(1+6i) - (3+6i) - (5+6i) -- G.0.0

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with distinctive formal names.
LET_1i=(1+6i) - LDN_i(3+6i) - LET_2i=(5+6i) -- G.0.0

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Where LET - Logical Eternal Triad (nodes) and LDN - Logical Dead Node .
LET's (Logical Eternal Triads Nodes), AND LDN name been chosen not by hazard , but by considering Theirs role in Collatz Conjecture New Structure brought by Group Theory Grouping in Grids . Logical Dead Nodes have double role, Logical closures and Connecting Down only and only A LET nodes , important for Logical Analytic.
Logical Eternal Triads Nodes (LET's) have triple role
1. Assure for Up Connection (first one Q₀) way back to Unity via Collatz Procedures.
2. Assure Eternal via infinity (not Infinity) Q_i (i >0) Collatz Conjecture State.
3. Even if Down Connection is offered by a Collatz Procedure, ante rules assure this connection will be only and only another LET.

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Now we are able to model a NEW Collatz Conjecture Structure which Respect Collatz Conjecture Procedures. This is absolute defined by Collatz Conjecture new Coordinate System Table2To3 . Table as we work with discrete values even if New Murgu LET's Formulas are Functions also.

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Table2To3 as Collatz Conjecture New Coordinate System.

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By hazard as work was going on, we defined
Axle Ox (X axis) as axle of all Odds Integer, then all LDN's and LET's.
Axle Oy (Z axis) as all LET"S included.

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By here -- Collatz Conjecture Murgu Inverse Method will play its role via New Murgu Formulas .
Murgu Table2To3 or Collatz Table2To3 are 9, but Important and
Collatz Conjecture Murgu LET's 2 New Formulas

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((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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Those are Murgu Table2To3 Formulas or Collatz Table2To3 Murgu Formulas

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Theirs Legacy over Collatz Conjecture is absolute and DEMONSTRATED by 2024.
As to follow graphic LET1's and LET2's will note those as T... . and will can write

((2^(2k+2)T_1i) - 1) = 3 Q_i     (C.E.-1.1)
((2^(2l+1)T_2j) - 1) = 3 Q_j     (C.E.-2.1)

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Those are Infinity and both Linear.
AND reveal
Infinity Murgu Table2To3

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as simple validator for Collatz Conjecture Unique Solution Murgu Table2To3 . (C.E.-1) and (C.E.-2) viewed analytic also as (C.E.-1.1) and (C.E.-2.1) are
Collatz Conjecture Unique Solution Murgu Table2To3

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Infinity Murgu Table2To3 Validate Murgu Table2To3 as Collatz Conjecture Unique Solution as SHOW Mathematical via k and l = 0 to Infinity that Cover as formulas all Collatz Conjecture Nodes.
Evens are covered by
(2^lT_1i) and (2^lT_2i)
- THEN - as answer -
(2^lQ₁) and (2^lQ_j) .
Murgu Table2To3 Formulas as Infinity Murgu Table2To3 Cover all Positive Integers.

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Collatz Unicity

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Collatz Unicity leaded a time to confusion Unique when mean
Infinity Murgu Table2To3 by theirs form show in fact every Q_i and Q_j for i=j=0
(First T_1i and T_2j connections are Unique.)
and assure into Collatz Pattern the way down to Unity .

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This Property may be understood easy if now imply Marker USA Murgu Arrow for k=l=0 .

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And Then Infinity Marker USA Murgu Arrows for k=l - 0 To Infinity.

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But here, first, need to demonstrate
Murgu Table2To3 or Collatz Table2To3 Formulas by theirs Structural Form
is a Collatz Conjecture Unique Solution for Mathematicians .

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For it to start this logical analytic.
As Linear Functions via Slopes for (C.E.-1.1) are Infinity neither one of those will intersect.
As Linear Functions via Slopes for (C.E.-2.1) are Infinity neither one of those will intersect.
Marker USA Murgu Arrows show clear for each Murgu Table2To3 for which k=l, 0 to Infinity, (C.E.-1) do not intersect (C.E.-2) if take each marker separately. But if take now all in the same Graph (Infinity Murgu Table2To3), any (C.E.-1) Intersect (C.E.-2) - see The Slopes.
This can mean Collatz Unicity exclusion.

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Then need a profound logical analytic for, and to do it for second form of (C.E.-1) and (C.E.-2) -- (C.E.-1.1) and (C.E.-2.1).
Intersection mean :
((2^(2k+2)T_1i) - 1) = ((2^(2l+1)T_2i) - 1)
then - (2^(2k+2)T_1i) = (2^(2l+1)T_2j)
then - (2^(2k+2-2l-1)T_1i) = T_2j
then - (2^(2(k-l)+1)T_1i) = T_2j
then (2^(2v+1)T_1i) = T_2j
v negative or positive doesn't matter.
Because, as we know T_1i , also T_2j are Odds Integers
Then :
All (C.E.-1.1) intersections with (C.E.-2.1) are not in Collatz Conjecture Nodal Points.

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Or never (C.E.-1.1) will Intersect (C.E.-2.1) into a Collatz Node or Integer.
Collatz Unicity became a Collatz Conjecture RULE.

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2024 Murgu Table2To3 - Collatz Conjecture Unique Solution.

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Collatz Conjecture Murgu LET's Formulas are also linear functions but into discrete domain.
For Collatz Conjecture Important are only linearity show hide convergence and then only Collatz Nodes reading. For us now Equation of Intersections
(C.E.-1.1) will Intersect (C.E.-2.1)

(2^(2v+1)T_1i) = T_2j - Eq.T.0

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Can be Write as:

(2^(2v+1)T_1i) ≠ T_2j - Eq.T.0.1

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Our 2 Times Infinity Linear Functions - Demonstrate Now
Do not Intersect into any Collatz Conjecture Nodes

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Is a Logical Truth only for Collatz Conjecture which express non validity as one T must to by Rational, then OUT of Domain.

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Logical treatment as Formulas and as Functions over new Group Theory Method Grids Grouping, using Collatz Conjecture Murgu New Formulas Legacy for, using Murgu Inverse Method Legacy for, Murgu Table2To3 Solved Collatz Conjecture

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And its TRUTH stand now CLEAR into MURGU LET's Formulas
((2^(2k+2)T_1i) - 1) = 3 Q_i     (C.E.-1.1)
((2^(2l+1)T_2j) - 1) = 3 Q_j     (C.E.-2.1)

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Eq.T.0 or Eq.T.0.1 clarify logic all as now can make a logical generalization.
If formal note LET_1i and LET_2j as T, also Q_1i and Q_2j as Q via rights offered by Eq.T.0, can express a absolute rule for Collatz Conjecture.
As every T to Q Connection is an Down to UP Collatz Connection , there each Q do not have second Down to Up Collatz Connection, when Q to another LET's or LDN are infinity(not Infinity).

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Eq.T.0 or Eq.T.0.1 via all Math Rigor join to above 2 Conditions
1. Linear function of Infinity don't Intersect as start from unity (T_1i or LET_1i) and have different slopes.
2. Linear function of Infinity don't Intersect as start from unity (T_2j or LET_2j) and have different slopes.
and solved in absolute mode Collatz Conjecture ISSUE
Each Q as LET or LDN have a Unique WAY to unity (via its T Connection )

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That mean Collatz apparent divergence is functional one.

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Murgu Table2To3 - Infinity Murgu Table2To3 - Infinity Marker USA Murgu Arrows

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Not Only Solved Collatz Conjecture - But extracted for 3 new Math Jewel.
1. Small one - Table2To3 - applicability .

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2. Collatz Conjecture Functional Divergence - victory may born a new Math Domain - Functional Divergence Study-.

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3. For a Time one of biggest Science Jewel
Murgu 1D Array Index Infinity_Per_6
Or Collatz 1D Array Index Infinity_Per_6.

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Where Infinity_Per_6 is a Reality.
Murgu 1D Array Index Infinity_Per_6 with its software Generator and Software Handler BECAME - for sure -
Eternal Computational Evolution METER and TESTER.

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Project - Murgu 1D Array Index Infinity_Per_6 - already started and wait for Science Community to Validate
2024 Promised Collatz Conjecture Pocket Map been upgraded to
Collatz 1D Array Index Infinity_Per_6

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Where Infinity_Per_6 a measure of our Computational Power.
Murgu Table2To3 Collatz Conjecture Unique Solution

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as to Validate our projects rights to truth.
If wish to see Collatz 1D Array Index Infinity_Per_6 at work, Click down
Murgu 1D Array - Collatz 1D Array, covering 220 000 Grids
Click here
Collatz Conjecture became now part of
Future Of Science

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Infinity_Per_6 Collatz Conjecture Legacy.

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Simple Now as been solved, Collatz Conjecture bring any Logical Beauty, and one may be reference to Infinity brought by Logical Dead Nodes, formal named also Collatz Conjecture Logical Closures.

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Infinity Per 6 is an Absolute Mathematical Reality of Exact Reference To Infinity.

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Infinity Per 6 was revealed by Murgu Inverse Method by checking Collatz Conjecture Murgu Primary form Formula deduced from Collatz Procedure (3x + 1 = 2^k Q) in inverse Form testing on 9,99,33,...
(2^k T -1 = 3Q)
then via Murgu not defined yet as Collatz Conjecture Murgu Logical Eternal Triads Formulas.
((2^(2k+2)3T_1i) - 1) = 3 Q_i     (C.E.-1.1)
((2^(2l+1)3T_2j) - 1) = 3 Q_j     (C.E.-2.1)

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3(((2^(2k+2)T_1i) - Q )) =1
3(((2^(2k+1)T_2j)- Q )) =1
Impossible into Integers.
As revealed, now, can be tested even using Collatz Procedure (3x +1 = T)
as (T even of form 3Q)
1 = 3(T-x) : Impossible

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For a time Collatz Conjecture became a long Story, but this include in a sense a logical beauty and will finalize in any util parts.
Now we need to understand Collatz Procedures between 2 Collatz Nodes define Connections Down, When Murgu Inverse Method Define Connections Up even apparently for (2^(2l+1)) for k=0 (Q_j<T_2j) , following The Logic of Collatz those are UP Connections .
So, all Positive Integers divided by 3 one time or mores , (and for us Odds specially) been named Logical Dead Nodes or Collatz Conjecture Closures, and Theirs Logical Proprieties
Reveal a new logical Collatz Conjecture Beauty

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Collatz Conjecture all Possible Patterns to Unity - Infinity Per 6 -

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DEMONSTRATION:
as LDN's do not have UP Connections mean, logical DOWN Proper Connection Must Be A Logical Eternal Triad (LDN).
AS a LET can't have a Down LDN Connection , THEN
1. Every Pattern starting from a LDN don't meet, via Collatz Procedure, all the way to Unity another LDN .
2. Every Pattern starting from a LET, do not meet an LDN on its way to Unity.
3. Each LDN have its unique way to Unity.
4. Each LDN may be considered as provided via Murgu Inverse Method by a LET and each LET have an image in a LDN via (3*(LET)).
Those 4 conditions demonstrate logical - a truth -
Collatz Conjecture Patterns into Positive Integers - Infinity Per 6 -

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Murgu 1D Array Index Infinity_Per_6 theoretical reality and eternal Provocation .

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Murgu 1D Array Index Infinity_Per_6 Is a Provocation as because of Infinity Paradox is a goal which never will be reach

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But for us will live as Murgu 1D Array Index infinity per 2

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Where this time infinity is our Computational Approach of BigInt evolving in time.
https://collatzconjectureend.com/inf6/usa_solved_collatz_conjecture_in_2024.html

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Collatz Conjecture Unique Solution Murgu Table2To3.

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Via Group Theory Grouping create into Positive Integers a New Analytic Structure and a New analytic Working Structure Working Collatz Domain which after Absolute Analyze can be Reintegrated into all Positive Integers via a very simple formula applied to each Odd.
2^kQ    where Q an Odd and k=1 to Infinity

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FOR Mathematicians
Murgu Table2To3 with is new structural concept and its LET's Formulas

((2^(2k+2)T_1i) - 1) = 3 Q_i     (C.E.-1.1)
((2^(2l+1)T_2j) - 1) = 3 Q_j     (C.E.-2.1)

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expressed as
((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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is an absolute theoretical Collatz Conjecture Solution, as (C.E.-1) , (C.E.-2) and the fact LDN's are closures contain also for all Math rigor analytical structure and logical incorporated.
Via its new structural representation, all Math Rigor Collatz Conjecture Murgu Inverse Method , Collatz Conjecture Mururgu Arrow analyzed as Infinity Marker USA Murgu Arrows,
Collatz Conjecture Won The Infinity
Collatz Won The Infinity

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as demonstrated
Infinity Pay Respect Collatz Conjecture

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Then, Infinity Pay Respect Collatz Conjecture, Certify PROOF Of All Profs

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Murgu 1D Array Index Infinity_Per_6

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Murgu 1D Array Index infinity_Per_2

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A Big H-Intel victory which leaded to important new Science Jewels : Table2To3 use for another Issues.
Collatz Functional Divergence as together with USA2015AEPDF started Functional Divergence Study.
Murgu 1D Array Index Infinity_Per_6 or Collatz 1D Array Index Infinity_Per_6

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for which maybe is the best to keep its provocative name confronting Infinity Paradox. Collatz Conjecture Murgu Method

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Collatz Conjecture Murgu Method Consist into:
1. Collatz Conjecture New Group Theory Structure - By grouping Positive Integers into Grids of 6 Integers.
2. As observed via Coming Murgu Inverse Method are Odd To Odd Formula and Then can by Considered also Functions
Even excluzion in interest of Logical Analitic and reinclude after.
3. Collatz Conjecture Murgu New Formulas (may be 9 but important are 2 LET's Formulas) 4. Collatz Conjecture Murgu New Logical Coordinate System - Table2To3.
5. Collatz Conjecture Murgu Inverse Method originating from Collatz Procedure - (3x+1) - and adapted for new coordinate system.
6. Math Rigor DEMONSTRATIONS for Collatz Conjecture Murgu Method Legacy over Collatz Conjecture Procedures . 7. Murgu Table2To3 as LET's Formulas derived into 2 Linear Functions whithout to affect formulas role.
8. Logical Observation - LDN do not accept Murgu Inverse Method - then are Logical Closures.
9. Marker USA Murgu Arrow as exposing Collatz Conjecture Murgu 2 Linear LET's Function Starting from 1 and second from 5 united into Arrow by Collatz Procedure (3*5+1)=16 . 10. Infinity Murgu Table2To3 as validarors .
11. Infinity Marker USA Murgu Arrows .
12. Proof Of Proofs or theoretical proof - Murgu 1D Array Infinity Per 6 .

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Those now can be named
Collatz Conjecture Logical Beauty Tools

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As not only SOLVED Collatz Conjecture - already declared unsolvable - but, extracted from any Science new Beauty included onbserved but not demonstrated proprietie
Collatz Conjecture Separe Positive Integers in 2 Parts

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for which I think Collatz Conjecture Logical Beauty Tools are ready to demonstrate it.

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Murgu LET's Formulas Legacy

Collatz Conjecture have a Unique Solution - Murgu Table2To3

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Murgu LET's As Functions and Formulas
Guarded by All Math Rigor Collatz Conjecture Murgu Method Legacy.
Maybe heavy Part Of understanding and clarify remain Murgu LET's Formulas Legacy and then Collatz Conjecture Procedures transfer to Infinity Collatz Conjecture Murgu Double Functions acceptance.
In 2024 Material this legacy been demonstrated using 4 Murgu Lemas - 2 connected To Murgu CVR and 2 to Collatz Conjecture:
1. Each Integer of Form (2^(2k+2)-1) is divided by 3 -- L.01
2. Each Integer of Form (2^(2k+1)+1) is divided by 3. --- L. 02
3. Each Integer of Form (2^(2k+2)+1) not divided by 3 -- L.03
2. Each Integer of Form (2^(2k+1)-1) not divided by 3. --- L. 04
((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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As Example for LET_1i = (1+6i)
((2^(2k+2)[1+6i]) - 1) =3Q
--- ((2^(2k+2)+ 2^(2k+2) 6i]) - 1) =3Q
(2^(2k+2) ) - 1 = 3(( Q - 2^(2k+3)*i]))
As (2^(2k+2) ) - 1 --- L.01 Lema - Then Confirm for LET1's (LET_1i) Formulas Legacy

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((2^(2l+1)[5+6j]) - 1) = 3 Q_j
((2^(2l+2)) - 1) = 3( Q_j - 2j(2^(2l+1))-^(2l+1))
- L.01 Lema - Then Confirm for LET2's (LET_2j) Formulas Legacy

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Seem forced maybe - but for both of them .

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Now, using Collatz Conjecture Murgu Method Group Theory Part
(1+6i) - (2+6i) - (3+6i) - (4+6i) - (5+6i) - (6+6i) -- G.0

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can demonstrate each LET's formuals will send (3Q) into a new grid and it will be only and only in
(3+6p) - or - (6+6p) -- G.0

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for,
((2^(2k+2)[1+6i]) - 1) = 3 Q_i That mean
((2^(2k+2)[1+6i]) - 1) = (3+6p)     where p= grid p or
((2^(2k+2)[1+6i]) - 1) = (6+6p)

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((2^(2k+2)[1+6i]) ) = (4+6p) or
((2^(2k+2)[1+6i])) = (7+6p) -- False
(7+6p)=Even+Odd - ODD
THAT mean - each
((2^(2k+2)[1+6i]) - 1) = 3 Q_i     (C.E.-1)
((2^(2k+2)[1+6i]) - 1) = 3 (1+2p)     Q_i= (1 +2p)
Then returning to Collatz Conjecture Procedure (3x+1)
3 (1+2p) + 1 = (2^(2k+2)[1+6i]) or
3 (1+2p) + 1 = (2^(2k+2)) LET_1i

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Collatz Conjecture Procedure for LET_1i -heavy to follow but Certify LET_1i Murgu Formulas Legacy over Collatz Conjecture.

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FOR ((2^(2l+1)[5+6j]) - 1) = 3 Q_j     mean
((2^(2l+1)[5+6j]) - 1) = (3+6p) or
((2^(2l+1)[5+6j]) - 1) = (6+6p)

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((2^(2l+1)[5+6j]) ) = 2(2+3p) - True or
((2^(2l+1)[5+6j]) ) = (7+6p) FALSE

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Then returning to Collatz Conjecture Procedure (3x+1)
3(p+1) + 1 = ((2^(2l+1)[5+6j]))
Collatz Conjecture Procedure for LET_2j -heavy to follow but Certify LET_2j Murgu Formulas Legacy over Collatz Conjecture.

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From this demonstration may derive any logical observation, part explaining, in part, Collatz Conjecture apparent divergence is functional one, for us now important is Murgu LET's Formulas have all rights for mathematic and logic image of Collatz Conjecture.

Collatz Unicity and Murgu LET's Formulas Explained

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Make possible as 2024 Collatz Conjecture Unique Solution to be Understood.

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For Collatz Unicity need to pay attention and treat
((2^(2k+2)T_1i) - 1) = 3 Q_i     (C.E.-1.1)
((2^(2l+1)T_2j) - 1) = 3 Q_j     (C.E.-2.1)

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As Infinity Double Linear Functions and Infinity Murgu Table2To3.

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Collatz Conjecture Functional State.

Demonstration of Collatz Conjecture Murgu Inverse Method and Murgu LET's Formulas Legacy brought a small victory.
Collatz Conjecture as functional 2 Functions
3 (1+2p) + 1 = (2^(2k+2)[1+6i]) CF.01
3(p+1) + 1 = ((2^(2l+1)[5+6j])) CF.02

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But those functions do not cover all Collatz Conjecture domains and include a lot of if's. Those do not cover (3+6i) and (6+6i). (6+6i) is included in (3+6i) and do not need demonstration,then need to include only (3+6i).
As Murgu Inverse Method revealed all Integers (3+6i) do not have UP Collatz Connection node, but only Down:
Collatz Conjecture Connections via Collatz Procedures are all Down connections and lead to Unity.
Also became clear when Murgu Inverse Method reveal UP Connections only, Collatz Procedures reveal DOWN Connections only.

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Then (3(3+6i)+1) = 2^kQ
(10+18i) = 2^kQ CF.03
CF.03 unfortunately remain at state of Collatz Procedures as need one a more divisions with 2. Also CF.01 and CF.02 imply lots of ifs for correct representation.
Collatz Conjecture Functions do not present any interest in solving Collatz Conjecture.

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Instead of Murgu LET's Formulas and Functions derivted by Formulas wich lead natural to
Infinity Murgu Table2To3
Infinity Marker USA Murgu Arrow
Murgu Collatz Unicity
VALIDATING

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Collatz Conjecture Unique Solution - Murgu Table2To3

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And then revealing 3 new small Treasure coming from Collatz Conjecture
1. Table2To3 as util even for Group Theory.
2. Functional Divergence Study.
3. Murgu 1D Array Index Infinity Per 6.
Murgu 1D Array Index Infinity Per 6 or Collatz 1D Array Index Infinity Per 6 for sure already won concurence for Eternal Computational Evolution METER and Tester.
Infinity_Per_6 is provocation as Infinity is Intangible, then Index will be all around infinity_Per_2 where infinity will be our computational aproach of Infinity via BigInt evolving (Infinity Paradox).

Murgu Inverse Method with its Murgu Table2To3 - Collatz Conjecture Key.

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For Collatz Conjecture All Math Rigor Solution was Important All :
Group Theory Method of grouping Positive Integers in Grids Of 6.
Into Negative Collatz Conjecture loss its sense as now we know Infinity Pay Respect For Collatz Conjecture- in negative not - see Murgu Conjecture Vicious Redundancy.
Also Collatz Conjecture Murgu Inverse Method is a key for but remarkable is its revelation Table2To3, or Murgu Table2To3, or Collatz Table2To3 which via theirs Formulas induced Functional Linear Properties Solved Collatz Conjecture as Unique 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)
Where [1+6i] are LET_1i and [5+6i] are LET_2j
Cover Infinity times axle OY as Double Linear Functions for k=l 0 To Infinity, and treated so reveal Collatz Unicity.
For k=l=0 Table2To3 reveal all LET_1i and [5+6i] are LET_2j PRIMARY Up Connections
And it make Possible and reveal Collatz Conjecture as Pocket Map and then
Proof Of Proofs Murgu 1D Array.

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(C.E.-1) and (C.E.-2) for Mathematician are Collatz Conjecture SOLUTIOn, and Unique. Even if change the names of Logical Dead Nodes and Logical Eternal Triads or even names of Murgu Methods , Mathematical, Logical and Scientific need to gain the same result as is Unique Solution.
Infinity Murgu Table2To3 is for analytic state via k=l=0 determine Primary Logical LET's Connections , for k=l=1 Secondary and so on for k=l=Infinity the Infinity.
Those show treated in general via theirs slope as Linear Functions do not Intersect or will Intersect into Rationals, then not into Positive Integers Collatz Nodes.
That mean, into Collatz Sense , not multiple or even double Collatz Way.

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Collatz Nodes have a single connection for its way to Unity.
Maybe we make Collatz story too long, but was for extracting logical its proper beauty and also any results from.
Theoretic now Collatz Patterns stand into software generator using Primary Murgu Table2To3 (k=l=0), Collatz Unicity. As a beauty of story need to remark how Logical Dead Nodes stand as a Fence Pillars watching Collatz Conjecture Divergence against falling into Infinity or redundant cycles.

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Collatz Conjecture Table2To3 Story have its beauty.

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Not forgot, (C.E.-1) and (C.E.-2) also as :
((2^(2k+2)T_1i) - 1) = 3 Q_i     (C.E.-1.1)
((2^(2l+1)T_2j) - 1) = 3 Q_j     (C.E.-2.1)

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Table2To3 as Collatz Conjecture new coordinate system is important step for Collatz Conjecture all Math rigor analytic but INTEGRATE All Positive Integers via
2^k Q     where Q Odd Positive Integer
For Odd Logical Eternal Triad 2^k Q evolve into Infinity and don't have a final closure, when for Logical Dead Nodes as are Collatz Conjecture Closures, (2^k Q Q a LDN) act as connecting wires with Infinity for k=1 To Infinity and for Collatz Conjecture remain under order of Collatz Conjecture Procedure of division with 2.
For Odd Logical Eternal Triad 2^k Q remain under order of both Collatz Conjecture Procedures, evolving to anothers LDN or anothers LET's . Truth as a Truth, Murgu Table2To3 Formulas and theirs Functional cloth Solved Collatz Conjecture by June 2024.
As an PDF as book can't afford Murgu Table2To3 examples accuracy :
Collatz Conjecture BOOK and Story are online html's.

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With all Science respect this BOOK can be considered also second affordable new
Science Domain - Functional Divergence -

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As Collatz Conjecture is solved via Murgu Table2To3 and now know Collatz Divergence is Functional ONE.

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Down can see a part of Murgu Table2To3 for k=l=0 AN D second for k=l=0 , 1 and 2 .

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