Collatz convergence is a Hydra game
CCollatz convergence is a Hydra Game
Alexander Rahn ∗ , Eldar Sultanow † and Idriss J. Aberkane ‡ January 23 rd Abstract
The Collatz dynamic is known to generate a complex quiver of sequences overnatural numbers which inflation propensity remains so unpredictable it could beused to generate reliable proof of work algorithms for the cryptocurrency industry.Here we establish an ad hoc equivalent of modular arithmetic for Collatz sequencesto automatically demonstrate the convergence of infinite quivers of numbers, basedon five arithmetic rules we prove apply on the entire Collatz dynamic and whichwe further simulate to gain insight on their graph geometry and computationalproperties. We then formally demonstrate these rules define an automaton that isplaying a Hydra game on the graph of undecided numbers we also prove is embeddedin 24 N −
7, proving that in ZFC the Collatz conjecture is true, before giving apromising direction to also prove it in Peano arithmetic.
The dynamical system generated by the 3 n + 1 problem is known to create complex quiv-ers over N , one of the most picturesque being the so-called ”Collatz Feather” or ”CollatzSeaweed”, a name popularized by Clojure programmer Oliver Caldwell in 2017. Theinflation propensity of Collatz orbits remains so unpredictable it can form the core of areliable proof-of-work algorithm for Blockchain solutions [1], with groundbreaking appli-cations to the field of number-theoretical cryptography as such algorithms are unrelatedto primes yet, being based on the class of congruential graphs, still allow for a wide diver-sity of practical variants. If Bocart thus demonstrated that graph-theoretical approachesto the 3 n + 1 problem can be very fertile to applied mathematics, the authors have also ∗ Nuremberg Institute of Technology, Keßlerpl. 12 90489 Nuremberg, Germany † Potsdam University, Chair of Business Informatics, Processes and Systems, Karl-Marx Straße 67,14482, Potsdam, GermanyCapgemini, Bahnhofstraße 30, 90402, Nuremberg, Germany ‡ Unesco-Unitwin Complex Systems Digital Campus, Chair of Prof. Pierre Collet, Pierre Collet,ICUBE - UMR CNRS 7357, 4 rue Kirschleger, 67000 Strasbourg, France. Corresponding author:[email protected] a r X i v : . [ m a t h . G M ] J a n ndeavored to demonstrate its pure number-theoretical interest prior to this work [2], [3],[4], [5].The definitive purpose of this article however is to establish fundamental properties ofthe ”Collatz Feather” and infer provable consequences of those properties to achieve apositive proof of the Collatz conjecture. Our methodology consists of using the completebinary tree and the complete ternary tree over 2 N + 1 as a general coordinate systemfor each node of the Feather. We owe this strategy to earlier discussions with Feferman[6] on his investigations on the continuum hypothesis, as it is known the complete binarytree over natural numbers is one way of generating real numbers. The last author’sdiscussion with Feferman argued morphisms, sections and origamis of n-ary trees over N could be a promising strategy to define objects of intermediate cardinalities between ℵ n and ℵ n +1 , in a manner inspired from Conway’s construction of the surreal numbers [7],which itself began by investigating the branching factor of the game of Go. The initialinterest therefore, was to investigate the branching factor of the Collatz Feather and todefine the cardinality of the set of its branches. Here we begin by identifying and provingfive arithmetical rules that apply anywhere on the Collatz dynamic, with the purpose ofdemonstrating that applying them from number 1 allows to generate the entire Feather. Note 2.1.
For all intent and purpose we will define
Syr(x) or the ”Syracuse action” as ”the next odd number in the forward Collatz orbit of x ” . Whenever twonumbers a and b have a common number in their orbit, we will also note a ≡ b , a relationthat is self-evidently transitive: ( a ≡ b ) ∧ ( b ≡ c ) ⇒ a ≡ c The choice of symbol ” ≡ ” is a deliberate one to acknowledge a kinship between our methodand modular arithmetic. Definition 2.1.
Actions G, V and S For any natural number a are specified as follows:1. G ( a ) := 2 a − S ( a ) := 2 a + 1 . The rank of a is its number of consecutive end digits in base .3. V ( a ) := 4 a + 1 = G ◦ S ( a ) The complete binary tree over odd numbers is defined as 2 N + 1 endowed with the following twolinear applications {· − · } . The complete ternary tree over odd numbers is defined as 2 N + 1endowed with operations {· − · · } efinition 2.2. Type A, B and C1. a number a is of type A if its base 3 representation ends with digit 22. a number b is of type B if its base 3 representation ends with digit 03. a number c is of type C if its base 3 representation ends with digit 1To remember which is which one need only remember the order of ABC: a + 1 is dividableby 3, and so is c − , thus A is on the left of B and C is on the right. G G V S V S S G V S S S S V S S S V G G G G Figure 1: Quiver connecting all odd numbers from 1 to 31 with the arrows of actions S,Vand G. The set 2 N + 1 is thus endowed with three unary operations without a generalinverse that are non commutative with G ◦ S = V . Whenever we will mention the inverseof these operations, it will be assuming they exist on N . Type A numbers are circled inteal, B in gold and C in purple. Theorem 2.2.
The following arithmetic rules apply anywhere over the system N + 1 endowed with the Collatz dynamic. • Rule One: ∀ x odd, V ( x ) ≡ ( x ) • Rule Two: ∀ x, k odd, S k V ( x ) ≡ S k +1 V ( x ) and ∀ x, k even, S k V ( x ) ≡ S k +1 V ( x ) • Rule Three: ∀{ n ; y } ∈ N , ∀ x odd non B, n x ≡ y ⇒ n (cid:86) i =1 ( V (4 i n − i x )) ∧ S ( V (4 i n − i x )) ≡ y • Rule Four: ∀{ n ; y } ∈ N , ∀ x odd non B, S (3 n x ) ≡ y ⇒ n (cid:86) i =1 ( S (4 i n − i x ) ∧ S (4 i n − i x )) ≡ y • Rule Five ∀ n ∈ N , ∀ y ∈ N , ∀ x odd non B where n x is of rank 1, a ≡ y , a = G (3 n x ) ⇒ n (cid:86) i =0 ( S i ( G (3 n − i x )) ∧ S i +1 ( G (3 n − i x ))) ≡ y
3n the following we will prove these five rules.
Note 2.3.
In reference to
Figure 1 we will call ”vertical even” a number that can bewritten V ( e ) where e is even, and ”vertical odd” if it can be written V ( o ) where o isodd. For example, is the first vertical even number and is the first vertical odd. If a is written 4 b + 1 then 3 a + 1 = 12 b + 4 = 4(3 b + 1) therefore a ≡ b . Lemma 2.4.
Let a be a number of rank thus with an odd number p so that a = G ( p ) then Syr ( S ( a )) = G (3 · p ) . Let a be a number of rank n so that S − ( n − ( a ) = G ( p ) then Syr n − ( a ) = G (3 n − · p ) Proof. If a = 2 p − p is odd, then it follows: S ( a ) = 4 p − · SS ( a )+12 = p − = 6 p − G (3 · p ) Syr ( S ( a )) = G (3 · p )Let’s generalize to the n . If Syr ( S ( a )) can be written G (3 · p ) it is also of rank 1, whereas S ( a ) was of rank 2, therefore, the Syracuse action has made it lose one rank. All we haveto prove now is that Syr ( S ( a )) = S ( Syr ( S ( a ))) under those conditions: · ( S ( a ))+12 = 6 a + 5 S ( Syr ( S ( a ))) = S (3 a + 2) = 6 a + 5 = Syr ( S ( a ))If a is of rank n > Syr ( a ) is of rank n −
1, and
Syr ( S ( a )) = S ( Syr ( a )) Note 2.5.
The n +1 action over an odd number, since it necessary yields an even one, isin fine equivalent to adding to it, then the half of the result, then − . How many timesone can add an half to an odd number +1 directly depends on its base 2 representation,and in particular its number of consecutive end digits 1. Let us take Mersenne numbersfor example, which are defined as n − . One can transform them consecutively in thisway a number of time equal to their rank-1, indeed, , which is written in base2 is of rank , because
32 = 2 so if one repeats the action ”add to the number+1 the alf of itself” this will yield an even result exactly four consecutive times . Thus, anystrictly ascending Collatz orbit concerns only numbers a of rank n > , and is defined by ( a + 1) · (cid:18) (cid:19) n − − Lemma 2.6.
Let a be an odd number of rank that is vertical even, then a is of rank2 or more, and a is vertical even. Let a be an odd number of rank that is vertical odd,then a is of rank or more, and a is vertical odd.Proof. If a is vertical even it can be written 8 k + 1 ∀ k : 3 a = 24 k + 3 and this numberadmits an S − that is 12 k + 1, which is an odd number, therefore 3 a is at least of rank 2.Moreover, 9 a = 72 k + 9 and this number admits a V − that is 18 k + 2, an even number.Now if a is vertical odd, it can be written 8 k +5 and ∀ k : 3 a = 24 k +15 and 9 a = 72 k +45.It follows that 3 a admits an S − and 9 a admits a V − , respectively 12 k + 7 and 18 k + 11and they are both odd. Lemma 2.7.
Let a be a number that is vertical even, then ( a ) ≡ S ( a ) and S k ( a ) ≡ S k +1 ( a ) for any even k. Let a be a number that is vertical odd, then S ( a ) ≡ S ( a ) and S k ( a ) ≡ S k +1 ( a ) for any odd k.Proof. If a is vertical even then it can be written as G ( p ) where p is necessarily vertical(odd or even). We proved that 3 p is then of rank 2 or more and also that we have Syr ( S ( a )) = G ( p ) so it is necessarily vertical odd (since 3 d is of rank 2 or more) so Syr ( a ) = V − ( Syr ( S ( a )) and therefore a ≡ S ( a ). This behavior we can now generalizeto the n , because if a is vertical even with a = G ( p ), then the lemmas we used alsoprovide that Syr n ( S n ( a )) = G (3 n · p ) and therefore Syr n ( S n ( a )) will be vertical even forany even n because 3 n · p will be vertical something (even or odd, depending on p only)for any even n .Now if a is vertical odd it can be written G ( p ) and p is necessarily of rank 2 or morebecause G ◦ S = V . Thus 3 p is vertical (even or odd) and therefore Syr ( S ( a )) = G (3 p )is vertical even. Note 2.8.
Observe that in the process of proving
Rule Two we also demonstrated thatany number of rank or more is finitely turned into a rank number of type A by theCollatz dynamic, and that any number x of rank or more so that x ≡ S ( x ) under RuleTwo is finitely mapped to a type A number that is vertical even, therefore proving theconvergence of such numbers is enough to prove the Collatz Conjecture . Lemma 2.9.
Let a be a vertical even number with a = G n +2 ( S ( b )) where n and b areodd, then a ≡ n +12 ( b ) . Let a be a vertical even number with a = G m +2 ( S ( b )) where m iseven (zero included) and b is odd, then a ≡ S (3 m ( b ))5 roof. If a = G n +2 ( S ( b )) by definition a = 2 n +3 b + 1. Then 3 · a + 1 = 3(2 n +3 b + 1) + 1) =2 n +3 · (3 b ) + 4 . As this expression can be divided by 2 no more than twice, we have
Syr ( a ) = 2 n +1 b + 1 = G n ( S (3 b )).Note that if n = 1 then V − ( Syr ( a )) = V − (2 · (3 a ) + 1) = 2 · · (3 b ) = 3 b which is ofcourse an odd number. Therefore Syr ( a ) is vertical odd and V − ( Syr ( a )) = 3 b thus wehave proven that a ≡ b .If n = 0 then a = 2 · b + 1 so 3( a + 1) = 2 · b + 4 therefore Syr ( a ) = S (3 b ) and thus a ≡ S (3 b ). From this we can generalise the progression of numbers that can be written G n ( x ) where x is of rank 2 or more. Let b be any odd number: • All ”Variety S” numbers above b are written V ( b · k − ) or S ( b · k ) = 2 k +1 · b +1and • all ”Variety V” numbers above b are written V ( b · k ) or equivalently S ( b · k +1 ) =4 k +1 · b + 1.Any number g that can be written G n ( V ( x )) with x odd and n > S (3 m x ) or V (3 m x ) by the repeated following transformation:( g − · (cid:18) (cid:19) k + 1Therefore we have indeed that, • for Variety S numbers: 2 k +1 · b · (cid:0) (cid:1) k + 1 = 2 b · k + 1 = S ( b · k ), which proves Rule Four . • for Variety V numbers: 4 · k · b · (cid:0) (cid:1) k + 1 = 4 b · k + 1 = V ( b · k ) which proves Rule Three because
Rule One already provides that V ( b · k ) ≡ b · k . Any type A number of rank 1 can be written a = G ( b ) where b is of type B. In proving Rule Two we showed that any number of rank n > G (3 n − · G − ( S − ( n − ( a ))), which combined with Rule Two itself gives
Rule Five . 6
35 7 151311917 19 21 23 25 27 29 3133 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63
65 67 12769 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 101 103 105 107 109 111 113 115 117 119 121 123 125 ≡ ⇒ ≡ ≡ ⇒ ≡ ≡ ⇒ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ ≡ Figure 2: Just a few applications of
Rules Three, Four and Five starting from 1 ≡ ≡ Rules One and Two are plotted in black. Whenevera number is connected to 1 by a finite path of black and/or gold edges it is proven toconverge to 1.
Definition 3.1. On { N + 1 ; G, S } where Rules One and Two are considered pre-computed (the black edges on Figure 2) the systematic computation of
Rules Three,Four and Five from number onward is called the ”Golden Automaton” . Our purpose is to develop an ad hoc unary algebra that could found a congruence arith-metic specifically made to prove the Collatz conjecture, and which we intend as an epis-temological extension of modular arithmetic, hence our use of the symbol ≡ in this articlerather than the usual ∼ which is seen more frequently in the Collatz-related literature.This ”Golden arithmetic” involves words taken in the alphabet { G ; S ; V ; 3 } , whichwe will call in their order of application, just like in turtle graphics. For example VGS3means 3 · S ◦ G ◦ V Rules 3, 4 and 5 may now be reformulated as such, without loss of generality as long as
Rules One and Two are still assumed: • Rule Three:
Let b be of type B, then b ≡ V GS − from b. We will call thisaction R b ( x ) = 16 x + 1 • Rule Four:
Let c be of type C, then c ≡ GS − from c. We will call this action R c ( x ) = x − Rule Five:
Let a be of type A, then a ≡ G − from a. We will call this action R a ( x ) = x − As Rules One and Two ensure that the quiver generated by the Golden Automaton isbranching, with each type B number that is vertical even providing both a new A typeand a new B type number to keep applying respectively rules 5 and 3, we may followonly the pathway of type A numbers to define a single non-branching series of arrows,forming a single infinite branch of the quiver. The latter, if computed from number 15,leads straight to 31 and 27, solving a great deal of other numbers on the way:15 ≡
81 Rule 381 ≡ Rule 3 ≡ Rule 5 ≡
607 Rule 2607 ≡
809 Rule 4809 ≡ Rule 5 ≡
319 Rule 2319 ≡
425 Rule 4425 ≡ Rule 5 ≡
377 Rule 4377 ≡ Rule 5 ≡
593 Rule 3593 ≡ Rule 5 ≡
233 Rule 4233 ≡ Rule 5 ≡
137 Rule 4137 ≡ Rule 5 ≡
161 Rule 4161 ≡ Rule 5 ≡
41 Rule 441 ≡ Rule 5
Again, it is in no way a problem, but rather a powerful property of the Golden Automatonthat this particular quiver branch already cover 19 steps (and actually more) because eachof them is branching into other solutions.We may follow another interesting sequence to show that in the same way that Mersennenumber 15 finitely solves Mersenne number 31, Mersenne number 7 solves Mersennenumber 127, this time we will follow a B branch up to
Syr (127) which we know can bewritten G (3 ) because 127 is the Mersenne of rank 7.By Rule 4 we have the first equivalence ≡ and ≡ ≡ .So by Rule 2 we also have ≡ .Rule 3 gives ≡ and again ≡ G (729) ≡ .8he cases of 15 proving the convergence of 31 and 27 and of 7 proving the one of 127naturally lead us to the following conjecture: Conjecture 3.1.
Suppose all odd numbers up to n are proven to converge to underthe Collatz dynamic, then the Golden Automaton finitely proves the convergence of thoseup to n +1 And indeed we already have that the Golden Automaton starting with 1 proves 3 by
RuleOne , then 3 proves all numbers from 5 to 15 which in turn prove all numbers from 33 to127. In the next subsection we render larger quivers generated by the Golden Automatonto provide a better understanding of their geometry and fundamental properties, and todemonstrate why it is so, and more generally, why it can be proven they can reach anynumber in 2 N + 1 in ZFC. The purpose of this subsection is to identify provable fundamental properties of the GoldenAutomaton by scaling it up on the full binary tree over 2 N + 1. To streamline its algo-rithmic scaling, we use the simplified rules we defined in the previous subsection, again,without loss of generality. Our precise purpose is to pave the way for a formal demon-stration that proving the convergence of odd numbers up to n is always isomorphic toa Hydra Game. In the next figures we color all the elements of 24 N − { , , , . . . } in red to as we demonstrate in the next section they precisely from the”heads” in the Hydra Game. 9igure 3: Golden Automaton confined to numbers smaller than 32Figure 4: Golden Automaton confined to numbers smaller than 6410igure 5: Golden Automaton confined to numbers smaller than 128Figure 6: Golden Automaton confined to numbers smaller than 256 In this section we prove that the Collatz conjecture holds in the Zermelo–Fraenkel settheory with the axiom of choice included (abbreviated ZFC). And we prove that Collatzconjecture also holds in the Peano Arithmetic.
Theorem 4.1.
In ZFC, the Collatz conjecture is true.
Definition 4.1. A hydra is a rooted tree with arbitrary many and arbitrary long finitebranches. Leaf nodes are called heads . A head is short if the immediate parent of thehead is the root; long if it is neither short nor the root. The object of the Hydra game is to cut down the hydra to its root. At each step, one can cut off one of the heads, afterwhich the hydra grows new heads according to the following rules: • if the head was long, grow n copies of the subtree of its parent node minus the cuthead, rooted in the grandparent node. • if the head was short, grow nothing Lemma 4.2.
The Golden Automaton reaching any natural number is a Hydra Game overa finite subtree of the complete binary tree over N − . roof. The essential questions to answer in demonstrating either a homomorphism be-tween a Hydra game and the Golden Automaton reaching any odd number, or that theGolden Automaton is playing at worst a Hydra Game are: • What are the Hydra’s heads? • How do they grow? • Does the Golden Automaton cut them according to the rules (at worst)?
Definition 4.2.
A type A number that is vertical even is called an A g . The set of A g numbers is N − . Type B numbers that verify b ≡ S ( b ) and type C numbers that verify c ≡ S ( c ) under Rule Two are called
Bups and
Cups respectively.
What are the Hydra’s heads? A g numbers are the heads of the Hydra. They are12 points apart on 2 N + 1 (24 in nominal value, e.g. 17 to 41) and any Bup or Cup of rank > Rule Five is smaller than them since action R a is strictlydecreasing so up to the n th A g there are 2 n (Bups + Cups) of rank 2 or more and half ofthem are equivalent to these A g (e.g. between 17 and 41 Bup 27 is equivalent to A g Rule FourHow do they grow?
Between any two consecutive A g in 2 N + 1 there are • • A g • • Let b be of type B, there are b numbers of type A g that are smaller than V ( b ) • Let c be of type C, there are S ( c )3 numbers of type A g that are smaller than V ( c ) • Let 3c be a type B where c is of type C, there are S ( c )3 numbers of type A g up to R b (3 c ) included • Let 3a be a type B where a is of type A, there are G ( a )3 numbers of type A g smallerthan R b (3 a )Which is defining the growth of the heads. Indeed, any supposedly diverging A g is forminga Hydra, as we have proven 24 N − Does the Golden Automaton play a Hydra game?
12t could be demonstrated that the Golden Automaton is playing an even simpler gameas it is branching and thus cutting heads several at a time and in particular cutting somelong heads without them doubling but as this is needless for the final proof we can nowsimply demonstrate that even under the worst possible assumptions it follows at least therules of a regular Hydra game.The computing of 15 ≡ . . . ≡
27 that we detailed in
Subsection 3.1 is one case of theplaying of a Hydra Game by the Golden Automaton; we underlined each use of
Rule 5 specifically so the reader can now report to it more easily, because each time this rule isused, a head ( A g ) has just been cut.The demonstration that 27 and 31 converge is the cutting of heads 41 and 161 respectively.This single branch of the Automaton having first cut head 17, reaches to the head 1025via B-typed numbers 15 and 81. It is therefore playing a Hydra with = 43 headsof which one (17) is already cut at this point and of which at least 8 are rooted (so cuttingthem does not multiply any number of heads). This process being independent of thetargeted number, we now have that the reaching of any number by the Golden Automatonis at least equivalent to the playing of a Hydra with n heads of which 0 < m < n arerooted. Even without demonstrating more precise limit theorems for each factors n and m (which could still be a fascinating endeavor) the road is now open for a final resolutionof the Collatz conjecture.From there indeed, we know with Goodstein [8] and Kirby and Paris [9] that assuming asystem strong enough to prove Peano arithmetic is consistent and that (cid:15) is well-ordered,no Hydra game can be lost. Since we have that the reaching of any number n is a HydraGame for the Golden Automaton, we have that the Golden Automaton cannot fail tofinitely reach any natural number. If it is now sure that any system strong enough to prove the convergence of Goodstein’sseries also proves the Collatz conjecture, it could very well be possible to prove it fromPeano arithmetic alone. In this final subsection, we intend to outline a strategy towardssuch a demonstration by defining a different game than the Hydra one and in particular,a zero-player game that is significantly simpler than John Conway’s Game of Life andplayed on the complete binary tree { N + 1; G, S } .In this cellular automaton, each cell is identified by a unique odd number and can onlyadopt three states: • Black , meaning the odd number is not (yet) proven to converge under the iteratedCollatz transformation or equivalently that it is only equivalent to another black In fact, the reason the Golden Automaton dominates 24 N − • Gold , meaning the odd number is proven to converge and the consequences of itsconvergence have not yet been computed, ie. it can have an offspring • Blue , meaning the number is proven to converge and the consequences of its con-vergence have been computed ie. its offspring has already been turned goldIn this ad hoc yet simpler game of life each gold cell yields and offspring then turns blue,and whenever a cell is blue or gold its odd number is proven to converge. Starting withone cell colored in gold at the positions 1, it applies the following algorithm to each goldcell in the natural order of odd numbers:1.
Rule 1 : if a cell on x is gold color cell on V ( x ) gold2. Rule 2 : if a cell on x is gold, color cell on S ( x ) gold depending on the preciseconditions of rule 23. If a cell on a of type A is gold, then color that on R a ( x ) in gold4. If a cell on c of type C is gold, then color that on R c ( x ) in gold5. After applying the previous rules on a gold cell, turn it blueNote that applying R b on a type B number being equivalent to Rule 1 then R c thealgorithm needs not implement a defined R b Whenever a complete series of odd numbers between 2 n + 1 and 2 n +1 − n +1 −
1, thus givinga clear measurement of the algorithmic time it takes the Golden Automaton to provethe convergence of each complete level of the binary tree over 2 N + 1. We then plot theevolution of this expense on a linear and a logarithmic scale.Figure 7: case n=6, illustrating the principles of the game we defined. On the middleimage, row { } has been solved , with an ”expense” of 8 numbers also solved above it.On the right image, row {
9; 11; 13; 15 } with an expense of 6. As number 1 is the neutralelement of operation R c we leave it in gold during all the simulation14igure 8: case 12, seventh row completedFigure 9: case 12, eighth row completed15o facilitate the observation of each row of the binary tree being covered by the GoldenAutomaton we here zoom into each of them individually:Figure 10: zoom of row 11 (going from 1025 to 2047: each line has about 100 dots)Figure 11: Row 10 (513 to 1023)Figure 12: zoom Row 9 (257 to 511)Figure 13: Row 8 (129 to 255)Figure 14: Charting expenses to proof the convergence of each row of the binary tree over2 N + 1. For example, the amount of gold and blue dots above the row going from 2049to 4095 (row 12) is slightly below 175000. The total expense emerges from the generatedtree’s height. The same plot against a logarithmic scale (Right) indicates a line-like shape.16rom there we can thus provide two strategies to finalise a proof of the Collatz conjecturewithout the axiom of choice (which is needed to demonstrate no Hydra Game can belost) and precisely within Peano arithmetic. The first strategy would consist of usingautomated theorem proving to single out the linear behavior we expose in Figure 14 as a provable property of our game. The second - and we believe the most promising -would consist of analysing the average reproductive rate of gold dots, and demonstratingat any level n of the binary tree they cannot fail to finitely take over any population ofblack dots below it. Definition 4.3. (Reproductive Rate)1. The reproductive rate of any golden dot on coordinate n is the number of goldenand blue dots it generates that are at or below coordinate V ( n ) .2. The average reproductive rate of all black dots converges to 3,5 new black dots gen-erated from x under or equal to V ( x ) . Indeed for any x , in the time to reach V ( x ) only the offspring below G ( x ) gets to generate new black dots under the rules ofthe binary tree, S ( x ) can only reproduce once by applying G ( S ( x )) , thus generating V ( x ) , and all numbers between S ( x ) and the next Mersenne number cannot repro-duce. More precisely, for any odd number x , there are x + G − ( x ) odd numbersbetween itself and V ( x ) included. We can now count the average or limit reproductive rate of each type of golden dot. • for type B numbers we always have V(b) in the offspring, and also S(b) one out oftwo times, so the average reproductive rate is 1,5 • for type C numbers we always have V(c) and R(c) in the offspring, and S(c) oneout of two times so the average reproductive rate is 2,5 • for type A numbers we have a more interesting converging series defining the averagenumber of gold dots generated below V(a)+2. To easily count the offspring we usethe property of the golden Automaton that it can only output A g through operations R b or R c , and can only output non-Aup through operations V and S.The net offspring below V ( A g ) + 2 of A g numbers is defined by a branching process: – R a ( A g ) outputs either an Aup (in proportion of ) or equally either a Bup ora Cup ( each) – then again R a ( Aup ) outputs either an Aup ( ) or equally either a Bup or aCup (again, each) – how long R a ( A g ) keeps rendering an Aup only depends on n where A g = G (3 n x ) with x non B (per Rule 5 )So the formula of the average offspring below V(a)+2 of all type A g number, whichcan be written G (3 n x ) with x non B is 3 n + 2 ,
5. Since one out of two A g can bewritten 3 x , a quarter can be written 3 x and so on we have the general formula17hat A g numbers up to G (3 n ) would have an average reproductive rate, which limitwe can now determine:lim n →∞ n (cid:88) i =1 i · (3 i + 2 ,
5) = 6 , V ( A g ) so the averagenet reproductive rate of A g numbers is converging to a strictly greater value thanthis limit.As we know that the reproductive rate of black dots below V ( x ) + 2 converges to 3 , , , , , = 3 , A g numbers and still below V ( A g ) in the computing of theirbirthrate, we can prove the average birthrate of golden numbers tends to equal 3 , (cid:15) with (cid:15) >
0, which finishes the Peano-arithmetical proof.
This work was supported by a personal grant to I. Aberkane from Mohammed VI Univer-sity, Morocco and by a collaboration between Capgemini, Potsdam University, The GeorgSimon Ohm University of Applied Sciences, and Strasbourg University. I. Aberkane cre-ated the framework of studying the Collatz dynamic in the coordinate system defined bythe intersection of the binary and ternary trees over 2 N + 1, identified and demonstratedthe five rules and predicted they would be isomorphic to a Hydra game over the set ofundecided Collatz numbers, which he defined as well, allowing for a final demonstration ofthe Collatz Conjecture; he also outlined and computed the strategy of using reproductiverates of dots to define a Peano-arithmetical proof. Contributing equally, E. Sultanow andA. Rahn designed and coded an optimised, highly scalable graphical implementation ofthe five rules and ran all the simulations, confirming the Hydra game isomorphism andcomputing the first ever dot plot of the Golden Automaton over odd numbers, whichthey optimised as well. They were also the first team to ever simulate the five rules tothe level achieved in this article, and to confirm their emerging geometric properties onsuch a scale, including the linearity of their logarithmic scaling and the limit reproductiverates of single dots of the golden automaton.I. Aberkane wishes to thank the late Prof. Solomon Feferman and Prof. Alan T. Water-man Jr, along with Prof. Paul Bourgine, Prof. Yves Burnod, Prof. Pierre Collet, Dr.Fran¸coise Cerquetti and Dr. Oleksandra Desiateryk.The authors dedicate this work to the memory of John Horton Conway (1937-2020),Solomon Feferman (1928-2016) and Alan T. Waterman Jr (1918-2008).18 eferences [1] F. Bocart. Inflation propensity of collatz orbits: a new proof-of-work for blockchainapplications. Journal of Risk and Financial Management , 11(4):83, 2018.[2] I. Aberkane. On the syracuse conjecture over the binary tree.
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