Mapping the Odd Integers >1,  the Trees,  and the List of Left Descent Assemblies Into One Another

The only way all the elements of this structure could possibly be brought into a cohesive whole is to have a firm mapping of the elements of each of the data structures encountered into the corresponding elements of each of the other data structures.  If this is achieved,  then the conclusions reached on the basis of one viewpoint can be applied to the whole system and there will be a much enhanced possibility of achieving a formal proof.

The first data structure is a general tree representation of the predecessor tree.  All the odd precursors of a given odd integer in the Collatz trajectories are collected as its children in order of increasing magnitude.

The second tree is a binary tree obtained from the previous,  general,  tree by the usual transformation.  The children of any node in the general tree are linked each to the next,  and the parent is linked only to that child which is smallest in magnitude.  There is clearly a 1:1 bi-directional mapping between the nodes in the general tree and those in the binary tree since there has been no change other than a rearrangement of the links which construct the tree.

The abstract predecessor tree is quite distinct from the two predecessor trees which provided the observations which suggested consideration of the abstract tree.  The abstract tree is a tree in which the nodes are infinite integer sets (not individual integers).  The mapping from the binary tree to the abstract tree is an infinity to one mapping.  Integers are located at a depth in the abstract tree identical to their depth in their left descent from their parental extensions.  (That is,  it ignores their depth from the root,  1,  of the predecessor tree and reflects only their depth in the left descent.)  Which set at its level an integer is found in is determined by the path down to it in its left descent (which I've been denoting by an e(sb)t notation).  Each kind of left descent found in the binary tree will map into a particular path in the abstract tree.  Every instance of a particular left descent in the binary tree will contribute its integer elements to the same series of sets in a path in the abstract tree.  Thus all elements of the successive instances of any kind of left descent in the binary tree will be found in the same path through the infinite sets in the abstract predecessor tree.

This is not a bi-directional mapping.  Left descents in the binary tree map with certainty into descents in the abstract tree.  The reverse is not true.  The descents in the abstract tree cannot be placed with certainty in the binary tree.  In fact,  the successive instances of a left descent from the abstract tree are scattered all over the binary tree with no apparent rhyme nor reason.  This is demonstrated for the ebbt left descent,  as an example.

The mapping from the abstract predecessor tree into the list of left descent assemblies is pictured more clearly than it can be explained.  The two are entirely equivalent,  mapping each into the other with complete fidelity.  One sees this by going through the process of teasing out of the initially very large sets in the abstract predecessor tree the series of subsets which may be assigned to a completed path from extension to leaf.  That teasing out process precisely defines the infinite set of instances of each particular left descent assembly in terms of the dn+c formulas which play a role in the depiction of the structure as a sieving operation performed by left descent assemblies.

One might feel that the inability to map from the completely subsetted abstract predecessor tree (equivalently the list of left descent assemblies)  back into the binary tree would be fatal to the possible use of these structures to reach a proof.  Fortunately there is an escape from this.  We have established a 1:1 bidirectional mapping between the integers and their position in the list of left descent assemblies.  Since it is the integers,  not their placement in the binary tree,  which are of interest in reaching a proof,  this alternative 1:1 mapping between the positive odd integers and the list of left descent assemblies should provide the needed linkage.


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