I have been ignoring the effect of the extensions on the shape of the predecessor trees. The story told on these pages seems to indicate that a treatment which considers only the left descent assemblies (l.d.a.s) may be enough to get at a proof for the Collatz conjecture. I.e., likely, some intrinsic property of the integers and the predecessor tree generation rules conspire to place the integers uniquely in the predecessor tree without any appeal to organizational assistance from the extensions.

Of course, it is certainly posible that essential information is passed along the extensions. Thus, it seems worthwhile to explore for cases where there is some information passed from a node to its extensions which specify or limit the values of their further descendents.

Certainly, a certain amount of control is exerted upon the
children of extensions by the predecessor tree element which heads the
extension set, although this may be little more than determining
which of the *b*, *s*, or *t* steps appears
first below the succession of extensions. Two cases are
presented; the first, involving the immediate children of
the extensions, is due to the invariant order (*b* then
*s* then *t*) of the first l.d.a. step under the
extensions, and is limited to those first steps because the deeper
steps in an l.d.a. vary from case to case. The second case
presented deals with five successive extensions for the six elements
which constitute the *ebssbt* l.d.a. early in the predecessor tree
which develops from the extension from 3.

The residues modulo 3 of successive elements in a series of
extensions cycle through the values 0, 1, and 2
indefinitely. Thus, these extension elements are,
respectively, leaves, departure points for a *b*
step, and departure points for an *s* step. Since these
edge types are fixed, and we know the relationships between the values
of the nodes at either end of all these edges, we can readily
calculate the relationships among the values in a sequence of
immediate children of an extension set.

The following diagram gives those relationships. The asterisks
represent the successive extensions, the first step of the left
descents are indicated as to types (*b*, *s*, or
*t*), and the formulas relating successive children's values
to that of the first are presented in a given row. Since the first
child of an extension set may be the result of either a *b* or an
*s* step, the diagram includes both possibilities. The
line with formulas involving *a* treat the case where the first
extension gives a *b* step and the line with formulas involving
*b* treat the case where the first extension gives an *s*
step. The relationships continue cyclically; to illustrate
that two additional lines, starting with *c* and
*d*, respectively, starting from the second instance of
the cycle are also shown. Comparison of rows 1 with 3 or 2 with 4 makes
the repeated cycle evident.

*------*------*------*---------*------*--------*------------*--|(b) |(s) t |(b) |(s) t |(b) |(s)| | | | | | a 2a+1 64a+49 128a+99 4096a+3185 8192a+6371 b 32b+17 64b+35 2048b+1137 4096b+2275 c 2c+1 64c+49 128c+69 d 32d+17 64d+35

The above relations can readily be verified for the extensions of
5, which start with an *s* left descent, where b is
3, c is 113, d is 227, and the higher first cousins
are 7281, 14563, 466033, and 932067 respectively. For an example
which starts with a *b* left descent, consider the extensions
of 273, where a is 1457, b is 2915, c is 93297,
and d is 186595.

The next table illustrates the behavior of the first five extensions
of the six elements of the *ebssbt* l.d.a. It is orthogonal
to the previous table in that the former compares only the children of
extensions whereas this next one compares the extensions themselves.

Successive extensions are formed *via* 4*n+1 from the element to
its left. The l.d.a. headed by each extension is given in
the line immediately below the extension values with the number of its
instantiation given in parentheses. The rotation through
*s*, *b*, and *t* is evidenced across the
rows by the second character in the strings describing the
l.d.a.s, but there is little regularity beyond that. In
particular, there is a great variety of descendent l.d.a.s and no clear
pattern beyond the first two characters which are enforced by the value
the extension modulo 3.

We focus on a given l.d.a. The relationship among the values of
the elements of each column of extensions may be calculated from the
known relationships on each extension edge and each l.d.a. edge.
The relationships in each column (corresponding to the first, then
the second, ... extensions) are given in the third and
successive lines under each extension value. Although certain
regularities are clear (i.e. the coefficients go up by 2 for an
*s* step and 4 for a *b* step, and the divisors go up by
3 for each step through the parental l.d.a.), it is difficult to
see at this level how the myriad of different l.d.a.s each giving rise
to its own table of this kind could possibly be employed to develop an
understanding of the overall conspiracy to develop each odd integer once
and only once in the binary predecessor tree.

**Table of extensions of 3 and the ebssbt l.d.a. from 13,
showing the resulting l.d.a.s, and the relationship among the extensions.**

3 e 13 53 213 853 3413 13653 essst(0) et(8) ebt(11) esbt(15) et(568) a1 b1 c1 d1 e1 b 17 69 277 1109 4437 17749 et(2) ebt(3) esbbssbt(0) et(184) ebbssbbsst(0) a2=(4a1-5)/3 b2=(4b1-21)/3 c2=(4c1-85)/3 d2=(4d1-341)/3 e2=(4e1-1365)/3 s 11 45 181 725 2901 11605 et(1) ebbt(0) est(10) et(120) ebsbbbst(0) a3=(2a2-3)/3 b3=(2b2-11)/3 c3=(2c2-43)/3 d3=(2d2-171)/3 e3=(2e2-683)/3 =(8a1-19)/9 =(8b1-75)/9 =(8c1-299)/9 =(8d1-1195)/9 =(8e1-4779)/9 s 7 29 117 469 1877 7509 esbbt(0) et(4) ebbst(0) est(26) et(312) a4=(2a3-3)/3 b4=(2b3-11)/3 c4=(2c3-43)/3 d4=(2d3-171)/3 e4=(2e3-683)/3 =(4a2-15)/9 =(4b2-55)/9 =(4c2-215)/9 =(4d2-855)/9 =(4e2-3415)/9 =(16a1-65)/27 =(16b1-249)/27 =(16c1-985)/27 =(16d1-3929)/27 =(16e1-15705)/27 b 9 37 149 597 2389 9557 ebbsbt(0) est(2) et(24) ebsssbt(0) esssst(4) a5=(4a4-5)/3 b5=(4b4-21)/3 c5=(4c4-85)/3 d5=(4d4-341)/3 e5=(4e4-1365)/3 =(8a3-27)/9 =(8b3-107)/9 =(8c3-427)/9 =(8d3-1707)/9 =(8e3-6827)/9 =(16a2-105)/27 =(16b2-409)/27 =(16c2-1625)/27 =(16d2-6489)/27 =(16e2-25945)/27 =(64a1-395)/81 =(64b1-1563)/81 =(64c1-6235)/81 =(64d1-24923)/81 =(64e1-99675)/81 t

Be careful to distinguish the preceding table from the following.
The preceding contains the successive ** extensions ** from the
elements of the *ebssbt* l.d.a. but the following contains the
successive ** instantiations** of that l.d.a. In the following
table, each column shows the prescribed behavior for the subject l.d.a.,
and each row contains the instantiations in arithmetic progression.
The minimal regularity in the previous table is in sharp contrast to
the complete regularity in the following table. This underscores the
idea that the structure of instantiations of l.d.a.s is far more successful
in understanding the predecessor tree than is any structure introduced by
the extensions.

n: 0 1 2 3 4 5 6 ex powers ------------------------------------------------------------------------ e 13 1957 3901 5845 7789 9733 11677 | 1944n+13 2^3*3^5 b | 17 2609 5201 7793 10385 12977 15569 | 2592n+17 2^5*3^4 s | 11 1739 3467 5195 6923 8651 10379 | 1728n+11 2^6*3^3 s | 7 1159 2311 3463 4615 5767 6919 | 1152n+ 7 2^7*3^2 b | t 9 1545 3081 4617 6153 7689 9225 | 1536n+ 9 2^9*3^1

This is an uncomplicated series of successive instantiations of the initial
*ebssbt* instance. As always, development of a column (an instance
of an l.d.a.) obeys the *b* or *s* path generation rules, and
the rows representing the successive instantiations of each element of
the l.d.a. obey the expression which describes all instances of the
particular l.d.a.

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