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.