The IDtheLDA program

The essence of the structure of the Collatz predecessor tree is that it is built of left descent assemblies interconnected by extensions.   Left descent assemblies  (l.d.a.s)  have a number of characteristics which,  while easily enough described,  are not so easily grasped and assimilated into a full picture of the characteristics they confer on the entire predecessor tree. Calculating the values of any particular left descent assembly is tedious at best using a hand calculator,  and becomes impossible if the precision of the calculator is insufficient to produce l.d.a.s containing large integers.

The proof proposed elsewhere in this work shows how the state transition diagram depicting the allowed transitions among the integers during the Collatz itineraries leads to the concept of  l.d.a.s.  An abstract predecessor tree makes it possible to construct all possible residue sets,  c[d],  which uniquely constitute the elements of the l.d.a.s.   Then,  using the density of the integers in these residue sets,   the contributions of the elements of all the l.d.a.s. can be summed.  The sum turns out to be 1, an excellent result.  But the result will seem less remote and abstract after some detailed experience with l.d.a.s is acquired.

A possible alternative approach to a proof of the Collatz conjecture is to argue that the numeric operations required to pursue a Collatz trajectory are simple ones (addition of 1,  multiplication by 3,  division by 2)  to which any conceivable integer is subject without question. Having a program which implements the identification of  l.d.a.s in detail could well lead to insights helpful in developing such an approach.

For these reasons a utility program was written for use as a tool in exploring their various properties.  The program accepts any positive odd integer  (or linear sequence thereof),  and,   looking only at the integer and its immediate neighbors without ever leaving that l.d.a.,  identifies the l.d.a.  it appears in,   the integer's position in its l.d.a.,  which instantiation of that l.d.a.  it is in,  and the values of the l.d.a.  elements in the zeroth instantiation of that l.d.a.   By showing that every odd positive integer can thus be placed precisely in the abstract predecessor tree,  we show directly that every one is present there,  thus bolstering the result of the infinite summations to the effect that all integers are present in the Collatz graph.  An extension, fullmap of this program has also been written to demonstrate how the Collatz trajectories make their transition from one l.d.a. to the next.

This page describes a program written in the Maple programming language which accepts any odd positive integer and determines its location in the abstract predecessor tree.  The program thus identifies the l.d.a.  in the abstract tree,  and the program name is IDtheLDA. The Maple programming language is for all practical purposes without limit as to the precision supported.   The program employs the formulas for Collatz successors and Collatz predecessors involving that simple arithmetic,  but to determine the instantiation of a given l.d.a.  instance a possibly lengthy multiple precision division followed by a subtraction is also required.

The program as formatted with statement numbers by the Maple system follows.  A few long statements have been split between lines to accommodate line length limitations.

Statements 1-4 and 62 are concerned with parameter definition,  variable definition,  creating loop control in accord with the parameters fed   and basic communication to files.

The central variables were named with "is" or "os" as initial letters according to whether they were conceived as dealing with activity on the "input side" or "output side".  The name were completed with "c",   "d",   or "w" according to whether they deal with the "c" or "d" of c[d],   or simply the "working" value (d*n+c) prior to its decomposition into "c" and "d".   A prefixed "h" is just a value held from its calculation until it value is needed later,   and a prefix "c" indicates a calculated value (as opposed to those developed step by step(which predominate in the code)).

Statements 5 and 61 perform the major program loop.

Statements 5-8 reject even integers offered as input or developed within a series specified as input.

Statements 9-10 perform some key variable initializations.

Statements 11-15 handle null l.d.a.s,   setting variables for eventual output by statement 60.

Statements 16-21 make the ascent from oddi (the odd integer input)  upwards (i.e. in the Collatz direction)  to the header of the l.d.a.  it resides in. The formulas used are the standard for the Collatz iteration,  and the header is recognized when reached by its presence in 5[8].   The position of oddi in its l.d.a.  is counted up in this passage also.

After some initiation (22-25), statements 26-40 pursue the l.d.a. downward (i.e. in the predecessor direction)  to its leaf node,  recognized by its membership in 0[3].  This code employs the formula for Collatz predecessors.  It also tracks in p3 the number of transitions in the l.d.a.  and accumulates in ldastr in left to right order corresponding to the head to tail order of predecessor exploration the string expression e{s|b}kt with the help of statement 41.

Statements 28-34 are concerned with b-steps in the l.d.a.s,  while 35-40 are concerned with s-steps in the l.d.a.s.  The order of testing is critical in the segments 29-32 and 35-38 where the choice is made among c,  c+d,  and c+2d as the point of departure for the next iteration.   If the choice for c is tested first,  an l.d.a. containing a series of early s-steps will cause the program to fail.

Statements 42 and 43 are the only exceptions to the "simple" arithmetic involving only addition of 1,  subtraction of 1,   multiplication by 2,  3,  or 4 and division by 2.  Statement 40 calculates the d of the c[d] value for the header of the l.d.a.   from the count in p3 of the steps in the l.d.a.   Then using the dn+c value for the header saved from statement 24,   the instantiation of the i.d.a.  in which oddi occurs can be calculated.

This bit of information allows statements 44-49 to initialize the variables for the next loop and output the complete information for the initial step at the l.d.a. leaf for the final,  upward (i.e. Collatz iteration direction).

The loop 50-59 performs that iteration with full output  (if requested by the parameter values)  of both the values in the nth instantiation of the l.d.a.  and the zeroth instantiation.

Finally, statement 60 provides a summary of the run on oddi.  Statement 2 has already provided a record of the run of which this report is one result,  so a run requesting a look at a series of 50 odd integers will provide 51 output lines,  one from statement 2,  and 50 from statement 60.   If fuller output is requested via the parameters,  there will an additional line for each element in each l.d.a.,  produced by lines 49 and 59.

```        idthelda := proc(start, step, num, plevel)
local cisd, endit, hisw, inst, isc, isd, isw, ldastr, oddi,
osc, osd, osw, pos, p3;
1    fopen(`\\maplev4\\mywork\\abtsite.lst`,APPEND,TEXT);
2    fprintf(`\\maplev4\\mywork\\abtsite.lst`,
`ID the LDA run with %d, %d, %d, %d\n`,
start,step,num,plevel);
3    oddi := start;
4    endit := max(start+num*step,oddi);
5    while oddi <= endit do
6      if `mod`(oddi,2) = 0 then
7        ERROR(`Sought integer must be odd. Quitting.\n`);
8        quit
fi;
9      isw := oddi;
10      pos := 1;
11      if `mod`(oddi,3) = 0 and `mod`(isw,8) = 5 then
12        ldastr := et;
13        osd := 24;
14        osc := 5;
15        inst := 1/24*oddi-7/8
else
16        if not `mod`(isw,8) = 5 then
17          while not `mod`(isw,8) = 5 do
18            if `mod`(isw,2) = 1 then
19              pos := pos+1;
20              isw := 3/2*isw+1/2
else
21              isw := 1/2*isw
fi
od
fi;
22        ldastr := e;
23        p3 := 1;
24        hisw := isw;
25        isd := 8;
26        while not `mod`(isw,3) = 0 do
27          p3 := p3+1;
28          if `mod`(isw,3) = 1 then
29            if `mod`(isw+isd,3) = 1 then
30              isw := -1/3+4/3*isw+4/3*isd
elif `mod`(isw+2*isd,3) = 1 then
31              isw := -1/3+4/3*isw+8/3*isd
else
32              isw := -1/3+4/3*isw
fi;
33            ldastr := cat(ldastr,b);
34            isd := 4*isd
elif `mod`(isw,3) = 2 then
35            if `mod`(isw+isd,3) = 2 then
36              isw := -1/3+2/3*isw+2/3*isd
elif `mod`(isw+2*isd,3) = 2 then
37              isw := -1/3+2/3*isw+4/3*isd
else
38              isw := -1/3+2/3*isw
fi;
39            ldastr := cat(ldastr,s);
40            isd := 2*isd
fi
od;
41        ldastr := cat(ldastr,t);
42        cisd := 8*3^p3;
43        inst := floor(hisw/cisd);
44        osw := isw;
45        osd := 3*isd;
46        isc := osw-inst*osd;
47        osc := isc;
48        if 2 < plevel then
49          fprintf(`\\maplev4\\mywork\\abtsite.lst`,
`n: %d d: %d n*d: %d c: %d: nd+c: %d  %d\n`,
inst,osd,inst*osd,osc,osc+inst*osd,osc)
fi;
50        while not osw = hisw do
51          osd := 3/2*osd;
52          osc := 3/2*osc+1/2;
53          osw := 3/2*osw+1/2;
54          if `mod`(osw,2) = 0 then
55            osc := 1/2*osc;
56            osd := 1/2*osd;
57            osw := 1/2*osw
fi;
58          if 2 < plevel then
59            fprintf(`\\maplev4\\mywork\\abtsite.lst`,
`n: %d d: %d n*d: %d c: %d: nd+c: %d c in 0th
instantiation:%d\n`,
inst,osd,inst*osd,osc,osc+inst*osd,osc)
fi
od
fi;
60      fprintf(`\\maplev4\\mywork\\abtsite.lst`,
`oddi:%d, lda:%s, isw:%d, osc:%d, osd:%d, pos'n:%d, inst'n:%f\n\n`,
oddi,ldastr,isw,osc,osd,pos,inst);
61      oddi := oddi+max(2,step);
od;
62    fclose(`\\maplev4\\mywork\\abtsite.lst`)
end

```

With that at hand,  let's illustrate with the fabulous "descent to 27"  often cited as an illustration of the difficulty of understanding the Collatz itinieraries.

From the invocation:

```     idthelda(27,0,0,0);

```

Comes the output lines:

```     ID the LDA run with 27, 0, 0, 0
oddi:27, lda:ebsssbssbsbbssssbst, isw:27, osc:445, osd:3099363912, pos'n:18, inst'n:0
```

This indicates that 27 appears as the 18th element   (counting from the header element)  in the zeroth instantiation of a very lengthy l.d.a. whose header element is 445[3099363912].  The leaf node on the input side contains the integer 27.

Like to see a little more detail?  Just raise the 4th argument to 3.

```ID the LDA run with 27, 0, 0, 3
n: 0, d: 201326592, n*d: 0, c: 27, nd+c: 27, c in 0th instantiation: 27
n: 0, d: 301989888, n*d: 0, c: 41, nd+c: 41, c in 0th instantiation: 41
n: 0, d: 226492416, n*d: 0, c: 31, nd+c: 31, c in 0th instantiation: 31
n: 0, d: 339738624, n*d: 0, c: 47, nd+c: 47, c in 0th instantiation: 47
n: 0, d: 509607936, n*d: 0, c: 71, nd+c: 71, c in 0th instantiation: 71
n: 0, d: 764411904, n*d: 0, c: 107, nd+c: 107, c in 0th instantiation: 107
n: 0, d: 1146617856, n*d: 0, c: 161, nd+c: 161, c in 0th instantiation: 161
n: 0, d: 859963392, n*d: 0, c: 121, nd+c: 121, c in 0th instantiation: 121
n: 0, d: 644972544, n*d: 0, c: 91, nd+c: 91, c in 0th instantiation: 91
n: 0, d: 967458816, n*d: 0, c: 137, nd+c: 137, c in 0th instantiation: 137
n: 0, d: 725594112, n*d: 0, c: 103, nd+c: 103, c in 0th instantiation: 103
n: 0, d: 1088391168, n*d: 0, c: 155, nd+c: 155, c in 0th instantiation: 155
n: 0, d: 1632586752, n*d: 0, c: 233, nd+c: 233, c in 0th instantiation: 233
n: 0, d: 1224440064, n*d: 0, c: 175, nd+c: 175, c in 0th instantiation: 175
n: 0, d: 1836660096, n*d: 0, c: 263, nd+c: 263, c in 0th instantiation: 263
n: 0, d: 2754990144, n*d: 0, c: 395, nd+c: 395, c in 0th instantiation: 395
n: 0, d: 4132485216, n*d: 0, c: 593, nd+c: 593, c in 0th instantiation: 593
n: 0, d: 3099363912, n*d: 0, c: 445, nd+c: 445, c in 0th instantiation: 445
oddi:27, lda:ebsssbssbsbbssssbst, isw:27, osc:445, osd:3099363912, pos'n:18, inst'n:0
```

Each element reached in the traversal of the l.d.a.  during assignment of their c and d values is listed in detail from the leaf at the top to the header node at the bottom.

What transpires if we select an oddi which is in a higher instantiation and which is not the leaf node of the l.d.a.?

```ID the LDA run with 529, 0, 0, 3
n: 1, d: 384, n*d: 384, c: 321, nd+c: 705, c in 0th instantiation: 321
n: 1, d: 288, n*d: 288, c: 241, nd+c: 529, c in 0th instantiation: 241
n: 1, d: 216, n*d: 216, c: 181, nd+c: 397, c in 0th instantiation: 181
oddi:529, lda:ebbt, isw:705, osc:181, osd:216, pos'n:2, inst'n:1
```

Now the column nd+c gives the elements of the first  (zero-origin indexing)  instantiation of the ebbt l.d.a.,  the oddi value appears in the second position of that instantiation, the isw value appears as the leaf node in the input instantiation,  and the final column gives the elements of the zeroth instantiation.

Let's explore how successive instantiations of the same l.d.a. behave.  To do so,  having picked the numbers 529 and 288 out of the middle line of the above output to focus on the behavior of the element in the second position counting from the leaf node,  we make the following run, requesting an extra determination to be run by specifying a non-zero third argument.

```ID the LDA run with 529, 288, 1, 3
n: 1, d: 384, n*d: 384, c: 321, nd+c: 705, c in 0th instantiation: 321
n: 1, d: 288, n*d: 288, c: 241, nd+c: 529, c in 0th instantiation: 241
n: 1, d: 216, n*d: 216, c: 181, nd+c: 397, c in 0th instantiation: 181
oddi:529, lda:ebbt, isw:705, osc:181, osd:216, pos'n:2, inst'n:1

n: 2, d: 384, n*d: 768, c: 321, nd+c: 1089, c in 0th instantiation: 321
n: 2, d: 288, n*d: 576, c: 241, nd+c: 817, c in 0th instantiation: 241
n: 2, d: 216, n*d: 432, c: 181, nd+c: 613, c in 0th instantiation: 181
oddi:817, lda:ebbt, isw:1089, osc:181, osd:216, pos'n:2, inst'n:2
```

By picking the d value of that second element we have selected a step size appropriate to it for examining the next instantiation  . We could as well have used 705,384 or 397,216,  the increments between the instantiations from the viewpoint of the first and the third elements of the l.d.a.   The major point to be noticed is that the distances between instantiations of correponding elements of an l.d.a. varies element by element.

You'd like to see a bigger example to explore that feature in more detail?   Well, be my guest.

```ID the LDA run with 445, 3099363912, 1, 3
n: 0, d: 201326592, n*d: 0, c: 27, nd+c: 27, c in 0th instantiation: 27
n: 0, d: 301989888, n*d: 0, c: 41, nd+c: 41, c in 0th instantiation: 41
n: 0, d: 226492416, n*d: 0, c: 31, nd+c: 31, c in 0th instantiation: 31
n: 0, d: 339738624, n*d: 0, c: 47, nd+c: 47, c in 0th instantiation: 47
n: 0, d: 509607936, n*d: 0, c: 71, nd+c: 71, c in 0th instantiation: 71
n: 0, d: 764411904, n*d: 0, c: 107, nd+c: 107, c in 0th instantiation: 107
n: 0, d: 1146617856, n*d: 0, c: 161, nd+c: 161, c in 0th instantiation: 161
n: 0, d: 859963392, n*d: 0, c: 121, nd+c: 121, c in 0th instantiation: 121
n: 0, d: 644972544, n*d: 0, c: 91, nd+c: 91, c in 0th instantiation: 91
n: 0, d: 967458816, n*d: 0, c: 137, nd+c: 137, c in 0th instantiation: 137
n: 0, d: 725594112, n*d: 0, c: 103, nd+c: 103, c in 0th instantiation: 103
n: 0, d: 1088391168, n*d: 0, c: 155, nd+c: 155, c in 0th instantiation: 155
n: 0, d: 1632586752, n*d: 0, c: 233, nd+c: 233, c in 0th instantiation: 233
n: 0, d: 1224440064, n*d: 0, c: 175, nd+c: 175, c in 0th instantiation: 175
n: 0, d: 1836660096, n*d: 0, c: 263, nd+c: 263, c in 0th instantiation: 263
n: 0, d: 2754990144, n*d: 0, c: 395, nd+c: 395, c in 0th instantiation: 395
n: 0, d: 4132485216, n*d: 0, c: 593, nd+c: 593, c in 0th instantiation: 593
n: 0, d: 3099363912, n*d: 0, c: 445, nd+c: 445, c in 0th instantiation: 445
oddi:445, lda:ebsssbssbsbbssssbst, isw:27, osc:445, osd:3099363912, pos'n:1, inst'n:0

n: 1, d: 201326592, n*d: 201326592, c: 27, nd+c: 201326619, c in 0th instantiation: 27
n: 1, d: 301989888, n*d: 301989888, c: 41, nd+c: 301989929, c in 0th instantiation: 41
n: 1, d: 226492416, n*d: 226492416, c: 31, nd+c: 226492447, c in 0th instantiation: 31
n: 1, d: 339738624, n*d: 339738624, c: 47, nd+c: 339738671, c in 0th instantiation: 47
n: 1, d: 509607936, n*d: 509607936, c: 71, nd+c: 509608007, c in 0th instantiation: 71
n: 1, d: 764411904, n*d: 764411904, c: 107, nd+c: 764412011, c in 0th instantiation: 107
n: 1, d: 1146617856, n*d: 1146617856, c: 161, nd+c: 1146618017, c in 0th instantiation: 161
n: 1, d: 859963392, n*d: 859963392, c: 121, nd+c: 859963513, c in 0th instantiation: 121
n: 1, d: 644972544, n*d: 644972544, c: 91, nd+c: 644972635, c in 0th instantiation: 91
n: 1, d: 967458816, n*d: 967458816, c: 137, nd+c: 967458953, c in 0th instantiation: 137
n: 1, d: 725594112, n*d: 725594112, c: 103, nd+c: 725594215, c in 0th instantiation: 103
n: 1, d: 1088391168, n*d: 1088391168, c: 155, nd+c: 1088391323, c in 0th instantiation: 155
n: 1, d: 1632586752, n*d: 1632586752, c: 233, nd+c: 1632586985, c in 0th instantiation: 233
n: 1, d: 1224440064, n*d: 1224440064, c: 175, nd+c: 1224440239, c in 0th instantiation: 175
n: 1, d: 1836660096, n*d: 1836660096, c: 263, nd+c: 1836660359, c in 0th instantiation: 263
n: 1, d: 2754990144, n*d: 2754990144, c: 395, nd+c: 2754990539, c in 0th instantiation: 395
n: 1, d: 4132485216, n*d: 4132485216, c: 593, nd+c: 4132485809, c in 0th instantiation: 593
n: 1, d: 3099363912, n*d: 3099363912, c: 445, nd+c: 3099364357, c in 0th instantiation: 445
oddi:3099364357, lda:ebsssbssbsbbssssbst, isw:201326619, osc:445, osd:3099363912, pos'n:1, inst'n:1
```

Since the d values for the elements of the l.d.a.s differ from one another,  the differences between the c values for two successive instantiations differ.  This feature certainly must have contributed to the difficulty of detecting the systematics of the Collatz predecessor tree.

Yet another application of this utility is to scan a series of integers looking for certain repetition phenomena.  One example is a run with values for the first and second arguments which do not exist together in an l.d.a.

```ID the LDA run with 529, 384, 40, 0
oddi:  529, lda:ebbt, isw:705, osc:181, osd:216, pos'n:2, inst'n:1            A
oddi:  913, lda:ebbsbbbbst, isw:1707, osc:685, osd:157464, pos'n:2, inst'n:0
oddi: 1297, lda:ebbbbbsbst, isw:2427, osc:973, osd:157464, pos'n:2, inst'n:0
oddi: 1681, lda:ebbt, isw:2241, osc:181, osd:216, pos'n:2, inst'n:5           A
oddi: 2065, lda:ebbsssst, isw:543, osc:1549, osd:17496, pos'n:2, inst'n:0
oddi: 2449, lda:ebbbt, isw:4353, osc:541, osd:648, pos'n:2, inst'n:2          B
oddi: 2833, lda:ebbt, isw:3777, osc:181, osd:216, pos'n:2, inst'n:9           A
oddi: 3217, lda:ebbst, isw:2859, osc:469, osd:648, pos'n:2, inst'n:3          C
oddi: 3601, lda:ebbbsbbbt, isw:10113, osc:2701, osd:52488, pos'n:2, inst'n:0
oddi: 3985, lda:ebbt, isw:5313, osc:181, osd:216, pos'n:2, inst'n:13          A
oddi: 4369, lda:ebbsbsbsbt, isw:4089, osc:3277, osd:157464, pos'n:2, inst'n:0
oddi: 4753, lda:ebbbbt, isw:11265, osc:1621, osd:1944, pos'n:2, inst'n:1      D
oddi: 5137, lda:ebbt, isw:6849, osc:181, osd:216, pos'n:2, inst'n:17          A
oddi: 5521, lda:ebbssbst, isw:2907, osc:4141, osd:17496, pos'n:2, inst'n:0
oddi: 5905, lda:ebbbt, isw:10497, osc:541, osd:648, pos'n:2, inst'n:6         B
oddi: 6289, lda:ebbt, isw:8385, osc:181, osd:216, pos'n:2, inst'n:21          A
oddi: 6673, lda:ebbst, isw:5931, osc:469, osd:648, pos'n:2, inst'n:7          C
oddi: 7057, lda:ebbbssbsst, isw:3303, osc:5293, osd:157464, pos'n:2, inst'n:0
oddi: 7441, lda:ebbt, isw:9921, osc:181, osd:216, pos'n:2, inst'n:25          A
oddi: 7825, lda:ebbsbt, isw:9273, osc:37, osd:1944, pos'n:2, inst'n:3
oddi: 8209, lda:ebbbbssbt, isw:11529, osc:6157, osd:52488, pos'n:2, inst'n:0
oddi: 8593, lda:ebbt, isw:11457, osc:181, osd:216, pos'n:2, inst'n:29         A
oddi: 8977, lda:ebbsst, isw:5319, osc:901, osd:1944, pos'n:2, inst'n:3
oddi: 9361, lda:ebbbt, isw:16641, osc:541, osd:648, pos'n:2, inst'n:10        B
oddi: 9745, lda:ebbt, isw:12993, osc:181, osd:216, pos'n:2, inst'n:33         A
oddi:10129, lda:ebbst, isw:9003, osc:469, osd:648, pos'n:2, inst'n:11         C
oddi:10513, lda:ebbbst, isw:12459, osc:109, osd:1944, pos'n:2, inst'n:4
oddi:10897, lda:ebbt, isw:14529, osc:181, osd:216, pos'n:2, inst'n:37         A
oddi:11281, lda:ebbsbbst, isw:11883, osc:8461, osd:17496, pos'n:2, inst'n:0
oddi:11665, lda:ebbbbbbbsst, isw:29127, osc:8749, osd:472392, pos'n:2, inst'n:0
oddi:12049, lda:ebbt, isw:16065, osc:181, osd:216, pos'n:2, inst'n:41         A
oddi:12433, lda:ebbssst, isw:4911, osc:3493, osd:5832, pos'n:2, inst'n:1      Z1
oddi:12817, lda:ebbbt, isw:22785, osc:541, osd:648, pos'n:2, inst'n:14        B
oddi:13201, lda:ebbt, isw:17601, osc:181, osd:216, pos'n:2, inst'n:45         A
oddi:13585, lda:ebbst, isw:12075, osc:469, osd:648, pos'n:2, inst'n:15        C
oddi:13969, lda:ebbbsbst, isw:14715, osc:10477, osd:17496, pos'n:2, inst'n:0
oddi:14353, lda:ebbt, isw:19137, osc:181, osd:216, pos'n:2, inst'n:49         A
oddi:14737, lda:ebbsbst, isw:11643, osc:5221, osd:5832, pos'n:2, inst'n:1     Z2
oddi:15121, lda:ebbbbt, isw:35841, osc:1621, osd:1944, pos'n:2, inst'n:5      D
oddi:15505, lda:ebbt, isw:20673, osc:181, osd:216, pos'n:2, inst'n:53
oddi:15889, lda:ebbssbbt, isw:16737, osc:11917, osd:17496, pos'n:2, inst'n:0
```

I've done a little column alignment and annotation.  Since these lines describe the leaf nodes of each l.d.a.,  while the integer involved in repetitive action is in position 2 in each case,  the table is pretty obscure.  But clearly,  quite a number  (unmarked in column 79)  of more lengthy l.d.a.s appear here in their zeroth instantiation and some relatively short l.d.a.s  (marked A-D)  appear repetitively.

From the sampling frequency  (384),  the number of sampling steps between appearances of instances of the same l.d.a.,  and the intantiation differences  (both the integer difference and the difference in instantiation number)  one soon determines how these repetitious appearance arise.  Note the plethora of powers of 2 and 3.

```  A  3*384 = 4* 288
B 18*384 = 8* 864
C  9*384 = 4* 864
D 27*384 = 4*2592
```

To see how the numbers go in a case like Z1,  where there is only a single instance available,  one could make an additional IDtheLDA run, as follows.

```ID the LDA run with 12433, 0, 0, 3
n: 1, d: 3072 n*d: 3072 c: 1839: nd+c: 4911 c in 0th instantiation: 1839
n: 1, d: 4608, n*d: 4608, c: 2759, nd+c: 7367, c in 0th instantiation: 2759
n: 1, d: 6912, n*d: 6912, c: 4139, nd+c: 11051, c in 0th instantiation: 4139
n: 1, d: 10368, n*d: 10368, c: 6209, nd+c: 16577, c in 0th instantiation: 6209
n: 1, d: 7776, n*d: 7776, c: 4657, nd+c: 12433, c in 0th instantiation: 4657
n: 1, d: 5832, n*d: 5832, c: 3493, nd+c: 9325, c in 0th instantiation: 3493
oddi:12433, lda:ebbssst, isw:4911, osc:3493, osd:5832, pos'n:2, inst'n:1
```

This output supplies all the data for position 2 of the l.d.a.,  from which we can calculate that 12433-7776  (4657)  will be the location of the zeroth instantiation and the next one will occur at 12433+7776 (20299),  respectively,  both outside the range examined in that earlier run.

This repetitive appearance of smaller l.d.a.s in a series occurs even with ridiculously large numbers and stepsizes.

```ID the LDA run with 13000001, 1000012, 50, 0
oddi:13000001, lda:ebbst, isw:8666667, osc:469, osd:648, pos'n:3, inst'n:11284
oddi:14000013, lda:et, isw:14000013, osc:5, osd:24, pos'n:1, inst'n:583333
oddi:15000025, lda:esbbssssbbsst, isw:3121479, osc:4120445, osd:4251528, pos'n:3, inst'n:3
oddi:16000037, lda:esssbbst, isw:5618667, osc:8693, osd:17496, pos'n:1, inst'n:914
oddi:17000049, lda:ebt, isw:17000049, osc:61, osd:72, pos'n:2, inst'n:177083
oddi:18000061, lda:ebt, isw:24000081, osc:61, osd:72, pos'n:1, inst'n:250000
oddi:19000073, lda:essssbsbbssbsbt, isw:16888953, osc:26188757, osd:38263752, pos'n:12, inst'n:2
oddi:20000085, lda:et, isw:20000085, osc:5, osd:24, pos'n:1, inst'n:833336
oddi:21000097, lda:ebsbbbbsbt, isw:33185337, osc:62149, osd:157464, pos'n:5, inst'n:84
oddi:22000109, lda:est, isw:14666739, osc:5, osd:72, pos'n:1, inst'n:305557
oddi:23000121, lda:ebbsbt, isw:23000121, osc:37, osd:1944, pos'n:5, inst'n:7487
oddi:24000133, lda:ebsbsbsbssssbt, isw:5919273, osc:11245549, osd:12754584, pos'n:1, inst'n:1
oddi:25000145, lda:ebsst, isw:11111175, osc:229, osd:648, pos'n:2, inst'n:28935
oddi:26000157, lda:et, isw:26000157, osc:5, osd:24, pos'n:1, inst'n:1083339
oddi:27000169, lda:esssbbt, isw:36000225, osc:4805, osd:5832, pos'n:5, inst'n:11718
oddi:28000181, lda:esbssbt, isw:14749065, osc:749, osd:5832, pos'n:1, inst'n:4801
oddi:29000193, lda:esbbbbt, isw:29000193, osc:245, osd:5832, pos'n:6, inst'n:2360
oddi:30000205, lda:ebbt, isw:53333697, osc:181, osd:216, pos'n:1, inst'n:138889
oddi:31000217, lda:esbst, isw:20666811, osc:533, osd:648, pos'n:3, inst'n:53819
oddi:32000229, lda:et, isw:32000229, osc:5, osd:24, pos'n:1, inst'n:1333342
oddi:33000241, lda:ebbst, isw:29333547, osc:469, osd:648, pos'n:2, inst'n:38194
oddi:34000253, lda:essbt, isw:20148297, osc:341, osd:648, pos'n:1, inst'n:52469
oddi:35000265, lda:essbt, isw:35000265, osc:341, osd:648, pos'n:4, inst'n:91146
oddi:36000277, lda:ebt, isw:48000369, osc:61, osd:72, pos'n:1, inst'n:500003
oddi:37000289, lda:essbbsbbsst, isw:19489863, osc:61685, osd:472392, pos'n:5, inst'n:99
oddi:38000301, lda:et, isw:38000301, osc:5, osd:24, pos'n:1, inst'n:1583345
oddi:39000313, lda:esbsbbbbsbst, isw:54782331, osc:1175789, osd:1417176, pos'n:5, inst'n:34
oddi:40000325, lda:est, isw:26666883, osc:5, osd:72, pos'n:1, inst'n:555560
oddi:41000337, lda:ebt, isw:41000337, osc:61, osd:72, pos'n:2, inst'n:427086
oddi:42000349, lda:ebsst, isw:24889095, osc:229, osd:648, pos'n:1, inst'n:64815
oddi:43000361, lda:ebsbssssbssst, isw:12740847, osc:1707637, osd:4251528, pos'n:9, inst'n:32
oddi:44000373, lda:et, isw:44000373, osc:5, osd:24, pos'n:1, inst'n:1833348
oddi:45000385, lda:ebbbt, isw:60000513, osc:541, osd:648, pos'n:3, inst'n:39062
oddi:46000397, lda:esbt, isw:40889241, osc:173, osd:216, pos'n:1, inst'n:212964
oddi:47000409, lda:esbt, isw:47000409, osc:173, osd:216, pos'n:3, inst'n:244793
oddi:48000421, lda:ebbst, isw:56889387, osc:469, osd:648, pos'n:1, inst'n:74074
oddi:49000433, lda:ebst, isw:32666955, osc:85, osd:216, pos'n:2, inst'n:170140
oddi:50000445, lda:et, isw:50000445, osc:5, osd:24, pos'n:1, inst'n:2083351
oddi:51000457, lda:esbssssssbssbbsbbst, isw:53728875, osc:827140349, osd:3099363912, pos'n:13, inst'n:0
oddi:52000469, lda:esst, isw:23111319, osc:197, osd:216, pos'n:1, inst'n:240742
oddi:53000481, lda:esbbt, isw:53000481, osc:29, osd:648, pos'n:4, inst'n:69011
oddi:54000493, lda:ebt, isw:72000657, osc:61, osd:72, pos'n:1, inst'n:750006
oddi:55000505, lda:ebsbsbst, isw:32592891, osc:7285, osd:17496, pos'n:4, inst'n:2652
oddi:56000517, lda:et, isw:56000517, osc:5, osd:24, pos'n:1, inst'n:2333354
oddi:57000529, lda:ebbbt, isw:101334273, osc:541, osd:648, pos'n:2, inst'n:65972
oddi:58000541, lda:est, isw:38667027, osc:5, osd:72, pos'n:1, inst'n:805563
oddi:59000553, lda:esbsbbbsssbt, isw:59000553, osc:32717, osd:1417176, pos'n:11, inst'n:75
oddi:60000565, lda:ebst, isw:53333835, osc:85, osd:216, pos'n:1, inst'n:277780
oddi:61000577, lda:essbsbbbssbbt, isw:48197985, osc:1367837, osd:4251528, pos'n:8, inst'n:15
oddi:62000589, lda:et, isw:62000589, osc:5, osd:24, pos'n:1, inst'n:2583357
oddi:63000601, lda:esbbt, isw:84000801, osc:29, osd:648, pos'n:3, inst'n:109376
```

The et l.d.a. appears frequently every sixth sample,  ebt  (position 1)  appears in every 18th sample,  but ebt  (position 2)  appears with a separation of 24 samples.  The est l.d.a.  (position 1)  appears at intervals of 18.  There is still a zeroth instatiation of a very long l.d.a.

Perhaps what one concludes from all this is that the intermeshing of the l.d.a.s to form the binary predecessor trees is an extremely complex and subtle phenomenon.  All the various views which can be gained by runs of IDtheLDA can help disentangle the various features,  but it is the abstract predecessor tree which presents the useful overview of the Collatz predecessor tree.