Mensanator's discovery
that the *shape* of the state transition diagram
for the Collatz trajectories is *identical* for
the positive and the negative domains led to his argument that the
state transition diagram cannot be trusted to indicate that Collatz trajectories
are loop free (excepting the trivial loop 4-2-1) since the negative
domain does exhibit non-trivial loops.

This is truly astounding because the Collatz trajectories in the
negative domain has loops and in the positive (presumably)
none other than the trivial one. The negative state transitions
show no sign of congruence sets (see below), while the
positive transitions produce nothing else. The positive transition
diagram has no edge which requires more than 2 divisions by 2,
while the negative one has cetain nodes every one of whose effluent
edges requires 3 or more divisions by 2. From the identical
*shape* he argues that one cannot conclude from the
*shape* of the state transition diagram that it excludes a
non-trivial cycle in the positive domain. Yet the behaviors of the
Collatz trajectories is profoundly different in the two domains.
We'll have to look elsewhere for the determining feature.

First look at a comparison of the state transitions for the negative
and positive domains starting from an arbitrary congruence set. The
sign of *1* in the exression *(3n+1)/2^i* is the same as
the sign of *3n* in the positive domain while it is opposite in
the negative domain. To illustrate the effect of this, consider
the Collatz successors of the series of instances the succesive integers
in the congruence set, *-3[24]*. In the negative
domain, the result exhibits a prolific variety of product states.
(The state names are assigned according to their congruences modulo
-24.)

-3 (-3*3+1)/2*i = -8/2^3 = -1 A -1 -27 (-27*3+1)/2*i = -80/2^4 = -5 C -5 -51 (-51*3+1)/2*i =-152/2^3 = -19 J -19 -75 (-75*3+1)/2*i =-224/2^5 = -7 D -7 -99 (-99*3+1)/2*i =-296/2^3 = -37 G -11 -121 (-123*3+1)/2*i =-368/2^4 = -23 L -23 -147 (-147*3+1)/2*i =-440/2^3 = -55 D -7 -171 (-171*3+1)/2*i =-512/2^9 = -1 A -1 -195 (-195*3+1)/2*i =-584/2^3 = -73 A -1 -219 (-219*3+1)/2*i =-656/2^4 = -41 D -7 -243 (-243*3+1)/2*i =-728/2^3 = -91 G -11 -267 (-267*3+1)/2*i =-800/2^5 = -25 A -1 -291 (-291*3+1)/2*i =-872/2^3 =-109 G -11 -315 (-315*3+1)/2*i =-944/2^4 = -59 F -9

Using the predecessor formula in the positive domain and operating on
elements of *5[8]*, we notably get elements of only three
congruence sets, a leaf set, *0[3]*, and two non-leaf sets,
*19[24]*, and *11[24]*. These results repeat
infinitely. Using any other *c[d]* congruence set will
yield a similar result (perhaps cyclically permuted in the sets of
3 predecessors). Note that the same result is obtained if
successors of *3[16]* are employed, a simple result of the
reversability of the edges in the state transition diagram among the
positive integers.

5 (2*5-1)/3 = 3 leaf in 0{[3]} 1st instance 29 (2*29-1)/3 = 19 in 19{[24]} 1st instance 53 (2*53-1)/3 = 35 in 11{[24]} 2nd instance 77 (2*77-1)/3 = 51 leaf in 0{[3]} 17th instance 101 (2*101-1)/3 = 67 in 19{[24]} 2nd instance 125 (2*125-1)/3 = 83 in 11{[24]} 3rd instance 149 (2*149-1)/3 = 99 leaf in 0{[3]} 33rd instance 173 (2*173-1)/3 =115 in 19{[24]} 3rd instance 197 (2*197-1)/3 =131 in 11{[24]} 4th instance

Additionally, supposing that a non-trivial loop might appear among large positive integers requires that some such numbers fail to obey the rules of the simple arithmetic operations, multiplying, adding , and dividing While very large numbers are wondrous in many ways , it cannot be expected that any behave in such a basically capricious way.

The question is now moot in any case. The program
*rsetprog* develops the congruence sets constituting the
Collatz predecessor graph *de novo* from the congruence set
*5[8]*. The *de novo* development avoids the use of
assumptions about the state transition diagram and the comprehensivess
of the abstract predecessor tree, but instead develops the whole
schema in which those features may be discerned.
Examination of
the membership of the congruence sets at the successive powers-of-2
levels allows calculations (two dimensional infinite
summations) which indicate that all the odd positive integers
would be produced in an infinite program run of *rsetprog*.
Detailed examination of the tree of congruence sets reveals the left
descent assemblies and confirms the validity of the abstract predecessor
tree construction. Unlike many computer programs which don't
*prove* anything, *rsetprog* effectively constitutes the
desired inductive constructive proof of the Collatz conjecture.