Why Twins (etc.) Arise

On the way down from the root node of the abstract predecessor tree, each generation produces disjoint sets in the succeeding generation. What we'll do here is show that an argument from the magnitude of the integers cannot show that the subsets are disjoint. As a byproduct, we'll see why there is such a predilection for closely spaced integers to arise in close proximity in the predecessor tree.

Each step in generating children forms new
coefficients and new constants in the resulting descriptive formula as
determined by the properties of the parental set. From the expression
for the set in the parental node, *dn*+*c*, the three subsets
whose values mod 3 will determine the directions to be taken to the next
generation are 3*dn*+*c*, 3*dn*+*c*+*d* and
3*dn*+*c*+2*d*. Since 3*dn* is congruent to 0 mod
3, the values of *c*, *c*+*d*, and *c*+2*d*
determine the residue mod 3 of each subset. Because *d* (a power
of 2) is relatively prime to 3 on the way down in the abstract
predecessor tree, *c*, *c*+*d*, and *c*+2*d*
each have a different residue mod 3, so one of the three subsets will be
leaves, another will take an *s* step, and another a *b* step.
The predecessor tree is grown by indefinite repetition of this process.

The *s* steps produce children with one of the constants
(2*c*-1)/3, (2(*c*+*d*)-1)/3, or
(2(*c*+2*d*)-1)/3 and the *b* steps with one of
(4*c*-1)/3, (4(*c*+*d*)-1)/3, or
(4(*c*+2*d*)-1)/3. If the -1 is omitted, and the floor
function employed instead, these become floor(2*c*/3),
floor(2(*c*+*d*)/3), floor(2(*c*+2*d*)/3),
floor(4*c*/3), floor(4(*c*+*d*)/3), and
floor(4(*c*+2*d*)/3), where the argument to the floor function
always has a fractional part equal to 1/3. Clearly whatever combination
of children arise, the children will be disjoint from the parent.

Since the tree starts with 8*n*+5, in which
*c*<*d*, the coefficients of the successive
generation are
2*d*/3 for the *s* step and 4*d*/3 for the *b* step,
and the expressions for the constants always result from applying a
floor function to a mixed number, the new *c* will always be less
than the new *d*. Applied iteratively, this shows that
*c*<*d* in all formulas for sets.

There are two possible regions of coincidence or overlap in the
children sets in the event that *c* closely approaches *d*, at
floor(4*c*/3) and floor(2(*c*+*d*)/3), and
between
floor(4(*c*+*d*)/3) and floor(2(*c*+2*d*)/3). The
latter is of more concern because there is a range of possible overlap
there. While it is often the case that children reached from
3*dn*+*c*+*d* are smaller than children reached from
3*dn*+*c*+2*d*, the values can occur in the inverse order
when 3*dn*+*c*+*d* takes the *b* step and
3*dn*+*c*+2*d* takes the *s* step. There are two
early occurrences of this in the predecessor tree: first, 15 as parent
gives 105 from 3*dn*+*c*+*d* using the *b* step and
95 from 3*dn*+*c*+2*d* using the *s* step and,
second, 3 as parent gives 25 from 3*dn*+*c*+*d* using the
*b* step and 23 from 3*dn*+*c*+2*d* using the
*s* step.

And what if, in two successive generations, one path goes *bs*
and another path goes *sb*? Both would reach a constant roughly
8/9 of the original. No such simple cases appear to exist, but the
multiple instances of closely spaced integers closely positioned in the
predecessor tree result from just such an interchange of steps except
that they are interspersed with extension steps. (E.g. 847 reaches
12049 by an *eseb* path but 12045 by a *bese* path, and 385 reaches 9733 by
an *esbbbe* path, but 9731 by a *bebbes* path.) Many more complex paths
with interchanged steps would lead to the same point if it were not for
the effect of the floor function at each generational level. (E.g. 583
reaches 7371 by an *esbebs* path, but 7369 by a *beessb* path. [I'm using *e*
here to denote a single step to a right child in the binary predecessor
tree, so *ee* means two steps through the right
descent.]) This phenomenon
is not limited to paths with equal numbers of *b* and *s*
steps.
(E.g. 1429
reaches 10161 by *beb*, but reaches 10163 by *eses*.)

The argument that all subsets are unique will have to depend on deeper issues than simple numerical ranges.

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