Bottoms and triplets

What follows is based on a limited partial tree in the Giraffe head*. Further confirmation is needed.

The coloring of the tuples (n mod 16) follows the color of the segment (n mod 12) the first number of a tuple belongs to. Bottoms - odd numbers not part of a tuple and facing rosa wallls" - are in black and pairs of predecessors are in light blue.

The partial tree on the left shows the numbers n, the tree in the center n mod 16 and the table on the right will be explained below.

It is worth reminding that n mod 16 are heavily involved in tuples:

• 4-5-6 form even triplets half of the time, except when they are involved in 5-tuples.
• 12-13-14 form even triplets more irregularly, being a composite of congruence classes with various incresing moduli.
• 1 is involved in odd triplets irregurarly. It is a bottom the rest of the time.
• 7 is a bottom when 4-5-6 triplets exist (half of the time).
• 9 and 11 are always bottoms.
• 15 is a bottom when 12-13-14 triplets exist.

This limited example seems to show that:

• Blue triplets and pairs of predecessors are always associated with a bottom. Yellow triplets don't.
• In mod 16, bottoms are associated with a specific type of triplet, as summarized in the table on the right (n mod 16 that are always bottoms are in black).
• Yellow triplets are not associated with a bottom.
• A sequence starts with a yellow even triplet, followed by blue even triplets, followed by a pair of predecessors that ends the sequence.

Rosa even triplets do not appear here, as the have a specific role post 5-tuples series ([Even triplets post 5-tuples series : r/CollatzProcedure](https://www.reddit.com/r/CollatzProcedure/comments/1ld1bp5/even_triplets_post_5tuples_series/)).

This is another example of how the segments involved in a tuple influence its role in the procedure.

https://preview.redd.it/esdhtpbipm9f1.jpg?width=1431&format=pjpg&auto=webp&s=b8a56378c37700fe4c1f131780564e0750d6d57a

[Overview of the project (structured presentation of the posts with comments) : r/Collatz](https://www.reddit.com/r/Collatz/comments/1k98c5a/overview_of_the_project_structured_presentation/)