Bridges domes: a preliminary synthesis (addendum)
Follow up to [Bridges domes: a preliminary synthesis : r/Collatz](https://www.reddit.com/r/Collatz/comments/1pgltvm/bridges_domes_a_preliminary_synthesis/).
The figure below is similar to the one in the post mentioned, except the central part, that is new.
It was difficult to accept the contrast between the two sides without further analysis. The central part presents the pairs at the bottom of each bridges series, one iteration after the black number. In that, it is similar to what exist on the left side.
On the left side, these pairs belong to the same classes mod 48 for a given m and are all yellow. On the right side, they alternate between green and yellow. Moreover, they form pairs with two other sets starting with the same color, once on top, once on bottom.
The brown and orange colors allow to see that there are only two sets of pairs, starting at a different point for each m.
The limited values of m analyzed so far only allow to suggest groups of 16 m's and classes of m mod 48.
In other words, bridges series on the right side control the fate of the sequences only up to the last orange / black number. After that, it seems that almost everything can happen.
Note that the numbering of the bridges series on the right should start from the right, as the last column is the series of m itself. It would better represent the infinite nature of the domes, developping on both sides of the orange-black triangle at the center of any dome.
[Updated overview of the project “Tuples and segments” II : r/Collatz](https://www.reddit.com/r/Collatz/comments/1ovicsc/updated_overview_of_the_project_tuples_and/)
Posted Dec 9, 2025, 9:34 AM UTC