How tuples merge continuously... or not
In my understanding. a tuple is made of consecutive numbers that merge continuously (a merge or a new tuple every third iteration at most).
The figure below contrast to potential tuples. The first was suggested in a comment, the second by me, as the longuest continuously merging tuple. This is due to the fact that it iterates into two other 5-tuples in the process. To facilitate the analysis, the starting 5-tuple is colored as such (brown, text in white), but the other two are decomposed into pairs and singletons, as detailed below. A quick way to identify 5-tuples, even if decomposed, is to search for the rosa odd singleton, part of an odd triplet, in which all known 5-tuples iterate into.
A quick reminder about how to analyze a tree, starting from any merged number and moving in retro-iterations: (3) final pair (dark orange, text in white), (4) pair of predecessors (dark blue, text in white), (5) preliminary pair 1 (orange), (6) even triplet 1 (light blue), (7) preliminary pair 2 (light green), (8) even triplet 2 (dark green, text in white). From there, the pairs and triplets can go on on some situations (not here), or change for (9 min) odd triplets and 5-tuples (10 min).
To differentiate a tuple from something else, here are the differences:
- The size of the tuple: so far, there are no known tuple larger than a 5-tuple.
- The order of the numbers: due to local order around merges, tuples are ordered.
- The density of the tuples: to merge continuously, tuples are present in a manner consistent with the definition.
- False pairs (red): pairs may occur, but may diverge. Non-ordrerd ones are a sure sign something is cooking.
Any tree is partial by definition. Other tuples in the upper tree won't change the fact that it merges discontinuously.
Posted Apr 26, 2025, 9:38 AM UTC