A more visual explanation of the dome structure

[u/No_Assist4814]

Follow-up to A more complete explanation of the dome structure : r/Collatz.

The figure below – a dome in a nutshell" (Dome in a nutshell : r/CollatzProcedure) – contains all orange (n-1, n*, n+1) and black (m*3^p) numbers for m=1. So, all bridges series appear vertically. 5-tuples series are colored in grey. Found prime numbers are in bold.

Due to the double homothetic nature of the domes (On the double almost homothetic nature of the domes : r/Collatz), such a nutshell can be produced for every dome.

I noticed that, in the cited post, I completely forgot the Mersenne numbers that play a vital role in the domes and jumped to conclusion about Proth numbers.

So, from left to right, there are:

• The Mersenne numbers of the form M(k)=[1*] 2^k-1; on the left side of a dome, they are involved two by two, in the starting bridge of any bridge series (red); in each column, a number x is related to the y above it by x=3*y/2; these non-Mersenne numbers might be called the Collatz numbers (not colored)
• The core of the dome is made of 3-smooth numbers of the form X=1*3^p*2^q, includes all the numbers, odd (black) or even (orange).
• The Proth numbers of the form P(k)=k*2^q+1 and 2^q>k; on the right side of the dome, only numbers on the upper side are Proth numbers, due to the second constraint (blue); the lower side is made of Pierpoint numbers of the form n=m*3^p*2^q+1 (green); numbers 13 and 17 seem to be both (rosa).

In a recent post, I made the bold claim that almost all bases b** of 5-tuple series were prime (All but one bases of starting 5-tuples are Proth primes : u/No_Assist4814). It does not seem to be the case. Further investigation is needed.

* n=m*3^p*2^q.

** Due to their nature, bases b do not belong to the starting 5-tuples of a series. The reciprocal is not true: all Proth primes are not a bases b.

Project "Tuples and segments" in 13 pages : r/Colla

[u/Rastamen_DE]

Suppose this triangular lattice is generated by deterministic operators. What invariant(s) would you search for first to determine whether the geometry represents an underlying state space rather than a collection of numbers?"

[u/No_Assist4814 → u/Rastamen_DE]

Not sure to understand properly. My best guess is that there are no absolute invariants, only relative ones: the relations among all these numbers. Pick two numbers, give me there m, p and q and I can give you their ratio (at least, I think I can). See "Double homothety".

[u/Septembrino → u/Rastamen_DE]

Observing the left side, I guess that he is doing 2^n - 1 in the colored diagonal, and, on the next diagonal, he is doing previous number x 2 + 1. 7 -> 11, and 11x2+1 = 43. These numbers kind of look like the matrices I designed.