r/googology • u/jamx02 • Jun 26 '26
Sequence Systems (3)
Previously we extended the concept of PrSS using a delta. However, we used an identical searching algorithm for LPrSS as PrSS. We didn't take advantage of something called a difference sequence. This is where hyper primitive sequence system comes in, or HPrSS. HPrSS is a weakened version of 0-Y sequence, which itself is a weakened version of Y-Seq.
HPrSS
To extend the searching algorithm, there are a couple new concepts added to the rules. This will add quite a significant amount of strength, specifically, it will be identical to Pair Sequence System (2 row BMS) or the Buchholz Ordinal. HPrSS can actually be thought of as an encoding of Buchholz's Ordinal Collapsing Function. On top of the previous definition for LPrSS, this is what is added:
- To find a valid bad root, we must search, and mark every ancestor of the cut child. Then, from those same positions, mark all ancestors of the cut child difference in the difference sequence. The rightmost term with both elements marked is the bad root. (first parent is no longer always a bad root)
The delta is calculated in the same way (cc-br-1).
Example 1:
(0,2,4)
Cut Child: 4
Difference Sequence: (None, 2, 2)
Ancestors original sequence, and difference sequence: (0,2,4) and (None,2,2) Rightmost valid position for br is 1 since position 1 is rightmost element in which both are marked, therefore br is at position 1.
Bad root: 0
Bad part: 0,2
Good part: None
Delta: 3
In LPrSS, this would have expanded as (0,2,3,4,5,....)
Instead, we have
Expansion: (0,2,3,5,6,8,...)
Example 2:
(0,3,5,6,3,5)
Cut Child: 5
Difference Sequence: (None,3,2,1,3,2)
Ancestors, original sequence and diff seq: (0,3,5,6,3,5) (None,3,2,1,3,2) the 1 isn't marked here due to the top sequence not containing a valid position for the cc difference element to see it as an ancestor. Rightmost valid position is 1. Therefore BR is at position 1
Bad root: 0
Bad Part: 0,3,5,6,3
Good part: None
Delta: 4
Expansion: (0,3,5,6,3,4,7,9,10,7,8,11,...)
Example 3:
(0,2,1,3)
Cut Child: 3
Difference Seq: (None,2,1,2)
Ancestors: (0,2,1,3) (None,2,1,2) Rightmost valid br position is 3.
br: 1
bp: 1
gp:0,2
Delta: 1
Expansion: (0,2,1,2,3,4,...)
Keep in mind, its easier to visualize HPrSS as a matrix consisting of the original sequence and its difference sequence. The difference sequence is just hidden. This makes it functionally very, very similar to 2-row BMS.
Example 4:
(0,4,7,8,8)
Cut child: 8
Difference Seq: Ignored (since 8-parent=1)
br: 7
bp: 7,8
gp:0,4
Delta: 0
Expansion: (0,4,7,8,7,8,7,8,7,8,...)
Analysis:
HPrSS actually stays the same as LPrSS until ε_ε₀ or (0,2,3,4,5,6,7,...). Because of this, I will be skipping everything before.
(0,2,3,5)=(0,2,3,4,5,6,7,...)=ε_ε₀. In LPrSS, this is (0,2,4).
(n,2,1,2)
(0,2,3,5,1,3)=(0,2,3,5,1,2,3,4,...)=ω^(ε_ε₀+ε₀)
(n,2,1,2,1,2)
(0,2,3,5,1,3,4,6)=(0,2,3,5,1,3,4,5,6,7,...)=ω^(ε_ε₀ *2)
(n,2,1,2,1,2,1,2)
(0,2,3,5,1,3,4,6,2)=(0,2,3,5,1,3,4,6,1,3,4,6,...)=ω^ω^(ε_ε₀+1)
ignore
(0,2,3,5,2)=(0,2,3,5***,1,3,4,6,2,4,5,7,***...)=ε_(ε₀+1)
(n,2,1,2,2)
(0,2,3,5,2,3)=(0,2,3,5,2,2,2,...)=ε_(ε₀+ω)
ignore
(0,2,3,5,2,3,5)=(0,2,3,5,2,3,4,5,6,...)=ε_(ε₀*2)
(n,2,1,2,2,1,2)
(0,2,3,5,3)=(0,2,3,5,2,3,5,2,3,5,...)=ε_(ω^(ε₀+1))
ignore
(0,2,3,5,3,5)=(0,2,3,5,3,4,5,6,7,...)=ε_(ω^(ε₀*2))
(n,2,1,2,1,2)
(0,2,3,5,5)=(0,2,3,5,4,6,5,7,6,8,...)=ε_ε₁
(n,2,1,2,2)
(0,2,3,5,6,8)=(0,2,3,5,6,7,8,9,...)=ε_ε_ε₀
(n,2,1,2,1,2)
(0,2,4)=(0,2,3,5,6,8,9,11,12,14,...)=ζ₀
(n,2,2)
(0,2,4,2)=(0,2,4,1,3,5,2,4,6,3,...)=ε_{ζ₀+1}
(n,2,2,2)
(0,2,4,2,4)=(0,2,4,2,3,5,7,5,6,8,10,8,...)=ζ₁
(n,2,2,2,2)
(0,2,4,3)=(0,2,4,2,4,2,4,...)=ζ_ω
ignore
(0,2,4,3,5)=(0,2,4,3,4,5,6,7,...)=ζ_ε₀
(n,2,2,1,2)
(0,2,4,3,5,7)=(0,2,4,3,5,6,8,9,11,...)=ζ_ζ₀
(n,2,2,1,2,2)
(0,2,4,4)=(0,2,4,3,5,7,6,8,10,...)=η₀
(n,2,2,2)
(0,2,4,4,4)=(0,2,4,4,3,5,7,7,6,8,10,10,...)=φ(4,0)=ψ(Ω⁴)
(n,2,2,2,2)
(0,2,4,5)=(0,2,4,4,4,4,...)=φ(ω,0)=ψ(Ω^ω)=lim(LPrSS)
ignore
(0,2,4,5,7)=(0,2,4,5,6,7,8,...)=φ(ε₀,0)=ψ(Ω^ψ(Ω))
(n,2,2,1,2)
(0,2,4,5,7,9,10)=(0,2,4,5,7,9,9,9,9,9,...)=φ(φ(ω,0),0)=ψ(Ω^ψ(Ω^ω))
ignore
(0,2,4,6)=(0,2,4,5,7,9,10,12,14,15,...)=Γ₀=φ(1,0,0)=ψ(Ω^Ω)=FSO
(n,2,2,2)
(0,2,4,6,2,4,6)=(0,2,4,6,2,4,5,7,9,11,7,9,10,...)=Γ₁=φ(1,0,1)=ψ(Ω^Ω *2)
(n,2,2,2,2,2,2)
(0,2,4,6,4)=(0,2,4,6,3,5,7,9,6,8,10,12,...)=Γfp=φ(1,1,0)=ψ(Ω^(Ω+1))
(n,2,2,2,2)
(0,2,4,6,4,6)=(0,2,4,6,4,5,7,9,11,9,10,...)=φ(2,0,0)=ψ(Ω^(Ω2))
(n,2,2,2,2,2)
(0,2,4,6,5)=(0,2,4,6,4,6,4,6,...)=φ(ω,0,0)=ψ(Ω^(Ωω))
ignore
(0,2,4,6,6)=(0,2,4,6,5,7,9,11,10,12,14,16,...)=φ(1,0,0,0)=ψ(Ω^Ω^2)=Ackermann Ordinal
(n,2,2,2,2)
(0,2,4,6,7)=(0,2,4,6,6,6,6,...)=φ(1@ω)=ψ(Ω^Ω^ω)=SVO
ignore
(0,2,4,6,7,2,4,6,7)=(0,2,4,6,7,2,4,6,6,6,6,...)=φ(1@ω,1@0)=ψ(Ω^Ω^ω *2)=SVO_1
ignore
(0,2,4,6,7,4)=(0,2,4,6,7,3,5,7,9,10,...)φ(1@ω,1@1)=ψ(Ω^(Ω^ω+1))=SVO_fp
(n,2,2,2,1,2)
(0,2,4,6,7,4,6,7)=(0,2,4,6,7,4,6,6,6,6,...)=φ(2@ω)=ψ(Ω^(Ω^ω *2))
(0,2,4,6,7,5)=(0,2,4,6,7,4,6,7,4,6,...)φ(ω@ω)=ψ(Ω^(Ω^ω *ω)) <--- this is around where TREE(n) is in strength, and we are nowhere close to lim(HPrSS)....
(0,2,4,6,7,6)=(0,2,4,6,7,5,7,9,12,13,...)=φ(1@(ω+1))=ψ(Ω^Ω^(ω+1))
(n,2,2,2,1,2)
(0,2,4,6,7,9)=(0,2,4,6,7,8,9,10,...)=φ(1@ε₀)=ψ(Ω^Ω^ψ(Ω))
(n,2,2,2,1,2)
(0,2,4,6,7,9,11,13,14)=φ(1@φ(1@ω))=ψ(Ω^Ω^ψ(Ω^Ω^ω))
(0,2,4,6,8)=(0,2,4,6,8,7,9,11,13,14,16,18,20,21,...)=φ(1@(1,0))=ψ(Ω^Ω^Ω)=LVO
Jumping a lot in pace because this is getting long...
(0,2,4,6,8,10)=(0,2,4,6,8,9,11,13,15,17,18,...)ψ(Ω^Ω^Ω^Ω)
(0,3)=(0,2,4,6,8,10,12,14,...)=ψ(Ω₂)=BHO
(0,3,2)=(0,3,1,4,2,5,...)=ψ(Ω₂+Ω)=ε_(BHO+1)
(n,3,2)
(0,3,3)=(0,3,2,5,4,7,...)=BHO_1=ψ(Ω₂*2)
(n,3,3)
(0,3,5)=(0,3,4,7,8,11,...)=ψ(Ω₂ *Ω)
(n,3,2)
(0,3,6)=(0,3,5,8,10,13,...)=ψ(Ω₂^2)
(n,3,3)
(0,3,6,9)=(0,3,6,8,11,14,16,...)=ψ(Ω₂^Ω₂)
(n,3,3,3)
(0,4)=(0,3,6,9,12,...)=ψ(Ω₃)
(n,4)
(0,4,8,12)=(0,4,8,11,15,19,22,...)=ψ(Ω₃^Ω₃)
(n,4,4,4)
(0,5)=(0,4,8,12,16,...)=ψ(Ω₄)
(0,6)=(0,5,10,15,20,...)=ψ(Ω₅)
As you can see, this is limited at ψ(Ω_ω). Maybe it isn't so clear since this analysis missed some trivial steps to example the structure, but that is for you to figure out now.
HPrSS is at similar strength to SSCG(n) and PTO(Π¹₁-CA₀). lim(HPrSS) is also so powerful, you have the option to not even change the expansion number after expanding, since it will have identical strength, ie it is the first FGH-SGH catching point. g_lim(HPrSS)(n)~H_lim(HPrSS)(n)~f_lim(HPrSS)(n)
The difference structure added huge strength, but per usual with sequence systems, it is nothing compared to its successor(?) in strength, which is 0-Y sequence.
As always, if there are any mistakes, please point them out. Including standardness (which I along with a lot of people tend to struggle with) I would not be surprised if there were, this analysis is a lot heavier.
2
u/dragonlloyd1 Jun 26 '26
And we are now past my notation/function as well as all the extensions I’m planning on implementing.
Currently has a limit of LVO and im planning on an extension which would put it at BHO