Now, it's time to be fancy with MiTerSkyArk's googology! Let's do a concise analysis of selected MiTerSkyArk's notations. I start analyzing the random notations first because I didn't want some complex notations yet.
GOSS(n) = Amount of possible n×n grids using n+1 colors.
The formula is: GOSS(n) = (n+1)^n^2
GOSS(1) = 2^1^2 = 2^1 = 2
GOSS(2) = 3^2^2 = 3^4 = 81
GOSS(3) = 4^9 = 262,144
GOSS(4) = 5^16 = 152,587,890,625
GOSS(5) = 6^25 = 28,430,288,029,929,701,376
GOSS(6) = 7^36 = 2,651,730,845,859,653,471,779,023,381,601
GOSS(10) = 11^100 ≈ 1.378061233982 × 10^104
GOSS(100) = 101^10000 ≈ 1.63582871119 × 10^20,043
GOSS(1,000,000) = 1,000,001^1,000,000,000,000 ≈ 10^6,000,000,434,294
GOSS(GOSS(5)) = GOSS(6^25) = GOSS(28,430,288,029,929,701,376) ≈ 10^(1.572412716802 × 10^40)
GOSS(GOSS(10)) = GOSS(11^100) ≈ 10^(1.97765965779 × 10^210)
MGOSS(n) is to apply GOSS to n, n times.
MGOSS(3) = GOSS(GOSS(GOSS(3))) = GOSS(GOSS(262,144)) = GOSS(262,145^68,719,476,736) ≈ 10^(1.29 × 10^744,718,683,926)
MGOSS(4) = GOSS(GOSS(GOSS(GOSS(4)))) = GOSS(GOSS(GOSS(152,587,890,625))) = 10^10^(2.603866176070 × 10^23)
MGOSS(10) = GOSS^10(10) ≈ 10^10^10^10^10^10^10^10^(1.97765965779 × 10^210) ≈ 10^^10
MGOSS(100) = GOSS^100(100) ≈ 10^^101
[x]GOSS(n)
[1]GOSS(n) = GOSS(n) = (n+1)^n^2
[2]GOSS(n) = MGOSS(n) = GOSS^n(n) ≈ (n+1)^^n
[3]GOSS(n) = [2]GOSS^n(n) = MGOSS^n(n) ≈ (n+1)^^^(n+1)
[4]GOSS(n) = [3]GOSS^n(n) ≈ (n+1)^^^^(n+1)
[5]GOSS(n) = [4]GOSS^n(n) ≈ (n+1)^^^^^(n+1)
[n]GOSS(n) = [n-1]GOSS^n(n) ≈ (n+1)^^^^...^^^^(n+1) with n arrows
[x, 2]GOSS(n) = [[[...[[x]GOSS(n)]GOSS(n)...]GOSS(n)]GOSS(n)]GOSS(n) with x nestings (x+1 GOSS's)
[x, a]GOSS(n) = [[[...[[x, a-1]GOSS(n), a-1]GOSS(n)..., a-1]GOSS(n), a-1]GOSS(n), a-1]GOSS(n) with x nestings (x+1 GOSS's)
Analysis of two-entry GOSS:
[1, 2]GOSS(n) = [[1]GOSS(n)]GOSS(n) = [GOSS(n)]GOSS(n) ≈ {n+1, n+1, GOSS(n)} in BEAF/BAN
I define this precise extension like this to avoid degeneration with the basic GOSS.
[2, 2]GOSS(n) = [[[2]GOSS(n)]GOSS(n)]GOSS(n) ≈ {n+1, n+1, {n+1, n+1, GOSS(n)}} in BEAF/BAN
[3, 2]GOSS(n) = [[[[3]GOSS(n)]GOSS(n)]GOSS(n)]GOSS(n) ≈ {n+1, n+1, {n+1, n+1, {n+1, n+1, GOSS(n)}}} in BEAF/BAN
[1, 3]GOSS(n) = [[1, 2]GOSS(n), 2]GOSS(n) ≈ {n+1, n+1, 1, 2} in BEAF/BAN
[2, 3]GOSS(n) = [[[2, 2]GOSS(n), 2]GOSS(n), 2]GOSS(n) ≈ {n+1, {n+1, n+1, 1, 2}, 1, 2} in BEAF/BAN
[1, 4]GOSS(n) = [[1, 3]GOSS(n), 3]GOSS(n) ≈ {n+1, n+1, 2, 2} in BEAF/BAN
[1, 5]GOSS(n) = [[1, 4]GOSS(n), 4]GOSS(n) ≈ {n+1, n+1, 3, 2} in BEAF/BAN
[1, n+1]GOSS(n) = [[1, n]GOSS(n), n]GOSS(n) ≈ {n+1, n+1, n-1, 2} in BEAF/BAN
Examples of two-entry GOSS:
[1, 2]GOSS(2) = [[1]GOSS(2)]GOSS(2) = [GOSS(2)]GOSS(2) = [81]GOSS(2) = [80]GOSS([80]GOSS(2)) ≈ 3^^^^...^^^^3 with 81 arrows = 3{81}3 in hyperoperator notation = {3, 3, 81} in BEAF/BAN
[1, 2]GOSS(3) = [[1]GOSS(3)]GOSS(3) = [262144]GOSS(3) ≈ 4{262,144}4 = {4, 4, 262144} in BEAF/BAN
[2, 2]GOSS(3) = [[[1]GOSS(3)]GOSS(3)]GOSS(3) = [[262144]GOSS(3)]GOSS(3) ≈ 4{4{262,144}4}4 = {4, 4, {4, 4, 262144}} in BEAF/BAN
[1, 3]GOSS(3) = [[1, 2]GOSS(3), 2]GOSS(3) = [[262144]GOSS(3), 2]GOSS(3) ≈ {4, {4, 4, 262144}, 1, 2} in BEAF/BAN
[1, 4]GOSS(3) = [[1, 3]GOSS(3), 3]GOSS(3) = [[[262144]GOSS(3), 2]GOSS(3), 3]GOSS(3) ≈ {4, {4, {4, 4, 262144}, 1, 2}, 2, 2} in BEAF/BAN
The limit of the GOSS function has the ordinal level of ω2 in the fast-growing hierarchy (FGH), with respect to the Wainer's hierarchy.
N-FOLD(n) takes n copies of n, and concatenates them together. It is also equal to n[n] in Copy notation.
Example: N-FOLD(11) = 11[11] in Copy notation = 1,111,111,111,111,111,111,111
Definition:
n⇶^(k)a = n⇶⇶...(k)...⇶⇶a
n⇶^(1)0 = n{n}n = {n, n, n} in BEAF/BAN (the ^1 can be omitted so ⇶^1 and ⇶ are basically the same, but in my system, ⇶ without ^1 is far more preferred)
n⇶^(k)0 = n⇶^(k-1)n⇶^(k-1)n...n⇶^(k-1)n⇶^(k-1)n with n n's
n⇶^(k-1)n⇶^(k-1)n...n⇶^(k-1)n⇶^(k-1)n is worked through in a similar way to arrow notation, right to left.
3⇶3⇶3⇶3 is the same as 3⇶(3⇶(3⇶3)), 4⇶⇶3⇶⇶5⇶⇶2 is the same as 4⇶⇶(3⇶⇶(5⇶⇶2)).
n⇶^(k)a = ((...(n⇶^(k)a-1)⇶^(k)a-1...)⇶^(k)a-1)⇶^(k)a-1 with n nestings (n ⇶'s)
n⇶[0] = n⇶^(n)n
n⇶[a] = ((...(n⇶[a-1])⇶[a-1]...)⇶[a-1])⇶[a-1] with n nestings (n ⇶'s)
Analysis:
n⇶0 = n{n}n [FGH level ω]
n⇶1 = ((((n⇶0)⇶0)...)⇶0)⇶0 with ⇶'s = ((((n{n}n){n}n){n}n...){n}n){n}n ≈ {n, n+1, 1, 2} [FGH level ω+1]
n⇶2 = ((((n⇶1)⇶1)...)⇶1)⇶1 with ⇶'s ≈ {n, n+1, 2, 2} [FGH level ω+2]
n⇶3 = ((((n⇶2)⇶2)...)⇶2)⇶2 with ⇶'s ≈ {n, n+1, 3, 2} [FGH level ω+3]
n⇶n = ((((n⇶(n-1))⇶(n-1))...)⇶(n-1))⇶n with ⇶'s ≈ {n, n+1, n, 2} [FGH level ω2]
n⇶n⇶n = ≈ {n, n+1, {n, n+1, n, 2}, 2} [FGH level ω2]
n⇶⇶0 = n⇶n⇶n⇶...⇶n⇶n⇶n with n n's ≈ {n, n, 1, 3} [FGH level ω2+1]
n⇶⇶1 = ((((n⇶⇶0)⇶⇶0)...)⇶⇶0)⇶⇶0 with n ⇶⇶'s ≈ {n, n+1, 2, 3} [FGH level ω2+2]
n⇶⇶2 = ((((n⇶⇶1)⇶⇶1)...)⇶⇶1)⇶⇶1 with n ⇶⇶'s ≈ {n, n+1, 3, 3} [FGH level ω2+3]
n⇶⇶n = ((((n⇶⇶(n-1))⇶⇶(n-1))...)⇶⇶(n-1))⇶⇶(n-1) with n ⇶⇶'s ≈ {n, n, n, 3} [FGH level ω3]
n⇶⇶⇶0 = n⇶⇶n⇶⇶n⇶⇶...⇶⇶n⇶⇶n⇶⇶n with n ⇶⇶⇶'s ≈ {n, n, 1, 4} [FGH level ω3+1]
n⇶⇶⇶1 = ((((n⇶⇶⇶0)⇶⇶⇶0)...)⇶⇶⇶0)⇶⇶⇶0 with n ⇶⇶⇶'s ≈ {n, n+1, 2, 4} [FGH level ω3+2]
n⇶⇶⇶n = ((((n⇶⇶⇶(n-1))⇶⇶⇶(n-1))...)⇶⇶⇶(n-1))⇶⇶⇶(n-1) with n ⇶⇶⇶'s ≈ {n, n, n, 4} [FGH level ω4]
n⇶⇶⇶⇶0 = n⇶⇶⇶n⇶⇶⇶n⇶⇶⇶...⇶⇶⇶n⇶⇶⇶n⇶⇶⇶n with n ⇶⇶⇶'s ≈ {n, n, 1, 5} [FGH level ω4+1]
n⇶⇶⇶⇶⇶0 = n⇶⇶⇶⇶n⇶⇶⇶⇶n⇶⇶⇶⇶...⇶⇶⇶⇶n⇶⇶⇶⇶n⇶⇶⇶⇶n with n ⇶⇶⇶⇶'s ≈ {n, n, 1, 6} [FGH level ω5+1]
n⇶^(k)0 = n⇶⇶⇶...⇶⇶⇶n with k ⇶'s ≈ {n, n, 1, k+1} [FGH level ω^2]
n⇶[0] = n⇶^(n)n = n⇶⇶⇶...⇶⇶⇶n with n ⇶'s ≈ {n, n, 1, n+1} [FGH level ω^2]
n⇶[1] = ((((n⇶[0])⇶[0])...)⇶[0])⇶[0] with n ⇶'s ≈ {n, n+1, 1, 1, 2} [FGH level ω^2+1]
n⇶[2] = ((((n⇶[1])⇶[1])...)⇶[1])⇶[1] with n ⇶'s ≈ {n, n+1, 2, 1, 2} [FGH level ω^2+2]
n⇶[n] = ((((n⇶[n-1])⇶[n-1])...)⇶[n-1])⇶[n-1] with n ⇶'s ≈ {n, n+1, n, 1, 2} [FGH level ω^2+ω]
But the last two lines aren't that complete. I am going to rewrite the [] part of the notation as follows:
n⇶[0]0 = n⇶^(n)n
n⇶[k]a = ((...(n⇶[k](a-1))⇶[k](a-1)...)⇶[k](a-1))⇶[k](a-1) with a nestings (a ⇶'s)
n⇶^b[k]0 = n⇶^(b-1)[k]n⇶^(b-1)[k]...⇶^(b-1)[k]n⇶^(b-1)[k]n with n n's
n⇶[k]0 = n⇶^n[k-1]0
And this analyzes:
n⇶[0]0 = n⇶^(n)n ≈ {n, n, 1, n+1} [FGH level ω^2. Equivalent to n⇶[0] in the original definition]
n⇶[0]1 = (((n⇶[0]0)⇶[0]0...)⇶[0]0)⇶[0]0 with n ⇶'s ≈ {n, n+1, 1, 1, 2} [FGH level ω^2+1. Equivalent to n⇶[1] in the original definition]
n⇶[0]2 = (((n⇶[0]1)⇶[0]1...)⇶[0]1)⇶[0]1 with n ⇶'s ≈ {n, n+1, 2, 1, 2} [FGH level ω^2+2. Equivalent to n⇶[2] in the original definition]
n⇶[0]n = (((n⇶[0](n-1))⇶[0](n-1)...)⇶[0](n-1))⇶[0](n-1) with n ⇶'s ≈ {n, n+1, n, 1, 2} [FGH level ω^2+ω. Equivalent to n⇶[n] in the original definition]
n⇶⇶[0]0 = n⇶[0]n⇶[0]...⇶[0]n⇶[0]n with n n's ≈ {n, n, 1, 2, 2} [FGH level ω^2+ω+1]
n⇶⇶[0]1 = (((n⇶⇶[0]0)⇶⇶[0]0...)⇶⇶[0]0)⇶⇶[0]0 with n ⇶⇶'s ≈ {n, n+1, 2, 2, 2} [FGH level ω^2+ω+2]
n⇶⇶⇶[0]0 = n⇶⇶[0]n⇶⇶[0]...⇶⇶[0]n⇶⇶[0]n with n n's ≈ {n, n, 1, 3, 2} [FGH level ω^2+ω2+1]
n⇶⇶⇶⇶[0]0 = n⇶⇶⇶[0]n⇶⇶⇶[0]...⇶⇶⇶[0]n⇶⇶⇶[0]n with n n's ≈ {n, n, 1, 4, 2} [FGH level ω^2+ω3+1]
n⇶[1]0 = n⇶^n[0]n = n⇶⇶⇶...⇶⇶⇶[0]n with n ⇶'s with n n's ≈ {n, n, 1, n+1, 2} [FGH level ω^2·2]
n⇶[1]1 = (((n⇶[1]0)⇶[1]0...)⇶[1]0)⇶[1]0 with n ⇶'s ≈ {n, n+1, 1, 1, 3} [FGH level ω^2·2+1]
n⇶⇶[1]n = n⇶[1]n⇶[1]...⇶[1]n⇶[1]n with n n's ≈ {n, n, 1, 2, 3} [FGH level ω^2·2+ω+1]
n⇶⇶⇶[1]n = n⇶⇶[1]n⇶⇶[1]...⇶⇶[1]n⇶⇶[1]n with n n's ≈ {n, n, 1, 3, 3} [FGH level ω^2·2+ω2+1]
n⇶[2]0 = n⇶^n[1]n = n⇶⇶⇶...⇶⇶⇶[1]n with n ⇶'s with n n's ≈ {n, n, 1, n+1, 3} [FGH level ω^2·3]
n⇶⇶[2]n = n⇶[2]n⇶[2]...⇶[2]n⇶[2]n with n n's ≈ {n, n, 1, 2, 4} [FGH level ω^2·3+ω+1]
n⇶[3]0 = n⇶^n[2]n = n⇶⇶⇶...⇶⇶⇶[2]n with n ⇶'s with n n's ≈ {n, n, 1, n+1, 4} [FGH level ω^2·4]
n⇶[4]0 = n⇶^n[3]n = n⇶⇶⇶...⇶⇶⇶[3]n with n ⇶'s with n n's ≈ {n, n, 1, n+1, 5} [FGH level ω^2·5]
n⇶[5]0 = n⇶^n[4]n = n⇶⇶⇶...⇶⇶⇶[4]n with n ⇶'s with n n's ≈ {n, n, 1, n+1, 6} [FGH level ω^2·6]
n⇶[n]0 = n⇶^n[n-1]n = n⇶⇶⇶...⇶⇶⇶[n]n with n ⇶'s with n n's ≈ {n, n, 1, n+1, n+1} [FGH level ω^3]
Examples:
3⇶1 = ((3⇶0)⇶0)⇶0 = ((3{3}3)⇶0)⇶0 = ((3^^^3)⇶0)⇶0 = ((3^^7,625,597,484,987)⇶0)⇶0 = ((3^^^3){3^^^3}(3^^^3))⇶0 = (3^^^3){3^^^3}(3^^^3){(3^^^3){3^^^3}(3^^^3)}(3^^^3){3^^^3}(3^^^3) ≈ 3{3{3^^7,625,597,484,987}3}3 = 3{3{3{3}3}3}3 = {3, 4, 1, 2} in BEAF/BAN
3⇶2 = ((3⇶1)⇶1)⇶1 ≈ (({3, 4, 1, 2})⇶1)⇶1 ≈ ({3, {3, 4, 1, 2}, 1, 2})⇶1 ≈ {3, {3, {3, 4, 1, 2}, 1, 2}, 1, 2} in BEAF/BAN ≈ {3, 4, 2, 2}
The limit of the original definition of the [] level is ω^2+ω, whereas my remake of the final level goes up to ω^3 in FGH.