User blog:B1mb0w/The HyperRex Function

The HyperRex Function
My new HyperRex function is compared here to recursive functions such as Veblen and my previous Hyper function. The HyperRex function is a set of two functions \(H\) and \(r\) and has a growth rate well beyond \(f_{\vartheta(\Omega\uparrow\uparrow\omega)}(n)\)

Notation Explained
The notation I use here is not in general use, but I find helpful. They are parameter subscript brackets, leading zeros assumption, recursion parameter subscript \(*\), and the decremented function \(C\).

Parameter Subscript Brackets, where:

\(M(a,0_{[2]}) = M(a,0,0)\)

\(M(a,0_{[2]},b_{[3]},1) = M(a,0,0,b_1,b_2,b_3,1)\)

Leading Zeros Assumption, where:

\(M(0_{[x]},0_{[2]},b_{[3]},1) = M(b_1,b_2,b_3,1)\)

Recursion Parameter Subscript \(*\), where:

\(M^2(a) = M^2(a_*) = M(M(a))\) and \(M(a,b_*) = M(a,b)\)

\(M^2(a,b_*) = M(a,M(a,b))\)

\(M^2(a_*,b) = M(M(a,b),b)\)

Decremented Function \(C\), where for any function:

\(M(a_{[b]},c + 1,d_{[e]})\) then \(C = M(a_{[b]},c_*,d_{[e]})\)

where parameter \(c\) is assumed to have the recursion parameter subscript \(*\).

Comparing Veblen, Hyper and other functions
For the comparison below, I will take some liberties with the definition of the Veblen Function. A separate comparison of Veblen function my Big number function is available here.

The Hyper function behaves like the FGH function up to a point. The p function acts like a stronger version of the Veblen function as show here:

\(H^h(g,n_*) = f_g^h(n)\)

\(H(p(0),n) = f_{\omega}(n) = f_{\varphi(1)}(n)\)

\(H(p_c(a_{[b]}),n) >= H(p(a_{[b]}),n) > f_{\varphi(a_{[b]})}(n)\) for any values of \(a, b, c\)

Growth rate of the Hyper function
The growth rate of the Hyper function is compared here against my Big number function and a calculated lower bound of FGH using Veblen function and collapsing ordinal function.

Further References
Further references to relevant blogs can be found here: User:B1mb0w