A9.1.1- ton, Ton and TON

I often face difficulty to save lengthy blog, hence this blog serves as summary for A9.1. Not sure if these are salad functions but they sure run beyond hyper-speed. Again they were named as such because they resemble TON 618.

Largest Ordinal and Fastest Function

1. The largest ordinal should be Σ, the last ordinal defined in ITTM.
2. The fastest function should be F₈(n), defined in the same way as F₇(n), but based on LNGN instead of Rayo(n).
3. I used SCG(n) as input simply because it gives the biggest value for n = 1 among all functions.

mini ton Function

For nested ordinal A, this is the simplest member of Ton family:

1. mton(n) = F₈(SCG(⍵))↥^AF₈(SCG(n)), where A = T(Σ, F₈(SCG(⍵))).
2. Refer to blog A3.1 and A9 for rules of T(#) and ↥^{#} expansion.

ton Function

For nested ordinals A, B and C:

1. ton(n) = [A](⍵)↥^A[A](n), where A = T(B, [B](⍵)).
2. [a+1](n) = [a](⍵)↥^B[a](n), where B = T(C, [a](⍵)).
3. [0](n) = F₈(SCG(⍵))↥^CF₈(SCG(n)), where C = T(Σ, F₈(SCG(⍵))).
4. Refer to blog A9.1 for rules on [#](n) expansion.

Ton Function

For non-negative integer b, with near infinite loops of ordinal nests, Ton gives much more juice than ton:

1. Ton(n) = [A](⍵)↥^A[A](n), where A = T(B(⍵), [B(⍵)](⍵)).
2. [a+1](n) = [a](⍵)↥^B(⍵)[a](n), where
   1. B(b+1) = T(B(b), [a](⍵)); and
   2. B(0) = T(C, [a](⍵)).
3. [0](n) = F₈(SCG(⍵))↥^CF₈(SCG(n)), where C = T(Σ, F₈(SCG(⍵))).

TON Function

For non-negative integers a, b and c, this is the humongous version:

1. TON(n) = [A(⍵)](⍵)↥^A(⍵)[A(⍵)](n), where
   1. A(a+1) = T(A(a), [B(⍵)](⍵)); and
   2. A(0) = T(B(⍵), [B(⍵)](⍵)).
2. [a+1](n) = [a](⍵)↥^B(⍵)[a](n), where
   1. B(b+1) = T(B(b), [a](⍵)); and
   2. B(0) = T(C(⍵), [a](⍵)).
3. [0](n) = F₈(SCG(⍵))↥^C(⍵)F₈(SCG(n)), where
   1. C(c+1) = T(C(c), F₈(SCG(⍵))); and
   2. C(0) = T(Σ, F₈(SCG(⍵))).

Mega TON Function

And finally this is the ultimate version:

1. MTON(n) = [A(⍵)](⍵)↥^U(A(⍵))[A(⍵)](n), where
   1. A(a+1) = U(A(a)); and
   2. A(0) = T(B(⍵), [B(⍵)](⍵)).
2. [a+1](n) = [a](⍵)↥^U(B(⍵))[a](n), where
   1. B(b+1) = U(B(b)); and
   2. B(0) = T(C(⍵), [a](⍵)).
3. [0](n) = F₈(SCG(⍵))↥^U(C(⍵))F₈(SCG(n)), where
   1. C(c+1) = U(C(c)); and
   2. C(0) = T(Σ, F₈(SCG(⍵))).

Giga and Tera TON Function

See blog A10.1

Examples

1. mton(1) = F₈(SCG(⍵))↥^AF₈(SCG(1)), where A = T(Σ, F₈(SCG(a))) = ...Σ↑^{Σ↑Σ↑ΣΣΣ}Σ... (with F₈(SCG(a)) floors, where a = F₈(SCG(1))).

Conclusion

Now these are real Infinity Scrapers !