Hyper-Alpha Notation Part IV: The Limit of My Understanding And Beyond

See:

Part I

Part II: Arrows to Alphas

Part III: Full Circle

This part is a part of a series of ideas I had. Reaching significantly past ψ(Ω_ω) is very hard.

EDIT: With some help with Googology Discord, I've improved my approximations.

EDIT 2: I just realized ψ(ε_Ω+1) is ψ(Ω₂) in psi. This is more powerful than I thought after all. Ignore the title.

EDIT 3: I need to stop jumping to conclusions about approximating this.

See the rules from Part 2:

A ↑^a+1 b = A ↑^a A ↑^a A ↑^a ... ↑^a A ↑^a A ↑^a A with b "A"s

A ↑ⁿ ★A = (A ↑ⁿ ★) ↑ⁿ A

UPGRADE:

<A,A> ↑^a+1 b = <A,A> ↑^a <A,A> ↑^a <A,A> ↑^a ... ↑^a <A,A> ↑^a <A,A> ↑^a <A,A> with b "<A,A>"s

<A,A> ↑ⁿ ★A = (<A,A> ↑ⁿ ★) ↑ⁿ A

Recursive progression

<A,A>^A ≈ ψ(Ω_ω)

<A,A>^AA ≈ ψ(Ω_ω+1)

<A,A>^AA^A ≈ ψ(Ω_ω2)

<A,A>^AA↑↑A ≈ ψ(Ω_ψ(Ω))

<A,A>^AA ↑^A A ≈ ψ(Ω_ψ(Ω^ω))

<A,A>^A<A,A>^A ≈ ψ(Ω_ψ(Ωω))

<A,A>^AA ≈ ψ(Ω_Ω)

<A,A>^AAA ≈ ψ(Ω_Ω+1)

<A,A>^AAA^A ≈ ψ(Ω_Ω+ω)

<A,A>^AA<A,A>^A ≈ ψ(Ω_Ω+ψ(Ωω))

<A,A>^AA<A,A>^AA ≈ ψ(Ω_Ω2)

<A,A>^AA ≈ ψ(Ω_Ωω^ω)

<A,A>^A↑↑A ≈ ψ(Ω_Ωψ(Ω))

<A,A>^<A,A>A ≈ ψ(Ω_Ω²)

<A,A>^<A,A>AA ≈ ψ(Ω_Ω^ω)

<A,A>^<A,A><A,A>A ≈ ψ(Ω_Ω^Ω)

<A,A>↑↑A ≈ ψ(Ω_Ω2)

(<A,A>↑↑A)^A ≈ ψ(Ω_Ω2+Ω)

(<A,A>↑↑A)^<A,A>↑↑A ≈ ψ(Ω_Ω22)

<A,A>↑↑AA ≈ ψ(Ω_{Ω2​​​​​²})

<A,A>↑↑A^A ≈ ψ(Ω_{Ω2​​​​​^ω})

<A,A>↑↑<A,A>^A ≈ ψ(Ω_{Ω2​​​​​^Ω})

<A,A>↑↑<A,A>↑↑A ≈ ψ(Ω_Ω2^Ω2)

<A,A>↑↑↑A ≈ ψ(Ω_Ω3)

(<A,A>↑↑↑A)↑↑A ≈ ψ(Ω_Ω3Ω2)

<A,A>↑↑↑AA ≈ ψ(Ω_{Ω3​​​​​²})

<A,A>↑↑↑<A,A>↑↑A ≈ ψ(Ω_{Ω3​​​​​^Ω2})

<A,A>↑↑↑<A,A>↑↑↑A ≈ ψ(Ω_{Ω3​​​​​^Ω3})

<A,A>↑↑↑↑A ≈ ψ(Ω_Ω4)

<A,A>↑↑↑↑↑A ≈ ψ(Ω_Ω5)

<A,A> ↑^A A ≈ ψ(Ω_Ωω)

So after that, <<A,A>,A> is easily definable similarly to <A,A> and can be iterated similarly.

<<A,A>,A>A ≈ ψ(Ω_Ωω)

<<A,A>,A><<A,A>,A>A ≈ ψ(Ω_Ωψ(ΩΩω))

<<A,A>,A>^A ≈ ψ(Ω_ΩΩ)

<<A,A>,A>↑↑A ≈ ψ(Ω_ΩΩ2)

<<A,A>,A>↑↑↑A ≈ ψ(Ω_ΩΩ3)

<<A,A>,A> ↑^A A ≈ ψ(Ω_ΩΩω)

<<<A,A>,A>,A>^A ≈ ψ(Ω_ΩΩΩ)

Limit: <<<...<<<A,A>,A>,A>...A>,A>,A> ≈ ψ(omega fixed point ) (commonly denoted as ψ(ψ_I(0)) but pbot doesnt like that smh)

This isn't over.