This is fucking beautiful! I've been working at this for three years now and you beat me to the punch! LOL!

Do you have a closed form expression for it yet? I have so many ideas involving rational operators.

And one thing that MUST be true is

x {1.5} 2 = x {0.5} x

If that's not true then the system doesn't meet requirements. But it wouldn't be hard to improvise a system since 0 <= q <= 1, {q} is defined.

Also, I wonder if we could try to implement the laws of logarithmic semi operators into this system:

q:log(x) = exp^[-q](x)

q:log(x {q} y) = q:log(x) + q:log(y)

q:log(x {1+q} y) = q:log(x) * y

And if we implement this we can feasibly solve for a new variation of tetration.

Since

x {0.5} y = arithmetic/geometric limiting algo

= -0.5:log(0.5:log(x) + 0.5:log(y))

= sexp(slog(sexp(slog(x)-0.5) + sexp(slog(y)-0.5)) + 0.5)

Holy jesus, yes!

And also, I was wondering what our identities are?, if S(q) is the identity function and x {q} S(q) = x, S(1) = 1, and S(0) = 0 obvi, but what is S(q)?

I found S(0.5) = 0.7019920407 for 2 {0.5} S(0.5) = 2. I used 1000 cycles so my numbers are probably more accurate.

However, sadly, S(0.5) for 3 {0.5} S(0.5) = 3 is a different number and therefore {q} has no identity. This is very sad indeed.

Do you have a closed form expression for it yet? I have so many ideas involving rational operators.

And one thing that MUST be true is

x {1.5} 2 = x {0.5} x

If that's not true then the system doesn't meet requirements. But it wouldn't be hard to improvise a system since 0 <= q <= 1, {q} is defined.

Also, I wonder if we could try to implement the laws of logarithmic semi operators into this system:

q:log(x) = exp^[-q](x)

q:log(x {q} y) = q:log(x) + q:log(y)

q:log(x {1+q} y) = q:log(x) * y

And if we implement this we can feasibly solve for a new variation of tetration.

Since

x {0.5} y = arithmetic/geometric limiting algo

= -0.5:log(0.5:log(x) + 0.5:log(y))

= sexp(slog(sexp(slog(x)-0.5) + sexp(slog(y)-0.5)) + 0.5)

Holy jesus, yes!

And also, I was wondering what our identities are?, if S(q) is the identity function and x {q} S(q) = x, S(1) = 1, and S(0) = 0 obvi, but what is S(q)?

I found S(0.5) = 0.7019920407 for 2 {0.5} S(0.5) = 2. I used 1000 cycles so my numbers are probably more accurate.

However, sadly, S(0.5) for 3 {0.5} S(0.5) = 3 is a different number and therefore {q} has no identity. This is very sad indeed.