Vittorio's limit formula:
Hey, everyone! So MphLee shared with me some of his work. And he explained a fixed point formula for Goodstein's equation. And I'd like to present it here as an algorithm which can be used to solve our problem.
Let's write:
\[
f(y,s) = x [s] y\\
\]
Where this \([s]\) is interpreted as the modified bennet operations. Now, we are going to look at the inverse in \(y\), and call it \(f^{-1}(y,s)\). So for \(s=0\) this becomes \(f^{-1}(y,0) = y-x\), and for \(s=1\) this becomes \(f^{-1}(y,1) = y/x\) and for \(s=2\) this becomes \(f^{-1}(y,2) = \log_x(y)\). Where, we have an analytic function \(f^{-1}(y,s)\) for \(s \in [0,2]\) and \(y>e\). (Remember \(x>e\)).
What Vittorio's limit formula describes is a very very weird way of identifying Goodstein's equation. In many ways, we are just searching for \(g\) such that:
\[
g(y,s) = g(g^{-1}(y,s+1) + 1,s+1)\\
\]
Now, this may seem obvious. But, when you attach it to the fact that we can use bennet to approximate these solutions--and that these solutions exist. Well, why not just use Vittorio's limit formula (which, although Mphlee didn't write it out perfect, this is exactly what he meant).
\[
g_n(y,s) = g_{n-1}(g^{-1}_{n-1}(y,s+1) +1,s+1)\\
\]
We are just solving this limit. And in solving this limit we have perfection...
Now, Mphlee mostly described this as an algebraic/categorical property any Goodstein looking sequence satisfies. I want to say, that this is a polynomial operation which converges. And it converges fast. And I'm naming it after Vittorio.
The main object we have to consider here are binary operations. And we have to create an iterative procedure off of binary operations. This is something I never would've thought of. Although this appears at face value, as something we've all seen. I believe this deserves to be called Vittorio's limit formula.
Which, I'll describe perfectly. Start with the modified Bennet operators:
\[
x\,[s]\,y = \exp^{\circ s}_{y^{1/y}}\left(\log^{\circ s}_{y^{1/y}}(x) + y\right)\\
\]
Call:
\[
x\,[s]^{-1}\, y\,\,\text{the inverse of}\,\,x\,[s]\,y\,\,\text{in}\,\, y\\
\]
Now, we are looking to solve \([1,2]\) while we solve \([0,1]\). This is done to solve the pasting issue. Now let's call:
\[
x\,[s]_1\,y = x\,[s+1]\,\left( (x\,[s+1]^{-1}\,y) +1\right)\\
\]
Now, we can continue this operation:
\[
x\,[s]_n\,y = x\,[s+1]_{n-1}\,\left( (x\,[s+1]_{n-1}^{-1}\,y) +1\right)\\
\]
The really weird part now, is that this solution doesn't work on its own. You have to solve for \([s+1]_n\) while you solve for \([s]_n\). The thing is... we can solve for:
\[
\begin{align}
x\,[s]_{n}\,y &= f(y)\\
x\,[s+1]_n\,y &= f^{\circ y}(q)\\
\end{align}
\]
You can actually do this pretty fucking fast... It just looks like iterating a linear function.
EDIT:ACk! made a small mistake here, this is an idempotent iteration like this, the actual iteration is a little more difficult, I'll write it up when I can make sense of controlling the convergence of this...
Ladies and Gents, I present
Vittorio's Limit Formula
And how you turn Bennet into Goodstein
PS: To Mphlee, the object \(x\,[s]\,y\) is in your attractive basin.... You wrote it, but you didn't see it as a polynomial. \(x\,[s]\,y\) is so close to \(x\,\langle s\rangle y\) that iterations of your identity, converge to it...... God, I hope you get this.....
Hey, everyone! So MphLee shared with me some of his work. And he explained a fixed point formula for Goodstein's equation. And I'd like to present it here as an algorithm which can be used to solve our problem.
Let's write:
\[
f(y,s) = x [s] y\\
\]
Where this \([s]\) is interpreted as the modified bennet operations. Now, we are going to look at the inverse in \(y\), and call it \(f^{-1}(y,s)\). So for \(s=0\) this becomes \(f^{-1}(y,0) = y-x\), and for \(s=1\) this becomes \(f^{-1}(y,1) = y/x\) and for \(s=2\) this becomes \(f^{-1}(y,2) = \log_x(y)\). Where, we have an analytic function \(f^{-1}(y,s)\) for \(s \in [0,2]\) and \(y>e\). (Remember \(x>e\)).
What Vittorio's limit formula describes is a very very weird way of identifying Goodstein's equation. In many ways, we are just searching for \(g\) such that:
\[
g(y,s) = g(g^{-1}(y,s+1) + 1,s+1)\\
\]
Now, this may seem obvious. But, when you attach it to the fact that we can use bennet to approximate these solutions--and that these solutions exist. Well, why not just use Vittorio's limit formula (which, although Mphlee didn't write it out perfect, this is exactly what he meant).
\[
g_n(y,s) = g_{n-1}(g^{-1}_{n-1}(y,s+1) +1,s+1)\\
\]
We are just solving this limit. And in solving this limit we have perfection...
Now, Mphlee mostly described this as an algebraic/categorical property any Goodstein looking sequence satisfies. I want to say, that this is a polynomial operation which converges. And it converges fast. And I'm naming it after Vittorio.
The main object we have to consider here are binary operations. And we have to create an iterative procedure off of binary operations. This is something I never would've thought of. Although this appears at face value, as something we've all seen. I believe this deserves to be called Vittorio's limit formula.
Which, I'll describe perfectly. Start with the modified Bennet operators:
\[
x\,[s]\,y = \exp^{\circ s}_{y^{1/y}}\left(\log^{\circ s}_{y^{1/y}}(x) + y\right)\\
\]
Call:
\[
x\,[s]^{-1}\, y\,\,\text{the inverse of}\,\,x\,[s]\,y\,\,\text{in}\,\, y\\
\]
Now, we are looking to solve \([1,2]\) while we solve \([0,1]\). This is done to solve the pasting issue. Now let's call:
\[
x\,[s]_1\,y = x\,[s+1]\,\left( (x\,[s+1]^{-1}\,y) +1\right)\\
\]
Now, we can continue this operation:
\[
x\,[s]_n\,y = x\,[s+1]_{n-1}\,\left( (x\,[s+1]_{n-1}^{-1}\,y) +1\right)\\
\]
The really weird part now, is that this solution doesn't work on its own. You have to solve for \([s+1]_n\) while you solve for \([s]_n\). The thing is... we can solve for:
\[
\begin{align}
x\,[s]_{n}\,y &= f(y)\\
x\,[s+1]_n\,y &= f^{\circ y}(q)\\
\end{align}
\]
You can actually do this pretty fucking fast... It just looks like iterating a linear function.
EDIT:ACk! made a small mistake here, this is an idempotent iteration like this, the actual iteration is a little more difficult, I'll write it up when I can make sense of controlling the convergence of this...
Ladies and Gents, I present
Vittorio's Limit Formula
And how you turn Bennet into Goodstein
PS: To Mphlee, the object \(x\,[s]\,y\) is in your attractive basin.... You wrote it, but you didn't see it as a polynomial. \(x\,[s]\,y\) is so close to \(x\,\langle s\rangle y\) that iterations of your identity, converge to it...... God, I hope you get this.....