Fish number 3

Fish number 3 (F3), is a number defined by Japanese googologist Fish in 2002. Its base is the same as in Fish number 1 and Fish number 2:
 * \(s(1)_1(0,y) = y+1\)
 * \(s(1)_z(0,y) = s(1)_{z-1}(y,y)\) for \(z ≥ 1\)
 * \(s(1)_z(x,0) = s(1)_z(x - 1,1)\)
 * \(s(1)_z(x,y) = s(1)_z(x - 1,S_z(x,y - 1))\)

The function \(s(1)_x(x,x)\) grows about as fast as \(f_{\omega^2}(x)\).

Then we get a \(s(x)\) map that diagonalizes over the xs in the \(s(1)\) function. We have also an \(ss(1)\) map that diagonalizes over the q in the \(s(q)\) map. There is one additional rule needed: The other rules are changed rules from the \(s(x)\) map. Fish number 3 is defined as \(ss(2)^63(3)\). Fish number 3 is comparable to \(f_{\omega^{\omega+1}\times63}(3)\)
 * \(s(1)_1(0,y) = y+1\)
 * \(s(q)_z(0,y) = s(1)_{z-1}(y,y)\) for \(z ≥ 1\)
 * \(s(q)_z(x,0) = s(1)_z(x - 1,1)\)
 * \(s(q)_z(x,y) = s(1)_z(x - 1,S_z(x,y - 1))\)
 * \(s(q)_1(0,y) = s(q-1)_{y}(y,y)\)
 * \(ss(1)_1(0,y) = s(y)_y(y,y)\)