User blog comment:Bubby3/Matrix system analysis (new blog post)/@comment-30754445-20181119174959/@comment-24725252-20181119210706

The effect of the (1,1,1) will always upgrade the (x,1,0), if an expression ends in it, even if that (x,1,0) is in the place of the W_x counter. That means that in (0,0,0)(1,1,1)(2,1,1)(3,1,0)(1,1,1), the (1,1,1) will also increase the 1 in the (3,1,0). This also applies to expressions that end in (1,1,1)(2,1,1), which replace W with W_(w^2). This effect keeps on iterating itself until we get to (0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,0,0). Then the function isn't so strong. That means that the (x,1,0's) get converted into psi_I(0) when a (1,1,1)(2,1,1)(3,1,0)(2,0,0) is placed after them. This continues until (0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,0)(3,2,1)(4,2,1)(5,2,0), which has level psi(W_(W_(psi_I(0)+1))). The upgrading effect continues when (3,2,1)'s are added to the seperator. Now, W_(psi_I(0)+1)'s are replaced with W_(psi_I(0)+w). This continueds on like before psi(psi_I(0)), and we get to   (0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,0)(3,2,1)(4,2,1)(5,2,0)(4,0,0), which has level psi(psi_I(1)). This pattern repeats w times, until we get to (0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,1), which does indeed have level psi(psi_I(w)). Can you read my analysis in detail before commenting that the function is weaker than I said so?