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

A more concrete objection to your specific list:

Let's assume for a moment that everything works as you've written up to (0,0,0)(1,1,1)(2,1,1)(3,1,0).

It is clear that at this point, the (2,1,1)'s are tallying ω powers in the subscript of Ω. So how (0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,1) is so large in your table?

comehere is my own analysis, assuming that this is correct:

(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,1) = ψ(Ω_(Ωω))

(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,1)(3,1,0) = ψ(Ω_(Ω^2))

(0,0,0)(1,1,1)(2,1,1)(3,1,0)(3,0,0) = ψ(Ω_(Ω^ω))

(0,0,0)(1,1,1)(2,1,1)(3,1,0)(3,1,0) = ψ(Ω_(Ω^Ω))

(0,0,0)(1,1,1)(2,1,1)(3,1,0)(3,1,0)(3,1,0) = ψ(Ω_(Ω^Ω^2))

(0,0,0)(1,1,1)(2,1,1)(3,1,0)(4,0,0) = ψ(Ω_(Ω^Ω^2))

(0,0,0)(1,1,1)(2,1,1)(3,1,0)(4,0,0)(5,0,0) = ψ(Ω_(Ω^Ω^ω))

(0,0,0)(1,1,1)(2,1,1)(3,1,0)(4,1,0) = ψ(Ω_(Ω^Ω^Ω))

(0,0,0)(1,1,1)(2,1,1)(3,1,0)(4,2,0) = ψ(Ω_(ψ1(0)))

(0,0,0)(1,1,1)(2,1,1)(3,1,0)(4,2,0)(5,3,0) = ψ(Ω_(ψ2(0)))

(0,0,0)(1,1,1)(2,1,1)(3,1,0)(4,2,1) = ψ(Ω_Ω_ω)

(0,0,0)(1,1,1)(2,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,0) = ψ(Ω_Ω_Ω)

(0,0,0)(1,1,1)(2,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,0)(7,3,1) = ψ(Ω_Ω_Ω_ω)

(0,0,0)(1,1,1)(2,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,0)(7,3,1)(8,3,1)(9,3,0) = ψ(Ω_Ω_Ω_Ω)

(0,0,0)(1,1,1)(2,1,1)(3,1,1) = ψ(ψi(0))

(0,0,0)(1,1,1)(2,1,1)(3,1,1)(2,1,1) = ψ(Ω_(ψi(0)*ω))

(0,0,0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,1) = ψ(Ω_(ψi(0)^2))

(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,0,0) = ψ(Ω_(ψi(0)^ω))

(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0) = ψ(Ω_(ψi(0)^Ω))

(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,1) = ψ(Ω_(ψi(0)^ψi(0))

(0,0,0)(1,1,1)(2,1,1)(3,1,1)(4,0,0) = ψ(Ω_(ψi(0)^ψi(0)^ω))

(0,0,0)(1,1,1)(2,1,1)(3,1,1)(4,1,1) = ψ(Ω_(ψi(0)^ψi(0)^ψi(0)))

(0,0,0)(1,1,1)(2,2,0) = ψ(Ω_(ψ_(ψi(0))(0)))

(0,0,0)(1,1,1)(2,2,0)(2,1,1) = ψ(Ω_(ψ_(ψi(0))(0)*ω))

(0,0,0)(1,1,1)(2,2,0)(2,1,1)(3,1,0) = ψ(Ω_(ψ_(ψi(0))(0)*Ω))

(0,0,0)(1,1,1)(2,2,0)(2,1,1)(3,1,1) = ψ(Ω_(ψ_(ψi(0))(0)*ψi(0)))

(0,0,0)(1,1,1)(2,2,0)(2,1,1)(3,2,0) = ψ(Ω_(ψ_(ψi(0))(0)^2))

(0,0,0)(1,1,1)(2,2,0)(2,1,1)(3,2,0)(2,1,1)(3,2,0) = ψ(Ω_(ψ_(ψi(0))(0)^3))

(0,0,0)(1,1,1)(2,2,0)(2,1,1)(3,2,0)(3,0,0) = ψ(Ω_(ψ_(ψi(0))(0)^ω))

(0,0,0)(1,1,1)(2,2,0)(2,1,1)(3,2,0)(3,1,0) = ψ(Ω_(ψ_(ψi(0))(0)^Ω))

(0,0,0)(1,1,1)(2,2,0)(2,1,1)(3,2,0)(3,1,1) = ψ(Ω_(ψ_(ψi(0))(0)^ψi(0)))

(0,0,0)(1,1,1)(2,2,0)(2,1,1)(3,2,0)(3,1,1)(4,2,0) = ψ(Ω_(ψ_(ψi(0))(0)^^2))

(0,0,0)(1,1,1)(2,2,0)(2,1,1)(3,2,0)(3,1,1)(4,2,0)(4,1,1)(5,2,0) = ψ(Ω_(ψ_(ψi(0))(0)^^3))

(0,0,0)(1,1,1)(2,2,0)(2,2,0) = ψ(Ω_(ψ_(ψi(0))(1)))

(0,0,0)(1,1,1)(2,2,0)(2,2,0)(2,2,0) = ψ(Ω_(ψ_(ψi(0))(2)))

(0,0,0)(1,1,1)(2,2,0)(3,0,0) = ψ(Ω_(ψ_(ψi(0))(ω)))

(0,0,0)(1,1,1)(2,2,0)(3,1,0) = ψ(Ω_(ψ_(ψi(0))(Ω)))

(0,0,0)(1,1,1)(2,2,0)(3,1,1) = ψ(Ω_(ψ_(ψi(0))(ψi(0))))

(0,0,0)(1,1,1)(2,2,0)(3,1,1)(4,2,0) = ψ(Ω_(ψ_(ψi(0))(ψ_(ψi(0))(0))))

(0,0,0)(1,1,1)(2,2,0)(3,2,0) = ψ(Ω_Ω_(ψi(0)+1))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(2,1,1) = ψ(Ω_(Ω_(ψi(0)+1)*ω))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(2,1,1)(3,1,0) = ψ(Ω_(Ω_(ψi(0)+1)*Ω))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(2,1,1)(3,1,1) = ψ(Ω_(Ω_(ψi(0)+1)*ψi(0)))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(2,1,1)(3,2,0) = ψ(Ω_(Ω_(ψi(0)+1)*ψ_(ψi(0))(0)))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(2,1,1)(3,2,0)(4,2,0) = ψ(Ω_(Ω_(ψi(0)+1)^2))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(2,1,1)(3,2,0)(4,2,0)(2,1,1)(3,2,0)(4,2,0) = ψ(Ω_(Ω_(ψi(0)+1)^3))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(2,1,1)(3,2,0)(4,2,0)(3,0,0)= ψ(Ω_(Ω_(ψi(0)+1)^ω))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(2,1,1)(3,2,0)(4,2,0)(3,1,0)= ψ(Ω_(Ω_(ψi(0)+1)^Ω))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(2,1,1)(3,2,0)(4,2,0)(3,1,1)= ψ(Ω_(Ω_(ψi(0)+1)^ψi(0)))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(2,1,1)(3,2,0)(4,2,0)(3,1,1)(4,2,0)(5,2,0)= ψ(Ω_(Ω_(ψi(0)+1)^^2))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(2,2,0)= ψ(Ω_(ψ_(ψi(0)+1)(0)))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(2,2,0)(3,2,0) = ψ(Ω_Ω_(ψi(0)+2))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(3,0,0) = ψ(Ω_Ω_(ψi(0)+ω))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(3,1,0) = ψ(Ω_Ω_(ψi(0)+Ω))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(3,1,1) = ψ(Ω_Ω_(ψi(0)*2))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(3,1,1)(4,2,0) = ψ(Ω_Ω_Ω_(ψi(0)+1))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(3,1,1)(4,2,0)(5,2,0) = ψ(Ω_Ω_(Ω_(ψi(0)+1)^^2))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(3,1,1)(4,2,0)(5,2,0)(5,1,1) = ψ(Ω_Ω_Ω_(ψi(0)*2))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(3,1,1)(4,2,0)(5,2,0)(5,1,1)(6,2,0) = ψ(Ω_Ω_Ω_Ω_(ψi(0)+1))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(3,2,0) = ψ(ψi(1))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(3,2,0)(3,2,0) = ψ(ψi(2))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(4,0,0) = ψ(ψi(ω))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(4,1,0) = ψ(ψi(Ω))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(4,1,1) = ψ(ψi(ψi(0)))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(4,2,0) = ψ(ψi(ψ_ψi(0)(0)))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(4,2,0)(4,2,0) = ψ(ψi(ψi(1)))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(4,2,0)(5,0,0) = ψ(ψi(ψi(ω)))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(4,2,0)(5,1,1) = ψ(ψi(ψi(ψi(0))))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(4,2,0)(5,2,0)(5,2,0) = ψ(ψi(ψi(ψi(1))))

(0,0,0)(1,1,1)(2,2,0)(3,2,0)(4,2,0)(5,2,0)(6,1,1) = ψ(ψi(ψi(ψi(ψi(0)))))

(0,0,0)(1,1,1)(2,2,0)(3,3,0) = ψ(I)

And I'll stop here.

As you can see, while your analysis claims that (0,0,0)(1,1,1)(2,2,0)(3,3,0) is way beyond weakly compact cardinals, the above progression clearly shows that we're still in the low inacessible range.

This is exactly what we'd expect from a hydra-based notation. It does grow a little bit faster then I thought it would (probably due to a nesting-effect that stems from the boost you've mentioned) but my point stands: Hydra-based notations do not give you Mahlo-level ordinals.

P.S.

I do not claim that BM4 or any other specific version of BMS indeed grows as fast as the above table states. It is simply the optimal growth rate possible, given the general way that the notation works.