Let $\mathcal{A}$ be the collection of all isomorphism classes of quotients of subalgebras of the classical Laver tables.

If $X\in\mathcal{A}$, then let $I_{X}$ denote the unique element such that $I_{X}*x=x,x*I_{X}=I_{X}$ for $x\in X$. We shall call an algebra $X\in\mathcal{A}$ critically simple if whenever $\simeq$ is a non-identity congruence on $X$ there is some $x\in X\setminus\{1_{X}\}$ with $x\simeq 1_{X}$.

If $X\in\mathcal{A}$, then let $X^{+}=X\cup\{c\}$ where we define $x*c=I_{X}$ for $x\in X\setminus I_{X}$, and $I_{X}*c=I_{X}$, and $c*x=x$ for each $x\in X$ and $c*c=I_{X}$.

Suppose $\lambda$ is an ordinal and $\phi:\mathcal{A}\rightarrow\lambda$ is a surjective function. Then we shall call $\phi$ a strength calibration function if it satisfies the following properties for $X,Y\in\mathcal{A}$:

1. If $X$ is a subalgebra or a quotient of $Y$, then $\phi(X)\leq\phi(Y)$.

2. If $f:X\rightarrow Y$ is a homomorphism such that if $f(x)=1_{Y}\rightarrow x=1_{X}$, then $\phi(X)\leq\phi(Y)$.

3. $\phi(X^{+})=\phi(X)+1.$

If the function $\phi$ is a strength calibration function, we shall call $\phi$ efficient if the problem of determining whether $\phi(X)<\phi(Y)$ from the multiplication tables of $X$ and $Y$ is solvable in polynomial time (I mean polynomial in terms of $|X|+|Y|$).

We shall call $\phi$ ultra fine tuning if whenever $X,Y$ are critically simple and non-isomorphic, then $\phi(X)\neq\phi(Y)$.

Does there exist an efficient ultra fine tuning strength calibration function $\phi:\mathcal{A}\rightarrow\lambda$?