The Laver tables give and will continue to give examples of sentences $\forall n P(n)$ which are known to be false under strong large cardinal assumptions but where no such $n$ where $P(n)$ is false has been explicitly found nor is known to exist under any weaker large cardinal assumptions. Since set theorists typically believe in either the consistency or the existence of large cardinals and since the algebraic structures that arise from the sufficiently strong large cardinal embeddings support the existence of such strong large cardinal hypotheses. Furthermore, since the classical Laver tables and similar structures are constructed using a double recursion that resembles the Ackermann function, one should expect the Laver tables to produce many examples of eventual counterexamples.

The Laver tables should be thought of as a combinatorial object that has many features such that under large cardinal assumptions, there will always come a point where these features no longer hold.

All these results on the final matrix and the Laver-like algebras are my own while the results on the classical Laver tables are not my own.

Classical Laver tables

The $n$-th classical Laver table is the unique algebra $A_{n}=(\{1,\dots,2^{n}\},*_{n})$ where $x*_{n}(y*_{n}z)=(x*_{n}y)*_{n}(x*_{n}z),x*_{n}1=x+1\mod 2^{n}$ for $x,y,z\in\{1,\dots,2^{n}\}$.

Under strong large cardinal hypotheses, the classical Laver tables should be thought of as algebraic structures consisting of patterns that will all eventually end but only after a very long time.

If $x\in A_{n}$, then let $o_{n}(x)$ be the least number $m$ with $x*_{n}2^{m}=2^{n}$. If $x\in\{1,\dots,2^{n}\}$, then let $\vartheta_{n}(x)$ be the least number $m$ with $x*_{n+1}m>2^{n}$.

Fact: (Randall Dougherty) If $n<\mathrm{Ack}(9,\mathrm{Ack}(8,\mathrm{Ack}(8,254)))$, then $o_{n}(1)<5$.

Eventual counterexamples: (Richard Laver) If for all $n$, there exists an $n$-huge cardinal, then $\lim_{n\rightarrow\infty}o_{n}(1)\rightarrow\infty$. No upper bound for the least $n$ with $o_{n}(1)=5$ has proven in ZFC. This fact the source of many eventual counterexamples one may have about the classical Laver tables.

Fact: If $n<9$ and $o_{n}(x)=o_{n+1}(x)=o_{n+2}(x)$, then $\vartheta_{n+1}(x)\leq\vartheta_{n}(x)$. Thomas Jech conjectured that this result holds for all $n$, but a suitably motivated human could even have computed a counterexample by hand, so this conjecture is easily disproven.

Eventual counterexample: $o_{9}(34)=o_{10}(34)=o_{11}(34)$, but $\vartheta_{9}(34)=5,\vartheta_{10}(34)=8$.

Eventual counterexample: A linear ordering $\preceq$ on $A_{n}$ is said to be compatible with $A_{n}$ if
$y\preceq z\Rightarrow x*_{n}y\preceq x*_{n}z$. If $n>0$ and $\preceq$ is a linear ordering on $A_{n}$ compatible with
$A_{n}$, then define a linear ordering $\preceq'$ on $A_{n-1}$ by letting $x\preceq^{'}y$ iff $x+2^{n-1}\preceq x+2^{n-1}$.

Let $P(n)$ denote the statement that for every linear ordering $\preceq^{\sharp}$ on $A_{n-1}$, there is a linear ordering $\preceq$ on
$A_{n}$ with $\preceq'=\preceq^{\sharp}$. Then $P(n)$ is true for $n\leq 17$, but $P(18)$ is false.

The final matrix

If $A$ is a set, then let $(A^{\leq 2^{n}})^{+}$ denote the collection of all non-empty strings from the alphabet $A$ with length at most $n$. Then $(A^{\leq 2^{n}})^{+}$ can be endowed with a unique operation $*_{n}$ that satisfies the following rules:

$\mathbf{x}*_{n}(\mathbf{y}*_{n}\mathbf{z})=(\mathbf{x}*_{n}\mathbf{y})*_{n}(\mathbf{x}*_{n}\mathbf{z})$ for $\mathbf{x},\mathbf{y},\mathbf{z}\in(A^{\leq 2^{n}})^{+}$.

$\mathbf{x}*_{n}\mathbf{y}=\mathbf{y}$ whenever $|\mathbf{x}|=2^{n}$, and

$\mathbf{x}*_{n}a=\mathbf{x}a$ whenever $|\mathbf{x}|<2^{n},a\in A$.

Let $FM_{n}^{-}:\{1,\dots,2^{n}\}\times\{1,\dots,2^{n}\}\rightarrow\mathbb{Z}$ be the mapping where if $r$ is the least natural number where $|a_{-1}\dots a_{-i}*_{n}a_{1}\dots a_{r}|=2^{n}$, then
$$a_{-1}\dots a_{-i}*_{n}a_{1}\dots a_{r}=a_{FM_{n}^{-}(i,1)}\dots a_{FM_{n}^{-}(i,2^{n})}.$$ We shall call $FM_{n}^{-}$ the final matrix.

The motivation behind the data $FM_{n}^{-}$ is that all the combinatorial complexity about the operation $*_{n}$ is coded inside the function $FM_{n}^{-}$.

For $n<7$, if $FM_{n}^{-}(x,y)>0$ and $y\leq 2^{n-1}$, then
$FM_{n}^{-}(x,y+2^{n-1})>0$ as well. However,
$FM_{7}^{+}(8,16)=1,FM_{7}^{+}(8,16+2^{6})=-4$.

If $n<9$ and $FM_{n}^{-}(i,j)>0,FM_{n}^{-}(i,j+1)>0$, then $FM_{n}^{-}(i,j+1)=FM_{n}^{-}(i,j)+1$. But

$FM_{9}^{-}(8,175)=1,FM_{9}^{-}(8,176)=3$,

$FM_{9}^{-}(8,191)=1,FM_{9}^{-}(8,192)=4,$ and

$FM_{9}^{-}(8,143)=1,FM_{9}^{-}(8,144)=1$.

If $n<11$ and $FM_{n}^{-}(i,j)>0,FM_{n}^{-}(i,j+1)>0$, then $FM_{n}^{-}(i,j+1)\geq FM_{n}^{-}(i,j)-1,$ but
$FM_{11}^{-}(8,255)=3,FM_{11}^{-}(8,256)=1$.

If $n<7,FM_{n}^{-}(x,y)>0$, then
$\gcd(2^{n},x,y)\leq\gcd(2^{n},FM_{n}^{-}(x,y))$, but
$FM_{7}^{-}(8,16)=1$.

The following counterexamples form $FM_{n}^{-}$ have only been established through large cardinal hypotheses.

Let $(x)_{a}$ be the unique natural number where
$x=(x)_{a}\mod a$ and $1\leq (x)_{a}\leq a$.

Let $ST_{0}=\{(1,1)\}$. Let $(x,y)\in ST_{n+1}$ if and only if
$((x)_{2^{n}},(y)_{2^{n}})\in ST_{n}$ and either $x\leq 2^{n}$ or $y>2^{n}$.

$\textbf{Hypothesis:}$ $(SE_{n})$

Suppose that $i,j\in\{0,\ldots,2^{n}-1\}$. Then

$FM_{2^{n}}^{-}(2^{i},2^{j})>0$ if and only if $i$ is even, $j$ is odd, and $(i+1,j)\in \mathrm{ST}_{n}$.

If $FM_{2^{n}}^{-}(2^{i},2^{j})>0$, then $FM_{2^{n}}^{-}(2^{i},2^{j})=2^{i}.$

Suppose that $FM_{2^{n}}^{-}(2^{i},2^{j})<0$. Then $FM_{2^{n}}^{-}(2^{i},2^{j})=-2^{k}$ for some $k$.
Furthermore,

i. if $j$ is odd, then $0<k\leq i$, and
$$FM_{2^{n}}^{-}(2^{i'},2^{j})=-2^{k}$$
for $k\leq i'\leq i$, and
$$FM_{2^{n}}^{-}(2^{k-1},2^{j})=2^{k-1}.$$

ii. if $j$ is even and $FM_{2^{n}}^{-}(2^{i},2^{j+1})>0$, then
$$FM_{2^{n}}^{-}(2^{i},2^{j})=-FM_{2^{n}}^{-}(2^{i},2^{j+1}).$$

iii. if $j$ is even and $FM_{2^{n}}^{-}(2^{i},2^{j+1})<0$, then
$$2\cdot FM_{2^{n}}^{-}(2^{i},2^{j})=FM_{2^{n}}^{-}(2^{i},2^{j+1}).$$
.
The following table gives a picture for the hypothesis $SE_{4}$.
The $(i,j)$-th entry in the following table is set to be $+\log_{2}(FM_{16}^{+}(2^{i},2^{j})$ whenever $FM_{16}^{+}(2^{i},2^{j})>0$ and it is set to be $-\log_{2}(-FM_{16}^{+}(2^{i},2^{j})$ whenever $FM_{16}^{+}(2^{i},2^{j})<0$.

$\begin{array}{r|rrrrrrrrrrrrrrrrr}
&0&1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16\\
\hline
0&-0&+0&-0&+0&-0&+0&-0&+0&-0&+0&-0&+0&-0&+0&-0&+0&4\\
1&-0&-1&-0&-1&-0&-1&-0&-1&-0&-1&-0&-1&-0&-1&-0&-1&4\\
2&-0&-1&-2&2&-0&-1&-2&2&-0&-1&-2&2&-0&-1&-2&2&4\\
3&-0&-1&-2&-3&-0&-1&-2&-3&-0&-1&-2&-3&-0&-1&-2&-3&4\\
4&-0&-1&-2&-3&-4&4&-4&4&-0&-1&-2&-3&-4&4&-4&4&7\\
5&-0&-1&-2&-3&-4&-5&-4&-5&-0&-1&-2&-3&-4&-5&-4&-5&8\\
6&-0&-1&-2&-3&-4&-5&-6&6&-0&-1&-2&-3&-4&-5&-6&6&8\\
7&-0&-1&-2&-3&-4&-5&-6&-7&-0&-1&-2&-3&-4&-5&-6&-7&8\\
8&-0&-1&-2&-3&-4&-5&-6&-7&-8&8&-8&8&-8&8&-8&8&11\\
9&-0&-1&-2&-3&-4&-5&-6&-7&-8&-9&-8&-9&-8&-9&-8&-9&12\\
10&-0&-1&-2&-3&-4&-5&-6&-7&-8&-9&-10&10&-8&-9&-10&10&12\\
11&-0&-1&-2&-3&-4&-5&-6&-7&-8&-9&-10&-11&-8&-9&-10&-11&12\\
12&-0&-1&-2&-3&-4&-5&-6&-7&-8&-9&-10&-11&-12&12&-12&12&14\\
13&-0&-1&-2&-3&-4&-5&-6&-7&-8&-9&-10&-11&-12&-13&-12&-13&14\\
14&-0&-1&-2&-3&-4&-5&-6&-7&-8&-9&-10&-11&-12&-13&-14&14&15\\
15&-0&-1&-2&-3&-4&-5&-6&-7&-8&-9&-10&-11&-12&-13&-14&-15&15\\
16&+0&1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16
\end{array}$

Theorem: If there exists a rank-into-rank cardinal, then there is a natural number $n$ where $SE_{n}$ is false.

I can currently calculate an arbitrary entry in the table $FM_{n}^{-}$ whenever $n\leq 48$. With modern computational technology, it should be possible to calculate arbitrary entries in $FM_{n}^{-}$ for $n\leq 96$. I have experimentally verified $SE_{n}$ for $n\leq 5$, and it is feasible to verify $SE_{6}$ as soon as someone gets the time to do so. I suspect that the least $n$ where $SE_{n}$ fails is exceedingly large. Perhaps, $SE_{n}$ is larger than the running time of any computer program of length at most $10^{100}$ characters which can be proven to terminate in "ZFC+There is a supercompact cardinal" with a proof of length at most $10^{1000}$.

$\textbf{Hypothesis:}$ ($EP_{n}$) Suppose that $r$ is a natural number and $x<2^{2^{r}}-1$. Then $FM_{n}^{-}(x,2^{y})=FM_{n}^{-}(x,2^{z})$ whenever $y=z\mod 2^{r}$ and $\max(y,z)<j\cdot 2^{r}<n$ for some $j$.

$\textbf{Theorem:}$ If there exists a rank-into-rank cardinal, then there is a natural number $n$ where $EP_{n}$ is false.

No known counterexample to $EP_{n}$ is known to exist in ZFC.

Laver-like counterexamples

Define the Fibonacci terms $t_{n}(x,y)$ for all $n\geq 1$ by letting $t_{1}(x,y)=y,t_{2}(x,y)=x,t_{n+2}(x,y)=t_{n+1}(x,y)*t_{n}(x,y)$.

We say that an algebra $(X,*,1)$ that satisfies the identities $x*1=1,1*x=x,x*(y*z)=(x*y)*(x*z)$ is permutative if for all $x,y\in X$, there is some $n$ where $t_{n}(x,y)=1$. Define $x^{0}*y=y,x^{n+1}*y=x*(x^{n}*y)$ for $n\geq 0$. Define $\mathrm{crit}(x)\leq\mathrm{crit}(y)$ if and only if there Is some $n$ with $x^{n}*y=1$. Then $\{\mathrm{crit}(x)\mid x\in X\}$ is always linearly ordered set. There is a unique operation $\circ$ on $X$ where $(X,\circ,1)$ is a monoid and $x\circ y=(x*y)\circ x$ for all $x,y\in X$.

$\mathbf{Theorem:}$ Suppose that all large cardinals exist. Suppose that $s,t,u$ are terms in the language $*,\circ$ with $s,t$ unary and $u$ binary. Suppose furthermore that
$A_{1}\models\mathrm{crit}(s(x))=\mathrm{crit}(t(x))=\mathrm{crit}(x)$. Then there is a finite permutative algebra $(X,*,1)$ along with $x,y\in X$

$s(x)*x=s(y)*y$

$u(x,y)\neq 1$

$\mathrm{crit}(x)=\mathrm{crit}(y)$

$\mathrm{crit}(r*r*s)\leq\mathrm{crit}(r*s)$ for all $r,s\in X$.

In the above theorem, for some very simple terms $u,s,t$, I have not been able to compute examples of such algebras.

Besides these counterexamples which are known to exist, there are many different kinds of Laver-like algebras which I conjecture to exist but for which I have no proof even if I assume large cardinals exist and for which I conjecture the first example of such algebra is extremely large.

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