Can we topple the surreal house of cards?

Question

In Winning Ways for Your Mathematical Plays , Volume 2 , there is a section "The House of Cards" which contains figure 18 "Plumtrees in the Uplands" which shows the game trees for (in respective order) $\text{off, oof, tiny, ace, duece & trey}$ .

We know that $\text{tiny} < \text{ace} < \text{deuce}$ ... & $\text{off} < \text{tiny}$ makes sense to me. But what about $\text{oof}$ ?

$$\text{oof}=\{0|\text{off}\}$$

The book says "the thermograph of $\text{oof}$ [..] shows it to be less than all positive numbers, but confused with all the others".

It seems like $\text{oof}$ is a positive value ( based on the left set being $0$ ), but the game tree seems to imply it is smaller than $\text{tiny}$ & this seems to be confirmed with the thermograph showing it to be less than all positive numbers.

Is $\text{oof}$ a positive game whose value is < $\text{tiny}$ ?

If so, this would topple ( Contradict ) the idea that $\text{tiny}$ is the smallest game value.

My suspicion is that $\text{tiny}$ is the smallest game value for some class of games (I'm not well versed in game classes) & $\text{oof}$ is the smallest in some other (loopy?) class.

Answer 1

OVER-VIEW :

We can see that $\text{oof}$ is less than all Positive Values , but it itself is not Positive.
The smallest Positive Value is $\text{tiny}$ , which is strictly/necessarily less than $\text{ace}$.
The House of Cards has not toppled !

I will use that Book itself to cover the various Points :

Page 353 :

$\text{tiny} = +_{on}$
tiny is the smallest positive value there is!
Of course , $\text{miny} = -_{on} = -\text{tiny}$ is the least negative game there is.

Page 357 :

[ Images have my high-lighting ]

We see here that $\text{oof}$ is very close to $\text{off}$ neither of which is Positive , while $\text{tiny}$ & upwards are Positive.
In other words , $\text{oof}$ is less than all Positive Values , though it itself is not Positive.

Page 360 :

Here we see that $\text{tiny}$ & $\text{miny}$ fall outside the fuzzy range , where the Values are neither Positive nor Negative.
It is in this range that we have $\text{oof}$ , which is less than all Positive Values.

Page 454 :

The Connection between $\text{off}$ & $\text{oof}$ & $\text{tiny}$ is given in the glossary :

SUMMARY :

We can see that $\text{oof}$ is less than all Positive Values , but it itself is not Positive.
The smallest Positive Value is $\text{tiny}$ , which is strictly/necessarily less than $\text{ace}$.
The House of Cards has not toppled !

Answer 2

less than all positive numbers, but confused with all the others

This means that if $x>0$ is a number then $\mathrm{oof}<x$, and that if $x\le0$ is a number then $\mathrm{oof}\parallel x$. In particular, this is saying that $\mathrm{oof}\parallel0$ so, no, $\mathrm{oof}$ is not positive.

It seems like $\mathrm{oof}$ is a positive value ( based on the left set being $0$ )

Even outside the context of loopy games, that can easily fail as soon as the game is not a number. For example, $*\cong\{0\mid0\}$ is not positive.