$c(G)+c(-H)\in\mathcal{L}\cup\mathcal{P}$. With that verified, Lemma 45
ensures that $c(G)+c(-H)\geqslant 0\iff c(G)-c(H)\geqslant 0\iff c(G)\geqslant
c(H)$. Suppose that Right moves to some $c(G^{R})+c(-H)$. Considering the game
$G^{R}-H\in\mathbbm{N}\mathbbm{p}^{\infty}\\!$, Left must have a winning move,
which can be either some $G^{RL}-H\geqslant 0$ or some $G^{R}-H^{R}\geqslant
0$. Concerning $\mathbbm{N}\mathbbm{p}$, by induction, in the first case, we
have $c(G^{RL})+c(-H)\geqslant 0$, and in the second case,
$c(G^{R})+c(-H^{R})\geqslant 0$. In other words, Left has a winning move in
$c(G^{R})+c(-H)$. If Right moves to some $c(G)+c(-H^{L})$, the inductive
argument works exactly the same way, and Left has a winning response. We can
conclude that $c(G)\geqslant c(H)$.

The proof for the second implication in the theorem can be carried out by
induction in a completely analogous manner. ∎

##  7 Classification of affine games

It is well-known that the games of the classical structure
$\mathbbm{N}\mathbbm{p}$ can be classified as cold, tepid, or hot. Moreover,
this classification is based on the concept of number and the related Number
Avoidance Theorem.

When a game is a number (cold), both players have a negative incentive,
meaning they are in _zugzwang_ 888Zugzwang (from German “compulsion to move”)
refers to a situation where a player is placed at a disadvantage because they
must make a move. in the sense that making a move leads to a “loss” in terms
of guaranteed moves (if there are any moves available at all). For example,
$\frac{1}{2}=\\{0\,|\,1\\}$; if Left plays, she reduces $\frac{1}{2}$ to $0$,
and if Right plays, he increases $\frac{1}{2}$ to $1$. In other words, both
players prefer the opponent to play rather than playing themselves. When a
game is hot, the opposite occurs. There is urgency in the sense that making a
move leads to a “gain” of guaranteed moves. An example of a hot game is $\pm
1=\\{1\,|\,-1\\}$. When a game is tepid, a move brings neither gains nor
losses in terms of guaranteed moves. Playing may be important for the sake of
making the last move _per se_ , but not in terms of gaining guaranteed moves
by having the right to move. An example of a tepid game is $\\{1\,|\,1\\}$;
Left already got one guaranteed move regardless of whether she moves or not.
Thus, intuitively, the Number Avoidance Theorem establishes that a player
should not play on numbers in order to avoid losing what is already
guaranteed; playing in hot games or tepid games avoids such a loss.

In this section, we extend these concepts, considering that in
$\mathbbm{N}\mathbbm{p}^{\infty}\\!$, infinities come into play. For this
purpose, it is important to start with the Number Avoidance Theorem, which
still holds in affine normal play.

######  Theorem 50 (Number Avoidance Theorem).

Let $G,x\in\mathbbm{N}\mathbbm{p}^{\infty}\\!$, and suppose $x$ is a number
and $G$ is not. If Left (resp. Right) has a winning move on $G+x$, then she
has a winning move of the form $G^{L}+x$ (resp. $G^{R}+x$).

###### Proof.

We may assume that $x$ is in the classical reduced form via Conway embedding,
since the outcomes of the considered sums do not depend on its form. If
$G=\infty$ or $G=\overline{\infty}$ there are no moves and there is nothing to
prove. Now suppose that $G+x^{L}$ is a winning move for Left. According to
Theorem 14, we have $G+x^{L}\geqslant 0$. But $G$ is not equal to a number,
thus $G\neq-x^{L}$ ($x^{L}$ is invertible), so in fact $G+x^{L}>0$ is
mandatory. Again by Theorem 14, Left has a winning move in $G+x^{L}$. By
induction, we may assume that it is some $G^{L}+x^{L}\geqslant 0$. Since $x$
is in the classical reduced form, we have $x^{L}<x$ and
$G^{L}+x>G^{L}+x^{L}\geqslant 0$. Hence, $G^{L}+x$ is a winning move for Left
in $G+x$. The proof from the Right’s perspective is analogous. ∎

The concept of _stop_ in classical theory is based on the assumption that two
players move in a component, and, considering the Number Avoidance Theorem,
stop playing on that component as soon as a number has been reached. Left
attempts to have this stopping number be as large as possible, and Right wants
it to be as small as possible. Our next step consists of defining this concept
for $\mathbbm{N}\mathbbm{p}^{\infty}\\!$, in order to classify affine games
similarly to that done in classical theory, with just a few additions due to
infinities. An obvious difference when comparing with the classical structure
is that the stopping point may not be a number, but rather $\infty$ or
$\overline{\infty}$. This is because there is the possibility for one of the
players to force a terminating move.

######  Definition 51 (Stops).

Left and Right _stops_ of $G$ are recursively defined by

$LS(G)=\begin{cases}\infty\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\,\,\textrm{
if }G=\infty\\\
\overline{\infty}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\,\,\textrm{
if }G=\overline{\infty}\\\
x\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\textrm{ if
}G=x\textrm{ is a number}\\\ \max\\{RS(G^{L})\,|\,G^{L}\in
G^{\mathcal{L}}\\}\quad\textrm{ otherwise}\end{cases}$
$RS(G)=\begin{cases}\infty\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\,\,\textrm{
if }G=\infty\\\
\overline{\infty}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\,\,\textrm{
if }G=\overline{\infty}\\\
x\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\textrm{ if
}G=x\textrm{ is a number}\\\ \min\\{LS(G^{R})\,|\,G^{R}\in
G^{\mathcal{R}}\\}\quad\textrm{ otherwise.}\end{cases}$

######  Definition 52.

If $G$ is an element of $\mathbbm{N}\mathbbm{p}^{\infty}\\!$, then

  * 1.

$G$ is _cold_ if it is a number;

  * 2.

$G$ is _tepid_ if the stops are finite and $LS(G)=RS(G)$, but $G$ is not a
number;

  * 3.

$G$ is _hot_ if $LS(G)>RS(G)$;

  * 4.

$G$ is _bubbling_ if both stops are $\infty$ or both stops are
$\overline{\infty}$.

######  Example 53.

The following list has an example of each of the four types of affine games.

  * 1.

$G=\\{\\{\infty\,|\,-1*\\}\,|\,0\\}$ is a number since $\\{\infty\,|\,-1*\\}$
is reversible through $-1*$, and $G$ reduces to $-\frac{1}{2}=\\{-1\,|\,0\\}$.

  * 2.

$G=\\{\\{\infty\,|\,0\\}\,|\,0\\}$ is tepid since both stops are finite and
$LS(G)=RS(G)=0$. It is a reduced form of a pressing game, as it is not
equivalent to any Conway form.

  * 3.

$G=\\{\\{\infty\,|\,1\\}\,|\,0\\}$ is hot since $LS(G)=1$, $RS(G)=0$, and
$LS(G)>RS(G)$. The game $G$ is a pressing game, as it is not equivalent to any
Conway form.

  * 4.

$G=\\{\infty\,|\,\infty\\}$ is bubbling since $LS(G)=RS(G)=\infty$. It is
inevitable that $G$ ends with Left delivering a checkmate, regardless of
whether she plays first or not. In this sense, $G$ has already reached an
irreversible boiling state, which explains the choice of the word “bubbling”.

The class of _mate games_ is a subclass of bubbling games. Its formalization
is needed given the frequency with which these situations occur. In an
$n$-Leftmate ($n$-Rightmate), Left (Right) is guaranteed the possibility of
checkmating in $n$ moves, without the opponent being able to defend. However,
the checkmate cannot occur more quickly.

######  Definition 54 (Mate Games).

Let $n$ be a nonnegative integer. An $n$-_Leftmate_ $\infty_{n}$ is
recursively defined by

$\infty_{n}=\begin{cases}\infty,\quad\quad\quad\quad\quad\textrm{ if }n=0;\\\
\\{\infty_{n-1}\,|\,\infty\\},\quad\textrm{ if }n>0.\end{cases}$

The definition of $n$-_Rightmate_ , denoted by $\overline{\infty}_{n}$, is
symmetric.

######  Observation 55.

The game $\\{\infty_{3}\,|\,\infty_{8}\\}$ is a bubbling game, but it is not a
$n$-Leftmate. The reason for this is that the inevitable checkmate can occur
at different speeds, depending on whether it is Left or Right who starts. If
Right starts, he can delay his fateful destiny in the component a little, and
this can be crucial in certain disjunctive sums.

Nomenclature: When a game is tepid and the stops are equal to zero, it is an
_infinitesimal_. When a game is tepid and the stops are not zero, it is a
_translation of an infinitesimal_. When a game is hot and at least one of the
stops is infinite, it is _scalding_.

######  Observation 56.

There are infinitesimals that are not Conway games, such as
$\\{\\{\infty\,|\,0\\}\,|\,0\\}$ already used in Example 53 and materialized
in Figure 9. On the other hand, there are no scalding games that are Conway
games. For example, $\\{\infty_{3}\,|\,0\\}$ is a scalding game. In this game,
Right is not yet condemned to being checkmated by the opponent as long as he
retains the right to make the move. The temperature is infinite, but the
boiling state of a bubbling game has not been reached yet, as in a scalding
game neither player is definitively condemned to be checkmated.

Now, let us move on to an important concept regarding affine normal play. If
we consider the classical structure, all reductions are based on the
fundamental idea of _moves that may never be made_ , given their uselessness.
A dominated option _never_ needs to be used. Regarding a reversible option,
since it is only chosen with the intention of continuing to play locally, all
moves that are not related to this local continuation are _never_ used. For
the affine structure $\mathbbm{N}\mathbbm{p}^{\infty}\\!$, something entirely
new happens. There are checks that must _always_ be used. In other words, as
long as the player is not mortally threatened in another component, the first
thing they must do is deliver a check in a specific component. In affine
normal play, in addition to “never”, “always” also comes into play. Thus,
there are forms that, although they cannot be reduced, already contain a
certain type of “intrinsic reduction”, since they will be the target of a move
at the first opportunity. In a way, these components are ephemeral. The
following definition formalizes this idea, capturing the concept of
_hammerzug_ 999Hammerzug (from German “move with a hammer”) is a move that,
when not under lethal threat, a player should always make, as it can only help
without having any negative consequences.. When a hammerzug is a component of
a disjunctive sum, a player must deliver a check in that component as long as
they are not threatened.

######  Definition 57 (Hammerzug).

A game $G\in\mathbbm{N}\mathbbm{p}^{\infty}\\!$ is a _Left-hammerzug_ if there
exists a Left-check $G^{L}=\\{\infty\,|\,G^{L\mathcal{R}}\\}$ such that
$G^{LR}\geqslant G$ for all $G^{LR}\in G^{L\mathcal{R}}$. The definition of
Right-hammerzug is similar. A game $G\in\mathbbm{N}\mathbbm{p}^{\infty}\\!$ is
a _hammerzug_ if it is both a Left-hammerzug and a Right-hammerzug.

######  Example 58.

The game $G=\\{\\{\infty\,|\,1\\}\,|\,0\\}=\\{\circlearrowright^{1}\,|\,0\\}$
is a Left-hammerzug since $1>G$. The game
$G=\\{\\{\infty\,|\,1\\}\,|\,\\{-1\,|\,\overline{\infty}\\}\\}=\\{\circlearrowright^{1}\,|\,\circlearrowleft^{-1}\\}$
is a hammerzug since $1>G>-1$.

######  Theorem 59 (Extreme Urgency).

If $G\in\mathbbm{N}\mathbbm{p}^{\infty}\\!$ is a Left-hammerzug, then, for
every Right-quiet game $X$, $o^{L}(G+X)=o^{R}(G^{L}+X)$ where $G^{L}$ is the
Left-check specified in Definition 57. There is a dual result in terms of
Right-hammerzugs.

###### Proof.

We prove that $o^{L}(G+X)\geqslant o^{R}(G^{L}+X)$ and
$o^{R}(G^{L}+X)\geqslant o^{L}(G+X)$. The first inequality is evident, since,
in a worst-case scenario, Left can move to $G^{L}+X$ in $G+X$. Regarding the
second inequality, we have $o^{R}(G^{L}+X)=o^{L}(G^{LR}+X)$ where $G^{LR}$ is
the best defense for Right against the Left-check. According to Definition 57,
$o^{L}(G^{LR}+X)\geqslant o^{L}(G+X)$, thus $o^{R}(G^{L}+X)\geqslant
o^{L}(G+X)$. ∎

Returning to the classical structure $\mathbbm{N}\mathbbm{p}$, there are
classes of infinitesimals of great relevance. The powers of up
$\uparrow\,=\\{0\,|\,*\\}$, $\uparrow^{2}\,=\\{0\,|\,\downarrow+*\\}$,
$\uparrow^{3}\,=\\{0\,|\,\downarrow+\downarrow^{2}+*\\}$, …, are
infinitesimals with respect to the previous ones. This means that
$n.\\!\\!\uparrow^{2}<\uparrow$, regardless of the number of copies $n$. The
same applies to $n.\\!\\!\uparrow^{3}<\uparrow^{2}$,
$n.\\!\\!\uparrow^{4}<\uparrow^{3}$, and so on. The tinies and minies exhibit
a similar behavior. The game tiny, denoted by
${\tikzpicture\clip(4.83,0.365)rectangle(5.92,1.452);[linewidth=2.pt,fill=black,fillopacity=1.0](5.212307692307691,0.3709401709401755)--(5.227098504388038,0.4109165222391566)--(5.244356740184811,0.4626804457451724)--(5.260695509797721,0.5186561018862608)--(5.273760554573898,0.5705732327687449)--(5.2837430321406975,0.616781456772672)--(5.293211594495013,0.6690752465306267)--(5.30077046042753,0.7213397005973032)--(5.305581482521969,0.7643279442575954)--(5.308600200833782,0.7997323673195916)--(5.310920237599428,0.8381937289079981)--(5.312211337771162,0.8750972065724719)--(5.312592592592592,0.9088319088319139)--(5.312211337771162,0.9425666110913559)--(5.310920237599428,0.9794700887558296)--(5.308600200833782,1.017931450344236)--(5.305581482521969,1.0533358734062324)--(5.30077046042753,1.0963241170665246)--(5.293211594495013,1.1485885711332011)--(5.2837430321406975,1.2008823608911556)--(5.273760554573898,1.2470905848950828)--(5.260695509797721,1.299007715777567)--(5.244356740184811,1.3549833719186553)--(5.227098504388038,1.4067472954246711)--(5.212307692307691,1.4467236467236524)--(5.531396011396009,1.4467236467236524)--(5.516605199315662,1.4067472954246711)--(5.499346963518889,1.3549833719186553)--(5.483008193905979,1.299007715777567)--(5.469943149129802,1.2470905848950828)--(5.459960671563002,1.2008823608911556)--(5.450492109208687,1.1485885711332011)--(5.44293324327617,1.0963241170665246)--(5.438122221181731,1.0533358734062324)--(5.435103502869918,1.017931450344236)--(5.432783466104271,0.9794700887558296)--(5.431492365932538,0.9425666110913559)--(5.4311111111111074,0.9088319088319139)--(5.431492365932538,0.8750972065724719)--(5.432783466104271,0.8381937289079981)--(5.435103502869918,0.7997323673195916)--(5.438122221181731,0.7643279442575954)--(5.44293324327617,0.7213397005973032)--(5.450492109208687,0.6690752465306267)--(5.459960671563002,0.616781456772672)--(5.469943149129802,0.5705732327687449)--(5.483008193905979,0.5186561018862608)--(5.499346963518889,0.4626804457451724)--(5.516605199315662,0.4109165222391566)--(5.531396011396009,0.3709401709401755)--cycle;[linewidth=2.pt,fill=black,fillopacity=1.0](5.909743589743589,0.7492877492877543)--(5.8697672384446085,0.7640785613681009)--(5.818003314938593,0.7813367971648741)--(5.7620276587975034,0.7976755667777837)--(5.7101105279150195,0.8107406115539613)--(5.663902303911093,0.8207230891207606)--(5.611608514153138,0.8301916514750758)--(5.559344060086461,0.837750517407593)--(5.516355816426169,0.8425615395020323)--(5.480951393364173,0.8455802578138449)--(5.442490031775766,0.8479002945794916)--(5.405586554111292,0.8491913947512248)--(5.371851851851851,0.8495726495726554)--(5.338117149592409,0.8491913947512248)--(5.301213671927935,0.8479002945794916)--(5.262752310339529,0.8455802578138449)--(5.227347887277532,0.8425615395020323)--(5.18435964361724,0.837750517407593)--(5.132095189550563,0.8301916514750758)--(5.0798013997926095,0.8207230891207606)--(5.033593175788682,0.8107406115539613)--(4.981676044906198,0.7976755667777837)--(4.92570038876511,0.7813367971648741)--(4.873936465259094,0.7640785613681009)--(4.833960113960112,0.7492877492877543)--(4.833960113960112,1.0683760683760717)--(4.873936465259094,1.053585256295725)--(4.92570038876511,1.0363270204989519)--(4.981676044906198,1.0199882508860423)--(5.033593175788682,1.0069232061098647)--(5.0798013997926095,0.9969407285430654)--(5.132095189550563,0.9874721661887502)--(5.18435964361724,0.979913300256233)--(5.227347887277532,0.9751022781617937)--(5.262752310339529,0.9720835598499811)--(5.301213671927935,0.9697635230843344)--(5.338117149592409,0.9684724229126012)--(5.371851851851851,0.9680911680911706)--(5.405586554111292,0.9684724229126012)--(5.442490031775766,0.9697635230843344)--(5.480951393364173,0.9720835598499811)--(5.516355816426169,0.9751022781617937)--(5.559344060086461,0.979913300256233)--(5.611608514153138,0.9874721661887502)--(5.663902303911093,0.9969407285430654)--(5.7101105279150195,1.0069232061098647)--(5.7620276587975034,1.0199882508860423)--(5.818003314938593,1.0363270204989519)--(5.8697672384446085,1.053585256295725)--(5.909743589743589,1.0683760683760717)--cycle;}_{1}$,
is the positive game $\\{0\,|\,\\{0\,|\,-1\\}\\}$. Similarly, the game miny,
denoted by
${\tikzpicture\clip(4.83,0.365)rectangle(5.92,1.452);[linewidth=2.pt,fill=black,fillopacity=1.0](5.909743589743589,0.7492877492877543)--(5.8697672384446085,0.7640785613681009)--(5.818003314938593,0.7813367971648741)--(5.7620276587975034,0.7976755667777837)--(5.7101105279150195,0.8107406115539613)--(5.663902303911093,0.8207230891207606)--(5.611608514153138,0.8301916514750758)--(5.559344060086461,0.837750517407593)--(5.516355816426169,0.8425615395020323)--(5.480951393364173,0.8455802578138449)--(5.442490031775766,0.8479002945794916)--(5.405586554111292,0.8491913947512248)--(5.371851851851851,0.8495726495726554)--(5.338117149592409,0.8491913947512248)--(5.301213671927935,0.8479002945794916)--(5.262752310339529,0.8455802578138449)--(5.227347887277532,0.8425615395020323)--(5.18435964361724,0.837750517407593)--(5.132095189550563,0.8301916514750758)--(5.0798013997926095,0.8207230891207606)--(5.033593175788682,0.8107406115539613)--(4.981676044906198,0.7976755667777837)--(4.92570038876511,0.7813367971648741)--(4.873936465259094,0.7640785613681009)--(4.833960113960112,0.7492877492877543)--(4.833960113960112,1.0683760683760717)--(4.873936465259094,1.053585256295725)--(4.92570038876511,1.0363270204989519)--(4.981676044906198,1.0199882508860423)--(5.033593175788682,1.0069232061098647)--(5.0798013997926095,0.9969407285430654)--(5.132095189550563,0.9874721661887502)--(5.18435964361724,0.979913300256233)--(5.227347887277532,0.9751022781617937)--(5.262752310339529,0.9720835598499811)--(5.301213671927935,0.9697635230843344)--(5.338117149592409,0.9684724229126012)--(5.371851851851851,0.9680911680911706)--(5.405586554111292,0.9684724229126012)--(5.442490031775766,0.9697635230843344)--(5.480951393364173,0.9720835598499811)--(5.516355816426169,0.9751022781617937)--(5.559344060086461,0.979913300256233)--(5.611608514153138,0.9874721661887502)--(5.663902303911093,0.9969407285430654)--(5.7101105279150195,1.0069232061098647)--(5.7620276587975034,1.0199882508860423)--(5.818003314938593,1.0363270204989519)--(5.8697672384446085,1.053585256295725)--(5.909743589743589,1.0683760683760717)--cycle;}_{1}$,
is the negative game $\\{\\{1\,|\,0\\}\,|\,0\\}$. In general, given a positive
integer $n$, we have
${\tikzpicture\clip(4.83,0.365)rectangle(5.92,1.452);[linewidth=2.pt,fill=black,fillopacity=1.0](5.212307692307691,0.3709401709401755)--(5.227098504388038,0.4109165222391566)--(5.244356740184811,0.4626804457451724)--(5.260695509797721,0.5186561018862608)--(5.273760554573898,0.5705732327687449)--(5.2837430321406975,0.616781456772672)--(5.293211594495013,0.6690752465306267)--(5.30077046042753,0.7213397005973032)--(5.305581482521969,0.7643279442575954)--(5.308600200833782,0.7997323673195916)--(5.310920237599428,0.8381937289079981)--(5.312211337771162,0.8750972065724719)--(5.312592592592592,0.9088319088319139)--(5.312211337771162,0.9425666110913559)--(5.310920237599428,0.9794700887558296)--(5.308600200833782,1.017931450344236)--(5.305581482521969,1.0533358734062324)--(5.30077046042753,1.0963241170665246)--(5.293211594495013,1.1485885711332011)--(5.2837430321406975,1.2008823608911556)--(5.273760554573898,1.2470905848950828)--(5.260695509797721,1.299007715777567)--(5.244356740184811,1.3549833719186553)--(5.227098504388038,1.4067472954246711)--(5.212307692307691,1.4467236467236524)--(5.531396011396009,1.4467236467236524)--(5.516605199315662,1.4067472954246711)--(5.499346963518889,1.3549833719186553)--(5.483008193905979,1.299007715777567)--(5.469943149129802,1.2470905848950828)--(5.459960671563002,1.2008823608911556)--(5.450492109208687,1.1485885711332011)--(5.44293324327617,1.0963241170665246)--(5.438122221181731,1.0533358734062324)--(5.435103502869918,1.017931450344236)--(5.432783466104271,0.9794700887558296)--(5.431492365932538,0.9425666110913559)--(5.4311111111111074,0.9088319088319139)--(5.431492365932538,0.8750972065724719)--(5.432783466104271,0.8381937289079981)--(5.435103502869918,0.7997323673195916)--(5.438122221181731,0.7643279442575954)--(5.44293324327617,0.7213397005973032)--(5.450492109208687,0.6690752465306267)--(5.459960671563002,0.616781456772672)--(5.469943149129802,0.5705732327687449)--(5.483008193905979,0.5186561018862608)--(5.499346963518889,0.4626804457451724)--(5.516605199315662,0.4109165222391566)--(5.531396011396009,0.3709401709401755)--cycle;[linewidth=2.pt,fill=black,fillopacity=1.0](5.909743589743589,0.7492877492877543)--(5.8697672384446085,0.7640785613681009)--(5.818003314938593,0.7813367971648741)--(5.7620276587975034,0.7976755667777837)--(5.7101105279150195,0.8107406115539613)--(5.663902303911093,0.8207230891207606)--(5.611608514153138,0.8301916514750758)--(5.559344060086461,0.837750517407593)--(5.516355816426169,0.8425615395020323)--(5.480951393364173,0.8455802578138449)--(5.442490031775766,0.8479002945794916)--(5.405586554111292,0.8491913947512248)--(5.371851851851851,0.8495726495726554)--(5.338117149592409,0.8491913947512248)--(5.301213671927935,0.8479002945794916)--(5.262752310339529,0.8455802578138449)--(5.227347887277532,0.8425615395020323)--(5.18435964361724,0.837750517407593)--(5.132095189550563,0.8301916514750758)--(5.0798013997926095,0.8207230891207606)--(5.033593175788682,0.8107406115539613)--(4.981676044906198,0.7976755667777837)--(4.92570038876511,0.7813367971648741)--(4.873936465259094,0.7640785613681009)--(4.833960113960112,0.7492877492877543)--(4.833960113960112,1.0683760683760717)--(4.873936465259094,1.053585256295725)--(4.92570038876511,1.0363270204989519)--(4.981676044906198,1.0199882508860423)--(5.033593175788682,1.0069232061098647)--(5.0798013997926095,0.9969407285430654)--(5.132095189550563,0.9874721661887502)--(5.18435964361724,0.979913300256233)--(5.227347887277532,0.9751022781617937)--(5.262752310339529,0.9720835598499811)--(5.301213671927935,0.9697635230843344)--(5.338117149592409,0.9684724229126012)--(5.371851851851851,0.9680911680911706)--(5.405586554111292,0.9684724229126012)--(5.442490031775766,0.9697635230843344)--(5.480951393364173,0.9720835598499811)--(5.516355816426169,0.9751022781617937)--(5.559344060086461,0.979913300256233)--(5.611608514153138,0.9874721661887502)--(5.663902303911093,0.9969407285430654)--(5.7101105279150195,1.0069232061098647)--(5.7620276587975034,1.0199882508860423)--(5.818003314938593,1.0363270204989519)--(5.8697672384446085,1.053585256295725)--(5.909743589743589,1.0683760683760717)--cycle;}_{n}=\\{0\,|\,\\{0\,|\,-n\\}\\}$
and
${\tikzpicture\clip(4.83,0.365)rectangle(5.92,1.452);[linewidth=2.pt,fill=black,fillopacity=1.0](5.909743589743589,0.7492877492877543)--(5.8697672384446085,0.7640785613681009)--(5.818003314938593,0.7813367971648741)--(5.7620276587975034,0.7976755667777837)--(5.7101105279150195,0.8107406115539613)--(5.663902303911093,0.8207230891207606)--(5.611608514153138,0.8301916514750758)--(5.559344060086461,0.837750517407593)--(5.516355816426169,0.8425615395020323)--(5.480951393364173,0.8455802578138449)--(5.442490031775766,0.8479002945794916)--(5.405586554111292,0.8491913947512248)--(5.371851851851851,0.8495726495726554)--(5.338117149592409,0.8491913947512248)--(5.301213671927935,0.8479002945794916)--(5.262752310339529,0.8455802578138449)--(5.227347887277532,0.8425615395020323)--(5.18435964361724,0.837750517407593)--(5.132095189550563,0.8301916514750758)--(5.0798013997926095,0.8207230891207606)--(5.033593175788682,0.8107406115539613)--(4.981676044906198,0.7976755667777837)--(4.92570038876511,0.7813367971648741)--(4.873936465259094,0.7640785613681009)--(4.833960113960112,0.7492877492877543)--(4.833960113960112,1.0683760683760717)--(4.873936465259094,1.053585256295725)--(4.92570038876511,1.0363270204989519)--(4.981676044906198,1.0199882508860423)--(5.033593175788682,1.0069232061098647)--(5.0798013997926095,0.9969407285430654)--(5.132095189550563,0.9874721661887502)--(5.18435964361724,0.979913300256233)--(5.227347887277532,0.9751022781617937)--(5.262752310339529,0.9720835598499811)--(5.301213671927935,0.9697635230843344)--(5.338117149592409,0.9684724229126012)--(5.371851851851851,0.9680911680911706)--(5.405586554111292,0.9684724229126012)--(5.442490031775766,0.9697635230843344)--(5.480951393364173,0.9720835598499811)--(5.516355816426169,0.9751022781617937)--(5.559344060086461,0.979913300256233)--(5.611608514153138,0.9874721661887502)--(5.663902303911093,0.9969407285430654)--(5.7101105279150195,1.0069232061098647)--(5.7620276587975034,1.0199882508860423)--(5.818003314938593,1.0363270204989519)--(5.8697672384446085,1.053585256295725)--(5.909743589743589,1.0683760683760717)--cycle;}_{n}=\\{\\{n\,|\,0\\}\,|\,0\\}$.
These are infinitesimals of a “magnitude” different from the powers of up.
Saying the same rigorously, in addition to the facts
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and so on,
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is infinitesimal with respect to $\uparrow^{k}$, regardless of $k$. Now, for a
better understanding of what follows, it is helpful to make a brief
observation about tinies and minies.

In a game like
$\\{0\,|\,-80\\}+{\tikzpicture\clip(4.83,0.365)rectangle(5.92,1.452);[linewidth=2.pt,fill=black,fillopacity=1.0](5.909743589743589,0.7492877492877543)--(5.8697672384446085,0.7640785613681009)--(5.818003314938593,0.7813367971648741)--(5.7620276587975034,0.7976755667777837)--(5.7101105279150195,0.8107406115539613)--(5.663902303911093,0.8207230891207606)--(5.611608514153138,0.8301916514750758)--(5.559344060086461,0.837750517407593)--(5.516355816426169,0.8425615395020323)--(5.480951393364173,0.8455802578138449)--(5.442490031775766,0.8479002945794916)--(5.405586554111292,0.8491913947512248)--(5.371851851851851,0.8495726495726554)--(5.338117149592409,0.8491913947512248)--(5.301213671927935,0.8479002945794916)--(5.262752310339529,0.8455802578138449)--(5.227347887277532,0.8425615395020323)--(5.18435964361724,0.837750517407593)--(5.132095189550563,0.8301916514750758)--(5.0798013997926095,0.8207230891207606)--(5.033593175788682,0.8107406115539613)--(4.981676044906198,0.7976755667777837)--(4.92570038876511,0.7813367971648741)--(4.873936465259094,0.7640785613681009)--(4.833960113960112,0.7492877492877543)--(4.833960113960112,1.0683760683760717)--(4.873936465259094,1.053585256295725)--(4.92570038876511,1.0363270204989519)--(4.981676044906198,1.0199882508860423)--(5.033593175788682,1.0069232061098647)--(5.0798013997926095,0.9969407285430654)--(5.132095189550563,0.9874721661887502)--(5.18435964361724,0.979913300256233)--(5.227347887277532,0.9751022781617937)--(5.262752310339529,0.9720835598499811)--(5.301213671927935,0.9697635230843344)--(5.338117149592409,0.9684724229126012)--(5.371851851851851,0.9680911680911706)--(5.405586554111292,0.9684724229126012)--(5.442490031775766,0.9697635230843344)--(5.480951393364173,0.9720835598499811)--(5.516355816426169,0.9751022781617937)--(5.559344060086461,0.979913300256233)--(5.611608514153138,0.9874721661887502)--(5.663902303911093,0.9969407285430654)--(5.7101105279150195,1.0069232061098647)--(5.7620276587975034,1.0199882508860423)--(5.818003314938593,1.0363270204989519)--(5.8697672384446085,1.053585256295725)--(5.909743589743589,1.0683760683760717)--cycle;}_{100}$,
Left, playing first, wins by playing in the infinitesimal component and not in
the hot component. This happens because Left can make a significant threat in
${\tikzpicture\clip(4.83,0.365)rectangle(5.92,1.452);[linewidth=2.pt,fill=black,fillopacity=1.0](5.909743589743589,0.7492877492877543)--(5.8697672384446085,0.7640785613681009)--(5.818003314938593,0.7813367971648741)--(5.7620276587975034,0.7976755667777837)--(5.7101105279150195,0.8107406115539613)--(5.663902303911093,0.8207230891207606)--(5.611608514153138,0.8301916514750758)--(5.559344060086461,0.837750517407593)--(5.516355816426169,0.8425615395020323)--(5.480951393364173,0.8455802578138449)--(5.442490031775766,0.8479002945794916)--(5.405586554111292,0.8491913947512248)--(5.371851851851851,0.8495726495726554)--(5.338117149592409,0.8491913947512248)--(5.301213671927935,0.8479002945794916)--(5.262752310339529,0.8455802578138449)--(5.227347887277532,0.8425615395020323)--(5.18435964361724,0.837750517407593)--(5.132095189550563,0.8301916514750758)--(5.0798013997926095,0.8207230891207606)--(5.033593175788682,0.8107406115539613)--(4.981676044906198,0.7976755667777837)--(4.92570038876511,0.7813367971648741)--(4.873936465259094,0.7640785613681009)--(4.833960113960112,0.7492877492877543)--(4.833960113960112,1.0683760683760717)--(4.873936465259094,1.053585256295725)--(4.92570038876511,1.0363270204989519)--(4.981676044906198,1.0199882508860423)--(5.033593175788682,1.0069232061098647)--(5.0798013997926095,0.9969407285430654)--(5.132095189550563,0.9874721661887502)--(5.18435964361724,0.979913300256233)--(5.227347887277532,0.9751022781617937)--(5.262752310339529,0.9720835598499811)--(5.301213671927935,0.9697635230843344)--(5.338117149592409,0.9684724229126012)--(5.371851851851851,0.9680911680911706)--(5.405586554111292,0.9684724229126012)--(5.442490031775766,0.9697635230843344)--(5.480951393364173,0.9720835598499811)--(5.516355816426169,0.9751022781617937)--(5.559344060086461,0.979913300256233)--(5.611608514153138,0.9874721661887502)--(5.663902303911093,0.9969407285430654)--(5.7101105279150195,1.0069232061098647)--(5.7620276587975034,1.0199882508860423)--(5.818003314938593,1.0363270204989519)--(5.8697672384446085,1.053585256295725)--(5.909743589743589,1.0683760683760717)--cycle;}_{100}$.
This threat is powerful enough to counteract the gain that Right can achieve
in the other component. However, in game practice, it is not common to have a
component with the value
${\tikzpicture\clip(4.83,0.365)rectangle(5.92,1.452);[linewidth=2.pt,fill=black,fillopacity=1.0](5.909743589743589,0.7492877492877543)--(5.8697672384446085,0.7640785613681009)--(5.818003314938593,0.7813367971648741)--(5.7620276587975034,0.7976755667777837)--(5.7101105279150195,0.8107406115539613)--(5.663902303911093,0.8207230891207606)--(5.611608514153138,0.8301916514750758)--(5.559344060086461,0.837750517407593)--(5.516355816426169,0.8425615395020323)--(5.480951393364173,0.8455802578138449)--(5.442490031775766,0.8479002945794916)--(5.405586554111292,0.8491913947512248)--(5.371851851851851,0.8495726495726554)--(5.338117149592409,0.8491913947512248)--(5.301213671927935,0.8479002945794916)--(5.262752310339529,0.8455802578138449)--(5.227347887277532,0.8425615395020323)--(5.18435964361724,0.837750517407593)--(5.132095189550563,0.8301916514750758)--(5.0798013997926095,0.8207230891207606)--(5.033593175788682,0.8107406115539613)--(4.981676044906198,0.7976755667777837)--(4.92570038876511,0.7813367971648741)--(4.873936465259094,0.7640785613681009)--(4.833960113960112,0.7492877492877543)--(4.833960113960112,1.0683760683760717)--(4.873936465259094,1.053585256295725)--(4.92570038876511,1.0363270204989519)--(4.981676044906198,1.0199882508860423)--(5.033593175788682,1.0069232061098647)--(5.0798013997926095,0.9969407285430654)--(5.132095189550563,0.9874721661887502)--(5.18435964361724,0.979913300256233)--(5.227347887277532,0.9751022781617937)--(5.262752310339529,0.9720835598499811)--(5.301213671927935,0.9697635230843344)--(5.338117149592409,0.9684724229126012)--(5.371851851851851,0.9680911680911706)--(5.405586554111292,0.9684724229126012)--(5.442490031775766,0.9697635230843344)--(5.480951393364173,0.9720835598499811)--(5.516355816426169,0.9751022781617937)--(5.559344060086461,0.979913300256233)--(5.611608514153138,0.9874721661887502)--(5.663902303911093,0.9969407285430654)--(5.7101105279150195,1.0069232061098647)--(5.7620276587975034,1.0199882508860423)--(5.818003314938593,1.0363270204989519)--(5.8697672384446085,1.053585256295725)--(5.909743589743589,1.0683760683760717)--cycle;}_{100}$.
This is because it would require a threat of $100$ in “territorial” terms to
be defended with a move to $0$, a situation that would require a ruleset with
very special characteristics. However, in
$\mathbbm{N}\mathbbm{p}^{\infty}\\!$, “powerful threat” can be replaced by
“terminal threat”, and given that mate threats are very frequent, there is a
certain type of games that are very common. The ultimate tiny (miny) can be
defined as follows.

######  Definition 60 (Pathetic Infinitesimals).

The game
$\\{0\,|\,\\{0\,|\,\overline{\infty}\\}\\}=\\{0\,|\,\circlearrowleft^{0}\\}$
is called _pathetic tiny_ and denoted by
${\tikzpicture\clip(4.83,0.365)rectangle(5.92,1.452);[linewidth=2.pt,fill=black,fillopacity=1.0](5.212307692307691,0.3709401709401755)--(5.227098504388038,0.4109165222391566)--(5.244356740184811,0.4626804457451724)--(5.260695509797721,0.5186561018862608)--(5.273760554573898,0.5705732327687449)--(5.2837430321406975,0.616781456772672)--(5.293211594495013,0.6690752465306267)--(5.30077046042753,0.7213397005973032)--(5.305581482521969,0.7643279442575954)--(5.308600200833782,0.7997323673195916)--(5.310920237599428,0.8381937289079981)--(5.312211337771162,0.8750972065724719)--(5.312592592592592,0.9088319088319139)--(5.312211337771162,0.9425666110913559)--(5.310920237599428,0.9794700887558296)--(5.308600200833782,1.017931450344236)--(5.305581482521969,1.0533358734062324)--(5.30077046042753,1.0963241170665246)--(5.293211594495013,1.1485885711332011)--(5.2837430321406975,1.2008823608911556)--(5.273760554573898,1.2470905848950828)--(5.260695509797721,1.299007715777567)--(5.244356740184811,1.3549833719186553)--(5.227098504388038,1.4067472954246711)--(5.212307692307691,1.4467236467236524)--(5.531396011396009,1.4467236467236524)--(5.516605199315662,1.4067472954246711)--(5.499346963518889,1.3549833719186553)--(5.483008193905979,1.299007715777567)--(5.469943149129802,1.2470905848950828)--(5.459960671563002,1.2008823608911556)--(5.450492109208687,1.1485885711332011)--(5.44293324327617,1.0963241170665246)--(5.438122221181731,1.0533358734062324)--(5.435103502869918,1.017931450344236)--(5.432783466104271,0.9794700887558296)--(5.431492365932538,0.9425666110913559)--(5.4311111111111074,0.9088319088319139)--(5.431492365932538,0.8750972065724719)--(5.432783466104271,0.8381937289079981)--(5.435103502869918,0.7997323673195916)--(5.438122221181731,0.7643279442575954)--(5.44293324327617,0.7213397005973032)--(5.450492109208687,0.6690752465306267)--(5.459960671563002,0.616781456772672)--(5.469943149129802,0.5705732327687449)--(5.483008193905979,0.5186561018862608)--(5.499346963518889,0.4626804457451724)--(5.516605199315662,0.4109165222391566)--(5.531396011396009,0.3709401709401755)--cycle;[linewidth=2.pt,fill=black,fillopacity=1.0](5.909743589743589,0.7492877492877543)--(5.8697672384446085,0.7640785613681009)--(5.818003314938593,0.7813367971648741)--(5.7620276587975034,0.7976755667777837)--(5.7101105279150195,0.8107406115539613)--(5.663902303911093,0.8207230891207606)--(5.611608514153138,0.8301916514750758)--(5.559344060086461,0.837750517407593)--(5.516355816426169,0.8425615395020323)--(5.480951393364173,0.8455802578138449)--(5.442490031775766,0.8479002945794916)--(5.405586554111292,0.8491913947512248)--(5.371851851851851,0.8495726495726554)--(5.338117149592409,0.8491913947512248)--(5.301213671927935,0.8479002945794916)--(5.262752310339529,0.8455802578138449)--(5.227347887277532,0.8425615395020323)--(5.18435964361724,0.837750517407593)--(5.132095189550563,0.8301916514750758)--(5.0798013997926095,0.8207230891207606)--(5.033593175788682,0.8107406115539613)--(4.981676044906198,0.7976755667777837)--(4.92570038876511,0.7813367971648741)--(4.873936465259094,0.7640785613681009)--(4.833960113960112,0.7492877492877543)--(4.833960113960112,1.0683760683760717)--(4.873936465259094,1.053585256295725)--(4.92570038876511,1.0363270204989519)--(4.981676044906198,1.0199882508860423)--(5.033593175788682,1.0069232061098647)--(5.0798013997926095,0.9969407285430654)--(5.132095189550563,0.9874721661887502)--(5.18435964361724,0.979913300256233)--(5.227347887277532,0.9751022781617937)--(5.262752310339529,0.9720835598499811)--(5.301213671927935,0.9697635230843344)--(5.338117149592409,0.9684724229126012)--(5.371851851851851,0.9680911680911706)--(5.405586554111292,0.9684724229126012)--(5.442490031775766,0.9697635230843344)--(5.480951393364173,0.9720835598499811)--(5.516355816426169,0.9751022781617937)--(5.559344060086461,0.979913300256233)--(5.611608514153138,0.9874721661887502)--(5.663902303911093,0.9969407285430654)--(5.7101105279150195,1.0069232061098647)--(5.7620276587975034,1.0199882508860423)--(5.818003314938593,1.0363270204989519)--(5.8697672384446085,1.053585256295725)--(5.909743589743589,1.0683760683760717)--cycle;}_{\infty}$.
The game $\\{\\{\infty\,|\,0\\}\,|\,0\\}=\\{\circlearrowright^{0}\,|\,0\\}$ is
called _pathetic miny_ and denoted by
${\tikzpicture\clip(4.83,0.365)rectangle(5.92,1.452);[linewidth=2.pt,fill=black,fillopacity=1.0](5.909743589743589,0.7492877492877543)--(5.8697672384446085,0.7640785613681009)--(5.818003314938593,0.7813367971648741)--(5.7620276587975034,0.7976755667777837)--(5.7101105279150195,0.8107406115539613)--(5.663902303911093,0.8207230891207606)--(5.611608514153138,0.8301916514750758)--(5.559344060086461,0.837750517407593)--(5.516355816426169,0.8425615395020323)--(5.480951393364173,0.8455802578138449)--(5.442490031775766,0.8479002945794916)--(5.405586554111292,0.8491913947512248)--(5.371851851851851,0.8495726495726554)--(5.338117149592409,0.8491913947512248)--(5.301213671927935,0.8479002945794916)--(5.262752310339529,0.8455802578138449)--(5.227347887277532,0.8425615395020323)--(5.18435964361724,0.837750517407593)--(5.132095189550563,0.8301916514750758)--(5.0798013997926095,0.8207230891207606)--(5.033593175788682,0.8107406115539613)--(4.981676044906198,0.7976755667777837)--(4.92570038876511,0.7813367971648741)--(4.873936465259094,0.7640785613681009)--(4.833960113960112,0.7492877492877543)--(4.833960113960112,1.0683760683760717)--(4.873936465259094,1.053585256295725)--(4.92570038876511,1.0363270204989519)--(4.981676044906198,1.0199882508860423)--(5.033593175788682,1.0069232061098647)--(5.0798013997926095,0.9969407285430654)--(5.132095189550563,0.9874721661887502)--(5.18435964361724,0.979913300256233)--(5.227347887277532,0.9751022781617937)--(5.262752310339529,0.9720835598499811)--(5.301213671927935,0.9697635230843344)--(5.338117149592409,0.9684724229126012)--(5.371851851851851,0.9680911680911706)--(5.405586554111292,0.9684724229126012)--(5.442490031775766,0.9697635230843344)--(5.480951393364173,0.9720835598499811)--(5.516355816426169,0.9751022781617937)--(5.559344060086461,0.979913300256233)--(5.611608514153138,0.9874721661887502)--(5.663902303911093,0.9969407285430654)--(5.7101105279150195,1.0069232061098647)--(5.7620276587975034,1.0199882508860423)--(5.818003314938593,1.0363270204989519)--(5.8697672384446085,1.053585256295725)--(5.909743589743589,1.0683760683760717)--cycle;}_{\infty}$.

It is trivial to verify that
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Furthermore, the name “pathetic” is easily explainable. As observed in the
previous paragraph, the presence of
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in a disjunctive sum tends to be ephemeral, as in most cases, Left uses her
threat to replace, to her advantage, a negative game by zero. Once again, in
$\mathbbm{N}\mathbbm{p}^{\infty}\\!$, “in most cases” can be transformed into
“always”, as the following theorem establishes – this is the first of three
notable facts about the pathetic infinitesimals. The “pathetic nature” of
these positive and negative values is due to their ephemerality. Both a
pathetic tiny and a pathetic miny have a very limited lifespan in a
disjunctive sum, as, at the first opportunity, one of the players makes them
disappear.

######  Theorem 61.

The pathetic tiny
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is a Right-hammerzug. Similarly, the pathetic miny
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is a Left-hammerzug.

###### Proof.

This is a direct consequence of Definition 57 and
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∎

The second notable fact concerns a certain “lack of continuity” exhibited by
$\mathbbm{N}\mathbbm{p}^{\infty}\\!$. The smallest positive element and the
largest negative element are effectively reached.

######  Theorem 62.

The game
${\tikzpicture\clip(4.83,0.365)rectangle(5.92,1.452);[linewidth=2.pt,fill=black,fillopacity=1.0](5.212307692307691,0.3709401709401755)--(5.227098504388038,0.4109165222391566)--(5.244356740184811,0.4626804457451724)--(5.260695509797721,0.5186561018862608)--(5.273760554573898,0.5705732327687449)--(5.2837430321406975,0.616781456772672)--(5.293211594495013,0.6690752465306267)--(5.30077046042753,0.7213397005973032)--(5.305581482521969,0.7643279442575954)--(5.308600200833782,0.7997323673195916)--(5.310920237599428,0.8381937289079981)--(5.312211337771162,0.8750972065724719)--(5.312592592592592,0.9088319088319139)--(5.312211337771162,0.9425666110913559)--(5.310920237599428,0.9794700887558296)--(5.308600200833782,1.017931450344236)--(5.305581482521969,1.0533358734062324)--(5.30077046042753,1.0963241170665246)--(5.293211594495013,1.1485885711332011)--(5.2837430321406975,1.2008823608911556)--(5.273760554573898,1.2470905848950828)--(5.260695509797721,1.299007715777567)--(5.244356740184811,1.3549833719186553)--(5.227098504388038,1.4067472954246711)--(5.212307692307691,1.4467236467236524)--(5.531396011396009,1.4467236467236524)--(5.516605199315662,1.4067472954246711)--(5.499346963518889,1.3549833719186553)--(5.483008193905979,1.299007715777567)--(5.469943149129802,1.2470905848950828)--(5.459960671563002,1.2008823608911556)--(5.450492109208687,1.1485885711332011)--(5.44293324327617,1.0963241170665246)--(5.438122221181731,1.0533358734062324)--(5.435103502869918,1.017931450344236)--(5.432783466104271,0.9794700887558296)--(5.431492365932538,0.9425666110913559)--(5.4311111111111074,0.9088319088319139)--(5.431492365932538,0.8750972065724719)--(5.432783466104271,0.8381937289079981)--(5.435103502869918,0.7997323673195916)--(5.438122221181731,0.7643279442575954)--(5.44293324327617,0.7213397005973032)--(5.450492109208687,0.6690752465306267)--(5.459960671563002,0.616781456772672)--(5.469943149129802,0.5705732327687449)--(5.483008193905979,0.5186561018862608)--(5.499346963518889,0.4626804457451724)--(5.516605199315662,0.4109165222391566)--(5.531396011396009,0.3709401709401755)--cycle;[linewidth=2.pt,fill=black,fillopacity=1.0](5.909743589743589,0.7492877492877543)--(5.8697672384446085,0.7640785613681009)--(5.818003314938593,0.7813367971648741)--(5.7620276587975034,0.7976755667777837)--(5.7101105279150195,0.8107406115539613)--(5.663902303911093,0.8207230891207606)--(5.611608514153138,0.8301916514750758)--(5.559344060086461,0.837750517407593)--(5.516355816426169,0.8425615395020323)--(5.480951393364173,0.8455802578138449)--(5.442490031775766,0.8479002945794916)--(5.405586554111292,0.8491913947512248)--(5.371851851851851,0.8495726495726554)--(5.338117149592409,0.8491913947512248)--(5.301213671927935,0.8479002945794916)--(5.262752310339529,0.8455802578138449)--(5.227347887277532,0.8425615395020323)--(5.18435964361724,0.837750517407593)--(5.132095189550563,0.8301916514750758)--(5.0798013997926095,0.8207230891207606)--(5.033593175788682,0.8107406115539613)--(4.981676044906198,0.7976755667777837)--(4.92570038876511,0.7813367971648741)--(4.873936465259094,0.7640785613681009)--(4.833960113960112,0.7492877492877543)--(4.833960113960112,1.0683760683760717)--(4.873936465259094,1.053585256295725)--(4.92570038876511,1.0363270204989519)--(4.981676044906198,1.0199882508860423)--(5.033593175788682,1.0069232061098647)--(5.0798013997926095,0.9969407285430654)--(5.132095189550563,0.9874721661887502)--(5.18435964361724,0.979913300256233)--(5.227347887277532,0.9751022781617937)--(5.262752310339529,0.9720835598499811)--(5.301213671927935,0.9697635230843344)--(5.338117149592409,0.9684724229126012)--(5.371851851851851,0.9680911680911706)--(5.405586554111292,0.9684724229126012)--(5.442490031775766,0.9697635230843344)--(5.480951393364173,0.9720835598499811)--(5.516355816426169,0.9751022781617937)--(5.559344060086461,0.979913300256233)--(5.611608514153138,0.9874721661887502)--(5.663902303911093,0.9969407285430654)--(5.7101105279150195,1.0069232061098647)--(5.7620276587975034,1.0199882508860423)--(5.818003314938593,1.0363270204989519)--(5.8697672384446085,1.053585256295725)--(5.909743589743589,1.0683760683760717)--cycle;}_{\infty}$
is the smallest positive game of all, i.e., if
$G\in\mathbbm{N}\mathbbm{p}^{\infty}\\!$ is such that $G>0$, then
$G\geqslant{\tikzpicture\clip(4.83,0.365)rectangle(5.92,1.452);[linewidth=2.pt,fill=black,fillopacity=1.0](5.212307692307691,0.3709401709401755)--(5.227098504388038,0.4109165222391566)--(5.244356740184811,0.4626804457451724)--(5.260695509797721,0.5186561018862608)--(5.273760554573898,0.5705732327687449)--(5.2837430321406975,0.616781456772672)--(5.293211594495013,0.6690752465306267)--(5.30077046042753,0.7213397005973032)--(5.305581482521969,0.7643279442575954)--(5.308600200833782,0.7997323673195916)--(5.310920237599428,0.8381937289079981)--(5.312211337771162,0.8750972065724719)--(5.312592592592592,0.9088319088319139)--(5.312211337771162,0.9425666110913559)--(5.310920237599428,0.9794700887558296)--(5.308600200833782,1.017931450344236)--(5.305581482521969,1.0533358734062324)--(5.30077046042753,1.0963241170665246)--(5.293211594495013,1.1485885711332011)--(5.2837430321406975,1.2008823608911556)--(5.273760554573898,1.2470905848950828)--(5.260695509797721,1.299007715777567)--(5.244356740184811,1.3549833719186553)--(5.227098504388038,1.4067472954246711)--(5.212307692307691,1.4467236467236524)--(5.531396011396009,1.4467236467236524)--(5.516605199315662,1.4067472954246711)--(5.499346963518889,1.3549833719186553)--(5.483008193905979,1.299007715777567)--(5.469943149129802,1.2470905848950828)--(5.459960671563002,1.2008823608911556)--(5.450492109208687,1.1485885711332011)--(5.44293324327617,1.0963241170665246)--(5.438122221181731,1.0533358734062324)--(5.435103502869918,1.017931450344236)--(5.432783466104271,0.9794700887558296)--(5.431492365932538,0.9425666110913559)--(5.4311111111111074,0.9088319088319139)--(5.431492365932538,0.8750972065724719)--(5.432783466104271,0.8381937289079981)--(5.435103502869918,0.7997323673195916)--(5.438122221181731,0.7643279442575954)--(5.44293324327617,0.7213397005973032)--(5.450492109208687,0.6690752465306267)--(5.459960671563002,0.616781456772672)--(5.469943149129802,0.5705732327687449)--(5.483008193905979,0.5186561018862608)--(5.499346963518889,0.4626804457451724)--(5.516605199315662,0.4109165222391566)--(5.531396011396009,0.3709401709401755)--cycle;[linewidth=2.pt,fill=black,fillopacity=1.0](5.909743589743589,0.7492877492877543)--(5.8697672384446085,0.7640785613681009)--(5.818003314938593,0.7813367971648741)--(5.7620276587975034,0.7976755667777837)--(5.7101105279150195,0.8107406115539613)--(5.663902303911093,0.8207230891207606)--(5.611608514153138,0.8301916514750758)--(5.559344060086461,0.837750517407593)--(5.516355816426169,0.8425615395020323)--(5.480951393364173,0.8455802578138449)--(5.442490031775766,0.8479002945794916)--(5.405586554111292,0.8491913947512248)--(5.371851851851851,0.8495726495726554)--(5.338117149592409,0.8491913947512248)--(5.301213671927935,0.8479002945794916)--(5.262752310339529,0.8455802578138449)--(5.227347887277532,0.8425615395020323)--(5.18435964361724,0.837750517407593)--(5.132095189550563,0.8301916514750758)--(5.0798013997926095,0.8207230891207606)--(5.033593175788682,0.8107406115539613)--(4.981676044906198,0.7976755667777837)--(4.92570038876511,0.7813367971648741)--(4.873936465259094,0.7640785613681009)--(4.833960113960112,0.7492877492877543)--(4.833960113960112,1.0683760683760717)--(4.873936465259094,1.053585256295725)--(4.92570038876511,1.0363270204989519)--(4.981676044906198,1.0199882508860423)--(5.033593175788682,1.0069232061098647)--(5.0798013997926095,0.9969407285430654)--(5.132095189550563,0.9874721661887502)--(5.18435964361724,0.979913300256233)--(5.227347887277532,0.9751022781617937)--(5.262752310339529,0.9720835598499811)--(5.301213671927935,0.9697635230843344)--(5.338117149592409,0.9684724229126012)--(5.371851851851851,0.9680911680911706)--(5.405586554111292,0.9684724229126012)--(5.442490031775766,0.9697635230843344)--(5.480951393364173,0.9720835598499811)--(5.516355816426169,0.9751022781617937)--(5.559344060086461,0.979913300256233)--(5.611608514153138,0.9874721661887502)--(5.663902303911093,0.9969407285430654)--(5.7101105279150195,1.0069232061098647)--(5.7620276587975034,1.0199882508860423)--(5.818003314938593,1.0363270204989519)--(5.8697672384446085,1.053585256295725)--(5.909743589743589,1.0683760683760717)--cycle;}_{\infty}$.
Similarly, the game
${\tikzpicture\clip(4.83,0.365)rectangle(5.92,1.452);[linewidth=2.pt,fill=black,fillopacity=1.0](5.909743589743589,0.7492877492877543)--(5.8697672384446085,0.7640785613681009)--(5.818003314938593,0.7813367971648741)--(5.7620276587975034,0.7976755667777837)--(5.7101105279150195,0.8107406115539613)--(5.663902303911093,0.8207230891207606)--(5.611608514153138,0.8301916514750758)--(5.559344060086461,0.837750517407593)--(5.516355816426169,0.8425615395020323)--(5.480951393364173,0.8455802578138449)--(5.442490031775766,0.8479002945794916)--(5.405586554111292,0.8491913947512248)--(5.371851851851851,0.8495726495726554)--(5.338117149592409,0.8491913947512248)--(5.301213671927935,0.8479002945794916)--(5.262752310339529,0.8455802578138449)--(5.227347887277532,0.8425615395020323)--(5.18435964361724,0.837750517407593)--(5.132095189550563,0.8301916514750758)--(5.0798013997926095,0.8207230891207606)--(5.033593175788682,0.8107406115539613)--(4.981676044906198,0.7976755667777837)--(4.92570038876511,0.7813367971648741)--(4.873936465259094,0.7640785613681009)--(4.833960113960112,0.7492877492877543)--(4.833960113960112,1.0683760683760717)--(4.873936465259094,1.053585256295725)--(4.92570038876511,1.0363270204989519)--(4.981676044906198,1.0199882508860423)--(5.033593175788682,1.0069232061098647)--(5.0798013997926095,0.9969407285430654)--(5.132095189550563,0.9874721661887502)--(5.18435964361724,0.979913300256233)--(5.227347887277532,0.9751022781617937)--(5.262752310339529,0.9720835598499811)--(5.301213671927935,0.9697635230843344)--(5.338117149592409,0.9684724229126012)--(5.371851851851851,0.9680911680911706)--(5.405586554111292,0.9684724229126012)--(5.442490031775766,0.9697635230843344)--(5.480951393364173,0.9720835598499811)--(5.516355816426169,0.9751022781617937)--(5.559344060086461,0.979913300256233)--(5.611608514153138,0.9874721661887502)--(5.663902303911093,0.9969407285430654)--(5.7101105279150195,1.0069232061098647)--(5.7620276587975034,1.0199882508860423)--(5.818003314938593,1.0363270204989519)--(5.8697672384446085,1.053585256295725)--(5.909743589743589,1.0683760683760717)--cycle;}_{\infty}$
is the largest negative game of all.

###### Proof.

If $G=\infty$, there is nothing to prove, so assume otherwise. According to
Theorem 14, $G>0$ implies the existence of some $G^{L}$ such that
$G^{L}\geqslant 0$ (Fact 1). On the other hand, the same theorem guarantees
that for each Right option $G^{R}\neq\infty$, there exists $G^{RL}$ such that
$G^{RL}\geqslant 0$. For these cases, it is straightforward to verify
$\mathbf{M}(G^{R},\\{0\,|\,\overline{\infty}\\})$, since for the Left option
to $0$ in $\\{0\,|\,\overline{\infty}\\}$, there exists some $G^{RL}\geqslant
0$. Consequently, according to Theorem 37, we have
$G^{R}\geqslant\\{0\,|\,\overline{\infty}\\}$ (Fact 2).

With the help of these two facts, let us check that the pair
$(G,{\tikzpicture\clip(4.83,0.365)rectangle(5.92,1.452);[linewidth=2.pt,fill=black,fillopacity=1.0](5.212307692307691,0.3709401709401755)--(5.227098504388038,0.4109165222391566)--(5.244356740184811,0.4626804457451724)--(5.260695509797721,0.5186561018862608)--(5.273760554573898,0.5705732327687449)--(5.2837430321406975,0.616781456772672)--(5.293211594495013,0.6690752465306267)--(5.30077046042753,0.7213397005973032)--(5.305581482521969,0.7643279442575954)--(5.308600200833782,0.7997323673195916)--(5.310920237599428,0.8381937289079981)--(5.312211337771162,0.8750972065724719)--(5.312592592592592,0.9088319088319139)--(5.312211337771162,0.9425666110913559)--(5.310920237599428,0.9794700887558296)--(5.308600200833782,1.017931450344236)--(5.305581482521969,1.0533358734062324)--(5.30077046042753,1.0963241170665246)--(5.293211594495013,1.1485885711332011)--(5.2837430321406975,1.2008823608911556)--(5.273760554573898,1.2470905848950828)--(5.260695509797721,1.299007715777567)--(5.244356740184811,1.3549833719186553)--(5.227098504388038,1.4067472954246711)--(5.212307692307691,1.4467236467236524)--(5.531396011396009,1.4467236467236524)--(5.516605199315662,1.4067472954246711)--(5.499346963518889,1.3549833719186553)--(5.483008193905979,1.299007715777567)--(5.469943149129802,1.2470905848950828)--(5.459960671563002,1.2008823608911556)--(5.450492109208687,1.1485885711332011)--(5.44293324327617,1.0963241170665246)--(5.438122221181731,1.0533358734062324)--(5.435103502869918,1.017931450344236)--(5.432783466104271,0.9794700887558296)--(5.431492365932538,0.9425666110913559)--(5.4311111111111074,0.9088319088319139)--(5.431492365932538,0.8750972065724719)--(5.432783466104271,0.8381937289079981)--(5.435103502869918,0.7997323673195916)--(5.438122221181731,0.7643279442575954)--(5.44293324327617,0.7213397005973032)--(5.450492109208687,0.6690752465306267)--(5.459960671563002,0.616781456772672)--(5.469943149129802,0.5705732327687449)--(5.483008193905979,0.5186561018862608)--(5.499346963518889,0.4626804457451724)--(5.516605199315662,0.4109165222391566)--(5.531396011396009,0.3709401709401755)--cycle;[linewidth=2.pt,fill=black,fillopacity=1.0](5.909743589743589,0.7492877492877543)--(5.8697672384446085,0.7640785613681009)--(5.818003314938593,0.7813367971648741)--(5.7620276587975034,0.7976755667777837)--(5.7101105279150195,0.8107406115539613)--(5.663902303911093,0.8207230891207606)--(5.611608514153138,0.8301916514750758)--(5.559344060086461,0.837750517407593)--(5.516355816426169,0.8425615395020323)--(5.480951393364173,0.8455802578138449)--(5.442490031775766,0.8479002945794916)--(5.405586554111292,0.8491913947512248)--(5.371851851851851,0.8495726495726554)--(5.338117149592409,0.8491913947512248)--(5.301213671927935,0.8479002945794916)--(5.262752310339529,0.8455802578138449)--(5.227347887277532,0.8425615395020323)--(5.18435964361724,0.837750517407593)--(5.132095189550563,0.8301916514750758)--(5.0798013997926095,0.8207230891207606)--(5.033593175788682,0.8107406115539613)--(4.981676044906198,0.7976755667777837)--(4.92570038876511,0.7813367971648741)--(4.873936465259094,0.7640785613681009)--(4.833960113960112,0.7492877492877543)--(4.833960113960112,1.0683760683760717)--(4.873936465259094,1.053585256295725)--(4.92570038876511,1.0363270204989519)--(4.981676044906198,1.0199882508860423)--(5.033593175788682,1.0069232061098647)--(5.0798013997926095,0.9969407285430654)--(5.132095189550563,0.9874721661887502)--(5.18435964361724,0.979913300256233)--(5.227347887277532,0.9751022781617937)--(5.262752310339529,0.9720835598499811)--(5.301213671927935,0.9697635230843344)--(5.338117149592409,0.9684724229126012)--(5.371851851851851,0.9680911680911706)--(5.405586554111292,0.9684724229126012)--(5.442490031775766,0.9697635230843344)--(5.480951393364173,0.9720835598499811)--(5.516355816426169,0.9751022781617937)--(5.559344060086461,0.979913300256233)--(5.611608514153138,0.9874721661887502)--(5.663902303911093,0.9969407285430654)--(5.7101105279150195,1.0069232061098647)--(5.7620276587975034,1.0199882508860423)--(5.818003314938593,1.0363270204989519)--(5.8697672384446085,1.053585256295725)--(5.909743589743589,1.0683760683760717)--cycle;}_{\infty})$
satisfies also the maintenance property as established in Definition 1, and,
therefore, by Theorem 37,
$G\geqslant{\tikzpicture\clip(4.83,0.365)rectangle(5.92,1.452);[linewidth=2.pt,fill=black,fillopacity=1.0](5.212307692307691,0.3709401709401755)--(5.227098504388038,0.4109165222391566)--(5.244356740184811,0.4626804457451724)--(5.260695509797721,0.5186561018862608)--(5.273760554573898,0.5705732327687449)--(5.2837430321406975,0.616781456772672)--(5.293211594495013,0.6690752465306267)--(5.30077046042753,0.7213397005973032)--(5.305581482521969,0.7643279442575954)--(5.308600200833782,0.7997323673195916)--(5.310920237599428,0.8381937289079981)--(5.312211337771162,0.8750972065724719)--(5.312592592592592,0.9088319088319139)--(5.312211337771162,0.9425666110913559)--(5.310920237599428,0.9794700887558296)--(5.308600200833782,1.017931450344236)--(5.305581482521969,1.0533358734062324)--(5.30077046042753,1.0963241170665246)--(5.293211594495013,1.1485885711332011)--(5.2837430321406975,1.2008823608911556)--(5.273760554573898,1.2470905848950828)--(5.260695509797721,1.299007715777567)--(5.244356740184811,1.3549833719186553)--(5.227098504388038,1.4067472954246711)--(5.212307692307691,1.4467236467236524)--(5.531396011396009,1.4467236467236524)--(5.516605199315662,1.4067472954246711)--(5.499346963518889,1.3549833719186553)--(5.483008193905979,1.299007715777567)--(5.469943149129802,1.2470905848950828)--(5.459960671563002,1.2008823608911556)--(5.450492109208687,1.1485885711332011)--(5.44293324327617,1.0963241170665246)--(5.438122221181731,1.0533358734062324)--(5.435103502869918,1.017931450344236)--(5.432783466104271,0.9794700887558296)--(5.431492365932538,0.9425666110913559)--(5.4311111111111074,0.9088319088319139)--(5.431492365932538,0.8750972065724719)--(5.432783466104271,0.8381937289079981)--(5.435103502869918,0.7997323673195916)--(5.438122221181731,0.7643279442575954)--(5.44293324327617,0.7213397005973032)--(5.450492109208687,0.6690752465306267)--(5.459960671563002,0.616781456772672)--(5.469943149129802,0.5705732327687449)--(5.483008193905979,0.5186561018862608)--(5.499346963518889,0.4626804457451724)--(5.516605199315662,0.4109165222391566)--(5.531396011396009,0.3709401709401755)--cycle;[linewidth=2.pt,fill=black,fillopacity=1.0](5.909743589743589,0.7492877492877543)--(5.8697672384446085,0.7640785613681009)--(5.818003314938593,0.7813367971648741)--(5.7620276587975034,0.7976755667777837)--(5.7101105279150195,0.8107406115539613)--(5.663902303911093,0.8207230891207606)--(5.611608514153138,0.8301916514750758)--(5.559344060086461,0.837750517407593)--(5.516355816426169,0.8425615395020323)--(5.480951393364173,0.8455802578138449)--(5.442490031775766,0.8479002945794916)--(5.405586554111292,0.8491913947512248)--(5.371851851851851,0.8495726495726554)--(5.338117149592409,0.8491913947512248)--(5.301213671927935,0.8479002945794916)--(5.262752310339529,0.8455802578138449)--(5.227347887277532,0.8425615395020323)--(5.18435964361724,0.837750517407593)--(5.132095189550563,0.8301916514750758)--(5.0798013997926095,0.8207230891207606)--(5.033593175788682,0.8107406115539613)--(4.981676044906198,0.7976755667777837)--(4.92570038876511,0.7813367971648741)--(4.873936465259094,0.7640785613681009)--(4.833960113960112,0.7492877492877543)--(4.833960113960112,1.0683760683760717)--(4.873936465259094,1.053585256295725)--(4.92570038876511,1.0363270204989519)--(4.981676044906198,1.0199882508860423)--(5.033593175788682,1.0069232061098647)--(5.0798013997926095,0.9969407285430654)--(5.132095189550563,0.9874721661887502)--(5.18435964361724,0.979913300256233)--(5.227347887277532,0.9751022781617937)--(5.262752310339529,0.9720835598499811)--(5.301213671927935,0.9697635230843344)--(5.338117149592409,0.9684724229126012)--(5.371851851851851,0.9680911680911706)--(5.405586554111292,0.9684724229126012)--(5.442490031775766,0.9697635230843344)--(5.480951393364173,0.9720835598499811)--(5.516355816426169,0.9751022781617937)--(5.559344060086461,0.979913300256233)--(5.611608514153138,0.9874721661887502)--(5.663902303911093,0.9969407285430654)--(5.7101105279150195,1.0069232061098647)--(5.7620276587975034,1.0199882508860423)--(5.818003314938593,1.0363270204989519)--(5.8697672384446085,1.053585256295725)--(5.909743589743589,1.0683760683760717)--cycle;}_{\infty}$.
For the Left move to $0$ in
${\tikzpicture\clip(4.83,0.365)rectangle(5.92,1.452);[linewidth=2.pt,fill=black,fillopacity=1.0](5.212307692307691,0.3709401709401755)--(5.227098504388038,0.4109165222391566)--(5.244356740184811,0.4626804457451724)--(5.260695509797721,0.5186561018862608)--(5.273760554573898,0.5705732327687449)--(5.2837430321406975,0.616781456772672)--(5.293211594495013,0.6690752465306267)--(5.30077046042753,0.7213397005973032)--(5.305581482521969,0.7643279442575954)--(5.308600200833782,0.7997323673195916)--(5.310920237599428,0.8381937289079981)--(5.312211337771162,0.8750972065724719)--(5.312592592592592,0.9088319088319139)--(5.312211337771162,0.9425666110913559)--(5.310920237599428,0.9794700887558296)--(5.308600200833782,1.017931450344236)--(5.305581482521969,1.0533358734062324)--(5.30077046042753,1.0963241170665246)--(5.293211594495013,1.1485885711332011)--(5.2837430321406975,1.2008823608911556)--(5.273760554573898,1.2470905848950828)--(5.260695509797721,1.299007715777567)--(5.244356740184811,1.3549833719186553)--(5.227098504388038,1.4067472954246711)--(5.212307692307691,1.4467236467236524)--(5.531396011396009,1.4467236467236524)--(5.516605199315662,1.4067472954246711)--(5.499346963518889,1.3549833719186553)--(5.483008193905979,1.299007715777567)--(5.469943149129802,1.2470905848950828)--(5.459960671563002,1.2008823608911556)--(5.450492109208687,1.1485885711332011)--(5.44293324327617,1.0963241170665246)--(5.438122221181731,1.0533358734062324)--(5.435103502869918,1.017931450344236)--(5.432783466104271,0.9794700887558296)--(5.431492365932538,0.9425666110913559)--(5.4311111111111074,0.9088319088319139)--(5.431492365932538,0.8750972065724719)--(5.432783466104271,0.8381937289079981)--(5.435103502869918,0.7997323673195916)--(5.438122221181731,0.7643279442575954)--(5.44293324327617,0.7213397005973032)--(5.450492109208687,0.6690752465306267)--(5.459960671563002,0.616781456772672)--(5.469943149129802,0.5705732327687449)--(5.483008193905979,0.5186561018862608)--(5.499346963518889,0.4626804457451724)--(5.516605199315662,0.4109165222391566)--(5.531396011396009,0.3709401709401755)--cycle;[linewidth=2.pt,fill=black,fillopacity=1.0](5.909743589743589,0.7492877492877543)--(5.8697672384446085,0.7640785613681009)--(5.818003314938593,0.7813367971648741)--(5.7620276587975034,0.7976755667777837)--(5.7101105279150195,0.8107406115539613)--(5.663902303911093,0.8207230891207606)--(5.611608514153138,0.8301916514750758)--(5.559344060086461,0.837750517407593)--(5.516355816426169,0.8425615395020323)--(5.480951393364173,0.8455802578138449)--(5.442490031775766,0.8479002945794916)--(5.405586554111292,0.8491913947512248)--(5.371851851851851,0.8495726495726554)--(5.338117149592409,0.8491913947512248)--(5.301213671927935,0.8479002945794916)--(5.262752310339529,0.8455802578138449)--(5.227347887277532,0.8425615395020323)--(5.18435964361724,0.837750517407593)--(5.132095189550563,0.8301916514750758)--(5.0798013997926095,0.8207230891207606)--(5.033593175788682,0.8107406115539613)--(4.981676044906198,0.7976755667777837)--(4.92570038876511,0.7813367971648741)--(4.873936465259094,0.7640785613681009)--(4.833960113960112,0.7492877492877543)--(4.833960113960112,1.0683760683760717)--(4.873936465259094,1.053585256295725)--(4.92570038876511,1.0363270204989519)--(4.981676044906198,1.0199882508860423)--(5.033593175788682,1.0069232061098647)--(5.0798013997926095,0.9969407285430654)--(5.132095189550563,0.9874721661887502)--(5.18435964361724,0.979913300256233)--(5.227347887277532,0.9751022781617937)--(5.262752310339529,0.9720835598499811)--(5.301213671927935,0.9697635230843344)--(5.338117149592409,0.9684724229126012)--(5.371851851851851,0.9680911680911706)--(5.405586554111292,0.9684724229126012)--(5.442490031775766,0.9697635230843344)--(5.480951393364173,0.9720835598499811)--(5.516355816426169,0.9751022781617937)--(5.559344060086461,0.979913300256233)--(5.611608514153138,0.9874721661887502)--(5.663902303911093,0.9969407285430654)--(5.7101105279150195,1.0069232061098647)--(5.7620276587975034,1.0199882508860423)--(5.818003314938593,1.0363270204989519)--(5.8697672384446085,1.053585256295725)--(5.909743589743589,1.0683760683760717)--cycle;}_{\infty}$,
there exists some $G^{L}\geqslant 0$ (Fact 1). For each Right option $G^{R}$,
we have $G^{R}\geqslant\\{0\,|\,\overline{\infty}\\}$ (Fact 2).

Proving that
${\tikzpicture\clip(4.83,0.365)rectangle(5.92,1.452);[linewidth=2.pt,fill=black,fillopacity=1.0](5.909743589743589,0.7492877492877543)--(5.8697672384446085,0.7640785613681009)--(5.818003314938593,0.7813367971648741)--(5.7620276587975034,0.7976755667777837)--(5.7101105279150195,0.8107406115539613)--(5.663902303911093,0.8207230891207606)--(5.611608514153138,0.8301916514750758)--(5.559344060086461,0.837750517407593)--(5.516355816426169,0.8425615395020323)--(5.480951393364173,0.8455802578138449)--(5.442490031775766,0.8479002945794916)--(5.405586554111292,0.8491913947512248)--(5.371851851851851,0.8495726495726554)--(5.338117149592409,0.8491913947512248)--(5.301213671927935,0.8479002945794916)--(5.262752310339529,0.8455802578138449)--(5.227347887277532,0.8425615395020323)--(5.18435964361724,0.837750517407593)--(5.132095189550563,0.8301916514750758)--(5.0798013997926095,0.8207230891207606)--(5.033593175788682,0.8107406115539613)--(4.981676044906198,0.7976755667777837)--(4.92570038876511,0.7813367971648741)--(4.873936465259094,0.7640785613681009)--(4.833960113960112,0.7492877492877543)--(4.833960113960112,1.0683760683760717)--(4.873936465259094,1.053585256295725)--(4.92570038876511,1.0363270204989519)--(4.981676044906198,1.0199882508860423)--(5.033593175788682,1.0069232061098647)--(5.0798013997926095,0.9969407285430654)--(5.132095189550563,0.9874721661887502)--(5.18435964361724,0.979913300256233)--(5.227347887277532,0.9751022781617937)--(5.262752310339529,0.9720835598499811)--(5.301213671927935,0.9697635230843344)--(5.338117149592409,0.9684724229126012)--(5.371851851851851,0.9680911680911706)--(5.405586554111292,0.9684724229126012)--(5.442490031775766,0.9697635230843344)--(5.480951393364173,0.9720835598499811)--(5.516355816426169,0.9751022781617937)--(5.559344060086461,0.979913300256233)--(5.611608514153138,0.9874721661887502)--(5.663902303911093,0.9969407285430654)--(5.7101105279150195,1.0069232061098647)--(5.7620276587975034,1.0199882508860423)--(5.818003314938593,1.0363270204989519)--(5.8697672384446085,1.053585256295725)--(5.909743589743589,1.0683760683760717)--cycle;}_{\infty}$
is the largest negative game of all is analogous. ∎

To better understand the third notable fact, recall that $\uparrow^{2}$ is
infinitesimal with respect to $\uparrow$, i.e., any number of copies of
$\uparrow^{2}$ is less than a single copy of $\uparrow$. But, it is important
to note that this is entirely different from saying that having two copies or
just one copy of $\uparrow^{2}$ is irrelevant. If only $\downarrow$ is
present, Left loses all disjunctive sums of the form
$n.\\!\\!\uparrow^{2}+\downarrow$. However, in other sums, having two copies
instead of one can be the difference between winning or losing. In fact, we
have $2.\\!\\!\uparrow^{2}>\uparrow^{2}$. Furthermore, in the classical
structure $\mathbbm{N}\mathbbm{p}$, if $G>0$, then $2.G>G$.

The terminating games $\infty$ and $\overline{\infty}$ bring the existence of
nontrivial finite indempotents. Having many or few copies of a pathetic tiny
is irrelevant since, at the slightest opportunity, Right disposes of them all
at once through a sequence of checks.

######  Theorem 63 (Pathetic Idempotents).

The games
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and
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are idempotents.

###### Proof.

We prove only the equality