The numbers it is confused with

Two positions are confused when neither is at least as good as the other. Between a position and a number that relation has a shape, because the numbers are totally ordered: as a number runs up the line, the position is greater than it, then confused with it, then less than it, and never anything else.

So a position picks out an interval of numbers, and the natural question is what the ends of the interval are.

The two coincide, and the way in which they fail to coincide is the whole of the essay.

What the stops already say #

Where the fight stops defines the pair. The left stop $LS(G)$ is what Left gets by moving first and both players fighting on until somebody faces a number; the right stop $RS(G)$ is the same for Right. The fight never runs backwards establishes that $LS(G) \ge RS(G)$, always, so the pair really is an interval and not a pair of numbers in some order.

The rule that suggests itself is immediate. If a number $x$ sits above the left stop, then Left cannot reach it even by moving first, so $G < x$. If $x$ sits below the right stop, then Right cannot reach it, so $G > x$. And between them neither player can force the comparison, so $G$ and $x$ are confused.

That is a claim with three clauses, and it has never been checked here.

The census #

Every value born by day three — 1,474 of them — against every quarter from $-3$ to $3$. Quarters, because every stop of a day-three value is a multiple of a quarter and a finer grid adds comparisons without adding cases. That is 36,850 comparisons, each run as a real comparison of games rather than as an application of the rule.

Four thousand and forty-six comparisons put the number strictly between the stops, and all 4,046 come back confused. Thirty thousand two hundred and eight put it strictly outside, and all 30,208 come back decided, and decided the way the rule says.

Not one exception in 34,254 comparisons. The remaining 2,596 are the ones where the number sits exactly on a stop.

The choice of quarters is the one arbitrary thing in that paragraph, so it is worth removing. Run the same census against every eighth and the comparisons nearly double — 72,226 of them — while the count that matters does not move at all.

The two ends #

At $x = LS(G)$ the rule has nothing to say, and the reason is worth stating precisely rather than as a caveat. Consider the difference $G - x$. Its left stop is $LS(G) - x = 0$. So the fight in $G - x$ ends level, and everything that decides the comparison lives underneath the number — in an infinitesimal, which is what a stop is constitutionally unable to see.

Both answers occur. Of the 2,596 edge comparisons, 2,151 come back confused and 445 come back decided.

$\{\ast 2 \mid 0\}$ has both stops at zero and is strictly less than zero. $\{\ast 2 \mid -1\ast\}$ has stops $0$ and $-1$ and is strictly greater than $-1$. $\{\ast 2 \mid -\tfrac12\}$ has stops $0$ and $-\tfrac12$ and is confused with $-\tfrac12$. Three positions with a $\ast 2$ in the same place, three different answers at the endpoint, and the pair of stops is identical in structure across all three.

So the confusion interval is the open interval $(RS(G), LS(G))$, and its closure is decided elsewhere.

Positions confused with nothing #

Eighty-one of the 1,474 values are confused with no number on the grid at all. Fifteen of those are numbers, which is unsurprising: a number is comparable with every number.

The other sixty-six are not numbers, and they are the interesting half. $\{\ast 2 \mid 0\}$ is one. So are $\{\uparrow \ast \mid \downarrow\}$, $2 \cdot \! \downarrow \ast$ and $-1\downarrow$. Each has its two stops at the same place, so the open interval between them is empty — and each is a genuine game rather than a number, distinct from the number its stops name, distinguishable from it only by something no number reaches.

That is what an infinitesimal is, stated as a fact about comparison rather than as a definition: a position whose confusion interval is empty and which is nonetheless not the number at its ends.

The width, and what it is #

The distance between the stops takes eight values across the day: 352 positions have a width of zero, 232 have a half, 581 have one, 94 have one and a half, 186 have two, twelve have two and a half, sixteen have three, and exactly one has four.

The width is not an accident of the construction. Compared against the temperature — the height at which the two walls of a thermograph meet — the relation is sharp:

$$LS(G) - RS(G) \;\le\; 2\,t(G)$$

on all 1,474, with equality on 1,124 of them and never a violation. The 350 where the interval is narrower than twice the temperature are the positions with follow-ups, where the walls bend before they meet and the mast is reached above the height a straight pair of walls would predict.

That gives a reading of the temperature nobody usually offers. It is half the range of numbers a position is confused with — half the width of the band inside which a player cannot say whether the position is better or worse than a plain score. A hot position is one that is confused with a lot of numbers, and a cold one is confused with few or with none.

The midpoint is not the mean, except when it is #

The interval has a centre, and the centre looks as though it ought to be the position’s mean value — the number the position is worth when it is played many times over, the height at which the two walls of the thermograph meet. It is a tempting identification because on a plain switch it is exact: $\{a \mid b\}$ has stops $a$ and $b$, mean $\tfrac{a+b}{2}$, and the midpoint of its confusion interval is the same number.

Run it over the day and the identification holds 1,124 times out of 1,474 and fails 350 times. The failures are large enough to matter: $\{2 \mid \{1 \mid -1\}\}$ has stops $2$ and $1$, so its interval is centred at $\tfrac32$, and its mean is $1$ — half a point out, on a position whose whole interval is only one wide.

The 350 are exactly the 350 from the previous section, and that is the fact worth carrying rather than the count. Setting the numbers aside, whose temperature the comparison is not defined for, a position’s midpoint is its mean precisely when its interval is as wide as its temperature allows:

$$LS(G) - RS(G) = 2\,t(G) \iff \tfrac{1}{2}\bigl(LS(G) + RS(G)\bigr) = m(G)$$

on all 1,459 non-numbers, with no exception in either direction.

Which says that the two readings of the interval are one reading. A position whose walls rise straight from its feet is summarised completely by its stops: the midpoint gives the mean, the half-width gives the temperature, and $\{m+t \mid m-t\}$ is the position back again. A position whose walls bend is not, and the bend is where the information goes — the mean sits off centre because a follow-up pulls one wall inward before the two meet, and no amount of looking at the feet recovers where it pulled to.

So the interval carries the mean when the position was a plain fight all along, and loses it exactly when there was something underneath worth knowing.

The practical reading is a warning about summaries. A player who records a position as its two stops has, on three quarters of the day, recorded its mean and its temperature as well and lost nothing worth having. On the other quarter the same note is quietly wrong about what the position is worth, and it is wrong in one direction only. In all 175 cases where the midpoint sits above the mean, Right holds an option that is not a number — the bend is on Right’s wall — and the summary flatters Left; in all 175 where it sits below, the follow-up is Left’s and the summary flatters Right. The error runs against the player whose side had the extra move in it, because a follow-up is a reason the fight does not end where the stop says it does.

The comparison itself, which is a search #

Every mark on every figure above is one comparison, and a comparison is not a look. $G$ and $x$ are compared by asking who wins $G - x$, which is a search over a game tree built from both — the same machinery that comparing two positions means playing a third describes.

The 36,850 comparisons here are therefore 36,850 searches, and the reason a rule is worth having is that the rule replaces every one of them with two subtractions. That is the practical content of the census: after it, a reader with the two stops in hand can answer is this position better than three quarters? without running anything, everywhere except at two numbers.

What the interval leaves out #

An interval of numbers is a poor summary of a position and the census makes the poverty measurable. Take the 352 values whose stops coincide: as far as the interval is concerned they are all the same object, and they include $0$, $\ast$, $\uparrow$, $\downarrow$, $\ast 2$ and hundreds more, of which four are wins for Left, four for Right and the rest are decided by whoever moves.

Take a wider case. $\{2 \mid 0\}$ and $\{2 \mid \{1 \mid 0\}\}$ have very different follow-ups and the second is worth strictly less than the first; both are confused with everything strictly between their right and left stops, and those intervals overlap almost completely. Two positions with nearly the same interval, and one is better than the other.

So the interval is a genuine invariant and a lossy one: it says exactly which numbers the position cannot be compared with, and nothing about which of two such positions to prefer.

Underneath, the interval is working at the level of the four outcome classes and no finer: it is the set of numbers x for which $G - x$ lands in the first-player-wins class, and a class is the coarsest summary of a position there is. Every distinction finer than that — which of two confused positions to prefer, what a follow-up is worth — is invisible to it by construction rather than by accident.

The endpoints, counted, and a symmetry that has to be there #

Taking every value of the day and each of its two stops in turn gives 2,948 endpoint comparisons. Of those, 2,422 come back confused, 30 come back equal — those are the values that are the number at their stops — and the remaining 496 are decided.

Split by direction, the 496 are 248 greater and 248 less.

That equality is not a coincidence and it is not evidence of anything either, which is worth separating. The day-three values are closed under negation: if $G$ is born by day three so is $-G$, because negating a game negates its options and changes no birthday. Negation exchanges the two stops, reverses every comparison, and therefore carries each of the 248 cases on one side to a case on the other. So the two counts are forced to agree, and their agreement is a check that the enumeration has not lost anything rather than a fact about confusion.

It is the kind of check worth building in. A census that came back 248 and 247 would be reporting a missing value, and nothing else in the run would have said so.

The widest interval on the day #

One value of the 1,474 has an interval four wide, and it is $\{2 \mid -2\}$ — the fight worth two points to whoever takes it, with nothing underneath. It is confused with every number strictly between $-2$ and $2$, which on the grid used here is fifteen of the twenty-five.

Sixteen values have an interval of three, twelve have two and a half, and 186 have two. Then the numbers climb steeply as the intervals shrink: 581 values have an interval exactly one wide, 232 have a half, and 352 have none at all.

The shape of that distribution is a fact about the construction rather than about confusion. Each day builds new values out of old ones, and a wide interval needs two options far apart, which needs a day to have produced them; so the widest positions are the ones assembled from the extremes of the previous day and there are correspondingly few of them. What the distribution says in practice is that most positions are confused with very few numbers, and a reader who thinks of confusion as the normal state of a comparison has the proportions the wrong way round.

What a reader guesses, and why it is nearly right #

The guess a reader makes on being shown the two stops is that the interval is closed: that the position is confused with everything from one stop to the other inclusive. It is a natural guess, because the stops are the numbers the fight actually reaches, and a position that can be driven to exactly $LS(G)$ looks as though it ought to be confused with it.

The guess is right 2,151 times in 2,596 and wrong 445 times, which is a hit rate high enough to survive any amount of casual checking and low enough to be useless as a rule. Worse, the failures are not spread thinly across odd positions: they cluster on exactly the positions whose stops coincide, which is to say on the infinitesimals — the class where a reader most wants a rule and where a rule of this shape cannot exist.

The opposite guess, that the interval is open at both ends, is wrong 2,151 times. So the honest statement is the one the census supports and neither guess does: the open interval is confusion, the exterior is decided, and the two endpoints are a separate question with a separate answer.

What the picture cannot show #

Every figure above draws numbers on a line and a position as a band across it, and the drawing invites a reading it cannot support: that the position is somewhere in the band, and that the band is a range of uncertainty about where.

It is not. A position confused with everything between $-2$ and $2$ is not a number that has yet to be pinned down; it is an object that is not a number at all and never will be, and running more computation narrows nothing. The band is the set of numbers a comparison refuses, not the set of numbers a better comparison would choose between.

The distinction has a practical edge. Two positions with the same band can be strictly ordered with respect to each other — the order among games is much finer than the order among the numbers they are confused with — so a player choosing between two hot fights gets no help from the bands and considerable help from comparing the two positions directly. The band is a statement about a position and the numbers; it is not a statement about the position and the rest of the board.

The convention, and the two theorems underneath #

Normal play throughout, and the stops are defined by the recursion halting at a number, which is not a convention but a theorem: a position with no options is zero, and the recursion that computes a stop is well-founded because play has to end.

The second theorem doing quiet work is the translation rule — that adding a number moves both stops by exactly itself. Without it the whole census would have to be run separately for every number rather than once per position, since $G$ compared with $x$ is $G - x$ compared with zero, and the stops of $G - x$ are the stops of $G$ shifted. Everything above is a statement about the interval around zero, translated.

Where the ladder goes next #

The stops anchor now has three rungs: what they are, that they never cross, and which numbers sit between them. The rung above is the closure. The two ends are decided by the infinitesimal part of $G - x$, and this essay reports the three answers without giving a rule that predicts which of them arrives. That rule exists — it is a comparison against zero of an all-small game — and it is the point at which the temperature theory hands over to the theory of the very small.

Two neighbours are worth the trip. What a number does to a fight is the same computation run the other way round, adding numbers rather than comparing with them, and it finds the same two endpoints failing for the same reason. And confused is not the same as unknown is where the fourth relation is established as a fact about a pair rather than a gap in the machinery — which is what makes an interval of confusion an object rather than an admission.