The fight never runs backwards

A fight has two ends. Left moves first, both players keep grabbing until neither wants to move again, and the score settles at a number; that number is the left stop. Right moves first and the same thing happens at the other end; that is the right stop. The first essay about them defined the pair and found them already drawn, as the feet of the two walls on every thermograph this site has printed.

This essay is about the one relation that holds between them everywhere, without a side condition:

The left stop is never below the right stop. Whatever the position, LS(G) ≥ RS(G).

That is not an observation about the positions people happen to write down. It is what makes a mean value definable at all, it is why a temperature is a height rather than a signed quantity, and it is the formal content of “both players are trying to improve their own position” — a sentence that sounds like a modelling assumption and is a theorem about every short game.

Why the inequality cannot fail #

The recursion is the one from rung one, and it is two lines. If a position is a number, both its stops are that number — nobody moves in a number when anything else is available, which is what numbers avoid numbers says, so the fight is already over. Otherwise LS(G) is the largest RS among Left’s options and RS(G) is the smallest LS among Right’s.

Notice what the two lines are made of. At every level Left takes a maximum and Right takes a minimum, over trees that interleave as the turn passes back and forth. The inequality is what survives that interleaving: a maximum taken by a player pushing the score up cannot land below a minimum taken by a player pushing it down, when both work through the same position with the same alternation.

The proof in the literature is an induction on birthday, and it has a second half worth stating in the same breath, because it is what this essay spends most of its length on. The two stops are equal exactly when the position is infinitesimally close to a number. So the case of equality is a complete characterisation of the positions with nothing at stake, and strict inequality a complete characterisation of the fights.

That gives the pair a job description. A position with LS > RS is hot — there is ground between the two ends and whoever moves first takes some of it. A position with LS = RS is cold: the ends coincide, the score is settled, and whatever is still going on there is smaller than every positive number. The survey below finds no third case over 1,780 values, and finds the boundary sitting exactly where a positive temperature does.

The plain case, where the position is its own stops #

The easiest positions to check are the ones where the notation hands the stops over. A switch is a position both of whose options are numbers: {a | b} with a above b. Left moves to a, a number, which stops immediately; Right moves to b, likewise. So LS = a and RS = b, and the inequality is the condition a > b that made it a switch in the first place.

For these the whole apparatus is a rearrangement of two numbers a reader can already see: gap, mean value, temperature. That is why switches and mean value could be written before stops were named, and why a switch proves nothing about the general claim. The interesting case is an option that is not a number — where the inequality still holds, and almost everything else a reader might expect stops holding.

Two routes, 1,780 values, and nothing that disagreed #

There are two ways to get a stop out of a position, and they were written independently.

The recursion. LS(G) = max over Left's options of RS(G^L), RS(G) = min over Right's options of LS(G^R), with a number returning itself — a search over the game tree that halts when it reaches numbers.

The thermograph. Build the wall diagram, which plots what each player gets when a tax of t is charged on every move, and read it across at t = 0. Charging nothing is charging nothing, so the foot of Left’s wall is what Left gets moving first. That is the same quantity by a completely different construction — a cooling argument rather than a playout.

Having both is the point: they can be run against each other. The machinery that draws the hero figure computes every bar’s endpoints by both and refuses to draw if any pair differs by so much as a rounding step. The survey ran the same comparison much wider. Its universe is 1,780 distinct values, deduplicated by canonical form: the 22 born by day two, the 1,474 born by day three, every {a | b} whose options are halves between −3 and 3, the ten small infinitesimals from up to down-star, the tinies and minies at 1, 2 and 4, every sum of two day-two values, and every infinitesimal added to a small switch.

Over all 1,780 the counts are these. Values with LS < RS: zero. Disagreements between the two routes: zero. Sensitivity of the wall feet to how high the thermograph is drawn: zero mismatches, at heights 4, 5, 8 and 16. Cold positions, where LS = RS: 460. Fights, where LS > RS: 1,320. The two add to 1,780, so there is no third case and no value the machinery declined to classify.

Those zeros are findings rather than an absence of news. The survey turned up no counterexample and no disagreement, which is what a check looks like when the thing being checked is true. A test that has never rejected anything proves nothing, so the same machinery is pointed at claims it does reject, one section below; a test that could have rejected something and did not is evidence, and 1,780 is how much of it there is.

One piece of housekeeping goes with that. stops() has been sitting in the site’s evaluator since it was written, with a docstring explaining what the pair cannot see, and nothing on the site called it — every earlier figure went to the thermograph directly. This essay’s figures are its first caller, which is how the two-route comparison became available at all, and a reminder that a function nobody calls is a function nobody has checked.

Two arithmetic shortcuts, and what breaks them #

There are two obvious things to do with a pair of numbers: subtract and average. Both have a familiar identity attached, both identities hold on switches, and both are false in general. The survey says by how much.

“The gap between the stops is twice the temperature.” It holds on 960 of the 1,320 hot positions and fails on 360 of them. Every failure has an option that is itself a fight.

The mechanism is visible in the drawing. A wall bends where a decision changes hands, and a bent wall does not reach the base line at the slope a straight one would; the gap at the bottom and the height of the meeting point are then two measurements rather than one.

The survey’s sharpest case has fights on both sides, and it is worth drawing rather than describing, because it is where the shortcut is furthest out.

“The position is worth the average of its stops.” It holds on 1,426 of the 1,780 and fails on 354. That is not a rounding error and it is not an approximation worth keeping; it is a fifth of the universe.

The mean value is always between the stops, and that follows from the inequality this essay is about: the interval is non-empty precisely because LS ≥ RS. Where the mean sits inside it is a separate question, and the midpoint is the answer only when the fight is plain. The largest gap anywhere in the survey is 6, at the switch {3 | −3}, whose walls are as straight as the {4 | 0} at the top of this page and whose every quantity can be read off its two options — that shape is what makes both shortcuts look like theorems.

Cold does not mean a number #

Now the equality case, where the pair of stops ceases to be a summary and becomes a blindfold.

A position is cold when its two stops agree. The natural conclusion is that it is a number: the fight has nowhere to go, the pair reports one value, and that value is what the position is worth. The survey refuses this flatly. Of the 460 cold positions, 27 are numbers and 433 are not.

The honest replacement is weaker. Every one of those 433 was tested against the number its two stops agree on, by playing the difference both ways: is G − x smaller than 1/1024, and is it larger than −1/1024? All 433 passed, and none failed. So a cold position here is a number plus something smaller than a thousandth — an infinitesimal — rather than a number. That is a test against one small number and not a proof about every one, which is exactly the gap between a check and a theorem.

The other half of the equality case is the one that closes the theorem. No cold position in the survey has a positive temperature, and no fight has a temperature of zero or below. Over all 1,780 values, LS > RS holds exactly when the temperature is positive — a biconditional, tested in both directions, with no exception on either side. So the pair of stops and the temperature are two readings of the same boundary, and a reader who has one has the other.

What no pair of numbers can see #

The sharpest statement of the blindness is not a list of values with equal stops but the observation that they are not even in the same outcome class as each other.

Take the four positions 0, ↑, ∗ and ↓ and they fall into four different outcome classes: nought is a second-player win, up is a win for Left whoever starts, star is a win for whoever moves first, and down is a win for Right whoever starts. Three of those four classes are comparisons with nought; the fourth — first player wins — is a position confused with nought, neither greater, smaller nor equal. All four have stops 0 and 0.

That last sentence is the whole of what a pair of stops cannot do. The stops are constant across the entire outcome classification — the same value on a second-player win, a Left win, a Right win and a first-player win. There is no coarser failure available: a summary that varies not at all across the property being asked about is not a weak summary, it is silent.

The instrument that does see the difference is comparison, and comparison is not an inspection. To ask whether G is at least H, both are played as one sum with H negated, and the question becomes who wins that third position — a search, and comparing two positions means playing a third is the essay about what it costs.

The two figures together are the argument. The stops declare four pairs identical and the difference game separates all four, and the separations are not subtle — they include a confusion, the relation with no counterpart in arithmetic at all, and the reason the order on games is partial rather than total.

Where the model stops #

The inequality is about short games. The recursion needs the options to be simpler than the position and eventually to be numbers. A loopy position has no such bottom and no stops in this sense, so the theorem has nothing to say there — not because it fails but because its terms are undefined.

Equality is characterised only up to a test. “Cold implies infinitesimally close to a number” is the theorem; what was run here is “cold implies within 1/1024 of a number”, on 1,780 values. A cold position a full thousandth away from its stops’ common value would have been caught, and none was — and a position outside the survey remains outside it.

And the pair does not survive addition. The stops of a sum are not the sums of the stops, because a player cannot move first in two components at once. The mean value adds and the stops do not, which is the division of labour that makes both quantities worth carrying.

What the picture cannot show #

Every figure above draws a stop as a mark on a horizontal line, and a mark on a line is a position on the number line — which is exactly what an infinitesimal is not. Up is above zero and below every positive number, and the drawing has no room for that anywhere. Placing it at zero is a compromise made knowingly, and it is why the figures that separate the infinitesimals do it by comparison and by canonical form rather than by position.

The second thing no picture shows is the quantifier. The claim is about every game, and twenty-two bars with 1,780 checks behind them are still a finite collection; what makes it a theorem is an induction no diagram contains. The figure’s contribution is that it is built to fail — both stops twice, and a throw rather than a drawing if a bar ever comes out backwards — so the picture is a running check and not an illustration.

The third is the most ordinary. Two bars of gap 1 look identical, and one may have temperature 1/2 while the other has temperature 1. The 360 positions where the gap is not twice the temperature are, in the hero figure, indistinguishable from the 960 where it is.

The convention, named #

Normal play throughout: the player who cannot move loses, and the fight ends when neither player wants to move rather than when the board is full.

The definition of a stop ends “until somebody is left facing a number, and the score is that number”, and facing a number is a normal-play idea: it rests on numbers being positions nobody wants to touch, which is a theorem of normal play and not a rule of the game. Under misère play the last move is a loss rather than a formality, numbers do not behave like numbers, and the whole account of stopping has to be rebuilt from the definition up. The inequality above carries no weight whatever on the other side of that convention.

The second convention is that a stop is a score, in a theory whose official answer is a winner. That is what makes stops useful for reading a Go endgame and it is also the seam: a genuine scoring game pays out on a count, and counting at the end changes everything.

Where the ladder goes next #

The mean value theorem. The mean value has been quoted throughout as what a position is worth once the fighting is over, and what actually defines it is the behaviour of n copies: the left stop of a pile of n grows like n times the mean plus a bounded error, and that bound is the theorem. The same fight, eight times over draws the pile; the rung that proves the bound, and measures how fast the error stops growing, is not written.

Confusion intervals. The pair of stops is one interval; a position is also confused with every number strictly inside another. The relation between “the stops straddle x” and “the position is confused with x” is a rung of its own — where confused is not the same as unknown meets the machinery of this essay.

Orthodox accounting. Stops are what a Go player computes under other names, and the orthodox account of a hot endgame is a sequence of stop calculations with the ambient temperature carried along. That is the applied rung: what a player does with the pair when it is one component of a board.

And stops under misère play. The convention section says the account has to be rebuilt; nobody here has rebuilt it. Whether a misère playout even has a well-defined pair of ends, and what an inequality between them would mean when misère play has no negatives, is where this anchor meets the part of the subject that does not work.