One king, and two files to be in

Both rungs below this one end by adding files up, and both of them say, in a sentence near the bottom, that adding requires the files not to interact. The first puts it this way: a move in one file has to change nothing in another; on a real board that holds when the pawns are blocked and the kings are elsewhere.

And the kings are not elsewhere. A pawn ending has two kings in it, they are the pieces doing the work, and a king is one piece. A king holding the a-file is not holding the h-file, and that is not a fact about chess so much as a fact about counting: there are several files and one king, and one does not go into several.

This rung is the price of the assumption, measured.

The model, stated exactly #

Nothing here plays chess, and being exact about the model is the whole of what makes the measurement mean anything.

A position is a set of files, each with a contested square: whichever pawn advances into it first stops the other one for good. That is the clause the rung below shows produces switches, and it is chosen here because a contested square is precisely the kind of file a king is needed for.

Each side has one king, standing at one of the files. Two clauses attach to it:

A side may advance in a file only while its own king is at that file. Otherwise the pawn is unsupported and the advance is not available.

A side may instead move its king to another file, which costs a move and does nothing else. The supply of such moves is stated and finite.

A side with neither an advance nor a king move loses, by the normal-play convention.

That last clause about the supply is the one modelling decision worth defending. Real kings walk about indefinitely, and a model in which they can would have a position graph with a cycle in it and no value at all — which is the next rung and not this one. Here the supply is finite and stated, and the point of stating several supplies is that the answer depends on it.

The control, which is the row that makes the rest a measurement #

Set the supply to nothing. Neither king may move; each stands where it stands; each side can advance only in the file its own king already occupies, for ever.

On all thirty-six configurations the sum of the parts is exactly right.

That is not a small observation and it is not a lucky one. A king that cannot move is not a shared resource — it is a permanent feature of the file it stands on, like the pawns. Every file can then be priced on its own, the prices add, and the ending is the sum of its files in the strict sense the theory requires.

So whatever goes wrong below is caused by the king having somewhere else to be, and not by the king existing. A reader who expected the failure to be about kings interfering with pawns has the mechanism wrong; the mechanism is that one piece cannot be in two places, and a piece that never moves is never asked to be.

That is worth separating from a second thing the control does not say. It does not say the files are simple when the kings are frozen — a file with both kings on it is still a fight, still a switch, still a position with a temperature — only that the fight can be priced without looking at the other file. Independence is not simplicity, and the two are confused constantly because the games this site can draw tend to have both.

What a reader would write for the sum #

The comparison needs a second object, and it has to be the object a player actually computes rather than a straw one. Here is the pricing rule the figures use, and it is the reading anybody would perform.

A file where only one king stands belongs to that side. Its pawn advances as often as the gap allows, the other pawn is unsupported and never advances at all, and the file is worth the count — an integer, which is exactly what the rung below says a file with one side stopped is worth.

A file where both kings stand is a fight. Both pawns are supported, the square is contested, and the file is worth a switch.

A file neither king has reached is dead. Nobody can advance, and it is worth nothing.

Those three cases are what a player means by “this file is mine, that one is a fight, and that one is nobody’s”. The pricing is exactly right about each file considered alone, and the whole question is what happens when three exactly-right prices are added together.

How wrong it gets #

With one spare king move each, twenty of thirty-six configurations still come out exactly. Sixteen do not, and six of the thirty-six name the wrong winner — the sum is not merely inexact, it puts the position in the wrong outcome class.

Raise the supply to two and nothing further changes: the same twenty, the same six. The damage is done by the king having a choice, and a second choice adds nothing to it.

Then take a third file. There are two hundred and forty-three configurations, thirty-nine of them come out exactly, and a hundred and twenty-six name the wrong winner — more than half.

The shape of that is worth naming. The failure does not creep in as positions get complicated; it arrives whole, at the first configuration where a king has a choice, and then gets worse as the number of files a king cannot be at grows. A reader looking for the boundary of the decomposition will not find a threshold, because there is not one.

The smallest position it gets wrong #

Two files with gaps of one and one, both kings at the first file, one spare move each.

The sum of the parts says: the first file has both kings on it, so it is a fight worth a move either way and comes to a star; the second file has neither king, so it is dead and worth nothing; total, a star, and whoever moves wins.

The joint position is worth nought, and whoever moves loses.

What the sum missed is a single line of play. The mover advances in the contested file, claiming the square; the opponent, unable to answer there, spends its spare move walking its king to the second file; and now the second file is not dead, because a king has arrived at it. The pricing called that file dead by reading the position as it stood, and a king move is exactly a move that makes a dead file live.

That is the whole mechanism in one sentence. A component’s price depends on which components the kings are at, and a move changes which components the kings are at. So a move in one part changes the value of another part, which is the definition of the parts not being independent.

It is worth noticing which direction the error runs, because a reader may expect a decomposition to be optimistic or pessimistic and it is neither. The sum called this position a first-player win and it is a second-player win; elsewhere in the table the error runs the other way. There is no correction to apply, no bias to subtract, and no sense in which the sum is a bound on the truth — which distinguishes it sharply from the approximations the complexity field prices, where a rule that is wrong in one direction is still worth something.

The failure is about the pricing, not about the theory #

It is worth being careful about what has been shown, because “the sum is wrong” is a sentence that can be read as an indictment of the disjunctive sum and it is not one.

The disjunctive sum is a theorem about games whose components do not interact. Nothing here contradicts it, and nothing could: the theorem’s hypothesis is false of these positions, so the theorem says nothing about them, and a false conclusion drawn from a false hypothesis is a mistake by the reader.

What is being measured is the cost of the reading — how much a player loses by treating a position as a sum when it is not one, on positions small enough to check both ways. That is a question about practice rather than about theorems, and it has a number for an answer.

The number is worth setting beside the one the sums field measures for a nearly independent split, where the components interact a little and the sum is usually close. Here the components interact through a single shared token and the sum is wrong about the winner on more than half of the three-file configurations. Shared-resource coupling is not a mild version of independence. It is a different regime.

What a joint solve costs instead #

The alternative to the sum is to solve the whole thing at once, and the reason nobody does that on a real board is arithmetic rather than principle.

The joint position’s state is: the gap in every file, which pawn has been stopped in every file, where each king stands, and how many waiting moves each side has left. Two files with gaps to three and two spare moves each is a few thousand states — instant. Eight files is that number raised to the fourth power with sixty-four king placements on top of it, and a real ending has more clauses in it than any of this.

That is the same trade the tree and the graph is about, and the same one that makes Zermelo’s theorem true of chess and useless about chess. A sum is cheap and is a claim; a joint solve is exact and is exponential; and the interesting question is never which is better but how much the cheap one costs on the positions anybody meets.

There is a third option that the theory does supply and that nothing here uses, and it is worth naming so it is not mistaken for an absence. A position that fails to decompose at this moment may decompose later: play a few moves, the kings commit, the spare moves run out, and what is left is a genuine sum. That is what finding the parts is about in Amazons, where the board is one fight until the arrows cut it and the cutting is something the play produces rather than the analyst assumes. A pawn ending does the same thing in reverse — it starts almost decomposed and the kings are what stops it being.

Why chess players decompose anyway #

A player looking at that table might reasonably conclude that pricing a pawn ending file by file is a mistake. Players do it constantly and are not usually wrong, and the reason is worth stating because it is the condition under which the reading is safe.

The reading is exact when the kings have nothing to decide. That is the control row, and it is not an artificial case — it is what a blocked ending is. If the pawn structure is locked and each king is tied to defending one part of it, neither king has a spare move that changes anything, and the files really are independent.

The reading fails when a king has a spare move and there is somewhere for it to go that matters. That is a position where the kings are manoeuvring, and a player who has reached that stage has stopped counting tempi and started calculating lines, which is the practical form of abandoning the decomposition.

So the folklore is calibrated. Endgame manuals say “count the spare moves” about locked positions and switch to variations the moment the kings are loose, and that is not two habits — it is one habit with the hypothesis attached.

What the manuals do not say, and what the table above supplies, is the price of getting the hypothesis wrong. A player who counts tempi in a position where a king still has a choice is not making a small error with a small consequence; on more than half the three-file configurations in range the count names the wrong winner. The habit is safe and the boundary of the habit is sharp, and knowing where a rule stops is worth as much as knowing the rule.

What the picture cannot show #

The model has one king each and a real ending has one king each with a geography. A king walks in two dimensions, reaches some files faster than others, and can occupy squares that support two files at once. None of that is here: a king is at a file or it is not, and moving between any two files costs one move. So the model gets the shape of the coupling right and the distances wrong, and every distance in a real ending is what the whole calculation turns on.

And the supply of waiting moves is stated rather than derived. In a real position the number of spare moves a side has is a fact about the pawn structure — pawns that can advance without weakening anything — and it changes as the game goes on. Here it is a number handed to the solver, and the measurement holds it fixed while it varies the rest.

The honest description: two hundred and forty-three configurations of a model with one shared token, solved twice, with the disagreements counted. Whether a real ending’s coupling has this shape is Elkies’s kind of claim and not this site’s.

The convention, named #

Normal play again: a side with no advance and no king move loses.

The clause deserves a second look here, because in this model a side can genuinely run out while pawns remain on the board. A side whose king is at a file where its own pawn has been stopped, with no spare moves left, has no legal move at all — and loses on the spot, though the position is full of pawns.

That is a zugzwang in the sharpest available form, and it is the mechanism the rung below is named after. What this rung adds is that the zugzwang can be arranged: the opponent’s king-walk is what leaves a side with nothing to do, so running out of moves becomes something one player does to the other rather than something the structure decides in advance.

Outside the model that substitution needs the same defence it always needs. Chess has no rule that a player with no good move loses; what it has is a class of position in which every legal move loses, and the normal-play convention is a model of that class and of nothing wider.

The surprise: the sum fails at nought files of coupling #

The result a reader would expect from a measurement like this is a curve — a sum that is exact on simple positions, drifts as the coupling grows, and becomes useless at the far end. There is no curve.

At zero spare moves the sum is exact on every configuration. At one spare move it is wrong on sixteen of thirty-six and wrong about the winner on six. At two spare moves it is wrong on exactly the same ones. The transition is a step, and the step happens at the first move that could have gone somewhere else.

The general shape is worth carrying past chess. A decomposition is not approximately valid when the parts are approximately independent. Independence is a property of the move rule — either a move in one component can change another or it cannot — and a single move that can is enough to make the sum a guess. The size of the coupling controls how badly the guess does, and the existence of the coupling controls whether it is a guess at all.

That is why splitting a position is a claim about the position rather than a convenience, and it is why the claim has to be argued for each position rather than assumed once for a game. Kōnane makes the same point from the other side: there no width of gap is a wall, because a stone can hop into the gap and out again, and the coupling is a property of the move rule rather than of how far apart the pieces are.

And it says what to look for in a game one has not met. Not “how strongly do the parts interact” — that question has no scale attached to it — but “is there a resource the whole position shares”. A shared token, a shared budget, a piece that has to be somewhere: any of those and the sum is a guess, however weak the interaction looks.

Where the ladder goes next #

The kings here have a stated, finite supply of waiting moves, and the reason for stating it is that without it the model has no values to compare at all.

That is the rung above. A king with unlimited waiting moves is a king that can shuffle for ever, a position graph with a cycle in it, and a recursion with nowhere to bottom out — which is a fortress, the one class of pawn ending every manual has a chapter about and no value theory reaches. Chess has a rule for it, the rule is a counter, and the counter changes the answer.