What has to break before a pawn is worth a number

The rung below this one evaluates a blocked pawn file and finds stars, zeros and multiples of up. It then proves, in about four lines, that nothing else was ever available: in that model both players can advance exactly when the other can, so the two sides run out of moves together, and a game with that property is all-small — smaller in absolute value than every positive number there is.

That proof is worth reading twice, because of what it is a proof about. It says nothing about pawns. It is a statement about a clause in the move rule, and the clause is one sentence long:

White may advance whenever the gap is at least one. Black may advance whenever the gap is at least one.

Those are the same condition. Everything exotic about the values — that they are infinitesimals, that no material count sees them, that a chess player meets them under the name tempo — follows from the two conditions being the same condition, and from nothing else.

So the way to find out what else a pawn structure can be worth is not to look for a cleverer file. It is to break that sentence, and a blocked pawn structure breaks it in exactly two ways.

The two clauses, and both of them are ordinary chess #

A pawn can be stuck. A pawn with a friendly pawn or piece directly in front of it cannot advance at all — doubled pawns produce this constantly, and so does a knight or a bishop parked on the square ahead. In the model that is a file where one side’s pawn is frozen and the other’s is not.

A square can be held. The empty square between the two pawns is not always neutral ground. If reaching it first means the other pawn can never pass, then whichever pawn gets there stops the other for good, and the file is a race for one square rather than a slow exchange of tempi.

Neither clause is a modelling flourish invented to produce interesting numbers. Both are things a reader who plays chess has seen, and neither is available in the rung below because the rung below evaluates a file in which the two pawns are symmetrical by construction.

What matters here is not that the clauses exist. It is that each of them removes the sentence the all-smallness proof depends on, and each removes it in a different way, and the two kinds of value that come out are as far apart as the theory’s vocabulary goes.

A frozen pawn is an integer #

Take a file with a gap of three where Black’s pawn cannot advance. White advances into the gap; Black cannot answer; White advances again; Black still cannot answer; White advances a third time and the gap closes.

That is three free moves for one player and none for the other, and a position with n free moves for Left and none for Right is worth exactly n. The value is an integer, and the integer is the count.

This is worth being slow about, because “an integer” sounds like the least interesting thing a position could be worth and here it is the whole finding. An integer is a number, and the entire point of the rung below is that a blocked pawn ending contains no numbers — that its values are quantities no material count can see at any resolution, which is why a chess player has to have a word like tempo for them at all.

One friendly pawn in front of another and the file is worth three. Three of what? Three moves, which is exactly what a chess player would say, and which the theory now agrees is a number rather than an infinitesimal.

The reason the shift is so complete is the reason all-smallness was there in the first place. All-small means the two sides run out together; and a side that has run out while the other has not is a side that is losing by an amount, rather than by a tempo. Freezing one pawn makes the ending asymmetric in the crudest way available, and crude asymmetry is what numbers measure.

A contested square is a fight #

The second clause produces something the first does not, and it is the more interesting half.

Give the file a square that can be held: whichever pawn advances into the gap first stops the other one permanently. Now both players want to move, because moving is what claims the square, and a position both players want to move in is what this site calls a hot position — one with a real quantity at stake rather than a spare move.

A switch is not a number and cannot be replaced by one. Its two ends are what the position becomes if Left moves and if Right moves, and the gap between them is what is at stake — a quantity with a name, a size, and a place in every account of an endgame anybody has ever written down.

The reason a contested file lands here rather than among the infinitesimals is a single line of the recursion. Advancing is no longer a move the other side can answer with a move of its own; it is a move that removes the other side’s move. So the two players do not run out together, the all-smallness argument has nothing to stand on, and what is left is a position with a size.

The sweep reports the hottest file in range at a temperature of four, which is to say four whole moves at stake on a single file. Set that beside the rung below, where the entire ending is worth a spare move or nothing at all, and the distance is the argument.

Where the two clauses put the values #

The dictionary is worth reading as a dictionary rather than as a list of results, because it says which structural feature of a pawn position produces which kind of value, and a reader who plays chess can look at a real board and use it.

Both pawns mobile, single steps only. Nimbers: nought and star, by the parity of the gap. This is the rung below in one line, and it is pure tempo.

Both pawns mobile, one of them on its starting rank. Infinitesimals: ups, multiples of up, ups with a star. Still all-small, because a double step is still a move the other side can answer with a move of its own — the extra option changes the arithmetic without changing who runs out first.

One pawn frozen. Integers, of either sign, and nothing else in range.

A contested square. Switches, hot, up to a temperature of four.

A contested square in a file where somebody is already stopped. Integers again, because the race has been decided and what is left is a count of free moves.

Every one of those cells is a count over an exhaustive sweep rather than a class of example, and the sweep produces twenty-five distinct values across a range in which the rung below produces two.

Why the second clause is the one that matters #

The two clauses look like a pair and they are not. Freezing a pawn is a way of describing a position that has already been decided in one file; contesting a square is a way of describing a position that has not.

That distinction has a name here, and it is the one the fourth field of this site is built on. A file worth an integer is a cold position — nobody wants to move in it, because moving cannot improve a count of free moves that is already fixed. A file worth a switch is a hot one, and the whole fourth field of this site exists because hot positions are what make an ending a contest rather than an addition.

So the second clause does not merely widen the range of values. It moves the ending into a different regime, one where the order in which the parts are played matters and where a player has something to decide beyond arithmetic. The rung below has a model in which the only decision is which spare move to burn. This one has a model in which two pawns are racing for a square, and racing for a square is what a pawn ending is mostly about.

The counting rule, one more time #

The rung below scores a chess manual’s tempo-counting rule against the computed outcome and finds it exactly right on plain files and hopeless as soon as one side gains an option the other lacks. The reason it gives is that plain files are impartial, so a sum of them is settled by an exclusive or, and an exclusive or over stars is a parity.

The clauses here make the same point from further away and it is worth stating in the stronger form.

A counting rule of any kind computes one number from a position and reads an answer off it. That is available exactly when the position’s value is determined by one number, which is to say when the values form a group that a single count can index — the nimbers, where the count is a parity, or the integers, where the count is a count of moves.

A frozen pawn keeps a counting rule available: a file worth an integer can be counted, sums of integers are integers, and a player adding free moves is doing arithmetic that is exactly right. That is why endgame manuals full of “White is two tempi up” are not lying.

A contested square destroys it. A switch is not a number, so a sum containing one is not settled by adding anything, and the position has to be played rather than counted — which is what the hottest-first rule is for, and why that rule comes with a guarantee rather than an answer. The distinction between the two regimes is not the same as the distinction between large and small, and a manual that sizes a move by how much changes hands is measuring the wrong one of them.

So the dictionary has a second column that nobody writes down. Beside every structural clause is not only the kind of value it produces, but whether a player can go on counting once it is on the board.

What a sum of these looks like #

The rung below adds up files whose values are stars and zeros, and the arithmetic is a parity. Adding these is a different exercise.

What is worth noticing is that the sum is computed by the same operation as before, and that comparing two of these endings is the same subtraction it always was. A file is a game; games add; the total is the value of the ending. Nothing about the arithmetic changed when the values stopped being infinitesimals, because addition is defined on games rather than on kinds of value and never depended on which kind was in the components.

That is the strongest thing this rung has to say for the theory as a whole. A reader who met the rung below might reasonably conclude that the apparatus is built for infinitesimals and happens to apply to pawn endings because pawn endings are infinitesimal. It is the other way round: the apparatus does not know what kind of value it is holding, and a sum of an integer, a switch and a star is computed exactly as a sum of three stars is.

There is one arithmetical fact in the mixed case that has no analogue in the rung below, and it is the reason a mixed ending is not simply a harder version of a pure one. A number beside an infinitesimal is decided by the number, so a file worth two swamps a file worth a spare move outright; and a switch beside a number is decided by neither, because a switch is not a number and the comparison comes back fuzzy. Three kinds of value in one ending is three different arithmetics deciding three different questions about it.

The all-smallness was doing more work than it looked #

There is a consequence here that the rung below states as a limitation and that turns out to be a statement about the model rather than about chess.

That essay observes that the theory is at its most useful when the material is level, and explains it: a model in which both sides always run out together is a model with the material question assumed away. True — but the sentence to carry is the one after it. Level material is not a lucky circumstance the theory happens to fit; it is the hypothesis the model was built with, and the hypothesis was in the move rule rather than in the position.

Once that is seen, the boundary of the theory moves. The apparatus is not restricted to positions where the material is level. It is restricted to positions that decompose, and a decomposed position may perfectly well contain files worth two and files worth a fight. What the rung below models is one corner of that, and the corner was chosen by the move rule rather than found in chess.

The measurement makes the point sharply. Over the whole sweep, twenty-four files of eighty are all-small and fifty-six are not — and the split follows the clause exactly, never the gap. A wider gap does not take a file out of the infinitesimals at any size; a single frozen pawn takes it out at every size.

What the picture cannot show #

The largest thing missing is the same thing missing from the rung below, and it has to be said again because these values are less obviously artificial than stars and therefore more inviting to over-read.

Nothing here establishes that a real pawn structure has these clauses in it. A file with a frozen pawn is a model of a position where a friendly piece blocks a pawn; whether the piece stays there, whether it can be driven away, whether moving it costs something elsewhere are all questions about a chessboard that this evaluation cannot ask. The same is true of a contested square: the claim that reaching it first stops the other pawn for good is a claim about the surrounding position.

The second clause has a shape no figure here draws. A contested square is modelled as a property of the file, and on a real board what makes a square contestable is usually a piece that could be somewhere else. That is a coupling between files, and coupling is exactly what a sum forbids — which is the rung above this one.

The honest description of what has been computed is: a model with two extra clauses, evaluated exhaustively over gaps up to five, producing twenty-five values where the model without them produces two. That is a claim about eighty files, and what “solved” means has to be said carefully every time — a complete answer for a stated model is not a complete answer for a game. The clauses are named after things that happen in chess and the evaluation knows nothing about chess.

The convention, named #

Everything above is normal play: the player who cannot move loses.

That convention is doing something specific here and it is worth separating from the clauses. A frozen pawn is a pawn that cannot move, and under normal play a side with no move at all has lost. So a file in which one side is frozen and the gap has closed is not a file where that side is merely worse off — it is a file that contributes nothing further, and the side that loses is the one with nothing left anywhere.

That is the same substitution the rung below makes and it is legitimate for the same reason and no further. In a blocked ending where running out of pawn moves means moving a king into a losing square, being unable to move is losing. Outside that class the substitution has nothing to attach to, and a chess player with no good move still has plenty of legal ones.

The other convention worth naming is the one not used. A pawn reaching the end of a file promotes, which in chess ends the discussion; nothing in this model promotes anything, and a file whose gap has closed simply stops offering moves. That is a real gap between the model and the game, and it is the gap the rung above this one walks into.

The surprise: the exotic values were the constrained ones #

The natural reading of the rung below is that a pawn ending is a strange corner of chess where the values turn out to be strange objects — infinitesimals, quantities no fraction measures, the parts of the theory that look most like inventions.

The sweep says the opposite. The infinitesimals are what a heavily constrained model produces, and the ordinary values arrive as soon as the constraint is lifted. Integers, switches and hot positions are the general case; nimbers and ups are the special case, and the special case is special because of a symmetry in the move rule that a real pawn structure breaks routinely.

That inverts what a reader is likely to take from the first rung, and it inverts it in a direction worth carrying past chess. A model that produces only exotic values is usually a model with something assumed away, and the thing assumed away is usually visible in the move rule rather than in the results. Here it was one sentence — both players may advance under the same condition — and every infinitesimal on the page was a consequence of it.

The same reading applies to Kōnane, where a game nobody designed produces the whole vocabulary and produces it narrowly, and the narrowness turns out to be a fact about the size of the board rather than about the game. A vocabulary is bounded by the model that generates it, and the model is in the move rule.

It also says where to look next in any game whose values come out uniformly strange. Do not ask which positions are unusual. Ask which clause of the rule makes the two players’ options match, and then ask what the game does when that clause is false.

Where the ladder goes next #

chess now has a dictionary from structure to value class, and every entry in it prices a file on its own.

The rung above is about the word own. Everything here, and everything in the rung below, adds files up — and adding requires that a move in one file changes nothing in another. A pawn structure does satisfy that; a pawn ending does not, because there are kings on the board and a king is one piece. That is the next rung, and its answer is a count of how often the sum is wrong.