A wall that bends

Three rungs below this one build a criterion. Every group must keep breathing sets up NoGo — Go with no captures, where a stone may be placed only if every group still has a liberty afterwards — and notes that this makes a move’s legality a fact about the whole board. When the regions add finds the condition under which two empty regions can be valued separately and added: no group breathes into both of them. How thick a wall has to be measures how well that works and finds that thickness was never the variable — a wall of four stones of one colour couples the two sides exactly as a single stone does, and what matters is whether the wall is one group or two.

All three are about strips. This rung asks the criterion of a board.

On the strips — 1 × 5, 1 × 6 and 1 × 7 with one or two stones — 93 boards split into two regions, 17 of them add, and the criterion holds on 9. Every one of those 9 adds. It is sound and it covers about half of what there is to cover.

On 3 × 3 boards with two, three or four stones, 227 split, 21 add, and the criterion holds on none of them.

Zero is a different result from a small number and it is worth being careful about which one this is. The criterion has not become a poor predictor. It has become impossible to satisfy.

What “add” means here, and how it is checked #

Before the mechanism, a word about the column headed regions that add, since everything rests on it.

A NoGo position whose empty squares fall into two regions is a candidate disjunctive sum: evaluate each region as a board of its own, add the values, and compare with the value of the whole position. When the two agree the regions add and the position is genuinely a sum; when they do not, the regions are coupled and valuing them separately gives the wrong answer.

That comparison is made directly, on every board in both populations. Nothing is inferred from the criterion — the criterion is one column and the addition is another, computed independently, and the whole point of the table is to set one against the other. So “17 of 93 add” is a measurement of the game, and “the criterion holds on 9” is a measurement of a proposed explanation, and the interesting number is where the two overlap.

On strips they overlap perfectly: every board the criterion covers is one that adds. That soundness is asserted rather than observed — the sweep stops if a board ever satisfies the criterion and fails to add — and it is what makes the criterion worth having on strips even though it explains only half of them.

Why a wall in two dimensions cannot be two groups #

On a strip, a wall between two empty stretches is a run of stones with one empty neighbour at each end. Two stones of different colours there are two groups: the left one has a liberty on the left and nothing but its neighbour on the right, and the right one is its mirror. Neither breathes into both stretches. That is the whole of the criterion, and it works because a strip is one-dimensional and a wall segment has exactly two ends pointing in exactly two directions.

On a board it does not work, and the reason is that a wall has to go somewhere. To separate a 3 × 3 board into two empty regions the stones have to reach from one edge to another — and a stone in such a chain has four neighbours, of which at most two are other wall stones. The other two are empty, and they are on opposite sides of the wall.

So that stone’s group breathes into both regions, by itself, regardless of colour. Making the wall two colours splits it into two groups and does nothing, because the problem is not that the groups are joined; it is that each of them individually sees both sides.

The simplest case is a corner. .b./b../... puts two stones diagonally, cutting the top-left square off from the rest of the board. The two stones are not orthogonally adjacent, so they are two separate groups — exactly the arrangement that works on a strip. Each of them touches the isolated corner and the large region. Two groups, both sharing.

That is checked rather than described. Every one of the 227 boards is examined for a group touching both regions, and every one has one; the sweep refuses to return a result if a single board does not. The worst case is b.b/.b./b.., which has four groups and all four of them share.

It is worth noticing what the mechanism does not depend on. It does not depend on the board being three by three, on the number of stones, or on NoGo’s rules at all — the argument is that a connected chain separating a grid region has a stone with empty neighbours on both sides, which is a statement about grids. NoGo enters only in what the consequence is: a group with liberties in two regions is a group whose legality constraints couple them, which is the thing the criterion was written to rule out.

So the result is stronger than the population it was measured on, and weaker than it might look. Stronger, because the geometry does not care about board size and a 5 × 5 sweep would find the same empty column if it could be run. Weaker, because “the criterion cannot hold” is not the same as “the regions never add” — twenty-one of them do, and the criterion simply has nothing to say about which.

A criterion is a shape, not a colour #

The rung below already found half of this without being able to say so.

Its finding was that thickness does not matter and colour does — a four-stone wall of one colour couples exactly as one stone does, and a two-stone wall of two colours does not couple at all. Stated that way it sounds like a fact about colour, and it is not. It is a fact about how many groups the wall is, and colour is the only thing that decides that on a strip because a strip gives the stones no choice about adjacency.

On a board, colour stops being the deciding variable, because two stones of the same colour that are not orthogonally adjacent are already two groups. The diagonal corner above is two groups whichever colours they are, and the census shows .b./b../... and .b./r../... and .r./r../... all behaving identically.

So the two dimensions do not weaken a colour rule. They reveal that there was never a colour rule — there was a group rule, which colour was standing in for, and the substitution stops working the moment adjacency has more than one direction to fail in.

Where the population stops #

The three-row sweep is 227 boards and it is the largest that could be run, which is worth stating because a reader is entitled to ask why not four rows.

A NoGo position’s value is computed by walking its game tree, and legality in NoGo is a whole-board condition: whether a stone may be placed depends on every group already down. So there is no per-region shortcut on the way in, and the tree is the whole cost. A 3 × 3 board with two to four stones takes four tenths of a second for all 227. A 3 × 4 board with three stones exhausts a six-gigabyte heap, and so does a 2 × 6 with two or three.

Two rows are enough to kill the criterion, which is the useful part of that limit. The 2 × 5 population — 148 split boards, of which five add — also has the criterion holding on none, so the empty column arrives with the second row rather than the third. That is consistent with the mechanism: a wall on a two-row board already has to reach both edges, and a stone in it already has neighbours above and below.

What the limit costs is generality of a different kind. Everything here is measured on boards where two, three or four stones are enough to separate the board, which means small boards and short walls. A long wall on a large board is exactly the case where a reader would most want to know whether some subtler condition works, and it is the case nothing here reaches.

What does not replace it #

With the criterion gone the obvious next move is to look for its two-dimensional cousin, and the honest report is that neither candidate is one.

Symmetry is the first thing to try because when the regions add is the essay that killed it on strips — it was the rung below’s guess, made from six examples, and fourteen symmetric boards do not add. In two dimensions it does even less: 2 of the 21 adding boards are symmetric against 30 of the 206 that do not, which is a separation of five points and is nothing.

The number of groups sharing breath does better. Seventeen of the 21 adding boards have exactly two groups touching both regions, against 96 of the 206 that do not — 81 per cent against 47, a separation of 34 points. That is a real signal and it is nowhere near a criterion: four of the boards that add do not have it and ninety-six that do not add do.

The sweep is written to notice if either ever became one. It refuses to return a result if a candidate separates the two populations by more than sixty points, because at that point the essay would be reporting a replacement and calling it a failure.

The same failure, named elsewhere #

What has gone wrong here has a name in another corner of this site, and putting the two together is the most useful thing this rung does.

Two clauses and a third question states the general condition for a component to carry its own rule: locality, that a component’s moves are a function of what it carries, and isolation, that a move in one component leaves the others alone. A ruleset satisfies both or the parts are not parts.

NoGo fails locality, and it fails it for the reason every group must keep breathing opens with: a move’s legality is a fact about the whole board, because placing a stone must leave every group with a liberty and some of those groups are elsewhere. So a NoGo region cannot say what its own legal moves are, and by the general condition it should not decompose at all.

The criterion is the repair. A group breathing into only one region cannot be the group that makes a move elsewhere illegal, so a board where no group breathes into two is a board where locality holds after all — region by region, for that board. It is not a general theorem about NoGo; it is a test for which positions of NoGo happen to satisfy the general condition.

Read that way, this rung’s result is not surprising and is still worth having. The criterion identifies the positions where a whole-board rule happens to act locally, and whether such positions exist is a question about the geometry rather than about the rule. On a strip they exist and are common enough to matter. On a board they do not exist at all, because the geometry never lets a wall be two one-sided things.

That is also why the twenty-one boards that add are not a contradiction. Locality failing does not force the sum to be wrong; it removes the guarantee. A position can satisfy the sum by accident — the coupling exists and never gets used, because no line of play reaches the position where it would matter — and twenty-one of 227 doing so is exactly what accidents look like. Independence is a claim is the essay about the difference between a sum that is guaranteed and a sum that happens to hold, and NoGo on a board is entirely the second kind.

What this closes #

The anchor ends here, and it ends with a negative result stated precisely rather than with a gap.

The criterion is correct and it is one-dimensional. Not approximately one-dimensional, not weaker in higher dimensions: its condition is satisfiable on a strip and unsatisfiable on any board with more than one row, for a reason that is a fact about grids rather than about NoGo. Anything with the shape no part of the boundary touches both sides is going to fail the same way, because a boundary that separates a plane region has an interior and the interior sees both sides.

Two dimensions are not more of the same. The rungs below treat the strip as a small version of a board, which is the usual and usually sound move on this site — amazons on one line is a strip standing in for a board, and it stands in well. Here it does not, and the failure is not statistical: a property that holds nine times out of ninety-three on strips holds zero times out of two hundred and twenty-seven on boards, and no amount of enlarging the strip population would have predicted that.

And twenty-one boards still add. Their regions really are independent, checked by evaluating both ways. Something explains them and nothing here does. That is the shortfall this rung records rather than fills, and the two candidates above are what it looked at before saying so.

That last figure is the ladder in one picture. On a strip, one stone in the middle is a wall. On a 3 × 3 board, one stone in the middle is a stone: the eight squares around it are still one region, and separating anything takes a chain long enough to reach two edges — which is exactly the chain that cannot avoid seeing both sides.

The criterion was never about walls. It was about a shape of board where a wall could be two things at once, and there is only one such shape.

There is a last thing worth saying about how the ladder got here, because it is a pattern rather than an accident. The first rung stated a condition. The second refuted a guess about which boards meet it. The third measured how the condition behaves as the wall changes and found that the variable everybody had been watching — thickness — was not the variable. This rung takes the condition to a board and finds it cannot be met there.

Each of those is a narrowing, and none of them is a repair. That is not a failure of the ladder; it is what a ladder built on measurement looks like when the object turns out to be smaller than the first rung hoped. The alternative — extending the criterion by weakening it until something is true of two-dimensional boards — is available and is worth nothing, because a condition that holds on 227 boards of which 21 add is not a condition at all. The two candidates above are exactly that temptation, measured and declined.

What is left is a clean statement with a boundary on it, and a shortfall recorded where the boundary is. On strips, NoGo regions add when the wall is two groups, and that is a theorem with 93 boards behind it. On boards, NoGo regions sometimes add and nothing here says when. Twenty-one positions are the whole of what a two-dimensional theory of NoGo separation would have to explain, and they are listed rather than accounted for — which is a smaller contribution than the rung was planned to make and is the true one.