When the regions add

Every group must keep breathing split 117 NoGo boards at their stones, evaluated the regions separately, added the values, and compared the total with the whole board. The regions add on twenty-four of them and fail on ninety-three. The essay described the twenty-four and flagged the description as unsafe:

The 24 boards that add have a description — symmetric walls — and the description is a guess made from six examples rather than a theorem. What is wanted is a statement of the form the regions add exactly when the wall’s liberty counts on the two sides are related in such-and-such a way, checkable against the 117.

The guess is refuted. Symmetry is neither sufficient nor necessary.

That is a better outcome than it sounds. A description drawn from six examples and checked against 117 either survives — in which case it was a lucky guess and there is still no argument for it — or it fails and the failure says where to look. This one fails in both directions at once, which rules out repairing it by adding a clause: a condition that is neither necessary nor sufficient is not a condition that is nearly right.

What replaces it #

The criterion that survives checking is not about the shape of the wall. It is about liberties, which is the only thing NoGo’s rule ever consults.

If no stone group has liberties in two different empty regions, the regions add.

The argument takes one line. A move in NoGo is legal when the new stone’s group breathes and no neighbouring group of either colour is left without a liberty — so the only way a play in one region can change what is legal in another is through a group that touches both. A group whose liberties all lie in a single region cannot be brought to its last liberty from anywhere else, and cannot make anything elsewhere illegal.

So no move in one region changes the move set of another, which is exactly what independence means. The regions are components and the values add.

It is sound and it is asserted rather than reported. On none of the 117 boards does the criterion hold while the regions fail to add — a board that did would break the argument above, and the figure refuses to draw rather than the caption softening.

The wall that satisfies it #

The clearest instance is a two-stone wall of different colours.

Take the row $.\,\mathrm{b}\,\mathrm{r}\,.\,.$ — a single empty point, a blue stone, a red stone, two empty points. The blue stone’s only empty neighbour is on the left; the red stone’s only empty neighbour is on the right. Neither group has a liberty in the other’s region, the criterion holds, and the regions add.

Change one letter to $.\,\mathrm{b}\,\mathrm{b}\,.\,.$ and it fails. The two blue stones are one group, that group breathes on both sides, and a stone played on the left can take away a liberty the right side was relying on. The regions do not add, and the error is $\{1\ast \mid \ast\}$ — a game, not a rounding.

That is the whole mechanism, and it explains why a two-coloured wall is special. Two stones of different colours are two groups, each breathing into one region only, so a two-coloured wall separates and a one-coloured wall of the same length does not.

The census bears that out with no exception. Of the single-wall rows in it, every one whose wall reads $\mathrm{br}$ or $\mathrm{rb}$ adds, and no one-coloured two-stone wall adds unless the two regions happen to be the same size — at which point it is adding for the other reason. One letter of the picture decides it, and the letter is not about where the wall is but about what it is made of.

Why a criterion of this kind cannot be complete #

The fifteen missed boards are not a gap to be closed by a better structural condition, and saying why turns an apparent shortcoming into a statement about what such conditions can do.

The criterion checks move sets. No group breathes into two regions implies that no move in one region alters what is legal in the other, and that is the hypothesis the disjunctive sum theorem needs — so soundness follows from the theorem, in a line, for every board and every size.

What the census measures is values. The regions add when the sum of the parts equals the whole, and that is a weaker condition than independence: two genuinely coupled regions can still add, provided the coupling never changes an optimal line.

Those are different questions, and the second is not a property of the picture. $.\,.\,\mathrm{b}\,.\,.$ has a coupling — the stone breathes both ways — and the values add because with two empty points a side neither player ever wants the move that would use it. Whether they want it depends on what the regions are worth, which is the evaluation the criterion was supposed to avoid.

So the criterion is exactly as complete as a structural criterion can be. It catches every board where the coupling is absent, and it must miss every board where the coupling is present and unexploited, because telling those apart requires playing the game.

The shape this keeps taking #

That is worth naming, because it is the second time on this site the same shape has turned up in a criterion transplanted or invented.

Bouton’s condition applied to a subtraction game is sound and incomplete in the same way: when it says a position is lost it is right, and it misses genuine losses, because the half of its argument that survived the transplant is the half establishing one direction only.

Here the surviving half is soundness, for the same reason: it is the direction that follows from a theorem rather than from a search. A criterion inherits exactly the directions its argument covers, and an argument of the form this structure implies that behaviour covers one direction by construction.

The practical reading is that a sound criterion is worth having and should be advertised as one. Applied to a NoGo board it either certifies the decomposition — in which case the parts may be evaluated separately, at whatever saving that buys — or it says nothing, and the board has to be solved whole. It never certifies a decomposition that fails, which is the only failure that would matter, since a wrong decomposition produces a confident wrong value with no symptom.

And the incompleteness is cheap. Missing fifteen boards of 117 costs fifteen searches that could have been avoided. Certifying one board wrongly would cost a value that is wrong and looks right, and every essay downstream of it. A criterion that errs only towards more work is the right one to have.

Why symmetry was the wrong guess #

Symmetry looked right on six examples and looks wrong on 117.

Fourteen symmetric boards do not add. $.\,\mathrm{b}\,.\,\mathrm{b}\,.$ has two blue stones symmetrically placed and three regions, and the regions do not add. $.\,.\,.\,\mathrm{b}\,.\,.\,.$ has a single stone dead centre with three empty points on each side, and the regions do not add either — the error is $1\ast$.

Sixteen boards that add are not symmetric. $\mathrm{b}\,.\,.\,\mathrm{r}\,.\,.$ adds and reads differently from either end; so does $.\,\mathrm{b}\,\mathrm{r}\,.\,.$, which is the cleanest case the criterion catches and has no symmetry in it at all.

The reason symmetry looked convincing is worth naming, because it is the same trap every small census sets. The six examples it was drawn from were mostly small boards, and on a small board a coupling between regions often has nothing to bite on: the regions are too short for a player to want the move that would exploit it. So symmetry correlated with adding, on a sample where the real cause was invisible.

The fifteen the criterion misses #

Being sound and being complete are different, and this criterion is only the first.

Fifteen boards add without satisfying it. They are $.\,.\,\mathrm{b}\,.\,.$ and its recolourings, $\mathrm{b}\,.\,.\,\mathrm{b}\,.\,.$ and its recolourings, $.\,.\,\mathrm{b}\,\mathrm{b}\,.\,.$, and a group of two-row boards. In every one of them a group does breathe into two regions — so the coupling exists — and the regions add anyway.

The reading is that a coupling can be present and never bite. On $.\,.\,\mathrm{b}\,.\,.$ the stone has a liberty on each side and either player could in principle use one side to threaten the other; with two empty points a side, neither ever wants to. The dependence is real, and the play never reaches it — which is a statement about how much room the regions have rather than about how they are joined.

That is the honest statement of what has been established. The criterion is a theorem about the rules; the residue is a fact about small boards. A criterion that captured the residue would have to be about the play rather than about the position, and would stop being checkable by looking — which is most of what makes this one worth having.

Sound, and why that is the right shape #

It is worth being explicit about the asymmetry between the two halves of the finding, because a criterion that is sound and incomplete is a different kind of object from a description that fits a sample.

The criterion is a theorem. Its proof is the paragraph above and it does not depend on the census at all: if no group breathes into two regions then no move in one region can change the legality of a move in another, so the regions are independent by the definition of independence. The 117 boards are a check on the implementation rather than evidence for the claim.

The symmetry description was never a theorem and could not have become one. There is no argument from the two sides have the same room to the two sides do not interact; the six boards it was drawn from happened to have both properties, and the census separates them.

That distinction is why the criterion is worth having even though it catches nine boards of twenty-four. Nine boards are settled for good, on any board of any size, by looking. The other fifteen are settled by computing, and no amount of looking will change that.

What this says about Go #

NoGo exists because Go’s endgame needs a whole theory of its own, and removing captures was supposed to put the game back inside normal play. It does, position by position. What it does not do is make a board a sum of its regions, and this page says which boards it does that for.

The criterion translates directly into Go player’s language: a wall separates two areas when every group in it has all its eyes on one side. A wall with liberties into both areas is a wall that can be attacked from either, and attacking it from one side is a move that changes the other — which is precisely the situation a Go player calls not settled.

So the ninety-three failures are not an artefact of a small model. They are the ordinary case, and the twenty-four are the exception. Independence is a claim treats decomposition sceptically in general; this is the sceptical treatment carried out on one game with a criterion attached.

The contrast with the rest of this site is worth stating plainly. Domineering decomposes by geometry: a region cut off by covered squares is a region no domino reaches into, and the claim needs no checking at any size. Amazons decomposes because the arrows burn the board, and once a region is walled off nothing crosses it. NoGo is the one game here where the wall is made of pieces that are still in play, and a piece still in play is a piece both players can push against.

That is the general lesson and it is not about NoGo. A decomposition is safe when the boundary is inert. Covered squares are inert; burnt squares are inert; a live stone with liberties on both sides is not, and the whole of this census is one game paying for that difference.

The two-row boards #

Six of the fifteen unexplained boards have two rows, and they are worth separating because they are the only place in the census where a wall can be attacked from more than one direction.

$.\mathrm{b}../.\mathrm{b}..$ is a two-row board with a blue stone in the second column of each row. The two stones are one group, that group breathes left and right, and the criterion fails — and the regions add anyway. Adding a second row to the wall has made it thicker without changing the count of regions it breathes into, and thickness is exactly the quantity the criterion cannot see.

That is the direction the rung below named as its second open question: a thick wall has more liberties on each side and might be expected to couple its sides more weakly. The six two-row boards here are the smallest evidence for it, and six is not a measurement. What they do establish is that the criterion’s yes-or-no shape is the wrong shape for the question, since a wall that fails it can be arbitrarily well separated.

What the census does not settle #

The boards are one and two rows, up to seven points wide, with one or two stones. That is a small population and it is small in the direction that matters: a thick wall — several stones deep, with more liberties on each side — is exactly the case a Go player would call settled, and it does not occur here at all.

The rung below named that as its second open direction and it stays open. A thick wall has more liberties on each side and might be expected to couple its sides more weakly; the criterion above says nothing of the kind, because it is a yes-or-no condition and thickness is a quantity.

The second limit is the residue and it is the one that matters. Fifteen boards add for reasons the criterion does not see, and characterising them is not attempted here. The obvious next measurement is whether the residue shrinks as the regions get larger — if a coupling that never bites on two empty points bites on five, then the criterion is not merely sound but asymptotically complete, and that would be a much stronger statement than anything on this page.

And the criterion is stated for empty regions rather than for arbitrary partitions. A board can be cut in places where the empty points are not separated at all, and nothing here applies to those.

The convention, named #

Normal play: the player unable to place a legal stone loses. NoGo’s rule is that a placement must leave every group of both colours with at least one liberty — the player is forbidden to capture and forbidden to be captured — which is what removes the scoring and puts the game inside this site’s theory.

Every value quoted was computed by the recursion on the board and on the regions separately, and a region is lifted to its own board with the stones that bound it kept in place, because a bare rectangle would give its points liberties into empty space that the original position does not have. That embedding choice is the one thing in the measurement that could be made differently, and it is the smallest one that preserves every legality question inside the region. A different embedding would change which boards add, and would be changing the question rather than the answer.

Where the ladder goes next #

nogo has two rungs: the game with its census of failures, and now a criterion that explains part of it.

The rung above is thickness. A wall one stone deep is the only kind here, and the interesting question is quantitative: how weakly does a wall couple its sides as it gets thicker? A criterion that answered it would be about liberty counts rather than about liberty regions, which is what the rung below asked for in the first place and what this page has not delivered.

Two neighbours are worth the trip. Every group must keep breathing is the rung below, whose guess this page refutes and whose census it re-runs. And the board falls apart is what a decomposition is worth when it is available, which is the reason a criterion for when it is available is worth having.