Seventy-two of them were not silence

Three rules and a tie-break found three rules answering 94.5 per cent of the 3,308 decisions a Domineering strategy has to store, and closed on the 181 they miss:

The rung above is the residue’s own structure … the question is whether the value of the position predicts which placement is right once the geometry has stopped distinguishing them — not the value naming the move, which is impossible, but the value naming which of two symmetric-looking placements to prefer. Twenty of the 181 are on positions worth star, which is one class large enough to answer that question on its own.

It does, on more than half. But the 181 has to be taken apart first, because the rung below’s description of it is wrong for forty per cent of it.

Two failures wearing one name #

The rung below states its residue’s mechanism plainly: in every one of the 181, the mobility count scores the candidates identically and the connectivity rule scores them identically too, so the list falls through to the arbitrary tie-break and takes the wrong placement. The rules are not being misled — they are being silent.

Run the two rules over the 3,308 decisions and record, for each, what they do rather than only whether the final answer is right:

• 3,034 are answered. Everything surviving mobility and then connectivity is a best placement, so a player following the two rules cannot go wrong.
• 72 are misled. The two rules narrow the candidates to exactly one placement, and that placement is worse than one they threw away. No tie-break was consulted, because there was nothing left to break.
• 202 are silent. Two or more candidates survive and at least one of them is worse, so the tie-break decides. It happens to pick a best placement in 93 of them and a worse one in 109.

And $72 + 109 = 181$. The rung below’s residue is two populations, and only the second is what it describes.

The distinction is not bookkeeping. A rule that is silent can be helped by adding another rule after it. A rule that is wrong cannot: whatever is appended, the wrong placement has already been selected and the appended rule never sees the alternative. So 72 of the 181 are unreachable by any extension of the list, and the rung below’s finding that the fourth and fifth rules add nothing is, for those 72, a theorem rather than a measurement.

That also sharpens what a repair would have to look like. Answering the 72 means changing one of the two rules, not adding a third — and since the rule that does the narrowing is mobility, the margin a count needs is where the fault is. The count settles the direction of a comparison only once the gap in replies is large enough, and here it is being asked to settle a gap of one.

Eleven tie-breaks on the decisions that are silence #

That leaves 202 decisions where two or more candidates genuinely survive, and a rule appended to the list would have something to choose between. Eleven tie-breaks are put to them — five reading the value of what a placement leaves, five reading only its shape, and one reading nothing at all.

The control is the rung below’s scan-order tie-break, which answers 93. That is the number every other rule has to be read against, and it is a high bar for a rule with no content because a decision often has more best placements than worse ones.

Only one rule beats it, by a single decision: leave a position that is a number, at 94. Every other rule in the panel is at or below the control — leave the largest single piece at 84, leave the position with the smaller birthday at 80, leave the colder position at 46.

So no single tie-break is the answer. Whatever the value predicts, it does not predict it uniformly, and a fourth rule appended to the list would buy about one decision. That is the negative half and it is worth having before the positive one, because a panel of eleven rules that all lose to a coin is what this residue is structureless would look like.

One rule per value #

The rung below’s question was not whether one rule answers the residue. It was whether the value picks — which means a different rule may apply in different value classes, and the value is what says which.

The 202 silent decisions carry 55 distinct values. Choose the best rule from the panel separately for each class and ask whether it answers every decision in that class, and 36 of the 55 classes come out answered, covering 118 of the 202 decisions.

A hundred and eighteen against the control’s 93 is not a landslide, and it is not nothing either: it is the difference between a residue that has structure and one that does not. And the three commonest classes are answered by three different rules, which is what makes it a value-indexed table rather than one rule in disguise:

• positions worth $\ast$ — leave the younger position;
• positions worth $\{1\ast \mid -1\ast\}$ and $\{\ast \mid -1\ast\}$ — leave the colder position;
• positions worth $\pm\tfrac12\ast$ — leave a number.

The star class, settled #

The rung below singled out the star class as one class large enough to answer that question on its own, and it is answered outright.

All 14 silent decisions on positions worth $\ast$ are settled by leave the position with the smaller birthday — and equally by leave the opponent fewer options, which on these positions is the same rule seen from a different side. Not most of them: all of them.

What the rule is doing is visible in the discards. The placements it keeps leave positions worth $0$; the ones it throws away leave $\tfrac12$ and $1\ast$. A region worth $\ast$ is a region with nothing at stake — the two players are level and whoever moves second wins — so the whole decision is about ending the region cleanly rather than about winning anything in it. A placement leaving nought has finished the argument; a placement leaving a half has handed the opponent a fraction of a move.

That is the tempo anchor’s own subject arriving where nothing else was left. What a value leaves out established that a value never names a move; what it does not say is that a value never narrows one, and on the class where the value is an infinitesimal the narrowing is total.

Where the value has nothing to say #

Nineteen of the 55 classes are answered by nothing in the panel, and the two largest of them are worth $\pm\tfrac74$ and the next two $\pm\tfrac14$. All four are plain numbers.

That is the right shape for the failure, and it is nearly a derivation. A position worth a number has no tempo in it: neither player wants to move, the value is settled, and every remaining question is about how the region will be cut up rather than about who is ahead. So the value has told the player everything it knows before the decision is reached, and what is left is geometry that the two shape rules already failed on.

The classes the value does answer are the ones with something infinitesimal or hot about them — a star, a number plus a star, a switch. Those are values that still describe a tension, and a rule that resolves the tension resolves the decision.

So the answer to the rung below is more precise than more than half. The value predicts where the value is not a number, and where it is, it does not.

Why an infinitesimal is the class that answers #

There is a pattern across the answered classes worth naming, because it is the tempo anchor’s own thesis turning up as a measurement rather than as an argument.

Every class the value settles is a class where the value is not a number: a star, a number plus a star, a switch between two number-plus-stars. Every class it fails on is a plain number. And the distinction between those two kinds of value is exactly the distinction between a position somebody still wants to move in and a position nobody does — which is what what a value leaves out calls tempo and what the number tree is the census of on the other side.

So the reading is: a value predicts a placement precisely as far as it is still describing a tension. Where the tension is an infinitesimal, the right placement is the one that discharges it, and leave the younger position is what discharging it looks like on a Domineering region. Where the tension is a switch, the right placement is the one that cools the position, and leave the colder position is that. Where the value is a number there is no tension left, the value is a summary of a settled account, and it has been silent about placements since the moment it became a number.

That also says why the two content rules and the value rules fail in the same places rather than in complementary ones. Both are reading the position for something that is not there: the shape rules look for an asymmetry that a symmetric-looking region does not have, and the value rules look for a tension that a number does not have. The residue’s hardest part is where the position has run out of both, and there the only thing left is the search the whole exercise was trying to avoid.

What this costs, and what it is for #

Every rule in the panel that reads a value has to evaluate the position each placement leaves, which is precisely the work a stored strategy exists to avoid. A table of 3,308 lines is expensive to hold and free to consult; a rule that computes a birthday is cheap to hold and expensive to consult.

So nothing here is a cheaper strategy, and the honest description is the one the rung below applied to an exception table indexed by value: this is a description of the residue rather than an answer to it. The rung below reached that conclusion from the outside — a table of values tells a player which positions are exceptional and not what to do in them — and this page reaches it from the inside, having built the table and found it needs an evaluator to read.

What the description buys is the shape. The residue was 181 decisions with no account; it is now 72 errors the list cannot reach at all, 93 lucky guesses, and 109 losses of which the majority sit in classes with a rule. That is three different things needing three different repairs, and the rung below’s single sentence about silence covered all three.

Two failures a residue can be made of #

The 181 turn out to be two populations and the distinction decides what a solver should do, so it is worth setting out what each failure looks like from inside a program.

Silence is recoverable. The rules score two candidates alike, the program sees the tie, and it can do something about it: evaluate both, apply a further rule, or fall back to a search. A silent rule is a rule that has told the truth about its own reach.

Being wrong is not recoverable. The rules score one candidate above another and prefer the worse one. The program has no signal, takes the answer, and the error propagates through everything built on it. There is no fallback, because a fallback needs a trigger and confidence is not one.

That is why the split matters more than the counts. A residue of 181 silences is a rule that answers 94.5 per cent and declines the rest, which is a usable object; a residue containing 72 confident errors is a rule that is wrong on 72 decisions out of 3,308, and a program using it is wrong on those and does not know.

And it changes what a repair has to do. Filling in silence means adding a rule; removing confident errors means changing an existing one, because the wrong preference is being produced by a rule that fires. The two repairs touch different parts of the strategy, and a page reporting only 181 unanswered would have sent the work to the wrong one.

What this does not say #

The panel is eleven rules, not all rules. A rule not in the panel might answer a class this page reports as unanswered, and one certainly does: play the placement leaving the position with the best stop, which is the definition of a best placement and answers everything by construction. The panel deliberately excludes rules that are the criterion in disguise, and the boundary between reads the value and is the criterion is a judgement rather than a definition.

Per-class fitting is fitting. Choosing the best of eleven rules separately in each of 55 classes is 55 choices made against the data they are scored on, and several classes hold only two or three decisions. The star class at fourteen and the two colder classes at ten and seven are large enough to mean something; a class of two answered by one of eleven rules is not evidence of anything, and the headline number of 118 includes those.

The population is regions of at most eight squares. The residue concentrates at seven and eight squares — 177 of the rung below’s 181 — so this is a measurement at the largest sizes the catalogue holds, and the trend says the residue would be larger on bigger regions rather than smaller.

And the 72 have not been repaired. They are diagnosed as a different failure, which is worth doing, and nothing here fixes them. A rule that answered them would have to be a better mobility count, and the margin a count needs is three rungs of evidence that no simple weighting of replies is one.

The convention, named #

Normal play throughout: Left places vertical dominoes, Right horizontal ones, and a player who cannot place loses.

A decision is a position and a player for whom some placements are best and others are not. A placement is best when it maximises the mover’s stop in the position it leaves, which is the rung below’s criterion and is what a strategy table would store.

The two content rules are the rung below’s: leave the opponent fewest replies, then leave the region in as few pieces as possible. Each is a filter rather than a chooser — it keeps every placement scoring highest and passes the rest on — so a rule that cannot tell two placements apart passes both.

A decision is answered when everything the filters keep is a best placement, misled when they keep exactly one placement and it is not best, and silent when they keep two or more and at least one is not best.

A rule reads a value when it evaluates the position a placement leaves; it reads a shape when it looks only at the cells. The birthday of a value is the day it is born on in the construction, and a position with fewer options has a smaller one, which is why the two rules that answer the star class are the same rule.

The population is every position reachable by play inside the catalogue of regions of at most eight squares, counted once however many regions it arises in.

Where the ladder goes next #

The tempo anchor has five rungs: what a value leaves out, how many moves are worth making, what a strategy has to remember, three rules and a tie-break, and now what the residue is made of.

The rung above is the seventy-two. They are the only part of the residue with a named cause — the mobility count narrowing to one placement and naming the wrong one — and they are a completely specified set of 72 positions with both placements in hand. The question is whether the count is wrong on them by one reply or by more, because a count wrong by one is a tie the rule broke badly and a count wrong by three is the rule being outside its own bound. The margin is recorded for each, and reading the distribution would say whether the fix is a tie-break rule inside mobility or a different rule altogether.

Two neighbours are worth the trip. What a strategy has to remember is where the 3,308 comes from, and it is the measurement everything on this ladder is a fraction of. And when a real board falls apart says which of these regions a played game actually produces, which is what decides whether a residue concentrated on eight-square regions is a small problem or the whole of one.