One number, stated two ways

How wrong a nearly-independent split is priced the mistake of treating a connected Domineering board as a sum of its two halves. Over every vertical cut of every small rectangle the error is a game rather than a number, it never favours Right, and it is bounded by twice the height of the cut — where the height alone, which is what the accounting suggests, fails on six.

It closed on the gap between those two:

The rung above is the smallest true bound. Twice the height holds and is attained nowhere; the height fails on six. Something between them is exact.

The smallest constant is three halves, and the reason nobody found it is that it is exact in a currency the census was not using.

Both columns change in the same place #

At $c = 1$ both columns say sixteen of twenty-two. At $c = 5/4$ both say eighteen. At $c = 3/2$ the stop column reaches twenty-two and the game column reaches nineteen — and above three halves both are complete.

So there is exactly one constant at which the two ways of stating the bound give different answers, and it is the constant the rung below was looking for.

That coincidence is not luck. The two tests differ only on cuts whose error settles exactly at the bound — a game strictly above the bound satisfies both tests, and one strictly below fails both. So the columns can only come apart at a value some error attains, and the smallest such value is the smallest bound. The constant that is exact and the constant where the two currencies disagree are necessarily the same number, which means the disagreement is a way of finding it rather than an obstacle to reporting it.

That also says the rung below could have found three halves from its own data without any new sweep, by asking where its table’s two readings stopped agreeing. It only ever computed one of them.

Two ways of stating a bound is worth setting out before the numbers, because the difference is easy to read past.

A stop is a number: the value a game settles at when both players play it out and neither has anything left worth fighting over. Saying the error’s right stop is at least $-x$ says the value cannot fall below $-x$ — a statement about where the game ends up.

A comparison is a search: $G \geq -x$ means Left wins $G + x$ moving second, which is decided by playing that difference to the end. Saying the error is at least $-x$ says the error could be substituted for $-x$ anywhere and never hurt Left.

On numbers the two coincide. On a switch they do not, and an error term here is a switch far more often than a number — which is what makes the choice between them a real choice rather than a matter of phrasing.

Exact, as a bound on the stop #

Every failing cut has $\mathrm{RS}(\text{error}) \geq -\tfrac{3}{2} \cdot \text{height}$, and three of them meet it exactly. That makes three halves the smallest constant of this shape: a smaller one fails on those three, and no cut goes below it.

Both halves of that are needed and they do different work. No cut goes below it is what makes it a bound — one counterexample would end the claim, and the sweep is exhaustive over the boards it covers. Three cuts meet it exactly is what makes it the smallest — without an attaining case, three halves would be one of infinitely many constants that happen to work, and there would be no reason to quote it rather than 1.6 or 1.9. A bound reported without its tight cases is a bound nobody has finished looking for.

And an infimum, as a bound on the game #

Stated as a comparison of games — is the error at least $-\tfrac{3}{2}\cdot\text{height}$? — three halves fails, on exactly the three cuts where the stop bound is tight. And $3/2 + 1/1024$ succeeds on all twenty-two.

So in this currency there is no smallest bound. There is an infimum, it is three halves, and nothing attains it.

The gap is one move #

The 2 × 2 cut down the middle has an error of $\{-1 \mid -3\}$ — a switch, mean $-2$, temperature 1. Its right stop is $-3$, which is exactly $-\tfrac{3}{2}$ times its height of two.

That the smallest board in the sweep is one of the three tight ones is worth a moment. A 2 × 2 has four squares, two vertical placements and two horizontal ones; cut down the middle it becomes two 2 × 1 columns, and every horizontal domino — both of them — is destroyed. So the cut takes away Right’s entire supply of moves on the smallest board where a cut is possible at all, and the constant that governs the whole sweep is set there rather than on anything large. The other two tight cuts are the 2 × 4’s, which are the same two-row situation with more room either side.

Is that switch at least $-3$? Add $3$ to it and ask whether Left, moving second, wins $\{2 \mid 0\}$. Right moves to $0$; Left is handed the empty game and has no move; Left loses. So the value cannot fall below $-3$ and the value is not at least $-3$, and both sentences are true of the same game.

That is the entire distinction. A stop is the value the game settles at when both players play it out; a comparison against a number asks whether Left survives being handed the difference. A switch settles exactly at its stops and is above neither of them.

Why the census missed it #

The rung below compared games, because comparing games is what this site does — every bound on the anchor is a ge against a number, and that is the right test when the question is may this be substituted.

It is the wrong test when the question is how much can be lost. The two coincide on numbers and come apart on switches, and an error term in Domineering is a switch far more often than it is a number.

So the factor of two was not a failure of searching. It is the smallest constant that works for the comparison the census was making, on a quarter-grid, and it is loose because the comparison it was making has no tight answer. A bound instead of an answer is this site’s account of how to hold a rule of thumb to a standard, and the standard it sets — know exactly where it fails and how thoroughly it was searched — is met here in a way that rung could not: the bound is now known to be unimprovable in one reading and improvable to an infimum in the other.

And the practical reading is the stop one. A player deciding whether to treat a board as a sum wants to know how much the estimate can be out by, which is a question about how far the value can fall — not about whether the true value could be substituted for an estimate in every sum. Three halves is the number they want, and it is a quarter of the height better than the one they had.

What the sweep says about the shape of the cost #

Two things about the twenty-two are worth having beside the constant, because they say what kind of quantity the error is.

It is never in Right’s favour. A cut down a column destroys horizontal dominoes, which are Right’s moves, so splitting can only overstate what Left has. The rung below established that and it holds on every one of the twenty-two — the error is at most nought, always, and the bound is therefore one-sided by construction rather than by accident.

And it is a game rather than a number on most of them. The 2 × 2’s error is the switch $\{-1 \mid -3\}$; the 2 × 4’s is a form several moves deep with that same switch buried inside it. Only a handful of the twenty-two have an error that is a number at all. That is the fact underneath this whole page: if the errors were numbers the two currencies would agree, the rung below’s comparison would have found three halves directly, and there would be nothing here.

So the accounting the rung below tried is not merely loose, it is the wrong shape. A cut through an $r$-row board destroys $r$ straddling dominoes, so the cost is $r$ is an argument about counting moves, and it produces a number. What the cut actually costs is a game — a position in which Left has been robbed of $r$ moves and the two players’ remaining prospects have shifted by different amounts. Its right stop is $\tfrac{3}{2}r$ rather than $r$ because losing a move in a hot position costs more than the move.

What a fleet reader should take from this #

The finding transfers past Domineering and past this anchor, and it is worth stating in the form that does.

A bound on a game and a bound on its stop are different bounds, and a census that tests one has not tested the other. This site tests comparisons everywhere — ge against a number is the workhorse of every ladder here — and every one of those tests is asking may this be substituted. Where the question is instead how much can be lost, the comparison is too strong: it fails on a switch that settles exactly at the bound, because settling at a number and being at least that number are not the same thing.

That is not a subtlety about Domineering; it is a fact about switches, and switches are what hot games are made of. So any ladder on this site that has bounded an error by comparison has a version of this page waiting for it — a constant that is loose by whatever the temperature of the error term is.

And the direction is always the same. The comparison bound is weaker, so the constant it produces is larger, so every such bound is an overstatement. A reader who has been told an error is at most twice something can suspect the truth is smaller and can suspect by how much: the gap is the error’s own temperature, which is a quantity the census already has.

Here that gap is exactly a quarter of the height — two against three halves — and the error at the tight cuts has temperature 1 against a height of 2, since $\{-1 \mid -3\}$ has stops $-1$ and $-3$ and a temperature of half their difference. Half the temperature is a quarter of the height, which is the gap. That is one comparison rather than a pattern, and it is the first thing a wider sweep should check.

What the solver computed, and how #

Every vertical cut of every Domineering rectangle up to fifteen squares — three rows, five columns, each cut position — which is the rung below’s sweep. For each, the whole board’s value is computed by the ordinary recursion and reduced to canonical form, and so is the sum of the two halves; the error is the whole less the parts, as a game.

Twenty-two of those cuts have the two disagree. For each, two quantities are taken: the error’s right stop, read off its thermograph, and the error itself as a game.

Each candidate constant $c$ is then tested twice against the same twenty-two cuts. As a bound on the stop it asks whether $\mathrm{RS}(\text{error}) \geq -c \cdot \text{height}$, an inequality between numbers. As a bound on the game it asks whether $\text{error} \geq -c \cdot \text{height}$ as games, which is decided by a search over the difference — the constant is built as an exact dyadic rational on a 1,024th grid so that a value just above three halves is representable.

Two things are asserted rather than reported. Three halves must bound every stop and be attained, or it is not the smallest constant. And it must fail as a bound on the game while every constant above it holds, since the gap between the two currencies is the page’s finding and a sweep where they agreed would make five of these figures argue the opposite.

Where the model stops #

Three rows and fifteen squares. The constant is set by three cuts, all of height two, and no cut of height one or three attains it. So three halves is the smallest constant on this sweep, and a wider one — cuts of height four and five, which need boards of twenty squares and a whole-board table too large to enumerate — could push it up. The rung below named that limit and it has not moved.

Vertical cuts only, and Domineering. A cut down a column destroys horizontal dominoes, which are Right’s, so the error never favours Right and every bound here is one-sided. The mirror sweep is the mirror result and has not been run; nothing here says what a diagonal or a ragged cut costs.

The infimum is not proved to be three halves. What is measured is that $3/2$ fails and $3/2 + 1/1024$ holds. Every constant in between is untested, and while there is no reason to expect one of them to fail, the statement the infimum is exactly three halves rests on the stop bound being attained rather than on a search of that interval.

Normal play throughout, and both the stop and the comparison are normal-play notions — the stop is where the recursion settles and the comparison is a search over a difference game.

And the figures cannot show a switch failing to be at least its own stop. Six tables of counts describe a comparison, and the object — a thermograph with its right wall meeting the axis at $-3$, beside the number $-3$ — is a picture of two things a reader would expect to be the same. What is at stake draws a thermograph and the numbers it is confused with draws the band a switch is confused with, and between them a reader has the drawing this page is counting.

Where the ladder goes next #

The disjunctive-sum anchor has six rungs: the sum as the object, which part to move in, the other ways to add, whether the parts are parts, what it costs when they are not, and now the smallest constant that cost obeys.

The constant is a fact about a comparison rather than about a game, which is why it sits beside the two questions this anchor asks with it. Which part to move in is the decision the bound is meant to inform, and the price of asking what the parts are is what finding the parts costs before any of it applies.

The rung above is the height-four cut. Three halves is set by three cuts of height two, and the sweep holds heights one, two and three — so the constant rests on one row and the row above it is the one nobody has reached. A 4 × 5 board is twenty squares and its whole-board table is out of reach of an exhaustive evaluation, but the cut only needs the whole board and the two halves rather than every position, which is three evaluations rather than a table. That is a much smaller computation than the rung below assumed when it named the limit, and it is the one thing that would say whether three halves is a constant of the family or of the small cases.

Two neighbours are worth the trip. Independence is a claim is where the split was first shown to be a claim rather than a fact, and it is the page this whole line of pricing descends from. And how often a board falls apart is the measurement of when the exact version is available — and reading it beside this page gives the whole trade: split when the board has split itself, and when it has not, know that the estimate can be out by three halves of the cut.