The cheap fights make the rule cheaper

The bound names the hottest part and the cost does not measures what moving in the hottest component costs against playing the whole board correctly, and finds the proved bound — the largest temperature on the board — attained and useless. The cost is nought on 90.8% of lines and otherwise a half or one, on boards whose largest temperature runs to three. What it is actually governed by is the coolest part: the cost exceeds the smallest temperature on 13 lines of 4,240 and never exceeds twice it.

It ends with a conjecture and the reason it cannot test it. If the cost is at most the coolest temperature times the number of components sharing it, then the bound to look for is a sum over the cool end of the board. On that range the number of components sharing the coolest temperature never exceeds two, so a constant of two fits the data exactly as well as a count does, and the two readings cannot be told apart.

Widening the boards to five parts tells them apart. The conjecture is false, and it is false in the direction opposite to the one it was framed in.

More cheap fights, and a cheaper rule #

Over 10,410 lines from four pools and boards of three, four and five parts, 761 cost anything at all. Sorted by how many components sit at the board’s lowest temperature, the lines that exceed it are all in the first two rows:

One copy: 14 lines exceed the coolest temperature. Two copies: 34. Three, four and five copies: none, out of 3,230 lines.

That is the opposite of what the conjecture predicts. A bound of coolest × copies rises with the copies, which implies there is something for it to accommodate — boards with several cheap fights on them where the rule goes further wrong. There are no such boards in the sweep. The ratio of the cost to the coolest temperature reaches 2 where one or two components are cheapest and never passes 1 where three or more are.

The lines with three or more copies are not scarce or strange, either. They are 3,230 of the 10,410, they come from all four pools, and 332 of them cost the rule something. The cost is there; it just never climbs out of the coolest temperature.

The boards that could hold the case #

The sweep the conjecture came from covers boards of two, three and four parts. A board of four drawn from a pool with two or three distinct cool components can hold at most two or three copies of the coolest temperature, and in practice held two. So the question was not merely unanswered there; it was unaskable, because the case the conjecture is about did not occur.

Five parts change that arithmetic. A five-part board can hold five copies of one temperature, and 454 lines here do. The extra part costs something — the sweep is 10,410 lines against 4,240, and the optimal play behind each is a minimax over a sum of five components — but it is the difference between a conjecture with no counterexample and a conjecture with fourteen.

It also matters which five-part boards. Drawing five parts from a pool whose cool components are all distinct in temperature produces one copy of the coolest every time, however wide the board is. The number of copies is a property of the pool, not of the board size, and getting four of them onto one board meant building a pool for the purpose.

A pool with four cheapest fights in it #

Four components at temperature one half — each a switch whose best reply is itself a fight rather than a number — and three hotter ones, at three quarters, one and two. A board of five from that pool can carry four or five components at the coolest temperature while still having something hot for the rule to prefer.

The first version of that pool had no follow-ups in it and measured nothing. Four plain switches at temperature one half, three plain switches above them, 1,512 lines swept — and not one line on which hottest-first lost anything at all. That is not a surprise in hindsight and it was worth finding the hard way: on a board of plain switches with distinct temperatures, playing the hottest one is optimal, so a pool of them is a pool on which the rule cannot be wrong. A rule that is never right and cannot be far wrong builds its own pool with follow-ups for exactly this reason, and the reason had to be rediscovered before this pool would say anything.

With the follow-ups in, the same 1,512 lines produce 204 that cost something — and, notably, none that exceeds the coolest temperature, at any number of copies. The pool designed to make the conjecture bite is the pool on which the rule is best behaved.

The same fight, several times over #

There is a case the count of copies was always going to be read against, and it is worth separating from the rest before the bounds are scored.

The same fight, eight times over is a board made of copies of one component, and it is where the arithmetic of many identical parts is worked out. What matters here is that identical and equally hot are different conditions, and the sweep uses the second. Two components of temperature one half can be quite different games — one a plain switch, one with a follow-up worth taking — and the rule, which reads temperature and nothing else, cannot tell them apart.

That is the whole mechanism by which cheap parts can cost anything at all. Playing the hottest states the rule and a rule that beats the hottest shows what a rule with more information does with the same boards; the gap between them is made of exactly these positions, where two parts look the same to the rule and are not.

So a board with four cheap parts is a board offering four ways to be confused, and it is the board on which the rule turns out never to pay more than the cheapest part is worth. Whatever is going on, it is not that the confusions add up.

Four bounds, scored against each other #

A conjecture that survives is worth much less than a conjecture that survives rivals, so four bounds were scored on all 10,410 lines at once.

The coolest temperature alone is broken on 48 lines, by a half at worst. That was known: it is what the bound names the hottest part and the cost does not reports as 13 lines of its own smaller sweep.

The coolest times the number of parts sharing it — the conjecture — is broken on 14 lines. Every one of them is a board with one component at the coolest temperature, where the conjecture reduces to the coolest temperature alone and inherits its failures. The first is a five-part board carrying a switch worth a quarter of a point, two components of temperature three eighths and two more above them, on which the rule costs a half.

Twice the coolest temperature is broken nowhere, on any line, in any pool, at any board size.

Every temperature but the largest, added is broken nowhere either, and it is the weaker statement — on a five-part board that sum is four temperatures rather than one, so it permits far more than twice the coolest does and rules out less.

So of the two bounds standing, the useful one is the constant. The cost is at most twice the smallest temperature on the board, and nothing in it counts anything.

Why the count was the natural guess and is wrong #

The reasoning behind the conjecture is good and worth stating properly, because what defeats it is not a flaw in it.

Hottest-first goes wrong by answering the wrong small fight. Late in a game the hot components are gone, several cheap ones remain, and the rule — which sees only temperature — picks among them by a criterion that no longer distinguishes anything. Each time it does that it can give away about what the cheap fight is worth. So a board with more cheap fights offers more chances to be made to answer the wrong one, and the cost should accumulate.

The measurement says the chances are there and the accumulation is not. What goes wrong with the argument is the word made. For the rule to lose, the opponent has to be able to force it into the wrong small fight, and forcing requires that the fights be distinguishable to the opponent and not to the rule. With two cheap components that is easy: one of them has a follow-up worth taking and the other does not, and the opponent can steer. With four, the cheap components are interchangeable to a degree that helps the rule rather than hurting it — whichever one it answers, the opponent’s advantage in the others is the same, and the parity of the remaining moves does the rest.

That is a claim about mechanism made after the fact, and this page cannot establish it. What it can establish is that the count does not appear in any bound the data supports.

What the cost is, whatever it is bounded by #

One thing the wider sweep does not change is the shape of the cost itself. Over boards of five parts the cost is still nought, a half or one — the same short list the earlier sweep finds, on boards with an extra component and a larger range of temperatures to be wrong about.

That is worth noticing beside the bound. The bound moves with the board and the cost does not. Twice the coolest temperature is a quantity computed from the position; on a board whose cheapest fight is worth a quarter it permits a half, and on one whose cheapest is worth two it permits four. The measured cost is a half or one, on every board in every pool.

So the honest summary of three measurements is that the rule is very nearly free and that nobody has a description of when it is not. Half of the smaller temperature is the shape a real answer would take — a quantity read off the position that equals the cost rather than bounding it — and nothing here reaches that.

It is also why the rule survives being wrong. Three different claims are all called solved separates knowing a winner from knowing how to play, and a rule costing a half on one line in fourteen is a rule that plays an endgame within a point of correctly while consulting one component at a time. That is the trade the whole question is about, and widening the boards has not moved it: the cost of the rule stayed at a half or one while the board it was measured on grew.

What the sweep cannot say #

Five parts is not many. A real endgame has a dozen components and the cost of playing one out optimally is a minimax over the whole sum, which is why the sweep stops where it does. Whether six or seven copies of the cheapest fight behave like five is unmeasured, and the conjecture could be about a threshold this range never crosses.

Four pools are not all pools. Every component here is a switch or a switch with a follow-up. Components with deeper structure — a follow-up with its own follow-up, or a cold part with a hot region inside it — are not in any pool, and the rule’s behaviour on them is untested. How cold a sum of hot games can be is the reminder that a sum of fights need not behave like a fight at all, and nothing here sweeps a pool chosen to make that happen.

And the mechanism is a story. The count does not appear in any surviving bound; why it does not is argued above and measured nowhere. The measurement that would test it is the one the last section describes and this page does not make.

The convention the cost is measured under #

Normal play, and the cost is measured in points rather than in wins. Each line is a whole game played out twice, once with each player moving first, and the cost is the difference between the final score under optimal play and the final score with the mover following the rule — with the sign taken so that a positive number always means the rule did worse.

Optimal play is a minimax over the whole sum, which is what makes the sweep expensive and what makes the comparison meaningful — and it is the reason a measurement of this kind has to be made on manufactured boards rather than real ones. The first time it told somebody something is a real endgame small enough to solve completely, and it is one position; a bound needs thousands. Hottest-first sees one component at a time: it takes the component of largest temperature and, inside it, the option of best mean. That is the rule as every account of it states it, and it is a local decision by construction — a rule that consulted the rest of the board would not be the rule under test.

A component of temperature nought is a number and is never chosen by anything, so the pools carry only fights.

The surprise: a conjecture can fail in a direction #

The usual way a conjecture fails is by being too strong: a bound turns out not to hold, a counterexample is found above it, and the repair is a larger bound. That is what the conjecture expects — its own words are that if the cost reaches three times the coolest temperature, the bound to look for is a sum over the cool end of the board.

This one failed by being too weak in the one place it was permissive and too strong nowhere at all. Where it permits the most — four or five cheap parts, where it allows four or five times the coolest temperature — the cost does not even reach one times it. Where it permits the least — a single cheap part, where it reduces to the coolest temperature alone — it is violated. The conjecture is wrong in both directions at once, and the quantity it is built on turns out to be anti-correlated with the thing it was supposed to predict.

That is a useful shape to recognise, and it comes from how the conjecture was formed. It was fitted to a range in which the count took two values, and the fit is a statement about that range: every line exceeding the coolest part had two copies of it. Read as a description of the range, true. Read as the copies are what allow the excess, false — and the second reading is the one that names a mechanism and generalises, which is exactly why it is the one that gets written down. A quantity that is constant across a range explains everything about the range and nothing outside it.

The same trap is why the sweep here scores four bounds rather than one. A single conjecture checked against data it fits is unfalsifiable in practice; four checked at once make the surviving one a choice rather than an assumption, and the one that survives — twice the coolest — survives because it mentions no count at all.

Still open: whether twice the coolest is attained, or merely never broken #

Twice the coolest temperature is unbroken over 10,410 lines, and unbroken is not the same as tight.

The lines that reach it are 48 of the 761 that cost anything, and every one of them sits on a board with one or two cheap parts. What is not known is whether the factor of two is attained for a reason or is an artefact of the temperatures the pools happen to carry: every pool here has its coolest component at a quarter, three eighths or a half, and a cost of a half against a coolest part of a quarter is a ratio of two arrived at by the cost being a half — which it is on nearly every lossy line, whatever the board.

The measurement that would settle it is a pool whose coolest component is worth considerably less than a half — a switch of temperature an eighth, say, with follow-ups to match — played on boards of four and five parts. If the cost on those boards is still a half, the ratio runs to four and the bound is not twice anything; it is the cost being a half and the coolest part being small. If the cost falls with the coolest temperature, then two is a real factor and the next question is what attains it.