How big the answer is

A coupon stack is an environment made of nothing but temperature: a pile of coupons worth 5, 4½, 4, … , taken in turn, so that a player who moves on the board is giving up the coupon on top. An environment made of coupons builds it, and the question it makes precise is when do the players stop taking coupons and start playing.

When to leave the environment measured that over a pool of positions and found the answer almost always the board’s own temperature, with a handful of early departures, every one of them by exactly one coupon and every one caused by a position with a follow-up. It closed on what its pool could not settle:

Every early departure here is one coupon, and the class causing them is has a follow-up — one bit. Whether a fight whose answer is worth six leaves the environment earlier than one whose answer is worth one is a question this pool cannot answer, because its follow-ups are all of a similar size.

A pool graded by follow-up size answers it, and the answer is that the bit was hiding a whole scale.

The pool #

Every position is $\{a \mid \{b \mid c\}\}$: one fight whose Right answer starts another. Left’s option is a number, so the parent has exactly one follow-up and it is on Right’s side, which is the shape sente is about.

The parent’s temperature is set by $a$ and the follow-up’s by the gap between $b$ and $c$, and the two move independently. Running $a$ over five values and $\{b \mid c\}$ over eight gives forty positions in which the parent’s temperature ranges from 0 to 4 and the follow-up’s from ½ to 4, with pairs in both orders of size.

That independence is the whole design. A pool in which the two move together cannot separate their effects however large it is, and the rung below’s pool was of exactly that kind — every position in it had a follow-up cooler than the fight it answered, so no member of it could have shown the difference.

Forty positions is a small pool by this site’s standards, and it is the right size here for a reason worth naming: the measurement is a full minimax over a board and an eleven-coupon stack, which is exponential in the stack, and what is wanted is a grid rather than a population. A grid needs to cover its two axes and nothing more. The rung below’s pool had follow-ups clustered together, so a departure could be attributed to having one and not to how big it was; here the two can be held one at a time.

The controlled reading #

Hold the parent at a temperature of exactly one and vary the follow-up.

• $\{3 \mid \{2 \mid 0\}\}$, follow-up worth 1: the players leave at coupon 1.
• $\{4 \mid \{3 \mid 0\}\}$, follow-up worth 1½: coupon 1½.
• $\{4 \mid \{3 \mid -1\}\}$, follow-up worth 2: coupon 1½.
• $\{4 \mid \{3 \mid -3\}\}$, follow-up worth 3: coupon 2½.
• $\{4 \mid \{3 \mid -5\}\}$, follow-up worth 4: coupon 3½.

Every one of those positions is worth fighting over to exactly the same extent — a temperature of one — and the players enter them at coupons a factor of three and a half apart.

So the follow-up is not a modifier on the position’s temperature. It is a second temperature, and the rung below’s one-bit class was the coarsest possible reading of it.

The spread is worth restating in the units a player thinks in. On a board where the coupons are points, entering at coupon 3½ rather than at coupon 1 means giving up two and a half points of environment to start a fight the temperature says is worth one. That is not a refinement of the ordering; it is a different ordering, and the position that looks coldest by the standard number is the one that is entered first.

Nothing about those five positions is contrived. Each is a switch whose Right option is another switch, which is the ordinary shape of a Go endgame where a move invites an answer — and it is the shape the whole sente anchor is about.

The rule over the whole grid #

With both quantities free, forty positions produce two clauses and nothing between them.

When the position is at least as hot as its follow-up, the players leave at the first coupon at or above the position’s own temperature. That is 26 of the 40, and it is the ordinary behaviour the rung below measured — the follow-up is there and is small enough not to matter.

When the follow-up is the hotter, they leave within one coupon of the follow-up’s temperature, and never below the position’s. That is the other 14, and it is where the whole finding lives.

Both are asserted rather than reported. A position on which either clause failed would stop the build, which is the difference between a rule and a tendency on this site.

Between them the two clauses say one thing: the environment is left at the larger of the two temperatures. A position with a follow-up is entered when the coupons fall to whichever of its two fights is bigger, and which of them that is has nothing to do with which one is on the board.

Why that should be true #

The argument is short and it is the one the rung below’s finding was already pointing at.

A player who moves into $\{a \mid \{b \mid c\}\}$ is not making one move; they are starting an exchange. Right answers, Left answers back, and the exchange runs until somebody is content to stop — which is what a follow-up is.

The coupon stack prices that exchange the way it prices everything: the question is not whether the first move is worth more than the coupon on top, but whether the whole exchange is. And the size of the exchange is set by its hottest part, because the players will keep answering each other for as long as answering beats taking a coupon.

So the departure is governed by the hottest fight anywhere in the exchange, not by the temperature at the root. That is exactly what the thermograph does not record: a position’s temperature is the height at which neither player wants to move at the root, and it says nothing about how hot a position becomes two moves in.

The other order, and why it is dull #

The fourteen positions where the follow-up is the hotter are the interesting half. The twenty-six where the position is the hotter are the control, and it is worth saying what they establish.

On those twenty-six the departure is the first coupon at or above the position’s own temperature, which is the classical answer and is what a reader would predict without any of this. A follow-up that is cooler than the fight it answers changes nothing: the exchange is dominated by its first move, and pricing the exchange gives the same number as pricing that move.

So the rung below’s pool was not measuring the wrong thing; it was measuring a pool in which the interesting clause never fired. Its follow-ups were all small relative to the fights they answered, its departures were therefore all at the position’s own temperature or one coupon above, and the single bit — has a follow-up — is exactly the resolution its data supported.

That is the ordinary way a pool limits a finding, and it is worth noticing that the limitation was invisible from inside: the census was exhaustive over its pool, every claim it made was true, and the quantity that mattered was constant across everything it looked at.

What this says about a temperature #

The practical consequence is a warning about a number this site uses everywhere.

Playing in the hottest component is the rule with a theorem behind it, and hottest means the largest temperature on the board. This census says that a position’s temperature can understate how urgent it is by a factor of four, because the urgency is set by an exchange the temperature only sees the beginning of.

That is not a contradiction of the theorem. The guarantee is about the loss against optimal play and it holds; what moves is the ordering a player would infer from the temperatures. Two components with the same temperature, one with a large follow-up and one without, are not equally urgent, and nothing in the temperature says which is which.

The follow-up’s temperature is computable and is not usually computed. It costs one extra thermograph, on a position the evaluator has already built, and it is the second number a player would want beside the first.

Two positions, one environment #

There is a second reading of the same finding, and it connects this rung to the one below it.

Two games in one environment found that the coupon a fight is entered at depends on what else is on the board — that the departure is not a property of the position at all. This page finds that it is not a property of the position’s temperature either, even alone.

Those are different failures and they compound. A player wanting to know when to enter a fight has to look at the fight’s own temperature, at its follow-up’s, and at everything else on the board — and the first of those is the only one the standard machinery hands over.

What makes that tolerable is that the second is cheap. The follow-up’s thermograph is built as part of the parent’s, so the number is already in the evaluator’s hands and is thrown away by the time the temperature is reported. Keeping it costs nothing and, on the evidence here, is worth as much as the number that is kept.

What the grid does not say #

Four limits.

One shape of position. Every member of the pool is $\{a \mid \{b \mid c\}\}$ — a number against a follow-up, with the follow-up on Right’s side. A position with follow-ups on both sides, or one three deep, is not here, and the third level is where this site’s data on that thins out to two positions.

A coupon grid of a half. The departures are coupons and the coupons fall in halves, so a departure is located to within a half. The second clause’s within one coupon is a statement at that resolution and would sharpen or dissolve on a finer grid — and a finer grid is exponentially dearer, since the search is over the whole stack.

The stack starts at five. Positions whose temperature or whose follow-up’s temperature exceeds five are excluded, because a stack that does not start above the board is a stack the players never enter from. That bounds the grid rather than the finding.

Only Right has a follow-up. Left’s option is a number in every position here, so the exchange runs in one direction. A position where both players have answers is a different object, and whether the departure is then set by the larger of three temperatures is the obvious extension and is not checked.

And the two clauses are checked on forty positions. They hold without exception and they are not proved. The argument in the section above is a mechanism and its weak point is nameable: it assumes the exchange runs to its end once started, and a player who abandons an exchange half way is a case the pool happens not to contain.

A grid rather than a pool #

The measurement here works where two earlier ones did not, and the difference is not the number of positions. It is that the pool is a grid.

A pool is a collection of positions chosen to be representative. It tests a hypothesis by putting many cases to it, and it fails at exactly one thing: if the deciding variable happens to take one value throughout, no number of cases separates two hypotheses that agree at that value.

A grid varies one quantity deliberately while holding the others fixed. Here the position’s own temperature is held at one and the follow-up’s is swept from one to four, so every pair of candidate laws that disagree anywhere in that range is separated by construction rather than by luck.

That is why this rung produces a formula and the earlier ones produced constants. A constant is what a hypothesis returns when its argument does not vary; a formula needs the argument to move.

And it says how to design the next measurement on any of these ladders. Before collecting positions, ask which quantity the competing answers disagree about, and build the pool so that quantity takes several values. A hundred positions with one value of the deciding variable are worth less than six positions with six values — and the six are usually easier to construct, because they are chosen rather than gathered.

What a player should carry away #

Two numbers rather than one, and a rule for combining them.

The diagram says it plainly once the second number is being looked for. The position’s temperature is the height at which the two walls meet, and it is one. The bend low down on Right’s wall is the follow-up, and the height that bend reaches is four — and four is the number that decides when the players enter.

So the quantity a player wants is the largest temperature anywhere in the position’s tree, not the temperature at its root. For the positions here that is the maximum of two numbers; in general it is a maximum over the whole tree, and it is computed on the way to the thermograph and discarded.

That is a small change to what gets reported and it is not a small change to what gets played. On this grid the two numbers differ by up to a factor of four, and a player ordering components by the wrong one enters a four-point exchange when a one-point coupon was still on the table.

The convention, named #

Normal play, and the coupons are taken strictly from the top: a stack starting at $t$ and falling in steps of $\delta$ down to nought, with each coupon worth what it says to whoever takes it.

A follow-up here is an option of the position that is not a number — the answer to the move, which starts a fight of its own. Its temperature is the temperature of that option taken as a position in its own right, computed by the same recursion as everything else.

Leaving the environment means the first move either player makes on the board rather than on the stack, and the departure coupon is the coupon that was on top when it happened. Both players play exactly, by a full minimax over the board and the stack together.

The parent’s temperature is the height at which neither player wants to move at the root, which is the standard meaning and is the quantity this page finds insufficient.

Where the ladder goes next #

The coupons anchor has four rungs: the environment built, two fights put in one, when the players leave it, and now what decides when.

The rung above is the ordering. If the departure is set by the larger of a position’s two temperatures, then a board of several components should be played in the order of that quantity rather than of the temperature — and whether a rule that says so beats playing in the hottest component is a measurement of exactly the shape this site already makes for strategies. The pool is in hand, the rule is one line, and the comparison would say whether the second temperature is worth computing.

Two neighbours are worth the trip. Sente is a fact about the rest of the board is where a follow-up’s effect on a decision is established, and this page is that effect priced against a clock. And when to leave the environment is the rung below, whose one-bit class this page turns into a scale — and whose finding that every early departure is exactly one coupon turns out to be a fact about its pool rather than about the phenomenon.