Half of the smaller temperature

A player in an environment of decreasing coupons leaves it for a local fight at some height, and that height is the crossover. For a plain switch the crossover is the fight’s own temperature: it is worth taking the fight while the fight is hotter than the coupon and not afterwards. For a fight whose answer starts another fight it is lower, and this ladder has now published two accounts of how much lower.

What the halving is a function of said half of the temperature. A subtraction, not a factor said the temperature less a half, and observed that the two agree exactly where the temperature is one, which nearly every position in the first pool had. It closed on the obvious next question:

The rung above is the size of the subtraction. A half is what the pool shows over temperatures from one to two and a quarter, and whether it is a constant or the first term of something is a question a pool reaching four or six would answer.

It is neither a constant nor the first term of a series. The subtraction is half the answer’s temperature, and a half is what that comes to when the answer’s temperature is one.

The law #

Write $T$ for the fight’s temperature and $T_f$ for its answer’s. Then the crossover is

$$T - \tfrac12 \min(T_f,\; T).$$

Three cases fall out of it, and each is a reading somebody has published:

• the answer is a number. Then $T_f = 0$, the correction vanishes, and the crossover is the temperature. That is the plain switch, where there is nothing waiting after the exchange.
• the answer is colder than the fight. The correction is half the answer’s temperature — a half when the answer is at temperature one, one when it is at two, one and a half at three.
• the answer is at least as hot as the fight. The correction saturates: $\min$ takes $T$ instead, and the crossover is half the fight’s own temperature. That is the factor of a half, arriving as the second clause of a subtraction rather than as a rival to it.

The two columns on the right are computed by completely different routes. The crossover is measured by putting the fight beside an environment and asking, coupon by coupon, whether the player takes the coupon or the fight — a play-out. The prediction is arithmetic on two temperatures, each read off a thermograph. Nothing in the code makes them agree.

The pool, and why the earlier ones could not see this #

The pool here is 128 fights of three shapes — a switch, a fight with one answer, and a fight whose answer starts a third fight — built so that the fight’s temperature and the answer’s move independently. Answers run from a number through temperatures of one, two, three, four, five and six; fights run up to eleven.

That independence is the whole methodological point, and it is the third time this ladder has had to learn it. The halving was measured on a pool where the answers were nearly all at temperature one, so a factor of a half and a subtraction of a half were the same number. The subtraction widened the pool in one direction — deeper fights — and left the answers’ temperatures clustered, so a subtraction of a half and a subtraction of half the answer’s temperature were the same number.

Each pool answered the question it could see. A pool that holds a variable fixed reports the special case as the law, and the failure is not visible from inside: every position agrees with the reading, the reading is asserted rather than reported, and the gate passes.

Where each earlier reading was true #

It is worth being explicit that neither earlier page was wrong about its own pool. The crossover is the temperature is exactly right for a switch. Half the temperature is exactly right when the answer is at least as hot as the fight. The temperature less a half is exactly right when the answer’s temperature is one.

What each of them was wrong about is the scope, and the scope is where the substance is: a rule of thumb is worth what it covers. The 83 fights outside all three regions are not exotic — they are fights with a moderate answer, which is what a Go endgame is full of — and no pool on this ladder had contained one until now.

The rung below’s twenty, re-read #

The twenty positions of the rung below are all inside this census’s first clause, and reading its table with the law in hand shows exactly what happened. Its fights were built to be deep — an answer whose own answer starts a third fight — and depth was the variable it varied. The answers’ temperatures were not varied, because nothing in the question suggested they mattered, and eighteen of the twenty sat at a temperature of one.

Two of its twenty missed the constant by one grid step, and that page reported them as misses rather than explaining them. They are the positions whose answer is at temperature one and a half, and the law puts their correction at three quarters — which is one grid step from a half. The two exceptions were this essay’s finding trying to arrive early.

Why half of it #

The mechanism is a comparison of two plans, and it can be done in three lines with the thermograph in front of one.

A player deciding between the coupon at height $t$ and the fight $G$ is comparing what each is worth once the other player replies. Taking the coupon banks $t$ and hands the opponent the move. Playing in $G$ banks the fight’s gain and hands the opponent a choice: answer in $G$ or take the coupon.

When the answer is colder than the coupon, the opponent takes the coupon, and the exchange in $G$ is complete; the fight is then worth its full temperature, and the crossover is at $T$. When the answer is hotter than the coupon, the opponent answers, and the player’s gain from opening the fight is reduced by half the answer’s temperature — half, because the answer is a move the opponent has to spend, and the two players split what a move in the environment is worth.

The saturation is the same argument with the roles reversed. Once $T_f$ exceeds $T$, the answer is no longer a follow-up to the fight so much as the fight is a prelude to the answer, and what the player is deciding about is a position worth $T$ whose whole value can be recovered later. The crossover then sits at $T/2$ and stops falling.

The split is worth one more sentence, because it is where the half comes from and a reader is entitled to be suspicious of a factor of a half that appears out of prose. Two players alternately taking coupons from a stack falling in equal steps each gain about half of what the stack holds; that is the standard accounting behind every mean-value argument in the subject, and it is why a move the opponent is forced to spend costs them half of what it would have gained them elsewhere. The answer’s temperature is what that move is worth, and half of it is what the exchange transfers.

That is an explanation and not a proof. It is stated here because the census supports it exactly and because it makes the two clauses one idea rather than two — but the honest status of the formula is 128 confirmations and an argument in prose.

What it changes for a player #

The crossover is not a curiosity: it is the correction to play in the hottest component, which is the one piece of advice this whole subject offers.

The plain rule says to compare temperatures and move in the largest. The correction says that a fight with a follow-up should be discounted by half its answer’s temperature before that comparison is made — so a fight at temperature 4 whose answer sits at 3 competes as though it were at 2½, and loses to a plain fight at 3 which the naive rule would have ranked below it.

That is a rule a person could apply at a board. It needs two numbers rather than one, and the second is a temperature the player has to look one move further to find; against that, it converts a comparison that is sometimes wrong into one that is right on every position measured here.

It also explains a habit of strong players that the plain rule cannot. A fight whose answer is large is worth less than its size suggests, because opening it invites a reply that is nearly as valuable as the fight — which is why an experienced player leaves such positions alone and plays the small clean exchange first.

Who built the environment #

The coupon stack is Berlekamp’s, and it was built for exactly this purpose: to turn how much is this worth into a question with an experimental answer. A player facing a board of many small fights has no way to say what any one of them is worth in isolation; put the same fight beside a ladder of moves of known size and the answer becomes the height at which the player switches.

The idea that a follow-up discounts a fight is older and belongs to Go, where it is practical rather than theoretical. A move whose answer is large is gote in disguise — it looks like a move that has to be answered and turns out to be one that hands the opponent a profitable reply — and the received advice is to leave such positions until the board has cooled. What this page adds is the number: the discount is half the answer’s temperature, and the position competes at that reduced size against everything else on the board.

Temperature theory came from Go and returned to it, which is unusual in this subject. Conway’s temperature is defined for its own reasons in On Numbers and Games; Berlekamp’s endgame work made it a method, and the coupon stack is the instrument that made the method measurable. Every number on this page is a reading off that instrument.

The law’s two halves are two different kinds of statement #

The formula has a linear part and a cap, and it is worth separating them, because only one of them is the finding and the other is forced.

The linear part — that the correction is half the answer’s temperature — is the measurement. Nothing about the construction predicts a half rather than a third or a whole, and the two earlier readings on this ladder are what a pool with too little spread makes of it. That half is the number this page exists to report.

The cap — that the correction saturates at half the fight’s own temperature — is not a finding of the same kind. It has to be there, and the reason is a sentence: a correction larger than that would put the crossover below nought, which would say the fight is answered at no ambient temperature at all, and a fight with a hotter answer than itself is not sente in the first place. So the cap is where the phenomenon stops rather than where the formula stops.

That distinction matters for what a wider pool could still overturn. A pool with answers far hotter than the fights would test nothing new: every one of those positions is past the cap and the formula returns the same number for all of them, correctly and uninformatively. What would test the law is more spread below the cap, and the pool here has that — answers running from a number up to a temperature of six against fights of their own — which is why it separates the factor from the subtraction and the two earlier pools could not.

So a reader auditing this result should ask how many of the 128 fights sit strictly inside the linear regime rather than how many there are. The saturated ones are confirmations of an inequality; the others are the measurement.

What the census does not say #

Four limits.

One environment. The crossover is measured against a coupon stack falling in steps of a quarter, which is the site’s standard environment. A real board is not a coupon stack, and how big the answer is is the page where the difference between an environment and a pool of real positions is taken seriously.

Constructed fights, not positions from a game. Every fight here is written down as a form, chosen to make the two temperatures independent. That is the right design for isolating a law and the wrong one for claiming a frequency: nothing here says what share of Go endgame positions have an answer colder than themselves.

A quarter-point grid. The crossover is read off a grid of coupons a quarter apart, so it is exact to a quarter and no finer. Every prediction in the census lands on a grid point, which is a check on the grid as much as on the law, and a law predicting eighths would not have been visible.

And three deep is the deepest. Fights whose answer’s answer starts a fourth fight are outside the pool. The formula uses only two temperatures, so the natural conjecture is that depth beyond the answer does not matter — but that is a conjecture, and it is precisely the shape of conjecture this ladder has now falsified twice.

The convention, named #

Normal play. An environment is a coupon stack: a set of moves worth $t, t - \tfrac14, t - \tfrac12, \dots$ down to nought, which either player may take.

The crossover is the largest coupon at which the player to move prefers the fight to the coupon, found by playing the sum out at every coupon in turn. A fight’s answer is its Right option — the move that replies to Left’s opening — and its temperature is read off its own thermograph, with a number counted as nought rather than as the $-1$ this site uses elsewhere, because what is wanted is the height of a fight and a number has none.

The law is stated with $\min$ rather than as two cases because the two agree at the boundary: when $T_f = T$ both clauses give $T/2$.

One consequence is worth stating for a reader who plays. A fight whose answer is cold behaves exactly like a switch — the correction vanishes — so the whole of this page is about positions with something waiting, and a player who can see that nothing is waiting can stop reading here.

Where the ladder goes next #

The sente anchor has seven rungs to here: what sente is, that double sente is a band rather than a property, what the reverse costs, that the rule has two cases, that the two cases are a feature of one wall, what the correction in the second case is, and now how large it is.

The rung above settles the depth question this page hands to the proof, and settles it without the proof. The two numbers at the top groups the positions by their two highest temperatures and finds the crossover single-valued on every group — however far apart the third temperature is, and at a fourth level of fight, which this page’s pool never reached. A quantity that is constant across everything below the top two numbers does not depend on anything below them, and that is the whole of the depth claim, established by construction rather than by argument.

That is worth noticing as a method. A proof would settle the depth question with it is true and it is not the only route: a grouping test settles the same question directly, needs no induction, and is available from data already computed. The proof is still wanted, and it is wanted for a different reason — to explain the formula rather than to establish its scope.

The premises an induction would need then says exactly how much of that proof is missing, which is more useful than a sketch would be. The law holds at five levels of fight, survives translation, survives heating, survives cooling — and none of those is the inductive step. What is left is four paragraphs about who spends what, and everything they would rest on has been checked.

Two neighbours are worth the trip. How big the answer is is the same quantity measured from the environment’s side, where the departure is set by the larger of two temperatures. And temperatures do not add is the standing warning against arithmetic on temperatures, which this page has just done a great deal of — legitimate here only because the two numbers belong to one position rather than to two components of a sum.