The two numbers at the top

Half of the smaller temperature closed the sente ladder’s sixth rung with a formula. The crossover of a fight — the highest ambient temperature at which a player still answers rather than taking a coupon — is

$$T - \tfrac{1}{2}\min(T_{\text{answer}},\, T),$$

exact on 128 positions. It closed on what a proof would settle:

The rung above is the proof. The formula is exact on 128 fights and the argument for it is four paragraphs of prose about who spends what; turning that into an induction on the thermograph would settle it, and would probably settle the depth question with it — a proof that uses only the two top temperatures says by its shape that the third one cannot matter.

The proof is not here. The depth question is, and it is settled by construction rather than by inference from the formula’s shape.

The difference between two claims #

There are two things the formula’s shape might be saying, and they are not the same.

The weak claim is that the formula, which mentions two temperatures, is exact on a pool. That is what the rung below established, and it leaves open the possibility that the third temperature does matter and simply happened to be correlated with something the formula does mention. A pool where every fight of temperature 4 had an answer at 2 and a third fight at 1 would confirm the formula and settle nothing.

The strong claim is that the crossover is a function of the two top temperatures: fix them, vary everything below, and the crossover does not move. That is the statement a proof would establish, and it is testable directly — build a pool where the third temperature varies a great deal within a fixed pair of top temperatures, and look.

The pool here is built for the second claim. There are 133 positions, 75 distinct pairs of top temperatures among them, and 25 of the 75 hold more than one third temperature. On every one of the 75 the crossover is single-valued.

One group, in full #

The richest group is six positions of temperature 3 with an answer at 2. Among them are a plain switch inside a switch — no third fight at all — and a fight three levels deep whose innermost fight is at temperature 4. Everything below the answer is different: the numbers, the depth, the size of the third fight. The crossover is 2 on all six.

$\{8 \mid \{4 \mid 0\}\}$ and $\{16 \mid \{12 \mid \{10 \mid 2\}\}\}$ are positions a player would describe entirely differently — one is a fight with a small answer, the other a fight with a large answer that starts another large fight — and a player facing either of them across a coupon stack stops taking coupons at exactly the same moment.

How far the third fight was varied #

The range is what makes the test able to fail. Eight distinct third temperatures appear, from nought to six, and the pool was built for that: the innermost fight is swept over seven different pairs of numbers while the outer two are held at values several positions reach.

A test that varied the third temperature between nought and one would have been consistent with the third temperature mattering a great deal — a quantity that moves by one cannot move an answer by much. Varying it by six and finding the crossover fixed is a much stronger statement, and it is the only reason the grouping is worth reporting at all.

And a level deeper than the sweep #

The pool also contains 24 positions with four levels — a fight whose answer has an answer whose answer has an answer — which is one deeper than anything the rung below swept. The formula is exact on all of them.

That is the prediction a two-temperature law makes and it is worth stating as a prediction. If the crossover depended on how deep the fight goes, the four-level positions are where it would first show, since they are the first with a fourth temperature to ignore as well as a third. It is not merely that they satisfy the formula: they satisfy it while carrying a quantity nothing in the pool below them had.

What a grouping test is, and why it is worth building for #

The method here is worth naming because it is cheap and this site has not used it before, and it answers a kind of question no amount of scoring can.

A fit takes a formula and measures how often it is right. It answers is this the rule? and it is what most of the ladders on this site do — a rule that beats the hottest scores four rules against perfect play, the margin a count needs scores one against a threshold. A fit cannot tell a rule from a rule with a hidden extra variable, because a variable that is constant on the pool is invisible to it.

A grouping takes the formula’s inputs, holds them fixed, and varies everything else. It answers are these the inputs? — and it is the question that actually matters when a formula is offered as a law rather than as an approximation. The cost is entirely in the pool: the positions have to be built so that the groups are populated and the free variables are genuinely varied, which is a design problem rather than a computational one.

The two together are what a claim of this kind needs. The rung below did the fit and got 128 of 128; this page does the grouping and gets 75 groups of 75. Neither on its own would be enough — a formula that fits and is not a function of its stated inputs is a coincidence, and a function of two inputs that is not this formula is a different law.

What is still missing #

The induction is still unwritten and it is worth being exact about what it would add.

The measurement says: over 133 positions and 75 pairs, the crossover is a function of the two top temperatures. A proof would say: over every position, it is — and it would say why, which is presumably that the two walls of the thermograph, at the height where the crossover sits, have forgotten everything below the answer. That is a plausible thing for walls to do, since a wall above a certain height is determined by the options that govern it there and by nothing lower down, and it is exactly the statement an induction on the thermograph would be about.

There is a second thing missing and this page has not touched it. The crossover is where the answer stops being worth taking; what the answer is worth at that moment is a different quantity, and nothing here says whether that depends on more than two temperatures. How big the answer is measures the same decision from the environment’s side, and the two readings together describe when a player leaves; neither describes what leaving costs.

Why two and not one #

It is worth saying why the answer’s temperature has to be in the formula at all, since a reader arriving at the crossover depends on two numbers might reasonably ask why it is not one.

A fight is worth answering while the ambient temperature is below what the fight is worth, and the naive reading is that the crossover is the temperature. It is not, and the rung below’s ladder has spent several rungs finding out why: answering a fight does not merely gain the fight, it hands the opponent the answer, and the answer is worth taking too. So the value of answering is the fight less some share of what it gives away, and the share turns out to be half.

That immediately explains why the third temperature cannot enter. The player deciding whether to answer is choosing between one move here and one coupon there; what happens after the answer is a decision for whoever moves next, and by the time it arrives the ambient temperature has fallen. The third fight is a fight for a colder board, and a colder board is a different question.

So the shape of the answer is not a coincidence of the pool. It is what the mechanism predicts, and the grouping is what says the prediction is right rather than merely available.

What this does not say #

The pool is built and not enumerated. Every position in it was written down to make a group populated, so the counts are a property of the design rather than a sample of anything. That is the right way to build a pool for a grouping test and it is the wrong way to build one for a rate, and no rate is reported here.

One hundred and thirty-three positions. Every fight in the pool is a nested switch built from integers, so the walls are as simple as walls get and every temperature is a multiple of a quarter. A position whose answer had two options, or whose temperatures were not commensurable, is outside the sweep entirely.

Twenty-five informative groups. Fifty of the 75 groups hold a single third temperature and prove nothing; the claim rests on the other 25, and the richest of them holds four. That is enough to make the test able to fail and it is not a large number.

The crossover is measured on a grid. It is found by playing the position against a coupon stack at ambient temperatures in steps of a quarter up to 24, and taking the highest at which the answer is still chosen. A crossover between two grid points would be reported at the lower one, and every crossover found here sits on the grid.

The four-level positions are twenty-four and they are built one way. Each is a chain of nested switches with a single option a side at every level, so their fourth temperature varies over two values rather than eight. They establish that the formula survives the extra depth and they do not group the way the three-level positions do.

And it is not the proof. Everything above is a measurement in the shape of a proof, which is a different object. A statement that holds on 133 positions with a mechanism behind it is what this site can produce; an induction on the thermograph is what would close the rung, and it remains the ladder’s outstanding debt.

Grouping is a test a sweep can run and a proof cannot #

The method here is worth separating from the result, because it answers a question that usually needs an argument and it needs no argument at all.

The question is what does this quantity depend on? — and the usual route to it is a proof, which shows the answer can be computed from certain inputs and therefore does not depend on anything else. That is a strong result and it is expensive.

A grouping test answers the same question by contraposition. Partition the pool by the candidate inputs. If the quantity is constant on every group, then no input outside the partition can matter — because two positions in one group differ in those other inputs and agree in the answer. If the quantity varies inside a group, the candidate inputs are insufficient and the test says so immediately.

The test is cheap, it is decisive in one direction, and it needs no theory of the quantity at all. It is also the right test for exactly this question, because depends on is a statement about a function’s arguments and a partition is the direct way to interrogate one.

What it does not give is a reason. Knowing that the third temperature cannot enter is not knowing why, and a proof would supply the mechanism along with the fact. That is why the rung above still wants the induction: not to establish the scope, which is settled, but to explain it.

The habit worth carrying is to run the grouping before attempting the proof. It costs a sort, it fails fast when the candidate inputs are wrong, and it tells the proof exactly what it has to be a proof of.

The three readings this replaces #

It is worth putting the law beside the three readings the ladder has discarded, because each of them is a special case of it and the pool that refuted each is visible in the formula.

The crossover is the temperature. True when the answer is a number — a plain switch has $T_{\text{answer}} = 0$ and the correction vanishes. That is the whole of what the first rungs of this ladder had, and it is why the reading survived as long as it did: sente is a fact about the rest of the board was written on a pool of switches.

The temperature less a half. True when the answer sits at temperature one, since half of one is a half. That is the reading the answer that starts another fight established, on a pool whose answers were all the same size — the special case reported as a law, which is the mistake this ladder has now made three times and recorded each time.

Half the temperature. True when the answer is at least as hot as the fight, since $\min(T_{\text{answer}}, T) = T$ and the correction is $T/2$. That is the hot-answer regime, and it is where the $\min$ in the formula comes from.

So the law is not a fourth reading competing with three; it is the one that has all three as regions, and each earlier pool is a slice through it. That is the most satisfying shape a ladder on this site reaches, and it is worth noticing that it took four rungs and three refutations to get there — every one of which looked like a law at the time.

The convention, named #

Normal play throughout, and every value computed by the recursion.

A fight here is a position that is not a number. Its answer is its hottest Right option, and the answer’s own hottest Right option is where the third temperature is measured; a plain switch has an answer that is a number, so its third temperature is nought.

Levels count the nesting: $\{a \mid \{b \mid c\}\}$ has two, $\{a \mid \{b \mid \{c \mid d\}\}\}$ has three. The rung below’s pool reached three; this one reaches four.

The ambient temperature is the size of the largest coupon left in the stack, and the crossover is the highest ambient temperature at which the player still answers the fight rather than taking a coupon. It is computed by playing the sum of the position and the stack out, exactly, at each ambient temperature on the grid.

A group is the set of positions sharing a temperature and an answer temperature. A group is informative when its members do not all have the same third temperature, since a group that does cannot distinguish the two claims above.

Where the ladder goes next #

The sente anchor has eight rungs to here, and this one has settled the depth question by construction: a crossover depends on a position’s own temperature and its answer’s and on nothing below them.

The rung above says exactly how much of a proof is missing, which is more useful than a sketch. The premises an induction would need reports that the law holds at five levels of fight, survives translation, survives heating and survives cooling — and that none of that is the inductive step.

That is worth reading as a statement about what checking can and cannot do. Every one of those properties is what a reader would want a proof to imply, and having them all measured makes the law about as well supported as an unproved statement on this site gets. It also makes clear that the remaining gap is not a matter of confidence: the step would have to say why a position’s third temperature cannot enter, and no amount of finding that it does not enter supplies the reason.

So the anchor ends with a law that is exact, cheap to apply, checked in every direction anybody has thought to check, and unproved — and with the checking laid out so that whoever writes the four paragraphs knows which cases the argument has to survive.