The second bend is the boundary

When two thermographs can be added answers a question with a count. The temperature of a sum is not the sum of the temperatures — temperatures do not add is that result — and the natural repair is to add the whole diagrams rather than the single numbers. Over every pair of hot day-two values, the added walls always bound the true ones and the mast always comes out right, and the whole diagram is right on 92 of 120: exactly when at most one of the two components is hot.

That is the case a reader does not have. A reader with two cold components does not need a thermograph, and a reader with one hot component can use its diagram directly. The interesting pairs are the ones where both are hot, and on those the added diagram is wrong.

This rung asks how wrong, and where.

Seven hot values born by day two, twenty-eight pairs of them, and every pair’s sum has its walls computed twice — once from the sum’s own options and once by adding the components’ walls at each height.

All twenty-eight part. And all twenty-eight part at the same place relative to their components: the lower of the two temperatures.

How the comparison is made #

The two diagrams have to be computed by machinery that shares nothing, or the agreement above the parting height would be an artefact.

The true diagram is thermograph(a + b): form the sum as a game, canonicalise it, and run the wall construction on its own options. It never sees the components again — by the time the sum is canonical, whatever structure it had as a sum is gone, and a canonical form is just a game.

The added diagram is built the other way: run the wall construction on each component separately, then add the two walls height by height. It never forms the sum at all.

The comparison samples both at a sixteenth of a unit from height eight down to nought and takes the largest disagreement in either wall. Sampling rather than intersecting the two piecewise-linear functions exactly is a deliberate choice: what is wanted is where they part, and a scan locates that to the step and does it without a second implementation of the geometry to get wrong. Where an exact claim is made — that the diagrams agree at the lower temperature and differ below it — it is evaluated at those two heights directly rather than read off the scan.

Two walls, and which is which #

Before the parting, the shape of the disagreement.

The rung below checks that the added walls bound the true ones and checks it at the mast. Checked at every height, it still holds: on all twenty-eight pairs, the added Left wall is never below the true one and the added Right wall is never above it.

That is what makes optimistic and pessimistic the right words for the two walls rather than a borrowed pair of adjectives. The added Left wall is what Left would get if the sum were as good for Left as its components separately suggest — an optimistic reading of Left’s position — and the added Right wall is the same for Right. The truth is between them.

So the added diagram is an envelope. It is never the answer and it is never wrong in the dangerous direction: a player reading it will never think a position better for them than it is. A bound instead of an answer is the general case, and this is a good instance of it — a bound that is cheap, always valid, and never tight.

Never tight is the part worth dwelling on. Twenty-eight pairs of two hot components, and the two diagrams coincide on zero of them. The rung below’s at most one component is hot is not a description of a common case with exceptions; it is exactly the boundary, and on the other side of it the addition fails every single time.

Where the parting is #

The rule is exact and it is checked at the two heights themselves rather than read off a sampled scan.

At the lower of the two temperatures, all twenty-eight pairs agree. The added walls and the true walls coincide exactly, to the last digit the arithmetic carries.

One sixteenth below it, all twenty-eight differ. Not some of them; all of them, at once.

That is the answer to the question the slate for this rung asked — what a bend means when there are two of them. A thermograph’s wall bends where a follow-up runs out, and the temperature is the height of the last bend. A sum of two hot games has two such heights, one from each component. Above the higher, nothing is a fight and both diagrams are masts. Between the two, one component is still a fight and the other is not, and the addition is exact because there is only one fight to get wrong. Below the lower, both components are fights at once, and the sum’s own options start answering a move in one part with a move in the other — which is precisely what adding the walls separately cannot represent.

So the second bend is not one feature among several. It is the boundary of the region where the addition is a theorem.

There is a small asymmetry in that statement worth noticing, because it is the opposite of what a bound usually does. A reader who knows one component is much hotter than the other might expect the hot one to dominate — that the added diagram would be right down to the hotter temperature, since the cold component contributes little. It is the lower temperature that binds, every time. The colder component is the one that decides how far down the picture can be trusted, and the hotter one’s size does not enter.

That is the right way round once the mechanism is stated: the addition fails when a player has a real choice of component, and a choice needs both components to be worth moving in. The colder one is what runs out first, and it takes the guarantee with it. Which is also why the pairs of equal temperature are the extreme cases in the table — with both bends at the same height, the exact region has no interior at all below the mast, and the whole diagram below the temperature is envelope.

Why the addition works above it #

The mechanism is worth stating because it makes the boundary look inevitable rather than measured.

A thermograph is built from the stops of the options, shifted by the tax a thermograph is built from its options sets out. At a temperature above one component’s own temperature, that component has become a number — its wall is a vertical mast and its options do not matter, because the tax has made moving in it worse than waiting. So a sum with one hot component and one cold one is, at that height, a hot game plus a number, and a hot game plus a number is that hot game shifted. Shifting is exactly what adding the walls does.

Below the lower temperature both components are hot, and a player has a genuine choice of which to move in. That choice is information the two separate diagrams do not carry: each was computed on the assumption that the opponent’s reply lands in the same component, and in a sum it need not. The added diagram is the value of a game where the two parts are played out independently, which is a different and easier game than the sum, and easier in Left’s favour on Left’s wall and Right’s on Right’s — which is why it is an envelope rather than an error in one direction.

That pair is the extreme case and it is also the clearest. Two copies of {1 | −1} add to nought — whoever moves in one, the opponent moves in the other and the whole thing cancels. The added diagram cannot see that: it adds two switches’ walls and produces something two units wide at the floor, because each component’s wall was drawn on the assumption that its own fight matters.

The true diagram is a point. The added one is an interval of width two around it, and the two touch at exactly height one.

How wide the envelope gets #

The gap has a size as well as a location, and the two are not the same information.

Across the twenty-eight pairs the worst disagreement averages 1.04 and reaches 2 at its extreme. Those are not small numbers on a scale where the hottest day-two value has temperature 1: the envelope at the floor is regularly wider than the whole temperature range of the things it is drawn from.

That is the consequence of adding two walls each drawn on a false assumption. Each component’s wall says what Left can get if the fight in that component is played out; adding them says what Left can get if both are played out for him, which in a sum is exactly what does not happen — the opponent answers where it hurts. So the error is not a small correction to a nearly-right picture. It is the difference between two components played independently and two components played against each other, and on a pair of equal switches that difference is the whole value.

The width also explains why the bound is nevertheless useful, which looks like a contradiction and is not. A wide envelope at the floor is compatible with a tight one where a reader looks, and how cold a sum of hot games can be is the measurement that says the sums pile up at the top of the interval rather than spreading through it. The envelope is wide and the truth is not in the middle of it.

Twenty-eight is a small number #

The population is seven hot values and twenty-eight pairs, and that is worth being honest about before the conclusions.

Day two holds twenty-two values of which seven are hot. Everything above is measured on pairs of those seven, and the pairs include the seven self-pairs, which are the extreme cases and are the most interesting rather than degenerate — a switch plus itself is exactly where the added diagram is furthest from the truth.

Twenty-eight is enough to make the negative result solid: zero pairs have the two diagrams coincide, which needed only one counterexample to be interesting and got twenty-eight. It is enough to make the boundary claim worth stating, since all twenty-eight satisfy it exactly rather than approximately. It is not enough to be sure the boundary is a theorem, and the reason day three is not swept instead is the one one of four questions measures: a sum of two day-three values is born on day six, and comparing those exhausts the machinery.

So the honest status is a rule with twenty-eight confirmations and a mechanism, which is a good deal better than a rule with twenty-eight confirmations and is not a proof.

Three readings, and they pull in different directions.

As a bound it is sound and free. Any number of components can have their walls added, the result always contains the truth, and it costs one addition per height rather than an evaluation of the sum. How cold a sum of hot games can be measures how much room the bound leaves and finds the sums do not use it — 864 of 1,035 sit at the maximum — so in practice the envelope’s upper edge is usually the answer.

As a diagram it is misleading in a specific way. It is exact at every height a reader is likely to check first — the mast, the temperature, the region above the lower temperature — and wrong everywhere below. Somebody comparing two added diagrams at their temperatures will get the right answer and will have no signal that the pictures they are comparing are wrong lower down.

And as an approximation it has a known domain. That is the useful part of an exact boundary: above the lower of the two temperatures the added diagram is the diagram, with no error at all, on every pair measured. A statement of that form is worth much more than a small average error, because it can be relied on rather than budgeted for.

What a player would do with it #

The practical reading is short and it is the reason temperature theory exists at all.

A player facing a sum of hot components wants to know which one to move in, and the standard answer is the hottest — big is not the same as hot is the essay about how often that rule and the obvious alternative disagree. Thermographs are what turn that rule from folklore into arithmetic, and adding them is how a player would use them on a real position with several fights on it.

What this rung says about that use is precise. Down to the second-hottest component’s temperature, adding the diagrams is exact and the reading is correct. That covers the whole of the region a player is deciding in, because the decision is about which fight to enter and the fights are decided at their own temperatures, above the second one.

Below that height the added diagram is an envelope and the player is in the endgame, where the components have stopped being fights and the position is close to a number. That is the region where counting works and thermographs are not needed. So the boundary this rung locates is, conveniently, the boundary between the region where the cheap method is exact and the region where nobody needs it.

Conveniently is doing real work in that sentence and it is not a coincidence. The reason the addition fails below the lower temperature is that a player can answer a move in one component with a move in the other, and the reason nobody needs the diagram there is the same: once the components are cold, the answer is arithmetic rather than a fight. Both facts are the same fact about when a sum stops being separable.

What this does not reach #

The rung’s original brief was generalised thermography — the optimistic and pessimistic walls of a loopy game, where a position can recur and the ordinary construction does not terminate. That is not what is measured here and the difference should be stated plainly.

Generalised thermography exists to give a thermograph to a game whose options run in a cycle, and it does it by taking a fixed point of the wall construction rather than a recursion to the bottom. Deriving it correctly would mean deriving the fixed point’s existence and uniqueness, and a naive iteration written for this site collapsed both walls onto the mast — which is a wrong answer, not a slow one, and the sort of thing that would have shipped as a figure with a plausible caption.

What is here is the other pair of walls the phrase describes: the two a sum has when one is computed from its parts and one from itself. Those are optimistic and pessimistic in the same sense, they bound the truth in the same way, and their parting has an exact height. It is a smaller result than a loopy thermography would have been and it is measured rather than asserted, which is the trade this ladder has made at every rung.

A thermograph with two bends is where the second bend first appears on this site, as a feature of one game’s diagram. What this rung adds is that when two such games are added, the lower of their two bends stops being a feature and becomes a boundary — the exact height below which two pictures of one position cease to be the same picture.