Cooling adds and heating does not

Cooling charges a tax on every move. Heating pays a bonus. The two are introduced together, written with matching notation, and described as operators that undo one another — and the resemblance is close enough that a reader can go a long way without noticing that only one of them has the property a sum needs.

The asymmetry comes from one clause, and the clause is the same one that makes cooling lossy. It is worth taking the definitions apart to see it.

The two definitions, side by side #

Cooling by $t$. For a position that is not a number,

$$G_t \;=\; \{\, G^L_t - t \;\mid\; G^R_t + t \,\}$$

with a number left unchanged, and with a position frozen to its mean once $t$ passes its temperature. Every move costs the mover $t$, all the way down the tree, so moving becomes progressively less attractive until nobody wants to and the position is a number.

Heating by $t$. For a position that is not a number,

$$G^t \;=\; \{\, G^{L,t} + t \;\mid\; G^{R,t} - t \,\}$$

with a number left unchanged. Every move pays the mover $t$, so moving becomes progressively more attractive, and a position that had nothing at stake acquires something.

The picture the pair is usually introduced with is a switch being cooled step by step: the two options move towards one another as the tax rises, and at the temperature they meet, so the position has frozen into the number it is worth on average and the fight is gone. Read right to left the same strip looks like heating, which is precisely the resemblance this essay is about. It survives contact with a single position and does not survive contact with a sum.

The two formulas are mirror images except for the clause about numbers, which is where the whole of the value theory hinges, which is identical in both — and identical is exactly wrong. Cooling a number leaves it alone because a number has nothing to tax. Heating a number leaves it alone because… there is no reason. The clause is there so that heating a cold board does not manufacture stakes out of nothing, which is a sensible thing to want and is not symmetric with anything. It is also the clause that makes cooling lose almost everything it touches.

Where the sum comes in #

The obvious explanation is that a sum can be a number when its parts are not — $\{2 \mid 0\}$ and $\{0 \mid -2\}$ are both hot and add to exactly zero — so the exempting clause fires once on the left of the equation and not at all on the right.

That explanation is wrong, and the sweep says so flatly: in all 263 failures, neither part is a number and neither is the sum. The clause never fires at the top level in a single one of them.

It fires two levels down, and the reason is a fact about what an option of a sum is. Left’s options in $G + H$ are $G^L + H$ and $G + H^L$: a part’s option plus the whole of the other part. So an option that is a number inside $H$ — and is therefore left untouched when $H$ is heated on its own — appears inside the sum as $G + H^L$, which is a number added to something that is not one, and is not a number. The clause exempts it on one side of the equation and heats it on the other.

The first failure in the sweep is small enough to follow all the way through.

$$\ast + \{1 \mid \ast\} \;=\; \{1\ast \mid 0\}$$

Heat the sum by one and Left’s option $1\ast$ is not a number, so the recursion goes into it: $1\ast$ heats to $\{2 \mid 0\}$, and the bonus makes it $\{3 \mid 1\}$. The whole comes out as $\{\{3 \mid 1\} \mid -1\}$.

Heat the parts instead and $\{1 \mid \ast\}$ has the Left option $1$, which is a number and is left alone; the bonus makes it $2$. Adding the heated $\ast$, which is $\pm 1$, gives $\{1, \{3 \mid 1\} \mid -1\}$ — one Left option more than the other side has.

What heating does to something infinitesimal #

The reason infinitesimals appear in 162 of the failures is that they are the positions on which heating does the most violent thing.

$\ast$ has both stops at zero and a temperature of zero: nothing at stake, and not a number. Heat it by one and the definition gives $\{0 + 1 \mid 0 - 1\}$, which is $\pm 1$ — a full one-point fight, manufactured out of a position worth less than every positive number and more than every negative one.

So heating is not merely non-additive; it is the operator that converts the finest distinctions the theory has into the coarsest. That is useful — it is how the atomic weight machinery gets started, by heating an all-small position until it has a temperature that can be measured — and it is why the operator cannot be expected to commute with anything.

The search for a cooling failure #

Cooling came through the sweep untouched: 1,768 pairs, no exceptions. That is a result about a search and it is stated that way on purpose, because cooling is not additive in general.

The general failure needs a position whose temperature is below the tax and which carries an infinitesimal underneath the freezing point. Cooling past the temperature replaces such a position by its mean, and the mean is a number; the infinitesimal is gone, and if the rest of the board still needed it the two sides come apart. The pools here do not produce one. Day two is small and its infinitesimals are shallow; the hot pool was built for temperature rather than for depth.

Doubling the tax narrows that window rather than widening it, which is worth seeing before the ladder goes looking in it.

Reporting the search size is the honest form of the claim, and the claim is still worth making. Cooling survived every pair a sweep designed to break heating could produce. Heating fell over on the first pair with a star in it. Whatever the general theory says, the two operators are not equally robust, and a reader who has been shown them as a matched pair has been shown something misleading.

How large the gap is #

A failure of additivity is worth more when its size is measured, and the size depends on the tax in a way worth reporting.

At $t = 1$ the two sides always differ by an infinitesimal: all 108 failures at that tax have a difference with both stops at zero, so no number reports the discrepancy and neither player is a point worse off. At $t = 2$ that stops being true — only 47 of the 155 failures have an infinitesimal gap, and the rest differ by something a number can see.

The pattern is what the mechanism predicts. The discrepancy is created by one option being heated on one side and not the other, and a single application of the bonus at a deep node moves the value by an amount bounded by the tax. Double the tax and the same structural difference becomes twice as visible, and at some point it stops hiding below every number.

The other operation called heating #

There is a second thing the word names, and conflating the two is easy because both turn a cold object into a hot one.

Given a number $x$ and a tax $t$, the switch built from them is $\{x + t \mid x - t\}$: a position with mean $x$ and temperature $t$. Every switch is one of these, and the correspondence is a bijection between pairs $(x, t)$ and switches.

That operation is invertible, and it is invertible on numbers, which is exactly where the operator above is not. So the sentence “heating undoes cooling” is true of one of the two things called heating and false of the other, and the two are separate functions here for that reason. A figure that used whichever one suited its caption would be claiming that heating is invertible on numbers, which it is not.

Why the asymmetry is not a defect #

It would be tidy if the two operators were inverse, and the reason they are not is a reason to want them anyway.

Cooling exists to answer “what is this position worth once moving is expensive?”, and the answer has to be a number eventually, because the whole point is to find the price at which the fight stops. An operator that never reached a number would never produce a temperature.

Heating exists to answer the opposite question — “what would this position look like if moving were cheap?” — and it is used almost entirely on positions that are already too small to measure. Atomic weight is defined by heating an all-small position until it has stakes a number can report, and the whole construction depends on heating doing something drastic to $\uparrow$ and nothing at all to $1$. That is the operator earning its clause rather than suffering from it, and the ups it measures are exactly the objects no temperature can see.

So the clause that breaks additivity is the clause that makes the operator useful. What the sweep establishes is that the price is real and can be counted, not that somebody made a mistake.

What the solver computed, and how #

Two pools. The first is the 22 values born by day two, enumerated rather than chosen. The second is twenty positions written down for the purpose: plain switches at several temperatures, switches with a follow-up, and the small infinitesimals.

For every ordered pair in each pool and each of two taxes, the sum is built and reduced, the operator applied to it, and the result compared for equality with the operator applied to each part and the two added. Equality is the ordinary comparison — the difference played out and its outcome read — so a failure is a genuine inequality of values rather than a difference of printed forms.

Failures are kept with both parts named and with a flag saying whether either is all-small, which is what turns “heating fails 263 times” into the sharper statement that 162 of the failures have an infinitesimal in them. The sweep also asserts on the other direction: a run in which heating turned out additive everywhere would mean the pools had stopped containing infinitesimals and the measurement had quietly become vacuous, and that is checked rather than assumed.

Only one of the two exemptions is an exemption #

The section on the definitions says the clause about numbers is “identical in both — and identical is exactly wrong”, and stops at heating a number leaves it alone because there is no reason. There is a sharper thing to say, and it is the cleanest account of the whole asymmetry.

Cooling’s clause is redundant. Delete it and cooling does not change. A number’s temperature is below every tax, so the freezing clause already applies to it, and a number’s mean value is itself — so the general rule returns exactly what the exemption returns. The clause is a convenience, stating a case the recursion would have got right anyway.

Heating’s clause is not. Delete it and heating changes everything. The recursion applied to a number $x$ would build $\{x + t \mid x - t\}$, a full fight with mean $x$ and temperature $t$, and it would go on doing that for ever, since the thing it builds is not a number either. Heating has no freezing clause and nothing to stop it, so the exemption is the only thing standing between the operator and an infinite regress.

So the two definitions look symmetric and are not. One of them describes an operator that reaches the numbers naturally and notes that it stops there. The other describes an operator that would never stop, with a rule bolted on saying where.

The condition an exempt set has to meet #

That reframing turns the additivity question into a question about a set, and the question has a clean answer.

An operator with an exempt set distributes over a sum only if the exemption fires in the same places on both sides of the equation. On the left it is consulted about the sum and about the sum’s options; on the right, about each part and each part’s options. And an option of a sum is a part’s option plus the whole of the other part. So the exemption is being asked about $G^L$ on one side and about $G^L + H$ on the other.

For those to agree, the exempt set must be closed under adding whatever the other part is. The numbers are not. A number plus a star is not a number; a number plus an up is not a number; a number plus almost anything that is not a number is not a number. The exempt set is as small as a set can be while still being the thing the operator is aimed at, and it leaks the moment anything is added to it.

That is the worked example above, stated as a property rather than as an incident. Left’s move to $1$ inside $\{1 \mid \ast\}$ is exempt on its own and is $1 + \ast$ inside the sum, which is not, so the bonus is paid on one side and withheld on the other. Nothing about stars or infinitesimals is essential to it; any component that carries a number as an option and sits beside a non-number will do, which is why 263 is a large count rather than a handful of curiosities.

And it says why cooling escapes. Cooling’s exemption is not a set the operator treats specially — it is a description of where the operator happens to land — so there is nothing to fire inconsistently. Cooling a number, cooling a number plus a star, cooling a number plus an up: all three go through the same clauses and come out at the mean, because the freezing rule reads a temperature rather than asking what kind of object it has been handed.

The general shape is worth carrying past this game. An operator defined by a recursion plus an exception distributes over a sum when the exception’s set is closed under the sum, and fails when it is not — and the failures are located exactly where an option crosses the boundary. That is a statement about definitions rather than about temperature, and it predicts where to look in any construction of this shape: at the smallest set the definition names.

The shape of the claim #

It is worth being explicit about what has and has not been shown, because two of the three statements on this page are of different kinds.

That heating is not additive is demonstrated: 263 counterexamples, each a pair of positions with the two sides of the equation printed and unequal. One would have been enough.

That the failures come from the exempting clause firing at different depths is explained and illustrated, not proved. The worked example above shows the clause doing exactly that, and the count of 263 out of 263 with nothing at the top level a number rules out the simpler story — but a general argument would need to show that no other mechanism can produce a failure, and none is offered.

That cooling is additive is not shown at all. It is false in general, and what is reported is that a search over 1,768 pairs did not find a counterexample. The difference between those two kinds of statement is the difference between this page and a proof, and it is the reason every count on it comes with the size of the search that produced it.

Where the model stops #

The pools are two, and neither is a random sample of the space of short games. 1,768 pairs is enough to make “cooling never failed here” a real observation and is not enough to make it a theorem — and the theorem is false, so the sweep is measuring the reach of the pools as much as it is measuring the operators.

The taxes are one and two. Fractional taxes were not swept, and a fractional tax is exactly where a position frozen just above its own temperature would appear. A sweep at $t = \tfrac12$ over positions of temperature $\tfrac14$ would be the natural place to look for the cooling counterexample this page could not produce.

Where the ladder goes next #

This rung establishes that the two operators differ where it matters for sums, and identifies the clause responsible. The rung above is the counterexample the sweep did not find: a pair on which cooling fails, which would need positions carrying infinitesimals below their freezing points and is a construction rather than a search.

Two neighbours sit alongside. Cooling by exactly one is the round-trip measurement — how much of a position survives being cooled and heated back — and this page is the same asymmetry seen from the side of sums rather than of single positions. And the operator chosen for one game is what happens when the tax is fixed at one and pointed at the game it was picked for.