Every essay — page 32
Temperature
How much is at stake, measured. Thermographs, cooling, and why a player moves where the game is hottest.
An environment instead of a stack
The guarantee behind playing the hottest component survives one step down a board's sorted temperatures and fails at two. Against a coupon environment as hot as the board it survives all of them — because the rule stops being approximate and starts being optimal, on every sum in the pool built to punish it.
The second bend is the boundary
Adding two thermographs wall by wall gives a diagram that is right at the mast and wrong below it. Over every pair of hot values born by day two, the added walls sit outside the true ones at every height — an outer envelope with the truth somewhere inside — and the two pictures separate at exactly the lower of the two temperatures, on all twenty-eight pairs. Above that height both components are still fights and the addition is exact; one sixteenth below it, every pair has parted.
The hotter residue survives
Charge a hot position its own temperature and subtract its mean, and an infinitesimal is left — its residue. Add two hot positions and the means add, the temperatures do not, and the residues do something stranger than either: on every one of 4,712 pairs whose parts differ in temperature, the residue of the sum is the hotter part's residue and nothing of the colder's. Cooling is additive, and a part cooled past its own temperature is a bare number. Residues add only when two parts are exactly as hot, and even then the fights usually cancel into something colder. The summary forgets; the game does not.
Colder exactly when the residues cancel
Add two fights of one temperature and the sum either stays that hot or grows colder, and which one happens is decided by the smallest thing either fight carries. The sum grows colder exactly when the two residues are negatives of each other — on all 3,142 equally hot pairs from days two and three and 3,000 sampled from day three, and by a two-line argument from cooling's additivity. What survives the cancellation is another matter. Group the colder sums by their parts' thermographs and the survivor differs in every group that holds more than one pair: {1 | 0} and {1 | 0, ∗} draw the same diagram and leave different infinitesimals behind. The diagrams fix only the survivor's temperature: nought, or exactly the lowest bend of either part.
Why cooling has to add
Cooling came through a sweep of 1,768 sums untouched while heating failed 263 times, and the sweep's own account was cautious: cooling is not additive in general, and a richer pool would break it. Aimed exactly where that account said to look — infinitesimals hidden under low temperatures, at taxes of a quarter and a half — 107,816 more pairs break nothing, because there is nothing to break. Cooling distributes over every sum of short games. Its freezing clause fires on one side of the equation and not the other in 6,032 of those pairs, and costs nothing every time, because what it hands back is the mean, and means add.
The bonus an integer is owed
Heating fails to distribute over a sum where cooling never does, and the repair proposed for it was to change what its exception returns: heat a number into a switch instead of leaving it alone. Run on the same 1,768 pairs, that repair fails on every one of them, beginning with nought plus nought. What adds is a heating that changes which numbers it exempts and pays the bonus on the moves an integer holds in reserve — Norton's product, and heating with no exception at all, which at a whole-number bonus is the same operator. Given heating's exception, cooling stops adding too.
An exemption that costs nothing below one
Heating that leaves integers alone fails to add at a bonus of one and adds on everything at a bonus of a half, and the natural guess was that a half is where it starts to go wrong, since that is where a fraction's moves become worth fighting over. It is not. Across twelve bonuses nothing fails at any value below one, and dozens of pairs fail at one exactly. The reason is the simplicity rule: below one, the recursion sends every integer to itself, so the exemption is not doing anything at all.
Values
What a position is worth — numbers, and the things that are not numbers, and how to find the simplest one.
Who moves last
The player who cannot move loses. That single convention generates the whole theory — and it produces four outcomes rather than three, because a position can be confused with zero rather than greater, smaller or equal to it.
The simplicity rule
When both players' options are numbers, the position is worth the simplest number strictly between them. Not the midpoint, not the average, and the difference between "simplest" and "middle" is the entire content of the rule.
Canonical form
Two positions are worth the same when neither player can tell them apart inside any larger game. Deciding that could be an infinite search. Instead there is a normal form — delete what nobody would play, bypass what backfires — and equality becomes a comparison of two small trees.
Infinitesimals
Some positions are positive — Left wins them whoever moves first — and smaller than every positive number, including a millionth and a millionth of that. They are the values that decide close games, and the smallest of them is a single move's worth of nothing.
The day a number is born
Start with a position in which neither player can move, apply one rule, and the numbers appear — but only the fractions with a power of two underneath, and only in a particular order. That order is what "simplest" means.
Numbers avoid numbers
In a position with a number in it and anything else, the number is never the right move. That is a theorem rather than a heuristic, and it is the closest this subject comes to advice a player can carry into a real game.
Worth nothing, and worth fighting for
A switch is a position both players want to move in. Its average value can be zero while the difference between getting there first and second is enormous, and that gap is a second number every position carries.
Two hundred and fifty-six ways to write twenty-two things
Every game whose options come from the four born on day one — there are 256 of them, and between them they carry 22 values. The reduction that collapses one to the other has choices in it at every step, and uniqueness is the claim that none of the choices matters.
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