Combinatorial Game Theory
who wins, and by how much
Series

Dissociation — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. A hot position, split into its mean and what is left. Each row is a position cooled by exactly its own temperature — the tax at which it stops being worth moving in. The result is the mean value with something small still attached, and the last column is that something, obtained by subtracting the mean from the cooled game rather than by inspection.

    A number and a fight

    Charge a position exactly what it is worth fighting over and the fight disappears, leaving the mean value — with something still attached to it. Over all 1,122 hot values born by day three the residue is smaller than every positive number, it is a star in 942 of them, and it is never nothing. So a hot game is its mean plus a fight plus a remainder that no number reports, and the remainder is what decides close games.

    part 1 · temperature
  2. The residue of a sum is the hotter part's. Every hot day-two value added to every hot day-three value, 7,854 sums in all, sorted by what happens to the residues: when the parts differ in temperature, on all 4,712 such pairs, the sum's residue is the hotter part's alone; when they are equally hot the residues add on 880, the sum grows colder on 2,237 and cancels to a number on 25.

    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.

    part 2 · temperature
  3. Colder exactly when the residues cancel. Equally hot pairs of hot values, 3142 from days two and three and 3000 sampled from day three, sorted by whether the sum stays as hot, grows colder or is a number, and by whether the two residues cancel. Every sum that stays as hot has residues that do not cancel, and every sum that grows colder or is a number has residues that do.

    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.

    part 3 · temperature

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