Context — where it appears
Named by 12 essays across 4 fields — each of them below, with the objects they name alongside it.
Equal in this company
Equality quantifies over every game there is, and the quantifier can be made smaller. Restricted to a company of nine numbers, the twenty-two values born by day two collapse to seventeen; restricted to four games — nought, one, minus one and star — they stay twenty-two, and no three of the four will do. The company that decides equality is tiny, and it has to contain a star.
How hot a background has to be
The reduced canonical form throws away infinitesimals, and the rung below asked for a bound: how hot must the rest of the board be for the discarded part not to matter? There is no such bound. Safety is not monotone in the background's temperature — an eighth is safe, a quarter is not, two is safe again — and the quantity that does decide it is not a temperature but a stop.
What restores the theorem
Fibonacci Nim breaks the recipe every impartial game is supposed to obey: one number per heap, exclusive-ored, gets a quarter of two-heap sums wrong. Index the recursion on the pair of heap size and cap instead and the recipe is exact on every pair and every triple — and the number a heap of nine carries turns out to be five rather than one.
Two clauses and a third question
A component can carry its own rule when two things hold: its moves are a function of what it carries, and a move in it leaves every other component alone. Two rulesets built to fail one clause each are both caught on a named witness. The four real games sort exactly — every one the recipe gets right fails no clause, every one it gets wrong fails one — and the two clauses still miss something, because Fibonacci Nim and a held pass fail the same clause and only one of them can be repaired.
A ko is won somewhere else
The rung below shows the ko rule buying finiteness by deleting one edge. What it buys with the same edge is a fight nobody can settle by looking at it — the prohibition forces a player to spend a threat, threats are counted on the rest of the board, and every decided cell of the sweep goes to whoever is ahead on a quantity that is not in the picture.
Nothing to subtract with
Comparison is defined by contexts and computed by subtraction, and the equivalence between the two is a theorem about groups. A scoring game is not one — sixty-six of eighty-one coin rows do not cancel against their own negatives — and the difference test then fails on a row compared with itself, which every context accepts and nothing certifies.
What two numbers cannot tell apart
A thermograph summarises a position in a mean and a temperature — nine characters against the brace form's twenty-two, and readable in a way the expression is not. It is also not exact: 1,454 of the 1,474 games born by day three share a pair with some other game, 291 of them share one pair, and adding a star to two of those gives different winners.
The restriction that buys the most
Four candidate classes of scoring game, scored on the same two families and the same three questions. The class everyone expects to be tiny — the rows that cancel against their own negatives — is empty on rows of three and the widest restriction on rows of four, where it holds fifteen rows against the hereditary class's twelve and gets all 225 of its comparisons right against 108 of 144. The trade everyone expected does not exist.
A cancelling pair is a zero
Two cancelling coin rows side by side cancel against their two negatives — all 190 pairs from rows of two, four and five coins, the odd row that cancels without pairing off included. And a cancelling row beside its negative is invisible next to any other row: in 741 tests against every row of one to three coins, neither score of the context moves. A non-cancelling pair moves a score in 452 of 780. Scoring games have no inverses in general; this class has them, and they behave as inverses must.
The same parts, different wholes
Equality is worth having because equal parts make equal wholes: a game may be replaced by an equal one inside any sum and any larger position. The relation that 184 contexts define — no context tells the two apart — is exact on day three and merges a third of a sample of day four. Put 567 of its merged pairs through negation, addition and forming options, and it survives two of the three. Adding the same game to both separates them 1,527 times.
What a part would have to report
Outcome classes leave four entries of the addition table open. Give each part its two stops as well and four inequalities settle 176 of the 369 open day-two sums, never wrongly — and every sum they leave has a bound sitting exactly on nought. Used as a lookup table instead, even the whole thermograph with the outcome beside it leaves 96 sums undecided. Four outcomes do what no thermograph can: who wins a day-two position alone, beside −1, beside 1 and beside a star names it, and no three questions do.
A universe grows by days, not by sums
A relation built from a sample of contexts can be repaired two ways: allow sums of more values, or allow values born a day later. On a sample of day four the first stalls — 147 pairs of different games stay merged from five summands to seven, through 25,696 contexts — while the 985 values of day three, a seventh as many, separate every pair. The reason is four lines long: the context that tells two games apart is always minus one of their options.
Named alongside it
The objects these essays reach for when they reach for this one.
Exhaustive searchEqualityComparisonOutcome classCanonical formCounterexampleDay threeDay twoDisjunctive sumIndistinguishabilityNegationAdditivity