Sente — the series · Combinatorial Game Theory

{5 | {4 | 0}} beside one other fight. A local position and a single switch, played out together at each of several ambient temperatures. The middle columns are what optimal play does: whether it opens the local fight, and whether it answers when the opponent opens it. The answer stops being forced at a temperature the local position alone does not name.

Sente is a fact about the rest of the board

A move that must be answered is called sente, and the word is used as though it described the local position. It does not. The same fight is answered while the rest of the board is quiet and ignored once it is busy, and the crossover — measured by solving the whole board at every temperature — sits at the follow-up's own temperature.

part 1 · temperature

Is {{5 | 3} | {2 | −4}} double sente?. One local fight with a follow-up on each side, played out inside a sum with a switch whose temperature rises. The two middle columns are what optimal play does when each player opens the fight. Whether the opponent has to answer is settled by the ambient temperature and not by the shape, so the same position is double sente, sente for one player, and gote for both, at three different ambients.

Double sente is not a property of the position

A fight with a threat on each side is called double sente, as though the phrase named a shape. Swept against a rising ambient temperature, one such fight — { {5 | 3} | {2 | −4}} — is double sente up to an ambient of 1, sente for one player only from 3/2 to 3, and gote for both from 7/2, and the two band edges are the temperatures of the two follow-ups.

part 2 · temperature

The move that stops the other one. Six local fights, each placed beside an environment of known temperature. The last two columns are the band over which taking the local move beats spending the move outside, and the band over which the opponent's move in the same fight has to be answered. They are not the same band.

What a move nobody makes is worth

If Right's move in a fight has to be answered, then Left's move in the same fight prevents an exchange Right was going to get for nothing. That is a reverse sente, and pricing it is the awkward case: over sixty measurements the gain matches the local temperature once and the follow-up's swing twice, and the band over which the move is worth taking is not the band over which the move it reverses is sente.

part 3 · temperature

Where sente stops, by what the answer costs. The largest ambient temperature at which a local move is still answered, sorted by how deep the fight below the answer runs. When the answer ends the fight the crossover is the follow-up's temperature; when the answer starts another fight it is exactly half of it, and a third level does not halve it again.

The answer that starts another fight

A local move is answered while the ambient temperature stays below the follow-up's — and that rule, which this site has carried since the anchor opened, is exact only when the answer ends the fight. When the answer starts another one the crossover is exactly half the follow-up's temperature, on every position tested, and a third level of fight does not halve it again.

part 4 · temperature

Two feet, and the one that halves the crossover. The thermographs of two follow-ups, drawn to the same scale. The first has a right wall that leans in from the axis and its move is answered up to the follow-up's full temperature; the second has a wall rising vertically first and its move is answered only to half of it.

What the halving is a function of

A move is answered while the board is cooler than the follow-up's temperature — or half of it, depending which of two classes the position is in, and the classes were stated in terms of forms. They are a feature of one wall: whether the follow-up's right wall rises straight up before it leans. Fifty-five positions, no exception, and the crossover becomes something a reader can see.

part 5 · temperature

The crossover by depth. How far the ambient temperature can rise with the move still answered, by how deep the fight goes. Where the answer settles the fight it is the follow-up's temperature; deeper it is that less a half.

A subtraction, not a factor

The crossover factor was a half on fights whose answer starts another fight, measured on a pool with two three-deep positions in it. A pool built to be deep gives twenty, and the factor does not survive them: the crossover is the follow-up's temperature less a half on eighteen of the twenty, and a factor of a half agrees with that only where the temperature is one — which nearly every position in the earlier pool had.

part 6 · temperature

Two clauses, one formula. The law split by which of the two temperatures is the smaller. When the answer is colder the correction is half of it; when it is hotter the correction saturates at half the fight's own temperature.

Half of the smaller temperature

The correction to the sente crossover has been priced twice — first as a factor of a half, then as a subtraction of a half — each time on a pool whose answers were all about the same size. Over 128 fights with answers from a number up to a temperature of six, the correction is half the answer's temperature, saturating at half the fight's own. Both earlier readings are regions of that one law.

part 7 · temperature

The third temperature is not consulted. Positions grouped by their two top temperatures, with the third temperatures they hold between them. The crossover is the same for every position in a group however far apart their third temperatures are.

The two numbers at the top

The rung below found the crossover of a sente fight to be its temperature less half its answer's, and said a proof would settle the depth question with it. The depth question is settled without the proof, by construction: group the positions by their two top temperatures and the crossover is single-valued on every group, however far apart the third temperature is — and the formula survives a fourth level of fight, which the rung below never reached.

part 8 · temperature

Five premises, and the step. The claims an induction would need, with what checks each. The last row is the step and nothing here checks it.

The premises an induction would need

The rung below settled by a grouping test that a position's crossover depends on its own temperature and its answer's and on nothing below them, and asked for the induction. The four paragraphs are not written here; the checking they would rest on is. The law holds at five levels, survives translation, heating and cooling — and none of that is the step.

part 9 · temperature

Three orders, one of them right. The holder's rank above the crossover under three different orderings of the options.

Which top is the top

The crossover law's proof rests on the walls above the crossover being governed by the top two options, and the check was never run. Run on 23,586 heights it holds exactly — but only when the options are ranked by mean value. Ranked by the temperatures the law is stated in, it fails on a fifth of them.

part 10 · temperature

Two claims read as one. The proposed geometry separated into the claim the rung below established and the claim it needs but did not.

The four paragraphs prove something else

Three rungs earned the right to write the crossover law's proof as two straight walls meeting where the law says. The walls are straight — one of them everywhere, for a trivial reason, and the other only above the answer's own temperature. The crossover sits below that height, so the geometry holds nowhere the law is about, and where it does hold it proves the temperature instead.

part 11 · temperature