Which top is the top

The premises an induction would need checked four premises of the crossover law — that it holds five levels deep, and survives translation, heating and cooling — found all four holding, and closed by naming the measurement it had not made and the place a proof would fail:

A proof that fails will fail at one identifiable place — the claim that the walls above the crossover are governed by the top two options — and a measurement that would test that directly is available … That is a figure this page could have drawn and did not, and it is the last measurement before the geometry.

The measurement has been made. The premise holds on every one of 23,586 heights, and it holds for a reason the law’s own statement does not name.

What holds a wall up means #

A thermograph’s left wall is an envelope. Each Left option $A$ contributes its own right wall shifted down by the tax — $\mathrm{RS}_A(t) - t$ — and the left wall of the position is the highest of those at each height. So at every height one option attains the wall, and which option is a question with an answer that can be computed.

That is the object the crossover law’s proof would have to reason about. The law says a position’s crossover is $t - \min(t_{\text{answer}}, t)/2$, a function of the position’s own temperature and its hottest answer’s; a proof would show that the walls above the crossover are made of the same small number of options that those two temperatures come from, so that nothing further down the option list can matter.

The census is every hot value born by day three — 1,122 of them — restricted to the 406 sides carrying three or more options, because on a side with one or two every option is in the top two by construction and the premise is vacuous. Above each position’s crossover, at an eighth of a move apiece, that is 23,586 heights.

Two conventions in that count are worth stating. A height is counted once per side, so a position with three Left options and three Right options contributes two sets of heights; and ties are counted as successes for every option holding the wall, which is the generous reading and the right one — a premise saying the wall is governed by the top two is not violated by the top two sharing it with a third.

The generosity matters more than it looks, because ties are common. Options of one side share a mean far more often than they share a temperature, and where all of a side’s options have the same mean, every one of them holds the wall at the top. On a population with many such sides, any ranking scores well at the top of the wall for free.

Ranked by temperature, the premise fails #

The natural reading of the top two options is the top two by the quantity the law is about, and the law is about temperatures.

Ranked that way the premise is false on a fifth of the heights. The hottest option holds the wall on $73{\cdot}3$ per cent of them and one of the two hottest on $82{\cdot}9$; on the rest the wall is held by an option that is third or lower by temperature. Ranking instead by the stop each option starts at gives $73{\cdot}9$ and $79{\cdot}6$ — no better.

So a proof written around the two hottest options would fail, and it would fail exactly where the rung below predicted a proof would fail. That is a useful thing to know before writing four paragraphs of geometry.

Ranked by mean, it is exact #

Rank the options by their mean value instead — what each is worth once its own fight is over — and the premise is not merely true but nearly trivial. The option with the largest mean holds the wall on 23,584 of the 23,586 heights. One of the top two holds it on all 23,586.

The mechanism is a sentence of thermograph geometry and it is why the mean is the right quantity. High on a wall, every option has been taxed past its own temperature: its contribution is flat at its mean less nothing, then falls at slope one, so at height $t$ every option contributes its mean minus $t$. Subtracting the same $t$ from every option leaves the largest mean on top. The temperature and the stops govern the bottom of a wall, where the fights are still live and an option’s own stake still matters.

So the premise is true because the crossover sits above the height at which a wall stops being about temperatures and starts being about means.

And it sits well above. For each side, the height from which the largest-mean option holds the wall for ever after is at or below the crossover on 404 of the 406, by seven eighths of a move on average and by as much as one and a half. The premise is not marginal; the crossover is comfortably inside the region where the wall is in its final regime.

The two exceptions #

Two heights in 23,586 have the wall held by the option with the second largest mean, and they are the same exception twice.

The two positions are $\{ {\uparrow}, {\uparrow}{\ast} \mid -1/2, \{0 \mid -1\}, \{ {\ast} \mid -1\}\}$ and $\{1, 1{\ast} \mid 1/2, \{1 \mid 0\}, \{1 \mid {\ast}\}\}$, and the second is the first plus one. Both fail on Right’s wall, at a height of $3/8$, a quarter above a crossover of $1/4$.

That the counterexample comes in a translated pair is not a coincidence and is mildly reassuring: the rung below established that the crossover law is invariant under translation, so anything that breaks it breaks its translate too. A counterexample that did not come with its translates would mean something worse than a counterexample — it would mean the invariance was wrong.

What the picture would be #

The measurement is a census and the object is a drawing, so it is worth saying what the drawing looks like even though this page cannot fit 406 of them.

A position’s left wall, drawn against three of its options’ contributions: three lines, each flat at that option’s right stop and then falling at slope one from the option’s own temperature onwards. The wall is the highest of the three at each height, so it starts as whichever option has the largest stop, may switch once or twice as the flat parts run out, and ends as whichever has the largest mean. The crossover is a horizontal line drawn across, and the claim is that every switch happens below it.

That is a legible figure and it is a figure of one position. Reading a thermograph draws the walls and a thermograph is built from its options draws the envelope being formed, and between them a reader has the picture. What neither can carry is the quantifier — every switch, on 406 sides — and that is what the tables here are.

What a proof now has to do #

The measurement narrows the geometry and it does not write it. Three things are now known that were not:

The premise is about means. A proof reasoning about the two hottest options is reasoning about the wrong two, and would fail on a fifth of the cases. A proof reasoning about the two largest means has a true premise.

The premise has margin. The largest-mean option takes over seven eighths of a move below the crossover on average, so a proof does not have to work at the exact boundary — it can afford a slack term.

And the crossover law is stated in the wrong currency for its own proof. The law reads two temperatures; the geometry that makes it work reads two means. Those are different quantities of a thermograph — the temperature is the height where the walls meet and the mean is the value they meet at — and a proof will have to convert between them at some point. That conversion is where the factor of a half in $t - \min(t_{\text{answer}}, t)/2$ presumably comes from, since a temperature is half a stop gap and the same number in two currencies is the site’s other example of the same exchange rate.

What the crossover is, in one paragraph #

The crossover is easy to lose sight of under this much arithmetic, and everything above is a statement about it, so it is worth restating.

A player with a hot position and an environment of coupons has a choice at every turn: answer the fight in front of them, or take a coupon from the stack. The crossover is the highest ambient temperature at which they still answer — above it the coupons are worth more than the fight and the fight can wait; below it the fight is the best thing on the board. It is a number attached to a position and it is what makes sente a fact about the rest of the board rather than about the position, which is the whole subject of this anchor.

Why the walls matter is that the crossover is read off them. A position’s behaviour at ambient temperature $t$ is its thermograph at height $t$, and which option governs the wall there is which move the player is really choosing between coupons and. So which option holds the wall above the crossover is the same question as which move a player is weighing when the fight is still worth answering, and the answer this page gives — the option with the largest mean — is a statement a player could act on: high in the game, the move to weigh is the one that leaves the most, not the one with the most at stake.

Why the temperature ranking is so poor #

Seventy-three per cent is a suspicious number and it is worth saying why it is not higher.

An option’s temperature and its mean are independent quantities. A cold option with a high mean sits high on the wall from the ground upwards; a hot option with a low mean starts low, rises relative to the others as its own fight gets taxed away, and never overtakes. So hottest and highest are simply different questions, and the wall is about the second.

Where the two rankings agree is where a population’s options have similar means, and on this pool they very often do — options of one side share a mean far more often than they share a temperature, which is what makes the worked example a tie at the top. So the temperature ranking scores 73 per cent by agreeing with the mean ranking by accident, and its failures are the positions where the two come apart.

That is the same shape as a threshold is a detection limit: a reading that scores well by correlating with the truth rather than by being it, and whose failures are the population where the correlation breaks.

A note on what has been established over ten rungs #

The sente anchor started with a word players use — sente, a move that must be answered — and has spent ten rungs turning it into arithmetic. It is worth writing down where that has arrived, because the numbers have accumulated quietly.

Sente is not a property of a move: it is a property of a move and the rest of the board, and the boundary is the crossover. The crossover obeys a law with two terms in it, both temperatures. The law holds five levels deep, survives translation, heating and cooling, and its geometry is about means rather than the temperatures it is written in.

None of that is a proof, and all of it is the kind of thing a proof is built from. What the anchor has that it did not have at rung one is a statement precise enough to be false — and nine rungs of trying to make it false.

What this does not settle #

It is a premise and not a proof. Four premises were checked on the rung below and a fifth is checked here, and the induction is still not written. What has changed is that the fifth one now has a statement — the walls above the crossover are governed by the options with the largest means — where before it had a plausible sentence with the wrong quantity in it.

One day of values. Every position is born by day three, which is where this ladder’s pool lives. The geometry argued above is about thermographs in general and the measurement is about 1,122 of them, and nothing here says a day-four value behaves the same way — though it is hard to see what about the day the argument could depend on.

Sides with three or more options. Two-option sides are excluded because the premise cannot fail on them, which means 716 of the 1,122 values contribute nothing. That is the right exclusion for testing the premise and it makes the population unrepresentative of the pool.

The mean ranking’s success is partly free. Where all of a side’s options share a mean, every one of them is the largest and the ranking cannot be wrong. How much of the 23,586 is that kind of height has not been separated out, and doing so would say whether the mean ranking is right because it is the right quantity or right because the population is full of ties. The temperature ranking’s failures are the answer to that in one direction — the two rankings differ on a fifth of the heights, so at least a fifth of them are not ties — and the full separation has not been made.

And the two exceptions are unexplained. They are a translated pair at a single height, a quarter above their crossover, and no account is offered of why the second-largest mean holds there. A proof would have to either explain them or be a proof of something slightly weaker than always the largest.

The grid is an eighth of a move. The heights are sampled rather than reasoned about, so a switch occupying less than an eighth of a move could be missed entirely. That is a real gap and it is bounded: the walls are piecewise linear with breakpoints at dyadic rationals, and on day-three values those breakpoints are eighths or coarser, so a switch narrower than the grid would need a breakpoint the pool does not produce. Would need is an argument rather than a check.

Normal play, short games, and thermographs computed by the ordinary recursion.

Where the ladder goes next #

The sente anchor has ten rungs: what sente is, that double sente is a band, what the reverse costs, the two cases, the wall behind them, half of the smaller temperature, what the halving is a function of, the two numbers at the top, what an induction would need, and now which options hold the walls.

The rung above is the proof, and for the first time everything it needs is stated in the quantities it will actually use. The law holds five levels deep, survives translation, heating and cooling, and its walls above the crossover are made of the options with the two largest means, with seven eighths of a move to spare. What is left is to write the geometry: two walls, each the envelope of a couple of means-less-tax lines, meeting where the law says they meet. It is four paragraphs and this anchor has spent three rungs earning the right to attempt them.

Two neighbours are worth the trip. The two numbers at the top is where the crossover was shown to depend on two quantities and nothing below them, and this page is what those two quantities turn out to be when the walls are looked at rather than the law. And reading a thermograph is where the walls are drawn and read directly, and it is the page whose picture this whole measurement is a census of.