A subtraction, not a factor

A move is sente when the opponent has to answer it, and sente is a fact about the rest of the board establishes that it is not a property of the local shape: the answer comes while the rest of the board is cold enough and stops coming when it is not, and the height at which it stops is the crossover.

What the halving is a function of measured the crossover against the follow-up’s temperature and found two cases. Where the answer settles the fight, the crossover is the follow-up’s temperature. Where the answer starts another fight, it is half of it. That page closed on the case its pool barely contained:

A follow-up whose own answer starts a third fight is a position three deep, and the census contains two of them; both have a factor of a half rather than a quarter, so the halving does not compound. Whether that is a fact or an artefact of two positions is the question a wider pool would settle.

A pool built to be deep settles it, and settles more than was asked: the halving does not compound, and it is not a factor.

The pool #

Forty-six positions with a hot follow-up: eighteen whose answer settles the fight, eight two deep, and twenty three or four deep against the earlier census’s two.

Each is swept against an ambient switch at every quarter-point up to sixteen, and the crossover is the highest ambient temperature at which the move is still answered. Every row is a full minimax over the local position and the ambient together, so the answer is played rather than predicted.

The pool is built to vary the follow-up’s temperature and not only its depth, and that is the whole difference from the earlier one. The follow-ups here run at temperatures of 1, 1½, 2 and 2¼; nearly every one in the census that found the factor was at a temperature of one.

The positions are written as nested switches with the numbers chosen so that the chain survives reduction — which most of them do not, and the ones that do not are reported in their own row rather than quietly dropped. Thirty-six positions were written to be three or four deep and twenty of them are, which is a yield worth knowing about for anybody building a pool of the same kind.

The two rules #

Where the answer settles the fight — the follow-up’s own options are numbers — the crossover is the follow-up’s temperature, on all eighteen. That is the rung below’s first case and it survives unchanged.

Where the answer starts another fight, the crossover is the follow-up’s temperature less a half, on 18 of the 20.

A factor of a half fits thirteen. And the thirteen are exactly the positions whose follow-up temperature is one, because $\tfrac12 \times 1$ and $1 - \tfrac12$ are the same number.

Where they part #

The positions that separate the two rules are the ones the earlier pool did not have.

A follow-up at a temperature of two is answered up to 1½, and that one row is enough to separate the rules. The factor predicts 1; the subtraction predicts 1½. A follow-up at 1½ is answered up to 1: the factor predicts ¾, which is not even on the quarter grid the sweep uses, and the subtraction predicts 1.

So the ratios the deep positions show are not one number. They are ½ on thirteen positions, ⅔ on two and ¾ on five — which is exactly what a fixed subtraction looks like when it is divided by a varying temperature.

That is the correction, and it is the ordinary shape of this kind of error: a pool with one value of a variable cannot distinguish a rule that multiplies from a rule that subtracts, and the rung below’s pool had one value of the variable.

It is worth naming the failure precisely, because it is not carelessness and it recurs. The earlier pool was built to vary the thing the question was about — the depth of the fight — and it held constant a thing the question was not about, the follow-up’s temperature. That is the right instinct and it is exactly how a confound is created: the quantity held fixed is the one the answer turns out to depend on.

The same shape has turned up twice more in this collection. A pattern over five hexadecimal codes held because the five shared a property nobody was varying, and two conditions on subtraction lists were indistinguishable until the range of the lists was widened. The remedy in all three is the same and it is not more care: it is a second axis.

What the eighteen settled positions establish #

The first class is worth a paragraph even though nothing about it changed, because it is the control and it does the work that makes the second class readable.

Where a follow-up’s own answer settles the fight, the crossover is the follow-up’s temperature — 18 positions, exactly, with no subtraction and no factor. That is the case the whole sente anchor is built on: a move is answered while the rest of the board is not worth more than the answer, and worth is the temperature.

So the second class is a departure from a rule rather than a different rule. The exchange goes one move deeper, and a move is spent, and the price of a move is a half.

Stated that way the two classes are one statement with a term that is nought in the first case, which is a better way to remember it than two cases with two formulas — and it is exactly the shape the translation ladder found, where three rungs turned out to be one bound with a quantity taking three values.

Why a subtraction should be the right shape #

The mechanism is worth stating even though it is a sketch, because a subtraction and a factor mean different things about the game.

A factor would say the exchange is discounted. The deeper fight is worth half as much attention, in some proportional sense, and a follow-up worth ten would be answered up to five.

A subtraction says a move is spent. Answering costs one move on the board and buys a move in the fight, and the accounting is in moves rather than in proportions. Half a unit of ambient temperature is the price of one move at the exchange rate a coupon stack fixes — and the coupon ladder has just found the same thing from the other side, where the departure is set by the larger of two temperatures rather than by a scaled version of one.

The two readings make different predictions at large temperatures and the pool here reaches 2¼, which is enough to tell them apart and not enough to be sure the subtraction does not itself drift.

What is left over #

Two of the twenty miss the subtraction, and both miss it by one grid step.

Their follow-ups are at a temperature of 2¼ and they are answered up to 1½ where the rule predicts 1¾. That is a quarter out, and a quarter is the resolution of the whole measurement: the ambient switch is built at quarter-points and at nothing finer, deliberately, so the crossover is located to within a quarter and no closer.

So the two misses are either real or invisible, and this census cannot say which. What can be said is that they are at the largest temperature in the pool, which is where a drifting rule would first show, and that the other eighteen are exact.

The two are also the same position twice, with the outer number changed from 12 to 16 — and the outer number is the one quantity the crossover provably does not depend on, since the sweep only ever plays the follow-up against the ambient. So the pool contains one exceptional shape rather than two exceptional positions, which makes the exception smaller and the sample behind it smaller too.

What it changes about the earlier reading #

The rung below’s finding was that the crossover factor is “a half on the positions whose answer starts another fight and one on the rest”, and that the wall reading predicts which — the foot of the wall is vertical or it is not, and there is no third state.

The two-state reading survives. There are two classes and they are told apart by whether the answer settles the fight, exactly as that page found.

What changes is the quantity attached to the second class. It is not a factor of a half applied to the follow-up’s temperature; it is that temperature with a half taken off. And the rung below’s prediction that the halving does not compound is confirmed in a stronger form: it does not compound because there is nothing to compound — the correction is one subtraction whatever the depth, and the four-deep positions behave exactly as the three-deep ones do.

What a player does with it #

The practical form is short and it is a change to a number a Go player would recognise.

A fight whose answer settles is worth answering while the board is worth less than the answer. That is the classical statement and it stands.

A fight whose answer starts another fight is worth answering half a point less than that. Not half as long — half a point less, which on a hot board is a much smaller concession than the factor suggests and on a cold one is a larger one.

At a follow-up worth two the two readings differ by a half, which is the size of the smallest ambient move on the grid: the factor says stop answering when the board reaches one and the subtraction says keep answering to one and a half. On a real board that is a move.

What the census does not say #

Four limits.

A quarter-point grid. Every crossover here is located to a quarter, because the ambient switch is built at quarter-points by a constraint the machinery states rather than assumes. A subtraction of a half is a claim at that resolution, and the two exceptions are one step out.

Forty-six positions of one shape. Every member of the pool is $\{a \mid \{b \mid \ldots\}\}$ — a number against a chain of fights on Right’s side. A position whose Left option is also a fight is a different object, and double sente is where that case is taken up.

One side of the position. Only Right’s follow-up chain is measured, because only Right’s move is the one being answered. A position with follow-ups on both sides has two crossovers and they need not agree, which is the whole content of the double-sente rung.

Temperatures to 2¼. The pool separates a factor from a subtraction because it contains follow-ups at four temperatures. It does not reach far enough to separate a subtraction from something that grows slowly — a correction of $\tfrac12 + \varepsilon t$ would look like a half over this range.

And the depth classification is by the form. How deep the fight goes is measured by walking Right’s options in the canonical form, so a position whose chain collapses under reduction is classified by what it reduces to rather than by how it was written. Four of the eight positions written three deep classify as two deep for that reason, and they are reported in their own row rather than counted as deep.

Two hypotheses that agree on a pool are one hypothesis #

The lesson here is worth separating from the correction, because it applies to every measured law on this site and it is what makes the correction possible at all.

A factor of a half and a subtraction of a half are different functions and they agree at exactly one input: a temperature of one. On a pool where every answer has temperature one they are the same hypothesis, and no amount of care in the measurement can tell them apart — not more positions, not more careful play-outs, not a tighter tolerance. The distinguishing information is not in the pool.

That is a different kind of limitation from a small sample, and it is invisible in the way a small sample is not. A small sample announces itself: the counts are low and the confidence is obviously thin. A pool with no spread in the deciding variable can be enormous, every measurement in it exact, and every conclusion drawn from it unidentified.

The check is to ask what the two candidate laws disagree about, and then whether the pool contains it. Here the answer is one line of arithmetic — they agree only at one — and the pool’s follow-up temperatures could have been listed in a sentence.

This page widens the pool along depth, which was the axis this ladder had been worrying about, and depth is not the axis that separates the two. That is why the finding here is a genuine improvement and still not the law: the right axis was the follow-up’s temperature, and the rung above is where it gets varied.

The eight in the middle #

One row of the table has been passed over and it is the least tidy, so it is worth saying what is in it.

Eight positions classify as two deep, and four of them are answered up to the follow-up’s temperature exactly — the first class’s rule — while the other four are answered lower.

They are the positions written three or four deep whose chain collapses under the canonical reduction: what looks like a fight two moves down turns out to be dominated away, and the position that remains is shallower than it was drawn. The classification walks the canonical form rather than the source, which is right, and it means a position’s depth is not visible from how it was written.

That is a caution rather than a finding. It says the pool has to be classified after reduction and not before, and it is the reason the deep row here has twenty members rather than the thirty-six positions written to be deep.

The convention, named #

Normal play. The ambient is a switch $\{t \mid -t\}$ standing for the rest of the board, built at quarter-points only.

A move is answered at ambient temperature $t$ when the full minimax over the local position and the ambient together has the opponent replying locally rather than taking the ambient. The crossover is the largest $t$ on the grid at which that happens.

The follow-up is Right’s option of the local position, and its temperature is its own, computed by the recursion. The depth is the length of the chain of Right options in the canonical form before a number is reached: one where the answer settles the fight, two or more where it does not.

Where the ladder goes next #

The sente anchor has six rungs to here, and this one has replaced a factor with a subtraction on a pool built to be deep. The rung above finds both to be regions of one law.

Half of the smaller temperature widens the pool again — 128 fights whose answers run from a number up to a temperature of six — and the correction comes out as half the answer’s temperature, saturating at half the fight’s own. Linear below the cap, constant above it.

That shape is exactly what produces the two earlier readings. On a pool whose answers all have temperature about one, half the answer’s temperature is a half, and the follow-up’s temperature less a half is also a half — so a factor and a subtraction make identical predictions and the pool cannot separate them. Both earlier answers were right about their own pools and neither was the law.

The instruction that follows is the one this page half states and the rung above makes precise. A pool for a question of this kind needs the quantity being measured to vary, not merely to be sampled many times. Twenty deep positions whose answers are all about the same size are twenty confirmations of a coincidence, and the depth this page adds was the wrong axis to have widened.

Two rungs further the dependence is settled entirely. The two numbers at the top groups positions by their two highest temperatures and finds the crossover single-valued on every group however far apart the third is — so nothing below the top two numbers enters, and the formula survives a fourth level of fight that this page’s pool never reached.