Half a follow-up out

A switch is summarised by two numbers: what it is worth, and how much is at stake in it. The stop reading computes both from the two stops — the mean is their midpoint and the temperature is half the gap between them — and the bend is the condition established exactly when that reading is right. It is right on a value whose thermograph walls are straight below the meeting point and wrong on one whose walls bend, on all 138 day-three values with one option a side, in both directions and with no exception.

That page closed on the quantity it had not measured:

Thirty-two values have bent walls and a wrong stop reading, and nobody has asked by how much — whether the error is bounded by something readable off the diagram, and whether it is ever large enough to change a decision.

The error is not bounded by such a quantity. It equals one.

Two sizes and no others #

The first surprise is how little variety there is.

Thirty of the thirty-two are out by a quarter and two are out by a half. Nothing is out by an eighth, or by three sixteenths, or by any of the dozen other dyadic rationals that day-three values are made of. The error takes two values because the follow-ups do: the hot option of one of these positions has temperature $\tfrac12$ in thirty cases and $1$ in two, and the error is half of that.

And the two errors are the same size. The mean is out by exactly as much as the temperature is, on every bent value — which is not obvious, because the two are computed from the two stops by different arithmetic: one is a sum halved and the other a difference halved. An error that entered the two stops asymmetrically would move the midpoint and the half-gap by different amounts.

It is worth doing one of them by hand, since the numbers are small enough. Take $\{\{1 \mid 0\} \mid -1\}$. Left moving first reaches $\{1 \mid 0\}$, where Right answers to $0$, so the left stop is $0$; Right moving first reaches $-1$, so the right stop is $-1$. The stop reading therefore gives a mean of $-\tfrac12$ and a temperature of $\tfrac12$. The recursion gives a mean of $-\tfrac14$ and a temperature of $\tfrac34$. Both are out by a quarter, and the hot option $\{1 \mid 0\}$ has temperature $\tfrac12$ — of which a quarter is half.

Why it is half the follow-up #

The mechanism is visible in one diagram.

A wall of a thermograph is what a player can guarantee at each tax. Left’s wall starts at the left stop and falls at slope $-1$: every unit of tax costs Left a unit, because Left is paying to move. That is the whole content of the stop reading — assume both walls fall at their fixed slopes until they meet, and the meeting point is then forced by the two stops.

A wall bends when one of the options is itself hot. Above the follow-up’s own temperature the option stops being worth moving in, so the wall stops falling and goes flat. The two walls then meet higher than the straight-line extrapolation predicts, and they meet at a point pushed sideways by half the flat’s height, because the other wall is still closing at slope 1.

Hence the two halves of the finding. The height of the flat is the follow-up’s temperature $t_f$; the meeting point moves by $t_f/2$; and both the mean, which is the height of the mast, and the temperature, which is where the mast starts, are displaced by that same $t_f/2$.

It also explains why one wall bending is the general case. There are no values in this sweep with both walls bent — a day-three value with one option a side has at most one hot option, because two hot options would need a day of construction the sweep does not reach.

The direction, which is usable #

The mean can be wrong in either direction. A bent Left wall means Left’s guarantee is better than the straight extrapolation, so the true mean is above the read one; a bent Right wall pushes it the other way. Sixteen of each, which is a fact about the symmetry of the population rather than a theorem.

The temperature is different: it is read too low in all thirty-two cases. That is the useful half of the result, because it makes the stop reading a floor rather than an estimate. A player who computes a temperature from two stops and gets $\tfrac12$ knows the position is worth at least $\tfrac12$ to move in, and may be worth more. It is never worth less.

That direction is not arbitrary either. A flat in either wall delays the meeting, and delaying the meeting raises the temperature; there is no way for a bend to bring two walls together sooner than the straight-line extrapolation does. So the stop reading under-rates a fight and cannot over-rate one — which is exactly the property a bound instead of an answer argues a heuristic has to have before it is worth anything.

Whether it changes a decision #

An error of a quarter is a number. What it costs depends entirely on what a player does with the temperature, and what a player does with it is play in the hottest component — a comparison rather than a measurement. So the question is not how large the error is but whether it reorders two positions.

Of 9,453 pairs, the reading gets 8,445 exactly right, reverses four, flattens 770 and invents a difference on 234.

The four reversals are the headline and the smallest part of it. The 770 are the effect a player would meet: two positions whose temperatures genuinely differ, read as equally hot, so the choice between them is made on some other ground or on none. A rule that says play in the hottest has nothing to say about a pair it cannot tell apart, and the theory it is standing in for did have something to say.

The 234 in the other direction are worth a sentence for the same reason. There the reading manufactures a preference between two positions that are genuinely equally hot — harmless in the sense that either choice is as good, and misleading in the sense that a player is being told something the position does not contain.

The four reversals #

The four are two positions each paired with two others, and the two positions are exactly the pair carrying the largest error in the sweep. $\{\{1 \mid -1\} \mid -2\}$ has a true temperature of $1$ and is read at $\tfrac12$; $\{2 \mid \{1 \mid -1\}\}$ likewise. Each is beaten in the reading by a straight-walled value at $\tfrac34$, which is read exactly.

So a reversal takes a specific ingredient: an error of a half, which takes a follow-up as hot as the position it sits inside. The thirty values with an error of a quarter reverse nothing, because a quarter is smaller than the gaps between the temperatures this population takes.

That is a satisfying shape for a heuristic’s failure to have. It fails when a position contains a fight nearly as large as itself, and that is precisely the position an experienced player would look at twice anyway.

There is a second reading of the same table, and it is the one that says how safe the rule is in practice. Because the error is always downward, a reversal needs the under-read position to be the genuinely hotter of the two — a position read at $\tfrac12$ beating one read at $\tfrac34$ is impossible, and what happens instead is that the true $1$ is dragged down to $\tfrac12$ and passes below a straight-walled $\tfrac34$ on the way. So every reversal is a position being under-rated past a rival, never a position being over-rated past one. A player following the reading is therefore always guilty of the same error — playing the fight without a follow-up first — and that is a systematic mistake rather than a random one, which is the kind that can be corrected without recomputing anything.

Who built the diagram, and when #

Thermographs are Conway’s, in On Numbers and Games, and the mean value they compute goes back further — to Milnor’s 1953 paper on positional games with a scoring rule, where the object is a number attached to a position and the question is how far a single move can move it. Milnor’s setting is not this one, and the quantity is recognisably the same.

The stop reading is older than either, in the sense that it is what anybody does before they know about thermographs. A player who says this is worth about a point and a half and it is worth a move to take it is quoting a midpoint and a half-gap, and for the overwhelming majority of positions that arise in play — three quarters of this very population — it is exactly right.

Berlekamp’s endgame work is where the difference started to matter, because a Go endgame is a sum of many small fights and the fights with follow-ups are the ones the reading mis-prices. That is also where the vocabulary of this page comes from: a follow-up is a Go word before it is a mathematical one, and it names the move that is waiting after the exchange rather than a feature of a diagram.

A correction whose size is a temperature #

The half of a follow-up’s temperature is a strange quantity to find in an error term, and it is worth asking what it means that the correction is measured in the same units as the thing being read.

A stop is a number: what the fight settles at. A temperature is also a number, but it measures something else entirely — how much is at stake in getting the next move — and the two are not usually interchangeable. So an error in a stop that comes out equal to half a temperature is a statement that the reading loses exactly one player’s share of one fight, valued at what that fight is worth.

That reading makes the half unsurprising once it is stated. A temperature $t$ is half the gap between two stops, so half a temperature is a quarter of a stop gap — but the fight in question is the follow-up’s, and what the stop reading fails to account for is that the follow-up gets fought over rather than conceded. The correction is the price of one contested exchange, and a contested exchange is worth half the difference to each side.

It also says why the correction is a subtraction rather than a factor, which is the question two other ladders on this site have had to settle by measurement. A quantity that is one fight’s worth is an absolute amount and does not scale with the position it sits in; a quantity that is a proportion of the position would. Everything on this page reports the first, and reporting it as the second would have made the same predictions on this pool and different ones elsewhere.

So the shape of the error is evidence about its cause, independently of the census. An error equal to a temperature is an unaccounted fight; an error proportional to a stop would be a mis-scaling; and the two are distinguishable by what happens when the position is translated by a number, which moves stops and leaves temperatures alone.

What the census does not say #

Four limits.

Day three, one option a side. The population is the rung below’s and it is chosen to isolate the reading rather than to be representative. A value with two options a side, or one born on day four, can have both walls bend, and then the two errors would not have to be equal and the mean’s sign would have no single wall to follow.

An equality on a population is not an identity. The error is half the follow-up’s temperature holds on thirty-two values here, and the argument in the third section says why it should hold whenever exactly one wall bends once. A wall that bends twice — which a thermograph with two bends shows is a real shape — has two flats, and this page has no case of one.

The decision census is a census of pairs, not of play. Two components in a sum are compared by temperature, and that is what has been measured. It is not a measurement of games lost: a mis-ordered pair costs whatever the eventual difference in play is worth, and the greedy rule’s own bound is the right instrument for that, on a pool built for it.

The correction is exact here and is not proved. Adding half the follow-up’s temperature repairs every value in the sweep, and the argument in the third section is a description of one bend rather than an induction over all of them. What would settle it is the general statement about a wall with a single flat, which is a short proof somebody should write down and which this page has not.

And the flattening is a property of the sweep’s arithmetic. Seven hundred and seventy pairs are read as level because these temperatures are coarse — quarters and halves on a small population, so an error of a quarter is enough to close a gap. A population with finer temperatures would flatten fewer pairs and reverse more.

The convention, named #

Normal play, day-three values in canonical form, computed by the recursion.

The stops are what each player gets moving first with no tax charged. The stop reading takes the mean to be their midpoint and the temperature to be half their gap. A follow-up here is a hot option — an option that is not a number — and its temperature is the height at which the wall above it goes flat.

A wall bends below the meeting point if a segment strictly below the temperature has a slope other than $-1$ on the left or $+1$ on the right. The meeting point itself is excluded, since every wall turns there.

The error is signed: true value less read value, so a positive temperature error means the reading is too low. Numbers are excluded from the population entirely, since a number has no fight in it to read.

One sentence is worth keeping from all of it. A reading that is always wrong in the same direction is a reading a player can correct, and the stop reading under-rates a fight by exactly half its answer’s temperature — which makes it a floor rather than a guess.

Where the ladder goes next #

The switches anchor reaches six rungs to here, and the two above take this page’s correction and make it both wider and cheaper.

A second level of stops answers all three of the questions this page leaves. The correction survives eleven times the pool, missing two values in 1,459. It needs no thermograph at all — the follow-up’s temperature is half its own stop gap, so the whole reading is arithmetic on stops rather than a measurement taken off a diagram. And it does not survive two bends, for a reason that is a fact about the pool rather than about the correction: day three contains no value with two of them, so the question cannot be asked at this depth.

That second point is the one worth carrying, because it changes what kind of object the reading is. A correction stated in temperatures needs a thermograph, which needs the recursion; a correction stated in stops needs two numbers a reader already has. The same law, re-denominated, stops being a measurement and becomes a formula.

The bend is in the stops then pushes that all the way. This ladder had reduced the whole stop reading to one question — does this wall bend? — and that rung answers it from the options’ stops in four lines, with no diagram anywhere. And it gets more than it asked for: on all 1,459 non-number values born by day three, the options’ stops determine the entire thermograph.

One day deeper it breaks, and the failures are exactly the values with a bent-walled option — which closes the loop this anchor has been running for four rungs. A bend is what makes stops insufficient, stops determine everything until a bend appears one level down, and the bend is the same feature that decided the stop reading in the first place.