A bound with one number too many

Adding a number to a position moves both its stops by that number and leaves the temperature alone. Adding an infinitesimal moves neither stop and changes the outcome anyway. What a fight does to a fight asked the third question — what a hot addend does — and found the answer to be a bound: over 720 sums the stops add on 330 of them, and where they do not, no stop is out by more than twice the smaller of the two temperatures, a bound 222 sums attain exactly.

Every addend in that pool was a plain switch, and the page said so:

Every addend here is a switch, whose thermograph is two straight lines, and the interleaving argument above is about two straight walls meeting. An addend whose own wall bends is a position with a follow-up … so the natural conjecture is that the bound becomes twice the smaller of three numbers rather than two.

The conjecture is false, and the bound it was meant to replace is not.

What a bent addend is #

An addend whose thermograph is two straight lines is a switch: $\{a \mid b\}$ with $a$ and $b$ numbers, and it offers each player one exchange with nothing waiting afterwards. An addend of the form $\{a \mid \{b \mid c\}\}$ is not: Right’s answer starts a second fight, the Right wall goes flat at that fight’s own temperature, and the diagram bends.

The pool here is twelve such addends, and it is built in two halves on purpose:

• six whose answer is about as hot as they are — $\{6 \mid \{4 \mid 0\}\}$ at temperature $2$ with an answer at $2$, and five like it;
• six whose answer is much colder — $\{8 \mid \{1 \mid 0\}\}$ at temperature $3\tfrac34$ with an answer at $\tfrac12$, and five like it.

The second half is what tests the conjecture, because that is where the third number is smaller than the other two and therefore where a bound containing it is a stronger claim.

The refutation #

Take $\{\ast 2 \mid -2\}$, whose temperature is $1$, and add $\{8 \mid \{1 \mid 0\}\}$, whose temperature is $3\tfrac34$ and whose answer $\{1 \mid 0\}$ has temperature $\tfrac12$.

A stop of the sum is out by $2$ from the sum of the stops. Twice the smaller of the two positions’ temperatures is $2\times 1 = 2$, so the older bound is satisfied — and attained. Twice the smallest of the three numbers is $2 \times \tfrac12 = 1$, so the conjecture is broken by a factor of two.

Sixty-six of the 1,440 sums do this, and they are exactly the pairs whose addend is from the second half of the pool. It is not a scattering of awkward cases: the failures are a class, and the class is the addend’s answer is much colder than the addend.

Why the follow-up cannot enter #

The mechanism is visible in the interleaving argument the rung below gave, once one asks what part of the addend the argument uses.

A stop of a sum is what a player banks by playing the sum out with both players moving first where it suits them — the quantity where the fight stops defines and shows not to determine a position. The stops fail to add because the players may interleave — one takes a move in the addend while the other is committed in the position — and the amount an interleaving can be worth is bounded by how much either component is worth moving in, which is its temperature. That is where the smaller comes from: an interleaving needs both players to want to be in the same component, and the component only pulls both of them while it is hotter than the other one.

The follow-up plays no part in that argument. It is a fact about what happens after the exchange in the addend, and the stops are read at the bottom of the diagram, where the exchange has already been made and the answer with it. So the answer’s temperature has no place to enter, and the conjecture is a quantity carried across from a different question.

That the answer’s temperature does matter to that different question is what made the conjecture tempting. Half of the smaller temperature finds a player leaving an environment at the fight’s own temperature less half its answer’s, so a follow-up genuinely halves a decision — and the sente ladder had just published that when the conjecture was made.

The bound is attained, which settles it #

There is a stronger reason the conjecture could not have been right, and it does not need a counterexample at all.

The two-number bound is attained on 358 of these 1,440 sums. A bound that is achieved exactly cannot be replaced by a smaller one, whatever extra quantities are put into the smaller one — and the three-number bound is smaller by construction, since the minimum over three numbers is at most the minimum over two.

So the conjecture was refutable by arithmetic the moment it was written: it proposes a bound below one already known to be attained, and it can only survive if the third number is never the strict minimum. It is not, and 66 sums show it.

The lesson is not that the conjecture was careless. It is that tightening a bound and generalising it are opposite operations, and a sentence that proposes to do both at once is proposing something that cannot be true unless the original bound was slack — which the rung below’s own attainment count had already ruled out.

The rung below, with its pool beside this one #

Setting the two censuses side by side is the cleanest way to see what the bend does and does not change. The share of sums whose stops add outright is 46 per cent there and 46 per cent here. The bound is attained on 31 per cent there and 25 per cent here. The worst stop movement in each is the same quantity for the same reason.

What the bend changes is nothing measurable about the sum. It changes the addend — its diagram, its behaviour in an environment, what a player should do with it — and none of that reaches the stops of a sum it sits in. That is a stronger statement than the census set out to make, and it is the one the two pools together support.

What survives, and what it is worth #

The rung below’s statement stands unchanged over a wider pool, which is worth stating as the positive finding:

A hot addend moves a stop by at most twice the smaller of the two temperatures, whether or not either position has a follow-up, and the bound is attained. That is a statement of the kind a bound instead of an answer argues is worth having: exact where it is exact, bounded where it is not, and with the bound achieved rather than assumed.

The mean still adds, on every one of the 1,440 sums, which is the control this anchor runs every time — and it is the reason the whole subject is worth doing: the mean of a sum is the sum of the means always, the temperature of a sum is the largest of the parts’ always, and the stops are the quantity in between, which fails and fails in a bounded way.

The stops add outright on 660 of the sums, a share of 46 per cent, which is close to the 330 of 720 the rung below measured on switches — and which end of the interval is open is where the same population’s stops are dissected for a different purpose. So the addend’s bend does not change how often the stops add either — a second reading of the same population, and the same answer.

Who bounds things this way #

Bounding the failure of an identity is Milnor’s habit before it is anybody’s in this subject: the 1953 paper on positional games with a score bounds how far a sum’s value can be from the sum of its parts’, and the bound is in terms of the largest single-move gain — which is what a temperature is.

Conway’s treatment replaces the bound with an exact theory wherever it can, and that is the shape this whole site inherits: compute the value, and where the value is out of reach, bound the error. The stops are an unusual case in that the exact answer exists and is simply not additive, so the bound is not a stand-in for an unavailable computation. It is a statement about how much information the pair of stops carries about a sum, which is a different kind of question and the reason this anchor exists.

And the interleaving argument is old, in the sense that every account of why the stops fail to add tells the same story: the two players do not have to take turns in one component. What the census adds is the size, and the size is the only part a player can use.

What a player should do with a bound like this #

Two things, and both are about when not to compute.

A player holding two components and wanting to know the sum’s stops has two options: add the stops, which is free and right 46 per cent of the time, or evaluate the sum, which is right always and costs a search. The bound is what makes the first option usable — it says the free answer is never out by more than twice the smaller temperature, so a player who is ahead by more than that can stop thinking.

That is the ordinary use of a bound in this subject, and it is worth saying that it is a use rather than a consolation prize.

And the smaller temperature is the cheap one to know. A player facing a large hot component and a small cold one does not have to measure the large one at all: the bound is set by the small one, so the error in adding the stops is small whatever the big component is doing. That is a genuinely practical reading and it survives the bend, which is what this page adds to it.

What a bound at the wrong grain looks like #

The refutation here is a specific instance of a general trap, and it is worth naming because the trap catches a reader in the direction of caution rather than of carelessness.

A bound with an extra term in it is not wrong. Adding a non-negative quantity to a correct upper bound gives another correct upper bound, so a reader who inserts the follow-up’s temperature “to be safe” has produced a true statement and can never be caught out by an example. The cost of a loose bound is invisible from inside, which is why loosening is the default instinct and why refuting a loosening needs an attainment argument rather than a counterexample.

That is the shape of the argument on this page and it is worth reading as a method. Showing the extra term is unnecessary means showing the bound without it is attained — that some sum sits exactly on it — because a bound nothing reaches could always have been smaller and no amount of testing would say so. Three hundred and fifty-eight sums sitting exactly on the smaller bound is what closes the question, and a sweep that reported only no violations would have closed nothing.

So the two halves of a bound claim need two different kinds of evidence. Soundness is refuted by one example and supported by many; tightness is refuted by many and supported by one. A page that reports a bound and its violation count has established half of what it needs, and the half it has established is the half nobody was in doubt about.

The habit that follows is small: whenever a bound is quoted here, the attainment is quoted beside it, and a bound with no attainment is described as a bound rather than as the answer.

What this does not say #

Four limits.

Twelve addends is a pool, not a population. They are constructed to separate the addend’s temperature from its answer’s, which is the right design for testing this conjecture and the wrong one for estimating how often anything happens in a game.

One kind of bend. Every addend here bends on the Right wall once, because its Right option is a fight and its Left option is a number. An addend with both walls bent, or one whose answer’s answer bends again, is untested — and if the bound survived the first two shapes it is reasonable to expect it to survive those, but reasonable expectation is what this page has just spent itself refuting.

The positions added are values rather than boards. Each is a day-three value written as a form, not a position of any game, so nothing here says what happens when a real board falls into components whose stops a player is actually adding.

Day three, first 120 values. The positions added to the addends are the first 120 values of day three, which is a sample of a day rather than a day.

And a bound is not a formula. The two-number bound says how far a stop can move, and the stops add on nearly half the sums. Nothing here says which sums those are; what a fight does to a fight leaves the same question open, and it is the one a player would actually want answered.

The convention, named #

Normal play. The stops are what each player banks by moving first with no tax charged, read off the feet of the two thermograph walls.

The temperature of a number is taken as nought here rather than the $-1$ this site uses elsewhere, so that a bound written as twice the smaller temperature says the right thing when one of the two positions is a number: nothing moves.

An addend’s answer is its Right option, and the addend’s wall bends when that answer is not a number. Much colder is not a defined term: the pool simply contains addends whose answers sit at $\tfrac12$ and addends whose answers sit at their own temperature, and the failures fall entirely in the first group.

One sentence is worth keeping from the whole census, and it is not about follow-ups. A bound that is attained cannot be tightened, whatever new quantity is offered — so an attainment count is the cheapest possible defence of a bound, and the rung below had already computed one.

Where the ladder goes next #

The translation anchor has four rungs to here: what a number does, what an infinitesimal does, what a fight does, and now that the last of those needs nothing added to it when the fight has a fight inside it.

The rung above supplies the description this page asks for, and it turns out to be short. Which end a sum lands at gives a four-line rule, exact on all 1,440 sums, saying which of the two ends a given pair falls on. Three of its four cases are decided by the value being translated alone — the other operand does not enter — and the property doing the deciding is the bend: whether the value’s wall rises straight before it leans, which is the feature the switches ladder isolated for a completely different question.

That two independent ladders converge on the same feature of a wall is the part worth carrying. The bend is not a curiosity of thermograph drawing; it is the structural distinction that decides when a reading taken at the feet of a diagram can be trusted, and it decides it here in a question about sums of stops rather than about switches at all.

The same number in two currencies then measures the other end. Bent-walled values fall strictly inside the bound, and the shortfall is the value’s own hottest follow-up’s temperature — exactly, on 400 of 408 pairs, and twice it on the remaining eight. So the whole error becomes one expression rather than a range, and the expression is the switches ladder’s constant seen through a different denomination: a half there and a whole here, because a temperature is half a gap between stops.

Two neighbours are worth the trip. Temperatures do not add is the companion failure at the top of the diagram, where the answer is a maximum rather than a bound. And half of the smaller temperature is where a follow-up’s temperature genuinely enters an answer, and it is the page whose finding this one had to decline to borrow — and which the rung above borrows after all, once the currency is right.