The bend is in the stops

A second level of stops got the stop reading down to one outstanding cost. The error is half the hottest follow-up’s temperature; that temperature is itself half a stop gap, so the correction needs no diagram; and what is left is a single question about the position in front of the reader. It closed on that question:

The rung above is the bend test itself. A bend on Left’s side happens when two Left options govern the wall at different heights, which is a statement about the options’ stops rather than about their diagrams. A test for a bend built from the options’ four stops would make this whole reading free.

The test exists and is four lines long. It gives more than the bend, and it is not free.

What an option contributes #

The construction comes straight out of the definition and is worth writing before anything is measured.

The left wall of $G$ at temperature $t$ is the highest of its Left options’ right walls, each shifted down by $t$. So one Left option $A$ contributes a function of $t$, and if $A$’s own walls are straight then $A$’s right wall is determined by its two stops: it rises at slope one from the right stop $RS_A$ until it meets the mast at $A$’s temperature, and is flat at $A$’s mean afterwards.

Shift that down by $t$ and the contribution is

$$f_A(t) = \min\left(RS_A,\; \frac{LS_A + RS_A}{2} - t\right),$$

flat at $A$’s right stop while $A$ is still hot, falling at slope one once it has cooled out. The corner sits at $A$’s own temperature, which is half its stop gap.

$G$’s left wall is the upper envelope of those. The right wall is the lower envelope of the mirrored ones. Where the two envelopes meet is $G$’s temperature, the height where they meet is its mean — and a wall bends exactly when its envelope is not one straight line of slope one.

Nothing in that reads a diagram. Two numbers per option go in, and the bend comes out.

The worked example is the smallest one that says anything. $\{\{0, * \mid -1\} \mid -2\}$ has one Left option, a fight with stops $0$ and $-1$; its temperature is a half and its mean is $-\tfrac12$, so it contributes $-1$ up to $t = \tfrac12$ and then falls. One option is enough to make a bend, which is worth noticing because the rung below’s sentence said two Left options govern the wall at different heights — and the more common cause is one option governing it at two different slopes.

Exact on the whole of day three #

Run it on every non-number value born by day three — 1,459 of them — and there is nothing to report except the count.

Both bends right on all 1,459. The temperature right on all 1,459. The mean right on all 1,459.

That is a stronger answer than the question. The rung below wanted a bend test so that the correction could be applied without a diagram; what the same four numbers give is the whole diagram — the two bends, the height the walls meet at, and the value they meet at. On day three a thermograph is a function of the options’ stops, and there is nothing a drawing knows that the stops do not.

And it is not free #

The rung below’s word was free, and it is the wrong word, which is worth saying before the result is enjoyed too much.

A stop is not a given. stops is defined by a recursion of its own — the left stop of $G$ is the best Left option’s right stop, and so on down to the numbers — so getting the options’ stops means walking the same tree a thermograph walks. Averaged over these values, the thermograph reaches 6.89 distinct subpositions and the test reaches 6.89.

What changes is the object carried at each level: a piecewise-linear wall with a point for every option that governs a stretch of it, against two numbers. Even that is a modest saving at this depth — a day-three thermograph’s two walls carry 2.9 points between them beyond the mast.

So the honest description is an equivalence in a different vocabulary. It is worth having for what it says — that a thermograph holds no information its options’ stops do not — and not for what it saves. This ladder has now made the same correction twice: half a follow-up out found an error term and called it a discovery about temperature, and it was a discovery about stops; this page finds a construction and would like to call it a shortcut, and it is a statement about what determines what.

One day deeper #

Build $\{A \mid B\}$ over the hot day-three values — 14,400 combinations, 944 of them distinct and not numbers — and the test stops being exact.

Both bends right on 750 of the 944. The left bend alone on 780, the right on 905, the temperature on 672. Four values in five rather than all of them, which is the difference between a construction and a theorem.

What breaks it is the one thing the derivation assumed #

The construction makes exactly one assumption: that an option’s two stops determine its wall. That is true when the option’s own walls are straight and false when they bend, because a bent wall has a corner the two stops know nothing about.

Split the deeper pool on that and the table has an empty cell. Of the 567 values whose options all have straight walls, the test is right on every one. Of the 377 with an option whose own wall bends, it is wrong on 194.

So day three is not exact because day three is small. It is exact because no value born by day three has an option with a bent wall — bends need depth, and a day-two value has none — so day three is precisely the population on which the assumption holds. The construction did not fail to generalise; it generalises exactly as far as its hypothesis does, and the census found the hypothesis rather than an exception list.

That is a better outcome than a rule with a residue, and this site does not often get it. The bend decides it is the other place on this ladder where a description turned out to be a definition, and the two readings now sit one inside the other: a wall bends exactly where the stop reading fails, and the bend is decidable from the options’ stops exactly where the options’ walls are straight.

How much a stop was thought to throw away #

It is worth pausing on the size of the claim in the middle of this page, because the whole ladder was built on the opposite assumption.

A stop is one number. A thermograph is a two-dimensional object with a mast, two walls and however many corners the position’s options put in them. Reading a thermograph is an essay about everything the diagram tells a reader that a pair of numbers does not, and switches and mean value opens the ladder by separating the two: the stops say what each player gets moving first, and the diagram says what happens as the rest of the board gets hotter.

The ladder then spent five rungs on cases where the stops are not enough. When a switch is not a switch found positions the stop reading misdescribes; a fight with no midpoint found the reading failing on a quarter of a census; the bend decides it named the condition. Every one of those is a true statement about a position’s own stops.

What this page says is that the stops of its options are enough — and that the two claims are compatible rather than in tension. $G$’s two numbers genuinely throw information away; the options’ two numbers each do not, on day three, because an option’s wall is straight and a straight wall is two numbers. The information a thermograph carries is the information about where corners are, and a corner in $G$’s wall is an option’s temperature, which is a stop gap one level down.

So the diagram is not richer than the numbers. It is richer than one position’s numbers, and it is exactly as rich as its options’, for as long as the options are simple. That is the reading the answer that starts another fight has been circling from the other side, where a position’s hardness turns out to live one level below where it is measured.

What the recursion would be #

The obvious repair names itself, and it is worth writing down because it is also the reason the repair is not a repair.

An option’s contribution is wrong when its own wall bends. Its own wall bends exactly when its options’ contributions do not make a straight envelope. So the test could be run one level down to decide each option’s bends, and one level below that, until it reaches numbers — which have no bends by definition.

That recursion is the thermograph. Every level of it carries the same information the wall carries and computes it the same way; writing it in stops rather than in walls changes the notation and nothing else. The construction here is the first level of that recursion, and its whole content is that on day three the first level is all of it.

Which puts a bound on what this ladder can hope for. There is no test for a bend that is not, at bottom, a computation of the wall — because a bend is a fact about the wall, and the wall is defined by a recursion over positions. What the four stops give is a way of writing one level of it in numbers, and one level is enough exactly as long as the options are simple enough that their own two numbers describe them.

Determining is not computing #

The result says the options’ stops determine the whole thermograph, and it is worth being exact about what that licenses, because a determination and a procedure are different things and this page supplies both.

Determination is a statement about information: two positions whose options have the same stops have the same thermograph, so nothing beyond those numbers is needed. That is a claim about what could in principle be recovered, and on its own it does not say by what means.

The four lines are the procedure, and they are the reason the result is usable rather than merely true. A determination with no known recovery would be an interesting fact and no help to a reader with a position in front of them; four lines of arithmetic on a handful of numbers is a computation anybody can run.

The two come apart in the direction that matters, and the failure one day deeper is where. At depth four the options’ stops stop determining the diagram, and the failures are exactly the values with a bent-walled option. So the determination has a boundary; the procedure inherits it; and neither degrades gently — inside the boundary the recovery is exact and outside it there is no recovery at all.

That is the honest shape of the claim. Not an approximation that gets worse with depth, but an exact reconstruction with a stated domain, and a reader applying it needs to check the domain rather than estimate an error. Which is the better arrangement of the two, and the rarer one.

What this does not say #

It says nothing about the correction’s accuracy. The rung below’s error term — half the hottest follow-up’s temperature — is unchanged and is still exact on 348 of 350 bent day-three values. This page supplies the missing test and touches neither the error nor its two exceptions.

The deeper pool is one construction, not a day. Day four proper is far past what an enumeration on this site can reach, and $\{A \mid B\}$ over the hot day-three values is a pool built to have the right property rather than a census of anything. The 194 failures are evidence that the assumption is what matters and not a measurement of how often it fails in the wild.

The temperature is found by bisection. The two envelopes are maxima and minima of two-piece functions, and where they cross is found by a hundred and twenty halvings rather than by solving. Every value here is dyadic and the bisection lands well inside the last bit, but the comparison against the diagram is made to a tolerance rather than to the bit, and that is a difference between this and the exact arithmetic the rest of the site’s claims rest on.

And the picture cannot show a proof. Drawing the envelope over the wall shows that they agree on one value, and a reader can check it by looking; 1,459 of them is a count and not a picture. What the drawing is for is the construction — that the corner in a contribution sits at an option’s own temperature — and that is a fact about one option which a single figure states completely.

The convention, named #

Normal play throughout, and every thermograph computed by the recursion.

A stop is what a player gets by moving first and playing on until somebody is left facing a number, with no tax charged. Every position has two, and this site writes them $LS$ and $RS$.

A thermograph plots, against a tax $t$ on every move, what each player gets from the cooled position. Its left wall is the Left scores, its right wall the Right scores, and above the height where they meet both are the mast. That height is the temperature and the value there is the mean.

A wall is straight when its slope is one throughout, below the meeting point — so the left wall falls at rate one and the right wall rises at rate one. Anything else is a bend, and the height of a bend is the temperature of whichever option stops governing the wall there.

A value is born by day three when it can be written with options born by day two, which is the largest complete census this site enumerates. The deeper pool is $\{A \mid B\}$ over the hot values of that census, which is a construction reaching one day past it and not a census of that day.

Exact here means to the last bit for the bends, which are booleans, and to a tolerance of a millionth for the temperature and the mean, which come out of a bisection.

Where the ladder goes next #

The switches anchor has eight rungs: the switch and its mean, what a player is really imagining, when a switch is not one, the fight with no midpoint, the bend as the condition, half a follow-up out, the second level of stops, and now the bend from the stops themselves.

The rung above is the depth of the reading. Everything here says the first level of the recursion suffices on day three because day three’s options have straight walls; the sharp version is a statement about how many levels a value needs, which is a number attached to a position and not a property of a day. A value whose options are straight-walled needs one; a value whose options have straight-walled options needs two; and the natural conjecture is that the number is the depth of the deepest bend in the position’s tree. Measuring it means running the construction to increasing depth on the deeper pool and recording where each value becomes exact, which is the same sweep with a counter in it.

Two neighbours are worth the trip. The bend decides it is where the bend became the condition rather than a symptom, and it is what makes a test for one worth eight rungs of work. And the two numbers at the top is the same kind of result on the temperature ladder — a quantity that turns out to depend on two numbers and nothing below them — and reading the two together is the clearest statement this site has of how shallow a hot position’s arithmetic usually is.