The bend is the condition

A switch is $\{a \mid b\}$ with $a$ and $b$ numbers and $a > b$, and every reader is taught two formulas for it: the mean is the midpoint of the options and the temperature is half their gap. Twenty-one values of day three satisfy that hypothesis. One hundred and forty-six have the same shape — one option a side — and do not.

A fight with no midpoint tested the obvious repair, which is to read the two formulas off the stops instead of off the options, since the stops are numbers whatever the options are. It found the repair exact on about three quarters of those 146, and closed by offering a description of the class it works on together with a route to proving it:

The stop reading is exact exactly when neither wall bends below the meeting point has been checked on 167 positions and argued in a paragraph, and turning it into a proof means showing that a wall bends below the meeting point precisely when some option is neither a number nor an infinitesimal.

Two claims. The first is exact. The second is not, and the way it fails says what a bend actually is.

The first claim, which holds #

Of the 146 values that break the textbook hypothesis, 138 are not numbers. On every one of those:

• if both walls run straight from their stops to the meeting point, the stop reading gives the mean and the temperature exactly — 106 values, no exception;
• if either wall bends anywhere below the meeting point, the stop reading is wrong — 32 values, no exception.

Both off-diagonal cells of that table are empty. A description checked on 167 positions and argued in a paragraph is now an equivalence checked in both directions, and the class the rung below could only point at has a definition: the values whose walls are straight below the meeting point.

The census asserts it. A single value in either empty cell would turn the finding back into a correlation and stop the build.

Why straightness is the right thing to ask about #

The mechanism is short once the diagram is in view.

A wall is built by taking each option’s opposite wall and shifting it — Left’s wall is the highest of Right’s walls of Left’s options, each pulled down by one unit of height per unit of tax. If an option is a number, its wall is a vertical line and the shift makes it a straight line of slope $-1$. If an option is anything else, its wall has structure, and the shift carries that structure up into the parent.

A straight wall is therefore a wall that has forgotten its option. It records only where the option’s own wall started, which is a stop — a number — and the stop reading is exactly the reading that uses only that. When the wall is straight, the two readings are the same reading.

A bend is the option’s structure surviving into the parent, and it is precisely the information the stops throw away. So the equivalence is not a coincidence about day three: it is the statement that the stop reading is right exactly when there is nothing else to know.

The eight that were not exceptions #

Eight of the 146 have straight walls and a stop reading that fails, and they are the reason this page had to look at the numbers separately.

$\{-\tfrac12 \mid 0\}$ is not a switch. Left’s option is below Right’s, so neither player wants to move, the simplicity rule applies, and the value is the simplest number strictly between them: $-\tfrac14$.

Its stops are both $-\tfrac14$, so the stop half-gap is nought. Its temperature, as this site reports it, is $-1$ — because a number is given $-1$ by convention, so that a sum’s stack of temperatures contains only its genuinely hot components and a cold part does not sit in the list at nought pretending to be a fight.

So the disagreement is between two conventions and not between a reading and a value. All eight are numbers, all eight disagree only in the temperature column, and every one of them has a stop half-gap of nought against a reported temperature of $-1$.

That is worth recording as a finding rather than as an aside. A convention adopted for a good reason elsewhere in the site produced eight false counterexamples to a theorem, and the theorem is exact once they are removed. The census asserts that too: if the straight-walled failures were not exactly the numbers, the exceptions would be real and this section would be wrong.

What the two halves of the census look like #

It is worth seeing the two classes side by side, because they are not distinguished by anything a reader would have guessed from the values.

The written value is no guide. $\{1 \mid \ast\}$ and $\{2 \mid \{1 \mid -1\}\}$ are both one option a side with an option that is not a number, they look equally awkward on the page, and the stop reading is exact on one and wrong on the other.

Nor is the option’s class a guide, which is the next section. What separates them is a feature of the picture, and the picture has to be built before the feature can be seen — so the condition costs a full thermograph to check and is then unambiguous.

That is an honest cost and worth stating. A criterion requiring the diagram is not a shortcut for computing the diagram. What it is good for is a reader who has the diagram in front of them, which is the situation the whole thermograph anchor is written for, and who wants to know whether the two numbers at the bottom are the whole story.

The second claim, which does not hold #

The route to a proof the rung below proposed was to characterise a bend by the kind of option that causes it: a wall bends below the meeting point precisely when some option is neither a number nor an infinitesimal.

One direction is right. All 32 bent values have such an option — no exception — so having only numbers and infinitesimals for options is enough to guarantee straight walls.

The converse is wrong by a wide margin. Eighty-one of the 138 values have an option that is neither a number nor an infinitesimal, and 49 of them have straight walls anyway.

The nearest better candidate is some option is hot, which reaches 118 of 138 and is also not it. Both are correlates, and the reason they can only be correlates is in the mechanism above: whether an option’s structure survives into the parent’s wall depends on whether that structure sits above or below the height at which the parent’s walls meet. An option can be as complicated as it likes and contribute nothing, if everything interesting about it happens above the parent’s temperature.

So a bend is not a property of an option. It is a property of an option relative to the parent’s temperature, which is why no classification of the options alone can capture it — and why the proof route the rung below proposed cannot be made to work as stated.

The 49 are the useful object here. Each has an option with genuine structure and a parent whose walls are straight anyway, so each is a small worked example of structure being clipped away — and 49 examples of a thing being invisible is a better starting point for a proof than a paragraph about when it ought to be. What they would have to show is a bound: that an option’s own breakpoints all lie above the parent’s meeting point, which is a statement relating two temperatures rather than classifying one value.

That reformulation is the honest residue of the rung below’s proposal. The route it named is closed; the route past it is a comparison of heights, and it is not a classification at all.

What is left of the proof #

The first claim being exact does not make it proved, and it is worth being precise about what would.

The equivalence is between two computed quantities on 138 particular values. The argument in the third section above is a mechanism, not an induction: it says why a straight wall carries no information beyond its stop, and it does not establish that a bent wall always carries information the stop reading needs. The direction that looks harder is the one this page has 32 examples of and no argument for.

What the census does supply is the class, stated in terms a proof can use. Both walls straight below the meeting point is a condition on the diagram, checkable in one pass over the wall’s breakpoints, and it does not mention the options at all. That is a better place to start than the option classification, which is now known not to work.

What this says about reading a diagram #

There is a practical reading of all this, and it is the reason the whole ladder exists.

A player looking at a thermograph wants two numbers: what the position is worth and what moving is worth. The stops give both, they are cheap, and this page says exactly when they are right — look at whether the walls are straight. A straight-walled diagram can be read off its base line; a bent one cannot, and the bend is visible at a glance.

That is a rare shape for a result in this subject. Most of the readings on this site are right most of the time with no way to tell which time, and the general account of such rules is that a bound without a warning light is worth much less than its hit rate suggests. Here there is a warning light, it is drawn on the figure, and it is exact.

An equivalence with nothing in the off-diagonal cells #

The result is stated as an equivalence and the evidence for it has a shape worth naming, because an equivalence needs two kinds of failure to be absent and a hit rate reports neither.

A rule of the form the reading is exact exactly when the walls are straight can fail in two independent ways. It can find a straight-walled position whose reading is wrong, which would mean straightness is not sufficient. And it can find a bent-walled position whose reading is right, which would mean straightness is not necessary — that some other property is doing the work and bending is merely correlated with it.

The second failure is the one a sweep is likely to miss and the one that turns a theorem into a correlation. A rule with a handful of bent-but-exact positions is still right most of the time, still reports a high agreement rate, and is no longer a characterisation of anything.

So the number to look at is not 138 of 138 but the two off-diagonal cells, both empty. 106 straight and exact, 32 bent and inexact, nothing anywhere else — which is what makes this a condition rather than a good predictor, and what a percentage would have concealed.

That also says exactly what would refute it and how cheaply. One position, in either off-diagonal cell, at any depth, and the equivalence becomes an implication. There is no margin, no tolerance and no room for a rate: an equivalence is the one claim on this site that a single counterexample settles, which is why it is worth stating as one when the data supports it.

What the census does not say #

Four limits.

One convention, doing real work. The eight exceptions were made by the site’s own choice to report a number’s temperature as $-1$ rather than as nought. That choice is right where it was made — a stack of temperatures should not contain cold components — and it produced eight false counterexamples here. Any other census that compares a computed temperature against a formula will meet the same eight.

Day three, one option a side. Every value here has exactly one Left option and one Right option, which is what makes the textbook formulas even askable. Whether the equivalence survives two options a side is a different census over a much larger population, and it is the obvious next check.

The equivalence is checked, not proved. One hundred and thirty-eight values in both directions is strong evidence and it is not a theorem. What has changed is that the class is now defined by a property of the diagram rather than described by a paragraph.

A bend is detected numerically. The test is whether any segment of a wall below the meeting point has a slope other than the $\mp 1$ a textbook switch’s walls have, with a tolerance of a billionth. Walls are piecewise linear with dyadic breakpoints, so the tolerance is generous rather than delicate, but it is a tolerance.

And nothing here says what a bend costs. The stop reading is wrong on the 32 and this page has not measured by how much. A thermograph with two bends is where the anatomy of a bend is taken on its own terms, and the size of the error the bend produces is a measurement neither page has made.

What it changes about the earlier rung #

The rung below reported its result as a proportion: the stop reading works on about three quarters of the values that break the hypothesis. That framing is now the wrong one.

Three quarters is not a hit rate. It is the share of day-three one-option values whose walls happen to be straight, and it would move with the population — a census of day-four values, or of positions from a real game, would give a different number and the same theorem. The reading is not right three quarters of the time; it is right on a class, and the class is 77 per cent of this particular sweep.

That distinction is the difference between a heuristic and a theorem with a precondition, and this site has spent several essays on the first kind. It is worth noticing when one of them turns into the second.

The convention, named #

Normal play throughout, and everything is day-three values in canonical form, computed by the recursion.

The stops are what each player gets by moving first and playing the fight out with no tax charged: $\mathrm{LS}(G)$ and $\mathrm{RS}(G)$, read off the feet of the two walls. The stop reading is that the mean is $(\mathrm{LS} + \mathrm{RS})/2$ and the temperature is $(\mathrm{LS} - \mathrm{RS})/2$.

A wall bends below the meeting point if any of its segments strictly below the temperature has a slope other than $-1$ for Left’s wall or $+1$ for Right’s. The meeting point itself is excluded, since every wall turns there by construction.

A number is given temperature $-1$ here, which is a convention of this site and is the whole of the eight exceptions above.

Where the ladder goes next #

The switches anchor has five rungs to here, and this one has reduced the whole stop reading to one question: does the wall bend? The three above measure what a bend costs, remove the diagram from the answer, and then answer the bend question from stops as well.

Half a follow-up out prices the error when the wall does bend, and it comes out at half the follow-up’s temperature — an unaccounted exchange, valued at what that exchange is worth, which is why the correction is a subtraction rather than a proportion.

A second level of stops then widens the pool elevenfold and removes the diagram. The correction survives, missing two values in 1,459; and it needs no thermograph at all, because the follow-up’s temperature is half its own stop gap — so the whole reading becomes arithmetic on stops rather than a height read off a picture.

The bend is in the stops finishes the job this page starts. The bend question itself is answered from the options’ stops, in four lines, with no diagram anywhere — and it delivers more than it was 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 every failure is a value with a bent-walled option.

So the anchor closes on itself. A bend is what makes stops insufficient; stops determine everything until a bend appears one level down; and the same feature that decided the stop reading at the start is what limits the stop reading at the end.