The bend above the top

A bend that never reaches the surface found the rule for how deep a thermograph’s recursion has to go before the stops suffice. It is not the depth of the deepest bend in the tree; it is the length of the longest chain of bends running unbroken down from the top. That is right on 2,400 of 2,403 values across two pools, and it closed by naming the three it is wrong about and the shape of the fix:

the fix is a predicate — does this option’s bend fall in the stretch of wall the envelope uses — which is computable from the same objects the chain reading already has. Three exceptions is a small target and that is the point.

The predicate works. It fixes all three, it breaks none of the 2,400, and it turns out to need only half of itself.

What the chain reading is counting #

A thermograph is built by a recursion. Each wall of a position is an envelope over its options’ walls — the Left wall is the upper envelope of the Left options’ contributions, the Right wall the lower envelope of the Right options’ — and the recursion bottoms out at numbers, whose walls are vertical lines.

Truncating that recursion at depth d means replacing every position d levels down with the switch on its own stops, which has straight walls. The question the last two rungs have been about is how small d can be before the truncation stops mattering.

The word mattering is doing precise work there and it is worth unpacking, because the whole ladder turns on it. Truncating always changes the tree — a position two levels down really is being replaced by a different game — and the question is never whether the tree changed but whether the diagram did. Two thermographs are the same when their walls agree at every height, their temperatures agree and their means agree, which is a comparison of pictures rather than of positions. A truncation that mangles a position whose contribution the picture never draws is a truncation that costs nothing, and that gap between what is in the tree and what is in the picture is where every exception on this ladder has lived.

The chain reading answers it by walking down from the top: if this position’s walls are straight, no levels are needed; otherwise one more than the longest chain of bends beneath it, stopping the moment a straight-walled position is reached. It is a good rule and it is a rule about the tree, and the three exceptions are where the tree and the diagram part company.

The three #

All three exceptions have the same shape. The smallest is $\{0, \ast \mid \{-1/2, \{1 \mid -1\} \mid -2\}\}$.

Its Right option, $\{-1/2, \{1 \mid -1\} \mid -2\}$, is genuinely bent — its walls depart from its stop lines — so the chain reading finds a chain two long and says two levels are needed. One is enough.

The reason is in the temperatures. The whole position’s temperature is a half. The option’s own temperature is one, and it does not begin bending until its wall has climbed past a half. So the position’s diagram is finished — the two walls have met at the mast — before the height at which the option bends. The bend is real, it is one level down, and the diagram never gets to it.

That is a different mechanism from the one the rung below was correcting for. There, the failure was that deepest bend counts bends under a straight wall, which the envelope discards by choosing some other option. Here the bend is not discarded by an envelope at all: it is simply above the top of the picture.

The other two exceptions are the same position with its Left options changed — $\{\ast \mid \dots\}$ and $\{0 \mid \dots\}$ against $\{0, \ast \mid \dots\}$ — which is what makes them a class rather than three coincidences. The Left side is doing nothing in any of them; the whole of the over-count comes from the one Right option, and varying the other side leaves it intact. That is a small thing to check and it is the difference between three exceptions and one exception counted three times.

It is also worth noticing what does not vary. All three have temperature a half, all three have the same Right option, and the Right option’s temperature is one in all three. An exception whose mechanism depended on the position’s own temperature being small would be a different finding; here the mechanism is entirely a comparison between two temperatures, and the three cases are the same comparison.

The predicate, and the half of it that matters #

The rung below proposed a two-part condition. The stretch of wall an envelope uses is bounded above by the position’s own temperature — nothing above the mast is drawn — and bounded sideways by which option is on top at each height. An option’s bend counts if it falls inside both bounds.

Count only the bends below the parent’s temperature and the reading is exact: 2,403 of 2,403. Add the second clause — the option must also be the one on the envelope where it bends — and it is still 2,403 of 2,403, because the second clause never fires. Not once, on either pool.

So the fix the ladder needed is a comparison of two numbers. Descend into an option only if the option’s bend begins below the parent’s temperature, which is a comparison of the option’s temperature against the parent’s plus a check of where the bend starts. The envelope is never consulted.

That is cheaper than what was asked for and it is also weaker, and both halves of that are worth saying. It is cheaper because the envelope is the expensive object — computing which option is on top at a height means having every option’s wall — where the two temperatures are already in hand. It is weaker because the envelope clause never fires here is a statement about 2,403 values and not a theorem, and a population with a bent option covered over by a hotter sibling below the mast would separate them.

Nothing in this sweep has that shape and nothing here rules it out.

There is a reason to suspect the population is not accidental about this, and it is worth stating without overclaiming. For the envelope clause to fire, a position needs two options on the same side, one of which is bent below the mast and the other of which is on top there. The deeper pool is built as $\{A \mid B\}$ over hot day-three values — one option a side, by construction — so it can never contain the case, and day three’s values have small option sets. The absence is therefore partly a fact about how these pools were built rather than a fact about hot games, which is the strongest thing this page can say against its own result.

What the reading costs to compute #

The two readings differ in price as well as in accuracy, and the difference runs the opposite way to what usually happens on this ladder.

Every previous refinement here has been more expensive than the thing it replaced. The correction to the stop reading needs the hottest follow-up’s temperature; the bend test that says when to apply it needed a thermograph until the rung that read it off the stops. Accuracy has been bought with computation each time.

Here it is not. The chain reading asks, for each option, is this option bent — which is a thermograph. The refinement asks the same question and then compares two temperatures, both of which it already has, and discards the option if its bend begins above the parent’s mast. The comparison is free.

What that means practically is that there is no reason ever to use the chain reading. A refinement that is exact where its predecessor is nearly exact, at no extra cost, replaces it outright rather than sitting beside it as the expensive accurate version — which is not the usual outcome and is worth marking when it happens.

Why the exactness matters more than the three #

Three values in 2,403 is a rounding error and this ladder has spent a rung on it, which needs justifying.

A reading that is right on 99.9% of a population is a good heuristic and it is not a description. It says the rule is nearly the right shape and leaves open whether the residue is noise, a second mechanism, or the thing the rule was actually about. Every rung of this anchor has been in that position: the stop reading is out by half a follow-up, the correction survives on 348 of 350, the chain reading misses three. Each of those numbers is small and each of them was hiding something.

Here the residue turned out to be a mechanism — bends above the top — and once it is named the reading is exact. That is a different kind of statement from 2,400 of 2,403: it says the rule is not nearly right, it is what the diagram does, and there is nothing left over to explain.

It is also falsifiable in a way a percentage is not. A single value in any future pool that needs a different number of levels would refute it outright, where a value that broke a 99.9% reading would merely lower the percentage.

The sweep is written to take that seriously. Two conditions are asserted rather than reported: the refinement must fix all three of the values the chain reading misses, and it must break none of the 2,400 it gets right. Either half failing would mean something different. A refinement that fixed two of three would say the residue is a third mechanism and this rung has found only part of the answer; one that fixed all three and broke four would say the predicate is not a refinement at all but a different rule with a different error, which is the outcome a fitted correction usually has.

What the reading now says #

Put the anchor’s rungs together and the account of a hot position’s arithmetic is complete for this population.

The stops give the mean and temperature exactly when both walls are straight. When a wall bends, the stop reading is out by half the hottest follow-up’s temperature, and the follow-up’s temperature is itself read off its stops, so the correction is a second level of stops rather than a diagram. How many levels are needed at all is the length of the chain of bends the diagram reaches — the count this page makes exact.

Every one of those is a statement about how shallow the arithmetic is. A thermograph is defined by an unbounded recursion and the population here needs one or two levels of it, with a rule for which. The whole of this anchor is an argument that the recursion is not the thing being computed; it is the definition, and the thing being computed is a small local reading of it.

That is worth setting beside what the reading does not give. None of these rungs computes a value from its position; they compute a thermograph’s shape from a game’s stops and its options’ stops, which is a statement about one object in terms of shallower versions of itself. Getting from a position on a board to those stops is the whole of the rest of the subject, and this anchor has never touched it.

What the shallowness does buy is a bound on how wrong a cheap reading can be. A player holding a hot position wants the temperature — the whole reason temperature exists is to say how much — and this ladder now says that reading it off the stops is exact unless a wall bends, that the error when a wall bends is half the hottest follow-up’s temperature, and that the follow-up’s temperature is itself readable off stops. Two levels of arithmetic, in a subject whose definitions are unbounded recursions.

Why a bend below the temperature is the only one that counts #

The correction is one clause and it is worth reading as a statement about diagrams rather than as a patch.

A thermograph’s walls rise from the stops, bend wherever a follow-up runs out, and meet at the temperature. Above the meeting point there is no diagram left — both walls have become the mast, and whatever bends the construction would have drawn up there are drawn on a picture the game never reaches. The chain reading counted them anyway, because it was reading the option structure rather than the diagram: an option with a follow-up contributes a bend whether or not the bend survives to be seen.

That distinction is invisible on almost every value, which is why the reading held for so long and why the three failures were so hard to place. A bend above the temperature requires a follow-up hotter than the position that contains it, and hot options inside a colder position are rare. When one occurs, the chain reading finds a bend the diagram does not have and adds a level that is not there.

Stating the fix as count only what the diagram reaches also says why it cannot over-correct. Every bend below the temperature is genuinely on the walls; discarding the ones above throws away nothing the reading needed. That is the difference between a repair that trades one class of error for another and one that removes a class outright, and it is why the corrected reading breaks none of the values the old one got right.

It also explains why three is the right order of magnitude for the failure count rather than a suspiciously small one. The clause only bites where a position holds an option hotter than itself, and a value born early enough to be in this population has very little room to arrange that. On a larger population the count would grow; what would not change is that every one of them is the same shape, because the clause names a configuration rather than a tendency.

Where the ladder goes next #

The switches anchor has ten rungs: the switches and their mean values, what a number does to a fight, the fight with no midpoint, the bend as the condition, half a follow-up out, the second level of stops, the bend from the stops themselves, how deep a reading has to go, and now which bends it has to go through.

The rung above is the envelope clause. This page shows it never fires on a population of 2,403 and cannot show it never fires. What would settle it is a pool built to contain the case: a position with two Right options, one of them hotter and straight-walled, the other bent below the mast and covered over by the first. The construction is direct — take a bent value and a hotter straight-walled one and put them on the same side — and the question is whether the resulting position needs the level the bent option is offering. If it does not, the envelope clause is real and this page’s simpler statement is a coincidence of the population; if it does, the simpler statement is the rule.

Two neighbours are worth the trip. The bend is in the stops is where the bend test stopped needing a diagram, and it is the reason this page’s predicate is cheap. And a second level of stops is the rung where a correction survived on 348 of 350, which is the same shape of near-miss this page turned into an exact rule — and the two exceptions there have never been explained.