A second level of stops

Half a follow-up out measured the error the stop reading makes on the values it is wrong about, and found it to be not a bound but an equality: on every value whose thermograph has a bent wall, the read temperature is too low by exactly half the temperature of the position’s hottest follow-up. Thirty-two values, no exception, and the error always in the same direction.

That page closed by naming a rung with a definite finish:

The rung above is the correction. If the error is half the follow-up’s temperature, then adding half the follow-up’s temperature to the read temperature is exact on every value in this sweep … Whether that survives two bends, and whether the correction can be stated without computing the follow-up’s temperature at all, is a rung with a definite finish and a real prize.

Three questions, and they go three different ways.

The pool, eleven times over #

The rung below’s population was the day-three values with one option a side, which is 167 values of which 138 are not numbers and 32 have a bent wall. That restriction was not arbitrary — a switch has one option a side, and the whole ladder is about what happens to the switch formulas when the hypothesis behind them is dropped — but it is a restriction, and an equality checked on thirty-two cases is a conjecture with thirty-two witnesses.

Dropping it gives 1,459 non-number values born by day three, of which 350 have a wall that bends below the meeting point. The correction is exact on 348 of them. Against the plain reading’s 1,109 exact values out of 1,459, the corrected reading gets 1,457 — which is to say that the two rungs below have between them turned a reading that is right three quarters of the time into one that is right on all but two values in the population.

That is a much stronger statement than the one it grew from, and it is worth being explicit about why. Eleven times the pool is not eleven times the confidence; what it is, is a population containing shapes the smaller one could not contain — positions with three Left options and one Right one, positions whose bent wall is governed by an option that is not the hottest, positions with an infinitesimal sitting next to a fight. The correction was written with none of those in view and holds on nearly all of them.

It also moves the reading out of the class it was invented for. Switches and mean value sets out the two formulas for a genuine switch, where both options are numbers and the mean and the temperature are the midpoint and the half-gap of those; a fight with no midpoint is where the hypothesis is dropped and the stops take over. A value with four options and an up in it is not a switch by any reading, and the corrected formula does not care.

The two it misses #

Both exceptions are the same position twice: $\{-1/2, \{1 \mid -1\} \mid -2\}$ and its mirror. Its Left side offers two options — the number $-1/2$ and the switch $\{1 \mid -1\}$, whose temperature is 1 — and the correction predicts an error of a half where the true error is a quarter.

The reason is visible in what a wall is. Left’s wall is the maximum, over Left’s options, of that option’s right wall shifted down by the tax; the hottest option governs the wall while its own diagram is still climbing, and when it stops the next option takes over. Here the switch governs the wall down to $t = 1/2$ and the number $-1/2$ takes it from there, so the bend is half as deep as the follow-up on its own would make it. The rung below’s pool had one option a side and therefore had no way to see this: with a single option there is nothing to take the wall over.

So the equality is not quite an equality. It is exact when the bent side offers one option that matters and approximate when two of them compete, and the population happens to contain only two positions where that competition changes the answer. The honest statement is that the correction is exact on 348 of 350 and that both exceptions over-charge, which makes the corrected reading an upper bound in the two places it is not exact — the opposite direction to the plain reading, which is always a floor.

What it costs to know when to apply it #

The correction is stated conditionally: add half the follow-up’s temperature where a wall bends. A reader with the thermograph in front of them can see whether a wall bends. A reader without one cannot, and the whole point of a stop reading is to be the thing that saves computing the diagram.

So the obvious question is what the correction is worth when applied blind — fired whenever the position has a hot option at all, which is a condition anyone can test.

It is a catastrophe. 818 of the 1,109 straight-walled values have a hot option, and the plain reading is already exact on every one of them, so a rule that fires on the presence of a hot option adds a spurious correction to three quarters of the values that were right. The blind reading is exact on 639, which is worse than doing nothing.

That is the real cost of this rung and it is worth stating plainly: the correction is cheap and the licence to apply it is not. Half a follow-up’s temperature is a small computation on one option. Knowing that this is one of the 350 rather than one of the 1,109 means knowing whether a wall bends below the meeting point, and the bend is the condition establishes what that test is — a claim about the slope of the wall everywhere below the temperature, which is the thermograph.

There is a cheaper test the rung below already ruled out. Some option is neither a number nor an infinitesimal is implied by a bend and does not imply one; it reaches 118 of 138 in the smaller pool and is exactly the kind of near-miss that would make this reading usable if it were exact. It is not, and nothing weaker than the diagram has yet been found that is.

Two bends, and the pool that has none #

The rung below’s second question was whether the correction survives two bends. It has an answer that arrived before any measurement did.

Not one of the 1,459 non-number values born by day three bends on both walls. The question the rung below posed had no instance in the population it was posed about — which is a fact about day three rather than about the correction, and is the kind of thing a page only discovers by trying to answer its own closing paragraph.

Two bends need a position whose two sides are each governed by a hot follow-up of their own, and day three is not deep enough for both to survive canonicalisation. Building one is easy enough: take $\{A \mid B\}$ with $A$ and $B$ each drawn from the hot day-three values, and 302 of the 14,400 constructions bend on both walls.

Two bends need two follow-ups #

Half the hottest follow-up is exact on 62 of the 302, and on 237 it reads too low — which is the same direction the plain reading errs in, and for the same reason. A wall that bends is a wall that has been under-read, and with two bent walls there are two under-readings and the rule is charging for one.

The generalisation is immediate and is exactly the rule the rung below stated, with the word hottest replaced by its own side’s: each bent wall contributes half of the follow-up on its own side. Where one wall is straight its side contributes nothing and the rule is the old one; where both bend, both contribute. That is exact on 237 of the 302, and 65 positions remain out of reach of it.

The 65 are the two-bend version of the two exceptions above, and there is a great deal more of them: with two sides competing, an option that takes the wall over below the bend is much easier to come by. So the shape of the answer is that the correction generalises in the obvious way and the generalisation degrades — 99.9 per cent of the population with one bend, 78 per cent with two — and the thing degrading it is competition among options rather than the number of bends as such.

The control in that table is doing real work and is worth a sentence. Half the smaller follow-up is exact on none of the 302, which is what says the pool can tell one reading from another: three quantities of the same order, scored on the same positions, and two of them fit while the third fits nothing. A pool on which every plausible arithmetic scored well would be measuring the arithmetic’s plausibility rather than the positions, which is the failure a rule with no promise at all is careful to avoid on its own pool.

The prize, collected #

The rung below’s third question was whether the correction can be stated without computing the follow-up’s temperature at all. It can.

On all 350 bent values, the hottest follow-up has straight walls — so by the rung two below’s own theorem its temperature is half its own stop gap, and no diagram is needed to get it. Substituting the one for the other changes nothing: 1,457 exact either way.

The corrected reading is therefore a second level of stops. Read the position’s two stops; read the hottest option’s two stops; the temperature is half the first gap plus a quarter of the second. That is four evaluations of a stop where the plain reading needs two, against a thermograph, which needs one per option all the way down the tree.

Which makes the state of this ladder a pleasing kind of awkward. The arithmetic of the correction is now as cheap as the reading it corrects. What is not cheap, and shows no sign of becoming so, is the licence — and the licence is a thermograph, which is what the reading was invented to avoid computing. A player who can afford the bend test can afford the true temperature and does not need the correction; a player who cannot afford it cannot use the correction either.

There is one setting where that trade comes out well, and it is the setting temperature is usually wanted in. A player comparing two components of a sum does not need either temperature exactly — temperatures do not add, so what decides which part to move in is which of them is larger, and an ordering survives an error that a value does not. Two stops apiece and one option’s stops apiece is enough to order most pairs correctly, and the rung below measured what the uncorrected reading costs in exactly those terms: seven hundred and seventy pairs read as level that are not.

Removing the diagram is the result #

The correction survives elevenfold widening and needs no thermograph, and of those two the second is the finding. It is worth saying why.

Surviving a widening is confirmation. The law was already stated, the wider pool tests it, and two failures in 1,459 is a good result of a familiar kind.

Needing no thermograph is a change of kind. A law stated in temperatures requires a diagram, a diagram requires the thermograph recursion, and the recursion is most of what evaluating a position costs. A law stated in stops requires two numbers that a player computing an endgame already has — so the same law moves from something a program computes to something a person can apply at a board.

The conversion is one line and it is exact rather than approximate: a follow-up’s temperature is half its own stop gap, so every appearance of a temperature in the correction can be replaced by a difference of two stops.

That is worth more than a sharper constant would have been. A correction with a better coefficient is still a correction requiring a diagram; a correction with the same coefficient in a currency the reader already holds is a usable rule. And it is the same move the switches ladder makes one rung later, when the bend question itself is answered from the options’ stops — which suggests the diagram was never carrying information the stops lacked, at this depth.

What this does not say #

The straight-walled follow-up is a fact about this population, not a theorem. All 350 bent values here have a hottest follow-up with straight walls, and nothing establishes that they must — a value born on day four could perfectly well have a bent-walled follow-up, in which case the stop substitution would need a third level and the recursion would run rather than stop. What has been shown is that on day three it stops at two.

The two-bend pool is built and not enumerated. $\{A \mid B\}$ over the first 120 hot day-three values is a construction chosen because it produces two bends, not a sample of anything. The 78 per cent figure is a statement about those 302 positions, and the right way to read it is as a demonstration that the generalised rule is not exact rather than as a rate.

And the mean has been left alone here. The rung below measured both errors and found them equal in size; everything above is about the temperature, because the temperature is the number a reading is usually wanted for. Whether the mean’s correction generalises the same way across two bends is the same sweep with a different column read, and it was not run.

The convention, named #

Normal play, canonical forms, computed by the recursion throughout.

The stops are what each player gets moving first with no tax charged, and the stop reading takes the mean to be their midpoint and the temperature to be half their gap.

A follow-up is an option that is not a number, and its temperature is the height at which its own walls meet. Numbers count as having no follow-up rather than a negative one, because what is wanted is the height of a bend and a number’s wall does not bend.

A wall bends below the meeting point when a segment strictly beneath the temperature has a slope other than $-1$ on Left’s side or $+1$ on Right’s. The meeting point itself is excluded, since every wall turns there.

Exact means agreeing with the thermograph’s temperature to within $10^{-9}$, and the comparison is always against the diagram rather than against another reading — two readings that agree with each other and not with the position would otherwise look like a result.

Where the ladder goes next #

The switches anchor has seven rungs: the class and its two formulas, the switch a player imagines when the position is not one, the edge of the regime, what the formulas become once the hypothesis is dropped, the condition that says when the replacement is exact, what it costs when it is not, and now what it takes to put the cost back.

The rung above is the bend test itself. Everything on this page turns on a single question — does this wall bend? — which is currently answered by building the whole diagram, and there is no reason in principle why it should need to be. 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, and the two exceptions above are precisely the positions where that happens without changing the temperature. A test for a bend built from the options’ four stops would make this whole reading free, and the 350 bent values against the 1,109 straight ones are the data it would be scored on.

Two neighbours are worth the trip. A thermograph with two bends is where the diagram’s shape is read as an argument rather than as a picture, and it is the page this one’s two-bend pool was built to test. And a rule that is never right and cannot be far wrong is the other way a reading can be worth having — a bound with a guarantee rather than an equality with a licence — and reading the two together shows the two kinds of thing an approximation on this site is allowed to be.