Which end a sum lands at

A bound with one number too many confirmed the translation anchor’s bound over a wider pool — adding a hot game to a value moves each of its stops by at most twice the smaller of the two temperatures — and refuted the conjecture that a third number would sharpen it. It closed on the other direction the bound could be improved in:

The rung above is … not a smaller bound, but a description of which sums attain it. Three hundred and fifty-eight of these 1,440 sit exactly on the bound and 660 sit at nought, so the population is not spread across the range — it clusters at the two ends, and a rule that said which end a given pair lands at would be worth more than any refinement of the bound itself.

The rule is four lines and it is exact on all 1,440.

The population, and what it is not #

Forty-six per cent of the sums move no stop at all, twenty-five per cent move one by exactly the bound, and twenty-nine per cent land somewhere between. A bound satisfied by a population spread evenly beneath it would be a bound that describes the population loosely; this one is either exact or vacuous three times in four, which is what makes the question of which worth asking.

The property that decides it #

Sort the values by their thermographs and three kinds appear. Cold values are numbers or infinitesimals — temperature nought, 55 of the 120. Straight values are hot and both walls run at slope $\mp 1$ all the way from the stop to the meeting point, 31 of them. Bent values are hot and some segment of a wall below the meeting point runs at some other slope, 34 of them.

That third category is not invented here. It is exactly the class the bend is the condition established for a question with nothing obvious to do with sums: whether a value’s stop reading — mean is the midpoint, temperature is half the gap — is exact. A bent wall means the reading is wrong; a straight one means it is right.

The table has an empty cell wherever it can have one. A cold value’s stops never move — 660 of 660. A bent-walled value’s stops never move the whole way — 408 of 408 land strictly inside. Only the straight-walled row has two entries in it, 358 attaining and 14 short.

The boundary of the third case #

The fourteen exceptions all have an addend less than half again as hot as the value: ratios of 1.00 and 1.33. Every pair whose ratio is 1.5 or more attains, and the census contains nothing between 1.33 and 1.5, so the threshold is sharp on this data and unlocated between those two numbers.

So the rule reads:

• the value is cold — the stops add exactly;
• the value is hot and a wall bends — the error is strictly inside the bound;
• the value is hot with straight walls and the addend is at least half again as hot — the error is exactly the bound;
• the value is hot with straight walls and the addend is not — strictly inside.

The same property, twice, on two ladders #

It is worth pausing on the coincidence, because it is not one.

The switches ladder asks whether a value’s two stops determine its mean and temperature. The translation ladder asks how far a value’s stops move when something hot is added to it. Those are different questions on different anchors, and the answer to both is decided by whether a wall bends below the meeting point.

The connection is that both questions are about what the stops know. A stop is the score a player gets moving first with no tax charged, so it is the bottom of one wall of the thermograph and it is set by whichever option governs that wall at height nought. The temperature is set by where the two walls meet, which is governed by whichever options are hottest. On a straight-walled value those are the same options and the stops and the temperature describe one shape; on a bent-walled one they are different options and the two numbers describe two shapes.

Every question about a value that is asked in terms of its stops therefore has the same fault line running through it. The stop reading is exact on the straight side and wrong on the bent side; the translation bound is attained on the straight side and unattained on the bent side. Two ladders, two questions, one property — which is the kind of thing worth recording as a fact about the subject rather than as an observation about two censuses.

Why a cold value cannot move #

Two of the four cases have arguments and the other two do not, and it is worth separating them.

The cold case is nearly a theorem already. If the value $G$ is a number, then $G + H$ has the same stops as $H$ shifted by $G$ — a number added to a game translates both its stops by exactly that number, which is what the translation anchor’s first rung establishes. So the error is nought by construction, and it is nought for the same reason the bound is nought there: twice the smaller temperature, where one of them is nought.

The infinitesimals are the interesting half of the cold row. An infinitesimal has temperature nought and is not a number, so nothing about shifting applies — and its stops still add exactly, on every one of its pairs. That is what an infinitesimal does to a fight measured from the other side: adding an infinitesimal to a hot game changes the value and leaves both stops where they were.

Why a bent wall cannot reach the bound #

The bent case has an argument too, and it is the same one the switches ladder found.

A bent wall is a wall governed by different options at different heights. At the very top it is governed by whichever option is hottest, and lower down some other option takes over — which is precisely why the stop reading fails on such a value: the stop is set by one option and the temperature by another, so the two numbers do not describe one shape.

The bound is twice the smaller temperature, and it is attained when a stop moves by the whole of what a fight can move it. On a bent-walled value the stop belongs to an option that is not the hot one, so the fight the temperature measures is not the fight the stop sits at the bottom of, and the movement stops short. A value whose stops do not tell the whole story is a value whose stops cannot be moved the whole way, which is a single sentence covering 408 sums.

Neither of those is a proof and both are the right shape for one. The third case — the ratio of one and a half — has no argument at all, and that is the honest state of it: it is a threshold read off a census with nine addend temperatures in it.

What a description is worth, against a sharper bound #

The rung below offered the choice between two ways of improving a bound and preferred this one, and having done it the preference is worth defending.

A sharper bound would say the error is at most some smaller quantity. It buys a reader a tighter guarantee and it costs whatever the new quantity costs to compute — and the rung below found that the obvious candidate, involving the addend’s follow-up, is not available: the two-number bound is attained, so nothing below it can be a bound at all.

A description of the ends buys something different. It says, before any arithmetic on the sum, which of three things will happen. A player holding a cold value knows the stops add and can compute the sum’s stops by addition; a player holding a bent-walled value knows the movement is real and incomplete; a player holding a straight-walled value against a much hotter addend knows the movement is exactly the bound and can compute it in one multiplication. In every case the description replaces evaluating a sum with looking at one thermograph.

And it is a stronger statement about the subject than a bound is. A bound says the sum cannot do more than something. This says the sum does one of three definite things and which one is settled by a property of one part — which is the kind of statement the sum is the object is about, and the kind that turns an inequality into a mechanism.

There is a cost, and it is the usual one. The bound needs two temperatures; the description needs a thermograph, because the bend is not readable off the stops. A second level of stops records exactly the same trade on the switches ladder, and for exactly the same reason: everything on this site that turns on a bend needs the diagram, and nothing cheaper has been found that decides one.

What this does not say #

The threshold is located between two numbers and not at one. The census has ratios of 1.00, 1.33 and then 1.50, so anything in $(1.33, 1.50]$ would fit the data equally well. A pool with an addend at 1.4 times a value’s temperature would narrow it, and the natural guess — that the true threshold is a nice number like $3/2$ — is a guess.

The bend is the value’s, not the addend’s. Every addend in the census is a switch with a follow-up on one side, so all twelve have the same shape, and nothing here says what happens when the addend’s walls bend. That is the same census with the two roles swapped and it was not run.

The classification is of a value, not of a pair — with one exception. On 113 of the 120 values the class is the same against every one of the twelve addends. The remaining seven are straight-walled values hot enough for the ratio to matter, and there the addend decides. So almost entirely a property of the value is exactly the right phrase and entirely would be wrong.

The infinitesimals are not separated from the numbers. Both are cold and both add exactly, so the cold row of the table is 660 sums of two rather different kinds behaving identically. That identical behaviour is a small finding of its own and it is stated here rather than measured — the census records a temperature of nought and does not record which kind of nought.

And the population is 120 values. They are the first 120 day-three values in the enumeration’s order, which is the rung below’s pool, chosen for cost rather than for coverage. Whether the three-way split of 55, 31 and 34 is representative of day three as a whole is a different question, and a second level of stops measures the bent-to-straight ratio over the whole day and gets a rather different one.

The convention, named #

Normal play throughout, and every value computed by the recursion and reduced to canonical form.

The stops are what each player gets moving first with no tax charged. A sum’s stops add when each of them is the sum of the corresponding stops of the parts, and the error is the larger of the two discrepancies, taken in absolute value.

The bound is twice the smaller of the two temperatures, which is the rung below’s result and is not re-derived here. It is nought whenever either part is cold, so attaining the bound is a statement only about pairs where both parts are hot.

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.

Cold means a temperature of nought, which covers numbers and infinitesimals together. The census keeps them apart in its rows and they behave identically here, which is itself worth noticing.

Why the errors cluster at the ends #

A bound with most of its cases sitting exactly on it or exactly at nought is an unusual distribution, and the shape says something about what the bound is measuring.

An error that arises from an accumulation of small effects is spread: many small contributions, a range of totals, and a bell shape in the middle of the permitted range. An error that arises from a discrete event — an exchange happening or not happening — is bimodal, because there are two cases and each has its own value.

The clustering at the two ends is the signature of the second. Nought means the exchange did not occur and nothing was lost; the bound means it occurred in full and cost exactly what an exchange costs. There is no mechanism producing a third of an exchange, so there is very little in the middle.

That reframes the four-line rule as a description of when the event happens rather than as a formula for a magnitude. It also explains why three of its four cases read only the value being translated: whether the exchange occurs is a property of that value’s own shape — whether its wall bends — and the other operand enters only in the one case where the two interact.

And it says what the strictly-interior cases are. They are the positions where the exchange happens partially, which on this population means where a bend takes back some of what the fight offered. That is a quantity rather than a switch, and it is exactly what the rung above measures.

What a player does with it #

The rule is about a census and the reason to want it is a board, so it is worth spending a paragraph on what it says to somebody adding two positions up.

A player who wants a sum’s stops has three choices: evaluate the sum, add the parts’ stops and accept the error, or add them and correct. The rule turns the second into the third for two of its four cases. Against a cold part, adding is exact — no correction and no evaluation. Against a straight-walled part and a much hotter addend, the movement is exactly twice the smaller temperature, so adding and then shifting by that amount is exact too, at the cost of one multiplication.

The remaining case is the bent one, and there the rule says only that the error is real and less than the bound. That is where the work is, and it is 408 of the 1,440 sums here — about three in ten. So the honest summary for a player is that seven sums in ten can have their stops added and corrected exactly, and three in ten need the diagram, which is the same proportion this site keeps arriving at from different directions.

Where the ladder goes next #

The translation anchor has five rungs to here, and this one has given a four-line rule for which end of the permitted range a sum lands at.

The rung above measures the cases the rule sends to neither end. The same number in two currencies finds the bent-walled values falling strictly inside the bound, and the shortfall is not a range: it 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 an interval with a rule for its ends, which is a considerably stronger result than this page’s classification and includes it.

The other half of that rung is the identification worth carrying. The constant it finds is the switches ladder’s constant, appearing here at twice the size — a half there and a whole here — and the conversion is one line: a temperature is half a gap between stops, so a quantity denominated in temperatures is half the same quantity denominated in stops.

Two ladders, two questions, one constant in two currencies. That is the sort of coincidence worth chasing rather than noting, and it is why the bend keeps turning up on both: a bend is where a wall gives something back, and what it gives back is the same exchange whether it is measured up the diagram or across it.