A cross in the table

Add then reduce again found that the arithmetic on reduced forms is add and reduce again: the sum of two reduced forms is already reduced on most pairs and needs a second pass on 431 of them. Two rungs since have described what the second pass does — that it deletes at most one option a side, that it is a pure deletion four times in five, and that a side empties exactly when the two parts’ option-sums are comparable.

None of them said which pairs it happens to, and the rung below named that as the anchor’s last unasked question:

The rung below counted 431 pairs needing a second reduction out of a much larger population, and the property that separates them from the pairs that reduce in one pass has never been looked for. That is a question about the pair rather than about a side.

Sorted by the two parts’ temperatures, the answer is a shape.

The cross #

Fifteen cells, and a second reduction happens in six of them.

The nine empty cells are the finding as much as the six live ones. A cell of 256 pairs with not a single second reduction in it is not a low rate; it is a rule, and it is what makes the cross a shape rather than a scattering.

It is worth naming the largest of them, because it is the one a reader would least expect to be empty. Two parts of equal temperature, both at a half — 256 pairs, the biggest cell in the table — and the second pass never fires. Two hot games of exactly matched temperature is the configuration where interference should be greatest, and temperatures do not add is the standing reminder that matched temperatures are where the arithmetic is least well behaved. Here they are the safest case there is.

And the same cell one row down, both parts at three quarters, fires 49 times in 100. So equal temperatures is not the variable either: it is empty at a quarter, empty at a half, half full at three quarters, and empty again at one.

Written out, the condition is:

A second reduction is needed only when the two temperatures differ by exactly a quarter, or the colder of them is exactly three quarters.

It is exactly necessary: no pair needing a second reduction fails it, on all 431. It is nowhere near sufficient: 477 pairs satisfy it and reduce in one pass.

What the rung below had #

The rung two below found one clause of this and could not have found the other.

The second pass is never needed when the two parts’ temperatures differ by three quarters or more — 196 pairs, no exception — is true, and it is a consequence of the cross. A gap of three quarters is not a gap of a quarter, and inside this pool a gap that large forces the colder temperature away from three quarters, so both arms of the cross are missed at once.

The clause nothing would have suggested is the other one. A colder temperature of exactly three quarters opens the door at every gap the pool reaches — including a gap of nought, where the two parts have the same temperature and where the whole rest of that column is empty. Three quarters is not an endpoint of the temperature range here, which runs from a quarter to five quarters; it is a value in the middle that behaves unlike its neighbours.

Inside the cells the shares run from 19 per cent to 100. So knowing the two temperatures usually narrows the question and does not settle it — with one exception.

Where the colder temperature is one and the gap a quarter, all sixteen pairs need a second reduction. That is the only cell in the table decided positively by the temperatures, and it is the smallest live cell, which is exactly the caveat a reader should attach to it.

What the cross is worth to a solver #

The one-sided condition is exactly the shape a solver can use, and it is worth pricing rather than only stating.

A program doing arithmetic on reduced forms adds two of them and then has to decide whether the result is reduced. Deciding it properly means computing the direct reduction as well and comparing, which is the whole expense the reduced form exists to avoid — add then reduce again is where that cost is established, and the reduction calls comparison, and comparison is a search.

The condition replaces that with two subtractions on numbers the program already has. Both parts’ temperatures are read off thermographs it computed when it built the forms; the gap is one subtraction and the comparison against three quarters is another. On 941 of the 1,849 pairs the answer is no second pass and it is certain, so the sum can be handed on without a second reduction and without checking.

On the remaining 908 the condition says nothing and the program does the work. So the saving is a little over half the pairs, bought for two comparisons, with no risk of a wrong answer in the direction that matters — the test never says no when the answer is yes.

That asymmetry is what makes a necessary condition worth having and a sufficient one not, here. A test that sometimes wrongly promised no second pass needed would hand a program an unreduced form it believed was reduced, and every later comparison against it would be against the wrong object. A test that sometimes wrongly says check costs a check.

What is left after the temperatures #

The obvious second variable is how large the two forms are, and it behaves the way every second variable on this anchor has behaved.

The share rises from 13 per cent at four options to 50 at eight — a real effect over a factor of four, and at no width does it reach nought or one. So width is a tendency inside a cell, and the pair is still not determined.

That is the second variable this anchor has failed to find, and the two failures have the same shape. The rung below tried eight rules for naming the option the second pass deletes and the best reached 79 per cent; this rung tries the widths and gets a monotone trend that never closes. In both cases a quantity moves in the right direction over the whole range and never becomes a decision.

Why the temperatures should matter at all #

It is worth saying why a temperature is a plausible variable here, because the cross is otherwise a coincidence with a good story.

The reduced canonical form throws away every infinitesimal, which is what what is left when the small change is thrown away establishes. So the reduction’s whole business is with the part of a value that is not small, and the temperature is the measure of how much of a value that is. Two parts whose temperatures are far apart are a hot game and a nearly-cold one, and adding a nearly-cold thing to a hot thing does not create new opportunities for one option to dominate another — how hot a background has to be is where that threshold was first measured on this anchor.

So a large gap means no second pass has a reason. What has no reason yet is the quarter and the three quarters. A gap of exactly a quarter being live while a gap of nought and a gap of a half are not is not the shape any smoothness argument gives, and neither is a single value of the colder temperature standing out from its neighbours on both sides.

The equal-temperature column makes that concrete and is the sharpest single row in the table. At a quarter it is 0 of 121, at a half 0 of 256, at three quarters 49 of 100, at one 0 of 16 and at five quarters 0 of 4. One value out of five, in the middle of the range, with hundreds of pairs on either side of it saying nothing happens.

Both suspicious values are small multiples of the day-three grid. Every temperature in this pool is a multiple of a quarter, because day-three values have stops on a quarter grid, so a gap of a quarter is the smallest non-zero gap available and three quarters is the third of five values. Whether the cross is about those positions in the grid or about the numbers themselves is a question this pool cannot answer, and it is the first thing a day-four sweep would settle.

Two negatives, one shape #

This anchor has now looked for a second variable twice and missed twice, and the two misses are worth setting beside each other because they are the same miss.

The rung below scored eight candidate rules for naming the option the second pass deletes — the hottest, the coldest, the widest, the narrowest and the four extreme stops — and the best reached 79 per cent. This rung sorts one live cell by the total width of the two forms and gets a share climbing from 13 per cent to 50. In both cases a quantity that plainly matters moves monotonically across the whole range and never reaches a decision at either end.

What that pattern usually means is that the quantity is a proxy. A variable that decides nothing but correlates with everything is generally standing in for something it is loosely tied to, and the thing it is tied to is what the anchor is looking for. Width correlates with having more options to dominate one another; temperature correlates with how much of a value survives the infinitesimal being thrown away. Neither is the mechanism.

And the mechanism has to be about a particular option, because that is what the second pass acts on: it deletes at most one option a side, and whether a form needs a second pass is whether such an option exists in the sum. So the right variable is presumably a property of the option lists rather than a summary statistic of the two forms — and every candidate this anchor has tried, on both rungs, has been a summary statistic.

That is a reading of two negative results rather than a new measurement, and it is the most useful thing the two of them say together.

What the solver computed, and how #

Every ordered pair of the first sixty values born by day three, kept when both parts are hot — 1,849 pairs. The pool and the filter are the rung below’s, so the 431 are the same 431 and the two pages are about one population.

For each pair the reduced canonical form is computed two ways: directly, by reducing the sum, and by adding the two parts’ reduced forms and taking the canonical form of that. A second pass is needed exactly when those two disagree, compared by identity rather than by value, so a form that differs only in the order of its options is not counted as a difference.

Each pair is then labelled by the two temperatures, read off thermographs rather than estimated, and sorted into the fifteen cells the colder temperature and the gap between them produce. Every temperature here is an exact dyadic rational and the cell boundaries are exact equalities, which matters — the whole finding is that a gap of exactly a quarter behaves unlike a gap of a half, and a tolerance would merge them.

Two things are asserted rather than reported. The condition must be exactly necessary, with no pair needing a second reduction failing it, and it must not be sufficient — a page about a one-sided rule needs both halves to hold. And one cell must be decided positively, since the page names it.

Where the model stops #

Day three, sixty values, and five temperatures. The whole cross is a statement about a table with fifteen cells, and two of its five rows and two of its five columns hold fewer than a hundred pairs. The colder-is-three-quarters row is 220 pairs in total, so the arm of the cross the rung below could not have guessed rests on a fifth of the population.

And the temperatures come from a grid. Day-three values have stops on a quarter grid, so a gap of a quarter is the smallest gap there is and every temperature in the pool is one of five numbers. A day-four sweep has stops on an eighth grid and would say at once whether the live column is the smallest gap or a gap of a quarter, which are the same thing here and are different things there. That is the single most informative thing that could be done to this page and it is out of reach: day four is over a million pairs and the reduction is not cheap.

The condition is one-sided and is not a rule. Half the pairs it admits do not need a second reduction. Quoted as a cheap test that rules the second pass out it is exact and useful, and quoted as a description of which pairs need one it is wrong nearly half the time, and the difference between those two sentences is the whole of what this rung established.

Normal play throughout, and the reduced canonical form is a normal-play object: both reductions call comparison, and comparison is a search over a difference game.

And the figures cannot show a pair. Six tables of counts describe a population; the object is two game forms, their sum, and the option a second pass removes from it — which the option nothing names draws, one pair at a time. What a reader would want here is one pair from a live cell beside one from a dead cell with the same widths, and the difference between them visible; the finding of this page is that it would not be.

Where the ladder goes next #

The reduced-form anchor has seven rungs: what is left when the small change is thrown away, how hot a background has to be, add then reduce again, the option nothing names, not a domination in that order, the side that empties, and now which pairs it happens to.

The rung above is the grid, and it is the question this page has left standing rather than one it has opened. Every temperature here is a multiple of a quarter and the live column is the gap of a quarter, which is simultaneously the smallest available gap and the number one quarter. Those are the same thing on day three and come apart on day four, where the grid is eighths — so a sweep of day-four pairs, even a sampled one, decides between them with a single column of the table. If the live column moves to a gap of an eighth the cross is about adjacency in the grid and has nothing to do with quarters; if it stays at a quarter, the number is real and the three-quarters row needs explaining alongside it.

Two neighbours are worth the trip. Add then reduce again is where the 431 were counted and where the arithmetic was found to need a second pass at all, and it is the page whose population this one sorts. And not a domination in that order is where the second pass’s shape was pinned down — at most one option a side, a pure deletion four times in five — and reading the two together gives what this anchor now knows: exactly when a second reduction can happen, and exactly what it does when it does, with nothing in between.