The option nothing names

Add then reduce again established what the arithmetic on reduced canonical forms actually is. Adding two reduced forms gives a form that is usually reduced and sometimes is not: on 431 of the 3,600 pairs it swept, the sum has to be reduced a second time, and every one of those pairs has both parts hot.

That page closed on a fork:

The rung above is the 431. They are in hand, each with the option the second pass deleted, and the question is whether that option can be named from the parts … If it can, the second pass becomes a lookup and the arithmetic becomes a formula; if it cannot, the reduced form is an object that adds and does not compose.

It cannot. The negative comes with two positives, and the positives are the useful half.

What the second pass does #

Before asking which option, it is worth being precise about the operation, because it turns out to be much more regular than the failure to describe it suggests.

The second pass deletes at most one option on each side. Two hundred and forty-eight of the 431 lose one Right option, 89 lose one Left, 94 lose one of each, and not one loses two on the same side. On 354 of them the reduction is a pure deletion: every option of the reduced sum was already an option of the unreduced one, so nothing has been rewritten, replaced or reached for.

That is a strong regularity and it says what kind of operation the second pass is. It is not a general reduction that happens to be applied twice; it is the removal of a single dominated move, once per side, of a form that is otherwise already reduced. The rung below asserted that it never widens a form — the reduction that puts options back is where widening is shown to be a real possibility for the ordinary canonical form — and the shape here is stronger than that: it does not merely fail to widen, it narrows by one.

And a test that rules it out #

The second pass is never needed when the two temperatures differ by three quarters or more — 196 pairs, no exception. And it is needed 46 per cent of the time when they differ by exactly a quarter, which is where most of the population sits.

That is a genuinely usable result, and it is cheap in exactly the way the rung below wanted. A player adding two reduced forms already has both temperatures, because a reduced form is what is left after the infinitesimals are thrown away and the temperature is what survives. Comparing two numbers already in hand rules the second pass out on a ninth of the hot pairs outright.

The other reading of the same population says the same thing from the other side: what makes a second pass likely is a hot colder part. Eight per cent at a quarter, 26 at a half, 68 at three quarters. Two parts of similar heat are two parts that genuinely interfere, and interference is what a dominated option in a sum is: an option of one part becomes worthless once the other part is added, because the other part offers something better.

Why interference is the right word #

The mechanism deserves a paragraph on its own, because once it is stated the two tables above stop being coincidences.

A dominated option in a sum is an option Left would never choose, and Left’s options in $G + H$ are $G^L + H$ and $G + H^L$. So a deletion in the second pass says: there is a move in one part that was worth making in that part alone and is not worth making once the other part is on the board. The other part offers a better move, and the two parts are competing for the same turn.

Whether that competition happens depends on how close the two parts are in temperature. If one part is much hotter, every player moves there regardless, and each part’s options survive on their own terms — the hot part’s because they are the ones being taken, the cold part’s because nothing about them has changed. If the two are within a quarter of each other, a move in either is a plausible use of a turn, and a comparison between them is possible for the first time. That is exactly what the gap table shows and it is why the far-apart case is empty rather than merely rare.

It also explains the second table. A pair with a hot colder part is a pair where both parts are worth moving in absolutely, not just relatively, so there is more for the comparison to bite on. Eight per cent at a colder temperature of a quarter and 68 at three quarters is the same effect measured in absolute terms rather than in difference.

None of that says which option. It says which pairs, and it does so with a mechanism rather than a fitted curve, which is why the negative result about the option is not the whole of the page.

The option itself #

Now the question the rung below asked. Given the sum’s option list and the option about to go, can the option be named?

Eight candidates were scored, each a rule that picks one option out of a list: the hottest, the coldest, the one with the most options of its own, the one with the fewest, and the four extreme stops. The best of them — the option with the largest left stop — names the right option on 354 of 448 deletions, which is 79 per cent. The worst reaches 7.

Four in five is close enough to be tantalising and it is not a rule, and the reason it is not is worth more than the number.

The census’s pool is the first sixty day-three values in the enumeration’s order, and it is not closed under negation. That matters because a rule about Left options and its mirror about Right options are then scored on different populations — the pool contains a value and not its negative, so it has more of one kind of asymmetry in it than the other.

Re-scored on a pool built to contain every value’s negative, the option with the largest left stop falls from 79 per cent to 44, and the ranking of the eight rules changes. So the leading rule was not a rule with exceptions; it was a fact about which values happened to be in the pool. The rule that survives best under the control — the option with the fewest options of its own — sits at 72 and 71 per cent, which is a stable tendency and still not a rule.

What the two horns mean #

The rung below stated the fork sharply and it is worth taking it at its word: if it cannot be named, the reduced form is an object that adds and does not compose.

That is where this leaves it, and the distinction is real. Adding means the operation is well defined on reduced forms — take two, add, reduce, and the answer does not depend on which representatives it started from. The rung below established that on all 3,600 pairs. Composing would mean the answer can be assembled from the parts by an operation on their descriptions: a formula, a lookup, a rule that says which option goes.

The reduced form does the first and not the second. So it is a quotient that carries an arithmetic and not a notation that carries one — the difference between a group and a numeral system. That is a familiar shape rather than a disappointing one: when two thermographs can be added found the same thing one summary coarser, where the diagram of a sum can be built from its parts’ diagrams only where nothing is at stake, and the reduced form is the object built to be better than a thermograph.

What is genuinely new here is that the failure is local. A second pass that could remove any number of options in any pattern would say the operation is unstructured; one that removes one option a side, on a fifth of the hot pairs, never when the temperatures are far apart, and by a mechanism whose likelihood climbs smoothly with the colder temperature, is an operation with a great deal of structure and one unnamed step.

What a control is for #

The control pool is the piece of method worth carrying away from this page, because it caught something no amount of care about the rule itself would have.

The option with the largest left stop scores 79 per cent, and there is a perfectly good story to tell about why it should: a Left option with a large left stop is one Left is likely to prefer, so it is a candidate for being the survivor rather than the casualty — and it is the Right options being deleted most often, where the same reading runs the other way. The story is coherent and the number supports it, and both are artefacts.

They are artefacts because the pool is not closed under negation. Negating every value in a game turns Left into Right and left stops into right stops, so a pool without its own mirror image measures a Left-side rule and a Right-side rule on populations of different shapes. Scoring the same eight rules on a pool built to contain every negative drops the leader to 44 per cent and promotes a rule that has nothing to do with stops.

The general form is one this site keeps meeting: a rule scored on a pool that is not symmetric under an operation the subject is symmetric under is measuring the pool. The margin a count needs has the same hazard on a different anchor and avoids it by construction, since a Domineering board and its transpose are both in the census. Here the pool came from an enumeration whose order has no reason to respect negation, and that was enough.

Why a pool closed under negation is the right control #

The score falling from four in five to under a half when the pool is closed under negation is the finding, and the reason that particular closure is the right test is worth stating, because it is the cheapest control available for a rule of this kind.

Every rule on this page is stated for one player. It names an option, and naming is not symmetric: a rule about Left’s options in a sum has a mirror image about Right’s, and the two are the same rule seen from opposite sides. So a rule that scores well ought to score identically on a pool and on that pool’s negation — anything else is the rule preferring one player, and every game has a negative is what makes the closed pool cheap to build.

A pool that is not closed under negation cannot detect that preference. If the pool contains more positions of one shape than its mirror, a rule biased toward that shape scores well, and the score is measuring the pool’s lopsidedness rather than the rule’s accuracy.

Closing the pool under negation costs nothing — negate every member and add it — and it removes the whole class of asymmetric artefacts at once. A rule that survives it is measuring something about games; a rule that halves under it was measuring the sample.

That is a control worth applying to every rule on this site that names a side. It is cheaper than widening a pool, it tests a property the mathematics guarantees rather than one anybody has to guess at, and — as here — it can turn an encouraging near-miss into a refutation in one run.

What this does not say #

Eight rules are not the space of rules. Every candidate here picks an option by a property of that option alone. A rule that compared an option to its siblings — the option some other option dominates in the sum but not in the part — is a different kind of object and is not tested; nor is anything referring to the two parts’ own option lists rather than the sum’s.

The 79 per cent is a real tendency and the control is what it is. The control pool is 79 values and the census’s is 60, and they are not the same population in any other respect either. What the control establishes is that the leader is unstable, not that a rule scoring 79 per cent on a fair pool would be uninteresting.

Only pure deletions are scored. The 77 pairs where the reduced sum contains an option the unreduced one did not are excluded from the rule scoring, because there is no deleted option to name on them. Whatever happens there is a different operation and it has not been described.

The infinitesimal difference is the rung below’s finding, not this page’s. Every second pass removes an option whose absence changes the value by an infinitesimal only, so nothing a number could see has moved — which is what makes the reduced form a legitimate object rather than a lossy one. That is asserted on every pair by the rung below’s own census and is repeated here without being re-derived. See what an infinitesimal does to a fight for what is being thrown away.

And the cheap test is one-sided. Temperatures three quarters apart implies no second pass is what the census shows; the converse is far from true, since 90 per cent of the pairs at a gap of nought need no second pass either. As a test it decides a ninth of the population and says nothing about the rest.

The convention, named #

Normal play throughout, and every form computed by the recursion.

A reduced canonical form is the canonical form with one clause added: an option is also deleted when it differs from another by an infinitesimal. It is what a position is worth once the small change is thrown away, and what is left when the small change is thrown away is where it is set out.

The census is the first sixty day-three values against themselves, ordered pairs, which is 3,600 sums and is the rung below’s population unchanged. Only the 1,849 with both parts hot can need a second pass, which the rung below established and this page assumes.

A pair needs a second pass when the canonical form of the sum of the two reduced forms is not the reduced form of the sum. A pure deletion is one where every option of the reduced sum was already an option of the unreduced one.

Temperature is the height at which a thermograph’s walls meet, and a number is given a temperature of $-1$ rather than nought so that a cold part is excluded rather than counted at the bottom of the scale. The gap and the smaller temperature are both computed over the two parts as they arrive, before any reduction.

Where the ladder goes next #

The reduced-form anchor has four rungs to here, and this one has scored eight rules for naming the deleted option and found the best of them a property of the population rather than of the reduction.

The rung above overturns the assumption underneath all eight. Not a domination in that order points out that every candidate rule here takes for granted that the second pass performs an ordinary domination — deleting an option because a survivor is at least as good — and on none of the 525 deletions is that true.

Read in the order the reduced form actually works in, where one option beats another when both its stops are at least as good, everything changes: every deletion with a survivor is a domination, the dominator is unique on all but twelve, and it always comes from the other part.

That last clause explains this page’s failure completely. Eight rules were scored on the parts, one at a time, looking for a property of the deleted option — and the thing that deletes it is in the other component. No rule reading one part could have named it, and the four-in-five score was the population supplying a correlation rather than the rule supplying an answer.

So the search was in the wrong order and looking in the wrong place, and the two errors are one: taking the operation to be a domination in the value order fixed both the relation to look for and the place to look for it.