Not a domination, in that order

The option nothing names scored eight rules for naming the option a second reduction deletes, got the best of them to 79 per cent, and closed by naming the object it had not looked at:

The rung above is the sibling rule. Every candidate here names an option by a property of that option, and the operation is a domination — one option going because another is at least as good — so the natural object is a pair, not an option.

The pair is the right object. The premise about it is wrong, and correcting the premise turns the near-miss into an exact account.

It is never a domination #

Take every one of the 525 single-option deletions the second pass makes across the 431 pairs it acts on, and ask whether any surviving option of the reduced sum is greater than or equal to the deleted one.

Not one. Zero of 525, on either side, on every pair.

That is not a near-miss to be explained; it is the relation the rung below assumed being absent from every case. The candidate rules it scored were looking for the better half of a pair that does not exist.

Many forms one value is where the ordinary reduction’s domination is set out, and the contrast is exact: there, an option goes because another is at least as good, and the relation is present on every deletion by construction. Here the same word is being used for something else.

It is a domination in the stop order #

Replace greater than or equal by both stops at least as good and the account becomes complete.

On all 448 deletions that leave a survivor, some survivor’s two stops dominate the deleted option’s — its left stop is at least as large and its right stop is at least as large. And the dominator is unique on 436 of the 448, so the object really is a pair rather than a set.

The other 77 deletions leave no survivor at all: they remove the last option on their side, so the reduced sum has no move for that player where the unreduced one had one. There is no pair in those, and nothing to name.

That is the whole census — 448 and 77, with no residue.

Why the ordinary order cannot see it #

The correction is not an accident of which order was tried; it is the definition of the object.

The reduced canonical form compares modulo infinitesimals: it is entitled to remove an option that another beats up to something smaller than every positive number. Two positions differing by an infinitesimal have exactly the same two stops and are incomparable in the ordinary partial order — neither is at least the other, because the difference is fuzzy with nought.

So the deletions are precisely the cases the ordinary order cannot see. A pair the ordinary order compared would already have been removed by the first reduction, inside the parts; what reaches the second pass is a pair the ordinary order calls incomparable and the stops call ordered.

Read that way, zero of 525 is not a surprising measurement. It is what the reduction being a second pass means: everything the ordinary order can settle has been settled, and what is left is exactly what it cannot. The measurement’s value is that it turns that from a plausible sentence into a count: not mostly, not usually, but nought of five hundred and twenty-five.

Always the other part #

The rung below’s second question was whether the surviving option is the one that came from the same part. It is not, and it cannot be.

On the 188 deletions whose provenance can be recovered, the dominator comes from the other part of the sum — 188 of 188, none from the same one.

The reason is a one-line argument rather than a measurement. Both parts are already reduced, so no option of a part beats another option of the same part, in any order the reduction uses. A dominator inside a part would mean the part was not reduced, which is a contradiction and not a finding.

So the pair the second pass acts on is always one option from each side of the sum, and that is what makes it a second pass rather than a repeat of the first. The first reduction can only compare options of one part; the second is the first chance the two parts’ options have to be compared with each other.

The other 260 are not counterexamples. An option of the sum is a canonical form, and recovering which part it came from means matching it against every option of both — which fails when two different constructions give the same game. The source column is a sample, and the claim about it is a claim about the sample rather than about the census. What would close it is a construction that carries provenance through the canonicalisation, which is a change to how the sum is built rather than a wider sweep.

Where nothing survives #

The 77 deserve a paragraph because they are the part no sibling rule will ever describe.

On them the second pass removes the only option a player had on that side. The reduced sum is then an end for that player where the unreduced one was not — a much larger change than deleting one move among several, and one a rule of the form this option goes because that one is better cannot express at all.

They are 15 per cent of the deletions, and the rung below’s eight candidate rules were scored on the ones with a single deletion and a survivor, so they never saw these. That is not an error in that page; it is a class its instrument could not reach, and it is where the second pass does its largest work. A player who has lost their only move in a component has lost more than a move; they have lost the option of playing there at all, which changes the sum’s whole outcome rather than its value by a little. The reduction that always shrinks is the standing fact that a reduction never adds an option, and a side emptying is that fact taken as far as it goes.

Why the rung below got so close #

Seventy-nine per cent is a suspicious number for a rule that is looking for a relation that never occurs, and the explanation is worth having because it is a general trap.

The rung below’s eight candidates named the deleted option by a property of the option itself — the hottest, the coldest, the widest, the narrowest, and the four extreme stops. Four of the eight are stop properties, and the relation that is actually doing the work is a stop relation. So the best candidates were reading the right quantity through the wrong lens: the option with the lowest left stop is often the stop-dominated one, because being dominated means having low stops.

That is why the best rule reached 79 per cent rather than 40. A rule correlated with the truth by construction is not a near-miss; it is a shadow of the truth cast on a smaller object. And the diagnostic the rung below reported — that a negation-closed control pool ranks the candidates differently — is exactly what a shadow does: the correlation between lowest stop and stop-dominated depends on the population, and the relation does not.

The general form is one this site meets often enough to name. A rule that scores well and cannot be made exact is often a projection of a relation onto one of its arguments, and the fix is to look for the relation rather than to add terms to the rule. The option nothing names added terms to the rule and got 79 per cent; asking what the pair is gets 448 of 448.

What this leaves the arithmetic #

The rung below’s closing dichotomy was that if the deleted option can be named, the second pass becomes a lookup and the arithmetic a formula; if it cannot, the reduced form adds and does not compose.

Neither is quite where this lands, and the middle is worth stating.

The relation is named. Delete an option whose stops are dominated by an option from the other part is exact on 448 of the 525 and is a rule about a pair, which is what was asked for.

The rule is not a shortcut. Applying it means computing the stops of every option of the sum, which is a recursion over the same tree the second reduction walks. So the arithmetic does not become a formula; what it becomes is describable, which is a different and smaller thing.

And 15 per cent of it is not a domination at all. The sides that empty are a second operation living inside the same pass, and a complete account needs both.

What is left when the small change is thrown away is where the reduced form is introduced and where the modulo-infinitesimals comparison is defined; reading it beside this page is the clearest statement of why an operation defined in one order looks like nothing at all when scored in another.

Two orders, and what each is for #

Standing back, this page is about a site-wide habit as much as about a reduction, and it is worth naming.

The subject has several orders on games and they are not interchangeable. The partial order — $G \ge H$ when $G$ is at least $H$ against every opponent — is the one comparison is a search is about, and it is the one every reduction on this site has used up to here. The stop order compares two numbers per game and is much coarser: it cannot tell two positions an infinitesimal apart from each other, and it orders pairs the partial order refuses.

Neither is more correct. What matters is which one an operation was defined in, and the reduced canonical form was defined in the second — it exists precisely to throw the infinitesimals away. Scoring its behaviour in the first is like measuring a temperature in the wrong units and reporting that the thermometer is broken.

The rung below made that substitution without noticing, and the site’s own vocabulary made it easy: dominated option is a term of art that means one thing in the canonical form and a coarser thing in the reduced one, and the word does not change. The reduction that puts options back is where the two reductions’ differences were first laid out, and it is the page that should have made this substitution visible.

The lesson generalises past this ladder and is short: when an operation is defined modulo something, every question about it has to be asked modulo that something. Otherwise the answer comes back as a count of nought, which is what happened here.

Two orders, and which one a reduction belongs to #

The correction here is that the operation is a domination in a different order, and it is worth setting the two orders side by side, because the whole anchor was searching in the wrong one.

The order on values is the subject’s standard relation: $G \ge H$ when $G - H$ is a second-player win. It is a partial order, it is decided by a search, and it is what dominated option means everywhere else on this site.

The order on stops is coarser and cheaper: one option is at least as good as another when both its stops are. It is a comparison of two pairs of numbers, it needs no search once the stops are known, and it is implied by the value order without implying it.

The reduced canonical form is defined by throwing away everything below every number, which is exactly the information the value order has and the stop order does not. So the reduced form’s natural comparison is the stop order, and looking for its deletions in the value order is looking in a relation the construction never uses.

That is why the search failed as completely as it did — no surviving option is ever value-greater than a deleted one, so the answer was not a rare case or a near-miss but a category error. And why the correct order works so cleanly: unique dominators on all but twelve deletions, every one of them from the other part.

The general instruction is to reduce in the order the reduction was defined in. A construction that discards a distinction has a coarser relation attached to it, and its operations are operations in that relation, whatever the ambient theory’s relation happens to be.

What this does not say #

Sixty values, and day three. The pool is the rung below’s — the first sixty day-three values, giving 431 pairs needing a second pass out of 3,600 — and every count here is over that.

Single deletions only. A pair where two options go from one side is not scored, because the pair the rung below asked about is a pair. The rung below established that at most one option goes from each side, so this is a restriction of the counting rather than of the population.

The stop order is not the reduction’s own order. Both stops at least as good is a test this page applies; the reduced canonical form is defined by comparison modulo infinitesimals, and the two agree on this census without that being checked in general. A pair that the stops ordered and the reduction did not would be a counterexample and none appeared.

And the provenance claim is a sample. A hundred and eighty-eight of 448, with the other 260 unmatched for a reason that has nothing to do with which part they came from — so the sample is not obviously biased, and is not obviously unbiased either.

The convention, named #

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

The canonical form of a game is what is left after removing dominated options and bypassing reversible ones. The reduced canonical form goes further: it compares modulo infinitesimals, so an option beaten by another up to something smaller than every positive number is removed too.

A position’s stops are what each player gets moving first and playing on until somebody faces a number. Two positions differing by an infinitesimal have the same stops, which is why the stop order sees comparisons the ordinary order calls incomparable.

The second pass is what the sum of two reduced forms needs when it is not already reduced: rcf(G) + rcf(H) against rcf(G + H). A deletion is an option of the first that is not an option of the second.

An option of a sum comes from a part when it is that part’s option added to the whole of the other. Recovering which is a matching against every option of both parts, and it fails when two constructions give one game.

The population is the first sixty day-three values, both parts hot — 431 pairs needing a second pass, carrying 525 single deletions between them.

Where the ladder goes next #

The reduced-form anchor has five 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, and now the pair it goes with.

The rung above is the seventy-seven. They are the deletions that empty a side, they are a fifth of the second pass’s work, and no rule on this ladder has looked at one — the rung below’s candidates were scored on the others by construction. A side emptying is a much stronger event than an option going, so the question is whether the pairs it happens to can be recognised from the parts: whether a player about to add two reduced forms can tell, from the two forms alone, that one of them is about to lose a move entirely.

Two neighbours are worth the trip. Add then reduce again is where the second pass was found and counted, and it is the page whose 431 everything here is over. And which option the reduction keeps is the same question asked of the first reduction, where the answer is exact and the order is the ordinary one — which is the contrast that makes this page’s correction worth a rung. How hot a background has to be is where the reduced form’s own justification lives, and it is worth reading beside a page about the order it works in.