No fifth value

The ordinal sum $G : H$ — play in $H$ while $G$ is untouched, and a move in $G$ wipes $H$ out — is famously an operation on forms rather than on values. When the nested sum only sees the value shows it failing: $0$, $\{\ast \mid \ast\}$ and $\{ \mid \ast\}$ are all worth nought and placed under a star they give $\ast$, $\ast 2$ and $\downarrow\!\ast$.

What the colon respects measured how often it fails, over every form whose two option lists are antichains of day-two values — 9,604 of them — and found something much narrower than the standard warning suggests: the forms of a value all give the same ordinal sum except at four values, and those four are nought, one, minus one and star, which are the values born by day one.

It closed on the obvious limitation:

Everything here is forms whose options are day-two values, which is depth two; the claim that the exception class is exactly day one is checked on those and on nothing deeper. A sweep over forms with day-three options would either confirm the class or find a fifth value.

There is no fifth value, and the reason turns out not to be about depth at all.

Building the forms #

A sweep over antichains of day three is not affordable. Day two has 22 values and its antichains are countable by machine; day three has 1,474, and the antichains of that are past anything a build could enumerate.

So the forms are built the other way round, by gift horses. Take a value’s canonical form and add one option that leaves the value unchanged: a Left option no greater than the value, or a Right option no smaller. That is the construction an option nobody would take is about, and it produces a different form of the same value with a day-three option in it — which is exactly the object the rung below could not reach.

One hundred day-three values, each given every gift horse from a pool of eighty day-three values that leaves it unchanged. Four followers — star, one, minus one and up. Eighteen thousand five hundred and sixty-four forms.

Every one is checked to be worth what it started as, by comparison rather than by construction, so a gift horse that turned out not to be one is not counted. That check is not a formality: a value no greater than the base is a gift horse by definition, and the definition is a comparison, so the test and the construction are the same operation run twice.

The result, and the control #

Not one of the hundred values has two forms giving different ordinal sums. Under any of the four followers.

An empty table is a weak finding on its own, because it is what a broken experiment produces. So the same construction is run on the four values the rung below identified.

They still disagree. Nought’s twenty forms give four different ordinal sums under the follower one and three under minus one, and minus one’s fifty-seven forms give two under minus one. The construction separates what it is supposed to separate, and it separates nothing among the day-three values.

The numbers are worth reading in the right order. The four controls have 170 forms between them and produce as many as four distinct sums from one value; the hundred day-three values have 18,564 forms between them and produce exactly one from every value. A hundred times the material and none of the effect.

That contrast is the finding. Going one day deeper does not widen the class; it narrows it, because a deeper value’s form is wide enough that a gift horse cannot reach the colon at all.

Why those four #

The mechanism is visible in the definition once the right thing is looked at.

$G : H$ has Left options $G^L$ together with $G : H^L$ — the base’s own options, unchanged, plus the ordinal sums with the follower’s options. So a gift horse added to the base becomes an option of the ordinal sum directly.

Whether it survives there is the question, and it survives only if nothing in the sum dominates it.

Nought is $\{ \mid \}$ — no options at all. One is $\{0 \mid \}$, minus one is $\{ \mid 0\}$, star is $\{0 \mid 0\}$. Every option any of them has is the zero game. A gift horse added to such a form is a genuinely new option of the ordinal sum, sitting beside a nought and the terms $G : H^L$, and it is not dominated by either.

A value with a richer form is different. Its options are already spread through the order, an added option is by construction no better than the value, and something in the existing list is at least as good — so the reduction deletes it and the ordinal sum never sees it.

So the exception class is about the width of the base’s canonical form, not about the depth of anything. That is why it does not grow with the day: the day-three values all have forms with something in them.

The word width is doing precise work there and it is worth pinning down. It is not that the four have few options — star has two, and plenty of day-three values have two. It is that every option they have is worth nought, so the order they span is a point. A form whose options span an interval has somewhere for a gift horse to be dominated; a form whose options are all one value has nowhere.

That also predicts what a fifth exception would have to look like: a value whose canonical options are all equal to each other and to nothing useful. There is no such value past day one, because a canonical form with two equal options has one of them deleted — which is the dominance reduction, and it is why the class cannot grow rather than merely happening not to.

The failure, drawn #

It is worth seeing the operation fail once, on the smallest case, before reading a table that says it mostly does not.

Four forms worth nought, one follower, four different ordinal sums. Nothing about that is subtle and nothing about it is rare at that base: nought has twenty gift horses in the pool used here and they produce four different answers between them.

The rung below’s contribution was to show that this is nearly the only place it happens. This page’s is to show that going deeper does not find more of them — and the two together turn a demonstration into a boundary.

There is a temptation, on seeing a failure this clean, to treat the operation as generally untrustworthy. The measurements say the opposite: it is trustworthy everywhere except at four bases, and those four are identifiable by inspection.

What that changes about the warning #

The standard statement is that the ordinal sum reads the form and equal values may not be substituted into it. That statement is true and this ladder has now measured what it is worth three times.

The rung below found the failure confined to four values out of 640. This page finds those four to be exactly the values with nothing in their forms, and finds the confinement stable as the forms get deeper.

So the practical rule is short. Substitute equal values into a colon freely, except under a base worth nought, one, minus one or star — and under those four, do not substitute at all.

That is a very much more usable statement than the operation reads the form, and it is the difference between a warning and a rule. A warning tells a reader to be careful everywhere; a rule tells them where.

Where it is used, and why it matters there #

The place the colon earns its keep on this site is Hackenbush, where a tree is reduced one branch at a time and each reduction is an ordinal sum of a trunk with a branch. The reduction is legitimate because the colon principle holds — for impartial games it reads the value after all — and for partizan trees it is legitimate branch by branch because the trunk is not one of the four.

A trunk worth nought is exactly the case the rule forbids, and it is not a rare case: a trunk of one blue edge and one red edge is worth nought, and a tree with that trunk cannot have its branches substituted. That is a real restriction on a real construction, and until this ladder measured the class it was buried inside a general warning that made every trunk look equally dangerous.

A different way to reach the same class #

There is a second reading of why those four, and it lines up with something the site has found elsewhere.

The four are the values whose canonical forms have no option worth anything — every option is the zero game. Equivalently, they are the values reached before any real option exists to be dominated by.

That is the same shape as the birthday rule for self-negative values, where being constrained turns out to be a property of how early a value arrives rather than of what it is. Here the constraint runs the other way: it is the unconstrained values — the early ones, with room in their forms — that misbehave, and the deeper ones that are safe.

Both are statements that a value’s day decides a structural property, and both were found by asking a question that looked like it was about something else. The colon question looked like it was about depth of options; it is about the base’s own day.

What the census does not say #

Four limits.

The argument in the last section is stronger than the census. If a canonical form with two equal options has one of them deleted, then no value past day one can be in the class, and the sweep is confirming a fact rather than discovering one. That is the better position to be in and it is worth saying which way round it happened: the census was run first, the empty table asked for an explanation, and the explanation turned out to close the question. A reader is entitled to know that the sentence came after the table.

A hundred values, sampled. The day-three values are sampled at random from the 1,474 rather than swept, so a value with an unusual form could have been missed. The finding is that no sampled value disagrees, and the sample is a fifteenth of the day.

Gift horses are one construction. Adding a single dominated option is not the only way to write a value differently — bypassing a reversible option produces forms too, and that is the other half of the canonical form. A form reached by bypassing rather than by adding is not tested here, and the rung below’s antichain sweep reached some of those and this construction does not.

Four followers. Star, one, minus one and up, chosen because the rung below found them to be the ones that separate anything. A follower with a richer form of its own would give a different set of $G : H^L$ terms for a gift horse to be dominated by, and it might give a different answer.

And the mechanism is a sketch. An added option survives only if nothing dominates it is a sentence about the reduction and the argument that a rich form always dominates a gift horse is not written out. What the census establishes is that nothing in eighteen thousand forms contradicts it.

A negative result whose value is in the design #

Eighteen thousand forms and no disagreement is a null result, and null results are worth exactly as much as the design that produced them. It is worth saying what this one was built to be able to find.

A sweep that finds nothing is uninformative if it could not have found anything. The obvious failure mode here would be a pool whose forms are all narrow — narrow forms are where the exceptions live, so a sweep over wide ones would report no disagreement and be measuring its own pool.

The design guards against that by keeping the control in the same run. The same treatment applied to nought still splits it four ways, which proves the machinery can detect a disagreement when there is one to detect. Without that line the result would be no disagreements found, which is compatible with the test being broken; with it, the result is disagreements found where they are expected and nowhere else.

The second thing the design has to get right is the axis it widens along. This page’s forms are a day deeper in their options, which is what the rung below’s limitation was about — it could only reach day-two options — and the null result is therefore a statement about depth specifically.

So what is established is narrow and clean: depth of options is not what creates the exceptions. That leaves width, and the rung below’s four exceptions are also the four narrowest values, so the two candidate explanations were indistinguishable there and are now separated by one of them being ruled out.

A null result that eliminates one of two hypotheses is worth as much as a positive one that confirms the other, and it is cheaper to obtain — which is why the sweep was worth eighteen thousand canonicalisations.

What eighteen thousand forms cost #

The construction is worth a note, because the reason it exists is that the obvious sweep does not.

Day two has 22 values; its antichains number in the hundreds and every pair of them is a form. Day three has 1,474 values, and the antichains of a 1,474-element partial order are past counting, let alone enumerating.

So the question had to be reshaped rather than scaled. A gift horse gives one extra form per value per horse, which is linear in both, and eighteen thousand of them are eight seconds of a build where the direct sweep is not finite in any practical sense.

What is lost by reshaping is coverage: the antichain sweep reaches every form of a value with day-two options and this construction reaches only the forms one gift horse away. What is gained is depth. Neither is a superset of the other, and the two together are the evidence — which is why the rung below’s result is quoted here rather than re-derived.

The convention, named #

Normal play, and all values are canonical forms computed by the recursion.

The ordinal sum $G : H$ is defined by its options: $G : H$ has Left options $\{G^L\} \cup \{G : H^L\}$ and Right options $\{G^R\} \cup \{G : H^R\}$. The base is $G$ and the follower is $H$, and it is the base whose form is read.

A gift horse for Left is an option $h$ with $h \le G$, added to Left’s list; for Right, an $h$ with $h \ge G$. Either leaves the value unchanged, which the census checks rather than assumes.

Born by day one means nought, one, minus one and star: the four values whose canonical options are all the zero game.

Where the ladder goes next #

The ordinal-sum anchor has four rungs: the operation and its Hackenbush reading, that it does not respect equality, how much of equality it respects anyway, and now that the exception class is stable under depth and is about width.

The rung above is the theorem. A gift horse added to a form with a non-nought option is dominated in the ordinal sum is a statement with a proof-shaped argument behind it and eighteen thousand confirmations, and turning it into an induction would replace the whole ladder’s measurements with a sentence. The shape it would take is visible: show that the base’s existing options are at least as good as any gift horse in the sum as well as in the base, which is where the ordinal sum’s own structure has to be used.

Two neighbours are worth the trip. What the colon respects is the rung below, where the exception class is found; this page is the class shown not to grow. And Hackenbush is a numeral is where the operation earns its keep, and where the four values this page forbids are trunks somebody might actually draw.