The wild side does not close

A function with no formula established that the genus of a sum of two wild heaps is determined by their two symbols — the composition is a function on the wild side as well as the tame one — and that no rule of the tame rule’s shape describes it. It closed by naming what to build:

The rung above is the table. If the composition is a function on the wild side and no rule describes it, the thing to build is the table — sixteen files, each with its answer — and then to ask what the table looks like: whether the wild symbols form a small closed set under the operation, and whether that set is a misère quotient in disguise.

The table is built, and both questions come out no.

Building it took a wider sweep #

The rung below swept heaps to nine counters, which across nine rulesets gives six wild heaps and five files — and four of those five are a symbol added to itself. That is a diagonal, not a table, and a function established on a diagonal is barely established at all: it says a symbol plus itself has one answer, which is nearly a tautology once the base is forced.

Sweeping to fourteen gives 23 wild heaps carrying ten distinct symbols, 59 wild pairs and 22 files, twelve of them holding two different symbols. That is a table.

Wildness is something a heap grows into. Not one wild heap in the sweep is smaller than five counters, and the great majority are above eight — which is exactly why the rung below’s sweep produced a diagonal and why the sweep had to be widened before the question could be asked. Kayles’ first wild heap is at five and its next at nine; ·6 has six wild heaps and every one is at seven or above.

The composition is still a function #

Twenty-two files over 59 pairs, and not one file holds two answers. That is the rung below’s finding confirmed on six times its data, and the confirmation matters more than it sounds: its own version rested on five files of which four were degenerate, so the claim that the composition is a function on the wild side was, until now, supported by one genuine instance.

The base of every answer is forced — normal-play Grundy values add by exclusive or, and the base of a genus symbol is the normal-play value — so what the table is actually about is the superscript, and the superscripts are where the irregularity lives. $3^{1431} + 3^{1431} = 0^{120}$ is a pair of hot-looking symbols giving something that looks like nothing; $3^{1431} + 4^{0564} = 7^{058\,75}$ is two symbols giving a longer one than either.

They are not closed #

Twelve distinct answers appear in the table, and none of them is a symbol any wild heap carries. Five are symbols carried by tame heaps; the other seven are symbols nothing in the sweep carries at all.

So the ten wild symbols do not form a closed set under addition. They do not nearly form one — the operation leaves the set on every single entry of the table, without exception. Whatever the wild side of the genus is, it is not a small algebra that multiplies among itself.

That is the strongest form the answer could have taken, and it is worth noticing which way it goes. A set that closed would have been an object with an arithmetic, and the ladder would have had somewhere to go. A set that leaves itself every time is a set that was never a self-contained thing: the wild symbols are what certain heaps happen to carry, not a subsystem.

And the second question falls out of the first. A misère quotient is a monoid — a set closed under the operation, with an identity — built by closing a family of positions under addition and reading off what is distinguishable. Closure is where the construction starts. The wild genus symbols are not closed, so they are not a quotient, and they are not a fragment of one that stops where they stop.

Wild plus wild is usually tame #

The most surprising number on the page is a different one. Of the 59 wild pairs, 39 are tame — 66 per cent. Adding two wild heaps gives a position that behaves like a Nim heap under misère play more often than it gives another wild one.

That is the opposite of how the tame side behaves. Tame and wild establishes that tameness is closed under addition: a sum of tame heaps is tame, always, and it is what makes the tame half of the theory usable — a player who has checked each part has checked the whole. On the wild side the property is not merely unclosed; it is lost more often than it is kept.

Together with the closure result that gives a fairly complete picture of what the wild side is. It is not a set with an arithmetic and it is not even a stable property: two positions that are individually intractable become, two times in three, a position the ordinary theory handles.

What a symbol that nothing carries means #

Seven of the twelve answers are symbols no single heap in the sweep carries, and it is worth being clear about what kind of object those are.

A genus symbol is not a name chosen from a list; it is computed from a position, and any position has one. So a symbol appearing only as a sum is not exotic — it simply means that among the 126 heaps swept, none happens to have it. Widen the sweep and some of them would presumably turn up.

But that is the point. The wild symbols were being asked whether they form a closed system, and the answer is that their sums keep landing on symbols the system does not contain — which is what not closed means and is not repaired by finding those symbols elsewhere. A system closed only after everything else is added to it is the set of all symbols, which is not a finding.

The five answers that are tame symbols are the more interesting group. A sum of two wild heaps carrying a tame symbol is a position the ordinary theory can handle, arrived at from two positions it cannot — and what a tame heap may be replaced by is where the licence that makes such a position usable is set out. So the wild side is not merely leaking; it is leaking back into the part of the theory that works.

What the two halves are doing #

It is worth putting the tame and wild sides side by side, because the contrast is sharper than either half alone.

The tame side is a subsystem. Tameness is closed under addition; the symbols compose by a rule stated in two cases; the composition is a function of the two symbols; and the whole thing is an algebra a player can carry. That is the rule the symbols follow, and it is what the genus was built to be.

The wild side is a residue. The symbols do not close, the composition is a function with no rule, and the property that defines them is lost by addition two times in three. Nothing about it is a system; it is what is left over when the system’s hypothesis fails.

That asymmetry is the honest reading of the whole genus anchor. The genus is not a theory with a tame case and a wild case. It is a theory with a domain, and outside the domain the symbols keep being computable and stop meaning anything the theory can use — which is exactly the situation misère quotients were invented to replace, and this page is the clearest measurement of why.

What the diagonal was hiding #

The sweep-widening deserves a paragraph of its own, because the difference between the two tables is a lesson about what a census has to contain before a question can be put to it.

At nine counters the table has five files. Four of them are $X + X$ for a single symbol $X$, and one is a genuine pair. A function on such a table is almost forced: $X + X$ has one answer because there is one pair of heaps producing it, and a file with one member cannot be ambiguous.

At fourteen the table has 22 files, twelve of them with two different symbols, and several files holding many pairs — $3^{1431} + 3^{1431}$ is fourteen different pairs of heaps, from four different rulesets, all giving the same answer. That is a function being tested: fourteen chances to disagree and no disagreement.

So the rung below’s claim was true and had almost nothing behind it, and this page’s contribution to that claim is worth as much as its two negative answers. The general form is one this site keeps meeting from different directions — the entry fee was the cap records the same shape on a different anchor, where a quantity measured at the edge of a sweep was read as a property of the thing swept. Here the edge of the sweep was hiding not a wrong number but an untested claim.

What this does not say #

Two thirds is a share of a small population. Thirty-nine of 59 is a clear majority and it is 59 pairs, drawn unevenly from six rulesets — ·6 contributes six wild heaps and ·007 contributes none. A different set of rulesets would move the share, and what it would not move is the direction: a sum of two wild heaps is often tame, which is enough to say wildness is not preserved.

Nine rulesets and heaps to fourteen. The table has 22 files and the whole wild population is 23 heaps. Every count here is small, and a wider sweep would add files — the point is what the table’s answers are, not how many there are, and the closure result would survive any number of new entries that behaved like these.

Pairs only. Every entry is a sum of two heaps. Whether the closure of the wild symbols under repeated addition is finite is a different question and it needs three-heap sums, which are out of reach at these heap sizes: a sum of three fourteen-counter heaps is a much larger computation than a sum of two.

The symbols are computed to six places of superscript. A genus symbol has an infinite tail in principle, and six is where the rest of this site truncates. Two symbols agreeing to six places are treated as equal, and a difference beyond the sixth would split a file that this page reports as unambiguous.

The closure result is about this table. Twelve answers, none of them a wild symbol, over 22 files — the claim is that the set of ten wild symbols is not closed, and one entry landing inside the set would not have overturned it either. What would overturn it is a sweep in which the answers systematically stayed inside, and nothing here suggests one exists.

And within one ruleset only. Every pair here is two heaps of the same game. A sum of a Kayles heap and a Dawson’s chess heap is a perfectly good position and the machinery does not reach it, which is a real restriction: the table is nine small tables rather than one.

What a class has to do to be worth naming #

Wildness fails a test that a useful class passes, and setting the test out says what the genus’s partiality really costs.

A class is worth naming when it is closed under the operation the theory is about. Numbers are closed under addition; nimbers are closed under addition; all-small games are closed under addition. Each of those classes supports statements of the form inside this class, such-and-such holds, and the statements survive the arithmetic because the arithmetic keeps every member inside.

Wildness is not closed, and it fails badly: two wild heaps added together are tame two thirds of the time. So the wild games is a set of exceptions rather than a subsystem, and there is no theory to be developed inside it — anything proved about wild heaps stops applying the moment two are added.

That has a specific consequence for what could ever be built here. A composition table needs its entries to be things the table can be applied to again, and not one of the twelve entries is a symbol any wild heap carries — so the table cannot be iterated, and a sum of three wild heaps is not answered by applying it twice.

So the genus’s partiality is not that it fails on a describable class; it is that the failures do not form a class at all. That is a worse situation than an exception list and a better one to know about, because it says what the missing work is not: there is no wild theory waiting to be written, and the next invariant has to be finer everywhere rather than special somewhere.

What is left of the genus #

Adding the three findings up gives a reading of the genus that is worth stating in one place, because the anchor has now measured every part of it.

The genus computes everywhere. Every position has one, the recursion is unremarkable, and nothing on this ladder has ever found a position it could not produce a symbol for.

It composes everywhere. The symbol of a sum is a function of the two symbols, on the tame side by a theorem and on the wild side by measurement over 22 files.

It means something only on the tame side. There the symbols are closed, the rule is two cases long, and the property they encode survives addition. On the wild side the symbols are computable, compositional and useless: they close under nothing, they follow no rule, and the property that named them is lost by the operation the theory is about.

So the genus is an invariant whose usefulness and whose definedness have different domains, and the gap between them is exactly what this anchor has spent six rungs measuring.

The convention, named #

Misère play throughout: the player who cannot move wins. That is the whole reason the genus exists — normal play needs only the Grundy value, and misère play needs more.

The genus of a position is its normal-play Grundy value — the base — with a superscript recording the misère Grundy values of the position plus increasingly many copies of a two-counter Nim heap. A position is tame when its genus is the genus of a Nim heap, and wild otherwise.

The superscript is computed to six places, which is where every genus on this site stops and is enough for every symbol here to have settled.

A file is an unordered pair of symbols; the table’s entry for that file is the genus of the sum of any two heaps carrying those symbols, and the file is ambiguous if two such sums differ. No file here is ambiguous.

Closed means every answer in the table is itself one of the symbols the table is indexed by. That is the property a quotient needs and it is the property this set does not have.

Where the ladder goes next #

The genus anchor has six rungs: what the genus is, what a tame heap may be replaced by, that the symbols compose, the function that composes them, that no rule of that shape reaches the wild side, and now what the wild table looks like.

The rung above is the closure itself. The wild symbols are not closed and their answers are twelve further symbols, so the obvious object is the set reached by closing the twenty-two of them under addition — is it finite, and if so how large? That needs sums of three heaps and more, which is a computation about an order of magnitude beyond this one and is the natural place for the anchor’s remaining effort. If the closure is finite it is a monoid after all, just not the one the rung below guessed; if it is not, the wild side has no algebra at any size and the anchor has its answer.

Two neighbours are worth the trip. Tame and wild is the classification everything here is bounded by, and it is where tameness is shown to be closed under addition — reading it beside this page is the whole contrast. And misère quotients is what the subject does instead of a general invariant, and this page is the measurement of why it had to: the genus’s own wild half is not the closed object a quotient needs.