The birthday is a floor

The cheapest way to show a value built a table of witnesses: for each of the 1,193 values that fifteen rulesets reach, the smallest position of any of them that is worth it. It set the size of that witness against the value’s birthday and reported a correlation — 0.31 over everything, 0.73 among the values that are not numbers — then closed on the residual:

If birthday predicts exhibit size at 0.73 among non-numbers, the residual is the interesting object: which values are much dearer to exhibit than their birthday suggests, and whether they have anything in common.

The residual is the interesting object and the correlation was the wrong instrument to find it with. The relation between the two quantities is not a trend at all.

Never below, and usually exactly #

On every one of the 728 non-number values the witness is at least as large as the birthday. Not approximately, not on average — on all of them, across fifteen rulesets and 11,397 positions.

And on 476 of them the witness is exactly the birthday. A value born on day five is exhibited on five pieces; a value born on day eleven, on eleven.

A correlation of 0.73 is what an inequality attained two thirds of the time looks like when a straight line is fitted through it. The line has a slope of 0.815 and an intercept of 2.08, and neither number means anything: the true relation is $\text{size} \ge \text{birthday}$, with equality common.

That is a considerably stronger statement than the one it replaces, and it is worth being clear about why it was not seen the first time. A correlation is a summary designed for two quantities that vary together with noise, and applying it to a bound reports the bound’s tightness as a slope and its exceptions as scatter. The rung below asked for the residual, which is exactly the instrument that would have revealed the shape — and asked for it as a next step rather than as the thing to compute first.

Why there should be a floor #

The mechanism is short and it is the reason to believe the inequality holds beyond the sweep.

A value’s birthday is the number of days of options underneath its canonical form: a value born on day $d$ has an option born on day $d-1$, which has one born on day $d-2$, and so on to nought. Those are $d$ distinct values in a chain.

A position exhibits that value only if its own game tree contains a position for each of them, and a move must be available to get from one to the next. In every ruleset in this sweep a move consumes something — a coin, an edge, a square, a counter — so a chain of $d$ moves needs $d$ pieces to consume.

So the floor is a statement about the games rather than about the theory, and it would fail for a ruleset with a move that consumes nothing. This site has one: a game whose positions can return to themselves has no such accounting, and none of the fifteen rulesets here is loopy.

Where the slack lives #

The interesting third is the 252 values that cost more than their birthday, and the rung below asked what they have in common. They have something very definite in common and it is not about them.

Hackenbush with green edges is exact on all 297 of its values. Every value it exhibits is exhibited on precisely its birthday’s worth of edges, with no exception anywhere.

Partizan subtraction pays four pieces on average, and its worst is nine. Toppling Dominoes pays 2.83, Toads and Frogs 2.25, Clobber 2.07 — and none of those three is exact on a single value.

End-Nim sits between: exact on 168 of its 214, and paying up to eleven on the rest.

Sorting the whole table by ruleset accounts for 40 per cent of the variance in the excess. What a value costs beyond its birthday is, to that extent, a fact about which game happens to be its cheapest exhibitor.

Why a game is exact or is not #

The two ends of that table are worth reading against each other, because the reason is visible in the rules.

A green Hackenbush edge is a whole move and nothing else: cutting it removes one edge and the position becomes another position of the same game. So the chain of $d$ options is a chain of $d$ cuts, and the position needs exactly $d$ edges. Hackenbush is a numeral is where that correspondence between drawing and value is set out, and this is the same correspondence measured from the other side.

A partizan subtraction heap is a single number, and the moves available from it are governed by two lists rather than by the pieces. A heap of eight with the lists $\{1,2\}$ and $\{1,3\}$ has a value whose form is deep, and the heap’s size is not the depth of that form — it is the length of the arithmetic that produced it. So the two quantities are counting different things, and the game pays a constant for the mismatch.

That is the transferable form of the finding. The witness size is not a property of the value; it is a property of the value together with an encoding, and the fifteen rulesets are fifteen encodings with fifteen exchange rates. Reading the pooled table as a fact about values was the rung below’s error, and it is the ordinary error of pooling a measurement across instruments.

The 476, and what they are #

Two thirds of the values are exhibited at exactly their birthday, and it is worth saying which two thirds rather than leaving it as a count.

They are 297 Hackenbush-with-green values, 168 End-Nim rows, four partizan subtraction heaps, four Snort positions and three Nim heaps. Two rulesets supply 465 of the 476.

That is the same finding as the section above, seen from the other side, and it says something about what the equality means. It is not that those 476 values are simpler than the other 252; it is that they happen to be reachable by a game whose pieces and whose days are the same currency. Move a value from the exact list to the dear list by deleting one ruleset from the sweep, and nothing about the value has changed.

So the right statement of the finding has two halves and only one of them is about values. The floor — never below the birthday — is about values, and holds for all 728. The attainment rate is about the pool of games, and would move if the pool did.

The numbers, which go under the floor #

The floor does not hold for values that are numbers, and the way it fails is the second half of the explanation.

An integer $n$ is born on day $n$: two is $\{1 \mid \}$, and one is $\{0 \mid \}$, and so on down. 286 of the 465 number-valued rows are exhibited below their birthday, and the extreme case is a Shove strip of six coins worth 21 — a day-21 value drawn on six pieces, fifteen days below the floor.

The chain argument shows why. Reaching 21 from a Shove strip does not require twenty-one distinct positions in a chain, because a single move can change the value by a great deal: shoving a coin along a strip moves several coins at once and the value jumps. A non-number cannot be reached that way, because each day of its form is a genuinely distinct option the position has to be able to move to, and the moves are what the pieces buy.

So the floor is a statement about a kind of value. Numbers compress and non-numbers do not, and the two correlations the rung below reported — 0.31 pooled and 0.73 among non-numbers — were the shadow of that division rather than two views of one trend.

What this changes about the pooled table #

The rung below’s table of witnesses by ruleset was read as saying which games are productive — how many distinct values each one reaches. That reading survives. What does not survive is any comparison of witness sizes across the rows.

Hackenbush with green supplies 297 of the cheapest exhibits and End-Nim 214, and those two numbers can be set beside each other: they count values. The mean witness sizes — 4.62 edges against 9.12 counters — cannot, because an edge and a counter buy different amounts of value.

That is not a defect of the earlier table; it is a limit on what a pooled table can be asked. The useful version of the question is asked within a ruleset, and asked that way the answer is sharper still: inside Hackenbush with green the correlation between birthday and size is exactly one, because the two quantities are the same number.

The tightness, stated properly #

A bound is worth as much as its attainment rate, and this one is attained on two thirds of the values it covers.

That is unusually high for a bound in this subject. A rule that is never right and cannot be far wrong is the general account of bounds here, and the standing observation is that a bound attained three per cent of the time is a bound whose number is right and whose typical case is elsewhere. Sixty-five per cent is a different regime: for most values the floor is the answer, and the question worth asking about a new value is not how far above the floor it sits but whether it is one of the exceptions.

The excess, when there is one, is small. A third of the 252 pay one or two pieces, most of the rest pay three or four, and only 21 values in the whole census pay more than five. The largest is eleven, on an End-Nim row.

What the census does not say #

Four limits.

Fifteen rulesets, and the floor is about all of them at once. The witness is the smallest position of any ruleset, so the floor says that no game in the pool beats the birthday. A sixteenth game with a cheaper encoding would lower some witnesses and could not lower them below the chain argument, but the argument is a sketch and the census is a check on it rather than a proof.

Sweeps that stop at six or seven pieces. Every witness is found inside a bounded sweep, so a value whose cheapest exhibit is a nine-piece position of some ruleset is recorded here with whatever larger witness the sweep did find, or not recorded at all. That biases the excess upward and leaves the floor untouched, which is the safe direction for a claim of this shape.

Forty per cent is not most of it. The ruleset explains 40 per cent of the variance in the excess, so 60 per cent is something else — and this page has not found what. The obvious candidate is the width of the form rather than its depth, which how wide a form can get measures and which nothing here has set against the witness.

The attainment rate is a fact about the sweep. Sixty-five per cent would fall if Hackenbush with green were removed and rise if a second game of the same shape were added. What would not move is the floor, and that is the part worth carrying.

And the numbers are not analysed. They break the floor, the mechanism is clear, and this page has not asked what governs how far below the floor a number goes. That is a question about how much a single move can move a value, which is a question about incentives rather than about realisability.

What makes a lower bound informative #

A floor that is attained two thirds of the time is a useful measure and a floor attained never would be a definition dressed as a result. It is worth setting out what separates them.

A bound is informative when it is attained often enough to constrain. If most values cost exactly their birthday, then the birthday is the cost for most purposes, and the third that exceed it are the interesting population — a small, identifiable set with something extra to explain.

A bound attained rarely is a different object. It is still true and it stops being a measure: the quantity being bounded is elsewhere, the bound is a formality, and the interesting question becomes what the real cost is rather than why some values exceed the floor.

Two thirds puts this one firmly in the first case, which is why the anchor above it goes looking for the residue rather than for a better bound. That is the right response when a bound bites: the exceptions are a population, not noise.

It also says what would have changed the anchor’s whole direction. Had the attainment rate been ten per cent, the right next question would have been what is the cost actually a function of, and the birthday would have been abandoned rather than corrected. The rate decides which of the two programmes is worth running, and it is the number to report first for exactly that reason — before any account of the exceptions, because it says whether the exceptions are the story.

What a reader should take from it #

The finding replaces a number with an inequality, and the difference matters for how it is used.

A correlation of 0.73 licenses nothing about any particular value. It says the two quantities move together and leaves every individual case open, which is why the rung below could only ask which values are dear rather than say.

The floor licenses a statement about every value at once: a value born on day $d$ cannot be shown on fewer than $d$ pieces of any game in this sweep, so a reader who knows a value’s birthday knows a lower bound on every position that could ever exhibit it. And because the bound is attained two thirds of the time, the natural first guess for a new value’s cheapest exhibit is the birthday itself — a guess that is exactly right more often than it is wrong at all.

The convention, named #

A witness for a value is the smallest position, over every ruleset in the sweep, whose canonical form is that value. Size is measured in the ruleset’s own pieces: edges for Hackenbush, coins for Shove and Toppling Dominoes, counters for End-Nim, stones for Clobber, and the heap for partizan subtraction.

The birthday is the depth of the canonical form: nought for the zero game, and one more than the deepest option otherwise. It is computed from the form rather than looked up.

Number means a value that is a dyadic rational — the class the simplicity rule produces — and every claim on this page separates the numbers from the rest, because the floor holds for one class and not for the other.

Where the ladder goes next #

The realisability anchor has three rungs: which values any game reaches, what the cheapest way to reach one costs, and now that the cost is bounded below by the birthday and equal to it two thirds of the time.

The rung above is the other 60 per cent of the excess. The ruleset explains 40, and the natural second variable is the width of the form rather than its depth — a value with many options at some day needs a position with many moves available there, and nothing in the chain argument charges for that. Setting the witness size against the widest level of the canonical form is a computation over data already in hand and would either close the account or say what else is in it.

Two neighbours are worth the trip. How old a value is is where the birthday is established as a measure and shown not to bound the width; this page is the same measure succeeding as a bound on a different quantity, and the contrast says what a birthday is and is not for. And how long a row a value needs is this question asked inside one ruleset, where the answer is a universality claim — and reading it beside this page shows the exchange rate at work, since a Toppling Dominoes row pays 2.83 pieces above the birthday on average and its universality claim is stated in rows rather than in days.