The entry fee was the cap

The birthday is a floor established that the cheapest position exhibiting a value is never smaller than the value’s birthday, that it is exactly equal to it on 476 of 728 values, and that on the rest the ruleset accounts for 40 per cent of what is left over. Wider costs less then tried the obvious second variable, the width of the value’s canonical form, and found it pointing the wrong way — leaving the ruleset in sole possession of the field.

That page closed by naming the constant nobody had measured:

The rung above is the entry fee. If most of the excess is a constant per ruleset, then the quantity to measure is the constant itself — what it is about a game’s rules that makes every value it produces cost two squares more than its birthday, when another game’s cost nothing. Toads and Frogs against green Hackenbush is the sharpest pair available … the difference should be readable off the two move rules rather than off the census.

It is not readable off the move rules, because it is not a property of them. It is a property of where the sweep stopped.

A fee would be flat #

The word fee carries a prediction, and it is easy to state: if every value a game produces costs two squares more than its birthday, then a value born on day three is exhibited on five squares and one born on day six on eight, and a line fitted through the pairs is horizontal.

It is not horizontal on any ruleset in the census large enough to fit a line to. End-Nim’s slope is $-0.36$ over 214 values, Toppling Dominoes’ is $-0.63$ over 84, Toads and Frogs’ is $-0.45$ over 48, Clobber’s is $-0.34$ over 44. Green Hackenbush’s is nought because its excess is nought everywhere, and partizan subtraction is the one ruleset with a positive slope and it has 24 values across four rule pairs, which is four populations rather than one.

So the excess is not a fee. On four of the six it is largest on the values born earliest and falls away steadily as the birthday rises — Toads and Frogs charges 2.43 squares on its day-three values and 1.00 on its day-six ones. Whatever it is that Toads and Frogs is charging for, it charges less for the more complicated goods.

What the excess is pressed against #

There is a ceiling on the excess, and it is not a finding but arithmetic.

A value with birthday $b$ appears in the census with excess $e$ exactly when some position of $b + e$ pieces exhibits it. Positions larger than the ruleset’s cap are not swept. So

$$e \le \text{cap} - b,$$

which is a hard bound built into the measurement, and it presses hardest exactly where $b$ is largest. That is the whole shape of the falling slope: not a game charging less for deep values but a sweep unable to see a deep value that costs much.

Between half and seven tenths of the values on the rulesets capped at seven pieces sit exactly on the ceiling: their cheapest exhibit is the largest position the sweep contains for them. A measurement whose values pile up against its own upper bound is a measurement that has been cut off, and the two rungs below both reported the resulting mean as though it were a property of the game.

The comparison the rung below called the sharpest available is the clearest case. Green Hackenbush is swept to five edges and pays an excess of nought on all 297 of its values; 211 of them are born on day five and exhibited on five edges, which is an excess of nought and a ceiling of nought. The zero fee and the 2.25 fee are the same ceiling read at two different caps.

Raising the cap raises the fee #

If the number is about the cap, moving the cap moves the number, and Toads and Frogs is cheap enough to move a long way. Every word over toad, frog and empty square up to ten squares is 88,572 positions and about twenty seconds of arithmetic.

The mean excess is 2.58 at the census’s cap of seven, 2.72 at eight, 3.10 at nine and 3.37 at ten. It rises at every step and nothing in the sequence suggests a limit. The slope against the birthday does not flatten as the ceiling recedes — it steepens, from $-0.47$ to $-0.83$ — because each new square admits a whole generation of newly-born values that are exhibited near their birthdays and a few expensive old ones, and the first outnumber the second.

The count of values makes the same point from the other side: 74 at seven squares, 1,472 at ten. The census reached a twentieth of what one ruleset produces at ten squares, and the quantity it measured on that twentieth was reported as a constant of the ruleset.

This is a slightly different population from the census’s, and the difference is worth naming rather than hiding: the birthday is a floor attributes each value to whichever of eleven rulesets exhibits it most cheaply, so a Toads and Frogs value also produced by a smaller End-Nim position is counted as End-Nim’s. The sweep here is Toads and Frogs alone. That is why 48 values become 74 at the same cap, and it does not touch the finding: the trend is measured within one population at four caps.

And the floor breaks #

At eight squares something else happens, and it is not a change of degree.

Values appear that are exhibited by positions smaller than their own birthday. Two of them at eight squares, twelve at nine, sixty-two at ten, and the worst by four days. The floor the anchor’s third rung established over 728 values — never below, exactly equal on two thirds — stops holding one square past where it was measured.

TTTF.FFF is eight squares with three toads and four frogs, and its value is $\{\{\{0 \mid \{0 \mid \dots\}\} \mid 0\} \mid -2\}$ — a canonical form nine levels deep, which is a birthday of nine. It is not a number, so the exclusion the floor was stated with does not cover it, and the floor says it cannot exist.

What the exclusion should have been #

The floor was stated for non-numbers because numbers break it trivially: an integer $n$ is born on day $n$, and a six-counter Shove strip can be worth 21, so a number’s birthday says nothing about how large a position it takes to draw.

Twenty-six of the 62 have a number among their options born later than the position has squares. $\{\{12 \mid 9\} \mid 8\}$ on a ten-square strip has a birthday of fourteen and it gets thirteen of those days from the integer 12 sitting three levels down. The value is not a number and it inherits everything the exclusion was written to keep out.

So the class the census should have excluded was not numbers but values with a large number in them, and that class has no clean boundary: every switch between two integers is one, and switches between integers are exactly what a long Toads and Frogs strip produces. There is no restatement of the floor that saves it and stays interesting — which makes the honest position that the birthday bounds the exhibit for values built out of small ones, and that the qualification is doing all the work.

What the rules do decide #

None of this says the rulesets are alike, and it is worth separating what has been taken away from what is left.

What has been taken away is the excess as a measure of a game’s extravagance. It is a difference between a value’s birthday and the size of its cheapest exhibit, and the second of those two numbers is bounded above by the sweep, so the difference inherits the bound.

What is left is everything about a ruleset that a bounded sweep can see honestly. How many values it produces at a given size is one: 74 Toads and Frogs values at seven squares against green Hackenbush’s 297 at five edges, and that comparison does not improve or degrade as the caps move, because both counts are complete for their size. Which values it produces is another, and it is what the values nobody’s game produces measured in the first place — a tenth of day three, which is a statement about presence rather than about distance. And the shape of what it produces is a third: green Hackenbush produces the numbers and nothing else, since a Hackenbush string is a binary numeral and the reading is exact, while the strip nobody has a formula for is Toads and Frogs, where every attempt at a closed form has failed and the values are switches between integers.

That last contrast is the one the rung below was reaching for, and it survives everything above. Green Hackenbush’s every value is a number and every edge is a move; Toads and Frogs’ values are fights and most of its squares are empty. Those are real differences in the rules and they have real consequences — they are simply not the ones the excess was measuring.

What a censored measurement looks like from inside #

The failure this page reports is worth generalising, because nothing about it is specific to entry fees and every sweep on this site is exposed to it.

A size cap does not remove positions at random. It removes the large ones, and largeness correlates with everything interesting — a value born late needs a big position to exhibit it, a wide form needs a position with many moves, a hot value needs room to fight in. So the sample a capped sweep produces is biased in a direction that is easy to state and hard to see: it is missing the expensive end of every quantity being measured.

The consequence is a specific and recognisable artefact. Any measure of excess — how much more a position costs than some lower bound predicts — is understated near the cap, because the positions that would have shown a large excess are exactly the ones excluded. Average the excess over a capped sweep and it comes out looking like a small constant. Average it per ruleset and it comes out looking like a small constant that varies by ruleset, because the rulesets hit the cap at different depths.

That is how a censoring artefact turns into a plausible finding: it produces a number, the number is stable, and it varies in a way that invites an explanation about the rules.

The diagnostic is the slope rather than the level. A genuine constant is flat against depth; a censoring artefact declines, because the deeper values in the sample are the ones nearest to being excluded. That check costs nothing — it uses data already collected — and it is the check the rung below did not run.

The habit worth taking from this: when a sweep with a cap reports a per-population constant, plot it against whatever the cap is a cap on before believing it. If it slopes, the constant is the cap.

What this does not say #

The census’s numbers are not wrong; they are about the census. Every figure on the two rungs below is a correct measurement of the population it was taken over, and none of them should be read as a statement about Toads and Frogs or about games in general. The mistake this page is about is one of interpretation, and it was made in the closing paragraph of a page rather than in its measurements.

One ruleset was extended, not eleven. Toads and Frogs was chosen because the rung below named it and because words over three letters are cheap. Whether End-Nim’s slope of $-0.36$ or Clobber’s of $-0.34$ would behave the same way under a raised cap has not been tested, and End-Nim’s ceiling share of 2 per cent suggests it might not — its cap is fifteen counters, which is far above its birthdays.

Sixty-two counterexamples are not a rate. They are 4 per cent of the 1,472 values at ten squares, and what they establish is that the floor is false rather than how often it fails. The share may well fall as the cap rises further, since the deep cheap values are the ones a wider sweep keeps finding.

And the excess may still converge. Four caps showing a rising mean is four points on a curve, and a quantity that rises from 2.58 to 3.37 could be heading for four or for infinity, and nothing here distinguishes them. What the four points do rule out is 2.25 being the answer.

Nor is the censoring an argument against bounded sweeps. Every measurement on this site is bounded somewhere, and what solved means is the page that says so plainly: the practical range of an exact evaluator is a few dozen moves and a figure that depends on smallness has to say so. The fault here is not that the sweep stopped; it is that a quantity pressed against the stopping point was read as though the stopping point were not there.

The convention, named #

Normal play, canonical forms, computed by the recursion.

A value’s birthday is the number of days of the construction it takes to appear, which is the depth of its canonical form; its exhibit is the smallest position producing it, and the excess is the second less the first. Size is counted in the pieces a reader can see — squares for Toads and Frogs, edges for Hackenbush, counters for End-Nim — and comparing an excess in squares with an excess in edges is comparing two units, which is why every fit here is inside one ruleset.

The cap is the largest position of a ruleset in the sweep, and the ceiling is the cap less the birthday: the largest excess a value of that birthday could possibly show.

Toads and Frogs is played on a strip; toads move right and frogs left, one square into an empty space or by jumping exactly one of the other kind. The sweep is every word over T, F and ., and positions with no piece at all are skipped.

A slope is least-squares, reported and not asserted, because a slope is a summary and the assertion in the code is the thing the page turns on: that the mean excess is larger at the widest cap than at the census’s, and that a value exhibited below its birthday exists.

Where the ladder goes next #

The realisability anchor reaches five rungs to here, and this one has just removed a quantity rather than adding one: the per-ruleset entry fee was an artefact of the size cap, so the anchor is short of anything a cap cannot censor.

The rung above supplies one and it is not the obvious candidate. The rate was the alphabet measures how many new values a ruleset produces per extra square, which is a slope rather than a level and is therefore insensitive to where the sweep stopped. That rate is honest, and what it turns out to measure is the notation: every ruleset grows at close to the number of symbols its positions are written in, and the seven of them span less than a factor of two.

That is a deflation of the same kind this page performs, one level up. A quantity that looked like a fact about a game — how fast its value set grows — is a fact about how many characters its positions are spelled with, and the games are barely distinguishable on it.

What does separate them is the yield, which spans a factor of a hundred and nineteen: how many values a ruleset produces in total, rather than how fast the count climbs. So the anchor ends with the two quantities properly separated — a rate that is about the alphabet and is nearly universal, and a yield that is about the rules and is not — and the entry fee this page dissolves was a confusion of the two seen through a cap.

Two neighbours are worth the trip. The values nobody’s game produces is where the yield question is asked directly, and where the same size cap is doing the same quiet work. And wider costs less is the rung below, whose per-ruleset constant this page removes.