A fraction does not reach

Two strips that end the same way ended by specifying an experiment and predicting its result, which is unusual on this ladder and is the reason this page can be short about the setup and long about the answer.

The experiment: take a Push strip built as a prefix, then a gap of i empty squares, then a tail. Widen the gap and watch the value. It converges, and the rate at which successive differences shrink is a power of two — the rate that first appeared when a cut in a strip invented a cliff. The previous two rungs established that this rate is a property of the tail and nothing in front of it, on seventy-eight of the eighty tails swept, and found two tails where it is not: LRRL and RLLR, each of which converges at a quarter behind sixteen of eighteen prefixes and at a half behind the other two.

The prediction, from that sample of four: the deciding bit is whether the prefix’s last coin can move alone — whether it stands isolated behind a gap.

It is worth being clear about what the rate is a rate of, because it is a second difference and second differences are easy to misread. Widen the gap by one square and the value moves; widen it again and the value moves less. The rate is the limit of the ratio of those successive movements, so a rate of a half means each extra empty square buys half of what the last one bought, and the value converges geometrically to something. The number the values converge to is not the rate and is not what any of these rungs measure — two strips can converge to quite different limits at the same rate, and two can share a limit and approach it differently.

That is wrong. It is wrong in both directions at once, and what is right is not a statement about the prefix’s shape at all.

What the census holds #

Prefixes here start and end with a coin, so that no gap in the prefix is confused with the widening one, and they run to five squares. There are a hundred and sixty-two of them, against the eighteen the previous rung had.

Thirty-eight of the hundred and sixty-two change the rate to a half, on each of the two tails, and the two counts are identical because the tails are mirrors — which is worth checking rather than assuming, since a sweep with an indexing error would break that symmetry first and loudest. So the four exceptional prefixes the rung below found were the four smallest members of a class with thirty-eight members, rather than four oddities — which is the first thing worth knowing, because a set of four with a description fitted to it is a coincidence waiting to be identified and a set of thirty-eight is a phenomenon.

The thirty-eight include all four of the originals. They also include twelve prefixes whose last coin is not isolated at all — L.LL, R.LL, L..LL, L.LLL, L.RLL and seven more, each ending in a run of two or three coins with nothing separating the last one from the coin before it.

And twenty-six prefixes have an isolated last coin and rate a quarter. Those are the ones whose isolated coin is Right’s, against the tail LRRL, and Left’s against RLLR. The shape rule takes no notice of colour, so it cannot see them.

Neither failure is a near miss. Twelve prefixes on one side and twenty-six on the other is thirty-eight disagreements out of a hundred and sixty-two, so the rule is wrong on nearly a quarter of the census — and it was right on all eighteen it was built from. That is the ordinary way a description fitted to a small sample fails: not by being approximately right on more data, but by being exactly right on the sample and unrelated to what is happening.

The number that decides it #

Group the hundred and sixty-two prefixes by their own value — the value of that prefix played as a Push strip on its own, with no gap and no tail — and the split is exact.

The rate is a half exactly when the prefix is worth at least one whole move to the player whose coin begins the tail. Against LRRL, whose first coin is Left’s, that means a prefix worth 1 or more to Left. Against RLLR it means 1 or more to Right. A hundred and sixty-two of a hundred and sixty-two, on both tails, and no value class splits.

The threshold is where the interest is. Every prefix worth three quarters of a move rates a quarter; every prefix worth one rates a half. There is nothing between them and nothing ambiguous at the boundary — the census contains prefixes worth ¼, ½, ¾, 1, 1½, 2, 3 and 4 in each direction, and the line falls exactly at one.

That is a sentence about reach. A widening gap is a gap that eventually exceeds any length anybody cares to name, and the question the whole ladder has been circling is what can still be felt across it. The answer here is that a fractional advantage cannot be. A prefix worth three quarters is genuinely worth something — Left really is better off with it than without it — and it is worth something that stops mattering once the gap is long enough. A whole move is not like that. A whole move is a move, and a move is available at any distance.

Two families make the difference concrete, and they are chosen to be as close as two families can be. Put L.L in front of the gap and the values run 0.5625, 0.6563, 0.7031, 0.7266, 0.7383, 0.7441, climbing towards three quarters. Put LL there instead — the same two Left coins, with the gap between them closed — and they run 0.3750, 0.4688, 0.4922, 0.4980, 0.4995, 0.4999, climbing towards a half.

The first differences are identical: 3/32 in both. It is the second difference where they part. L.L’s differences halve — 3/32, 3/64, 3/128 — and LL’s quarter — 3/32, 3/128, 3/512. One empty square inside the prefix, five squares and then a growing gap away from the tail, changes what the far end of the strip does to every subsequent square, and it does so from the second square onwards.

And L.L is worth one to Left where LL is worth nought, which is the whole of the difference between them by the criterion. LL has two Left coins and no room to push either without pushing the other, so it stores no move at all; L.L has a square to push into.

Why one and not some other number #

The threshold’s being exactly one is worth another paragraph, because “at least one” is doing more work than “large enough”.

In Push a coin is pushed by its owner, and pushing moves that coin and everything already in front of it one square along. What a prefix worth v contributes to a longer strip is, loosely, a stock of moves — and the stock is only usable if the moves are there to be made. A prefix worth three quarters is a position in which Left is ahead by less than one move: whatever advantage it holds is a fraction that arises from an option Right can answer, and answering it is a local business that the far end of the strip never learns about. A prefix worth one is a position in which Left can simply move, once, and be no worse off. That move is a move in the whole strip, and it is still a move when the gap between the prefix and the tail is a hundred squares long.

So the reason the boundary is at one, rather than at a half or at two, is the same reason integers matter anywhere in this theory: an integer is a count of moves and a fraction is a description of a fight. A fight resolves locally. A count does not have to.

The colour coupling falls out of the same reading and is worth checking against a case. The prefix L.R is worth one — to Right. Against the tail RLLR, whose first coin is Right’s, it rates a half. Against LRRL it rates a quarter, and its values run 0.4063, 0.4297, 0.4355, 0.4370 towards 7/16, quartering all the way. A stored move only reaches across the gap when it is the right player’s move to make, which is why the criterion has a sign in it and why a rule stated on shapes could never have found it: the two tails are mirror images, and a shape rule sees mirror images as the same shape.

Against prefixes it has never seen #

A criterion read off a hundred and sixty-two prefixes and reported as exact on those hundred and sixty-two is a description of a hundred and sixty-two prefixes. Something has to be done about that, and the ladder’s own history says what — the shape rule looked exact on four.

So the criterion is applied, unchanged and with nothing refitted, to every prefix of exactly six squares. That is three hundred and twenty-four more, twice the census it came from, and none of them in it. It is right on every one, on both tails.

It also predicts both classes rather than one, which is the part a test like this can quietly fail. A criterion that said every prefix rates a quarter would be right on ninety per cent of any sample and would be saying nothing; here it names ninety-four of the three hundred and twenty-four as changing the rate, and ninety-four is what changes it.

The share shifts between the two sweeps and that is a small point in its favour rather than against it. Thirty-eight of a hundred and sixty-two is 23%; ninety-four of three hundred and twenty-four is 29%. A longer prefix has more room to store a whole move, so the class should grow with the prefix length — which is what a criterion about value predicts and what a criterion about the last coin’s shape does not, since the proportion of words ending .L is the same at every length beyond three.

The half of the statement that is easy to drop #

Say the criterion aloud and it sounds like a law about Push: the rate changes when the prefix is worth a move. It is not a law about Push. It is a law about two tails.

The same hundred and sixty-two prefixes — ranging from four moves ahead for Left to four ahead for Right, with seventy-six of them worth at least one whole move — leave the rate completely alone on six ordinary tails. Behind LR every one of them gives a half. Behind LLR every one gives a quarter. Behind L.R the differences vanish outright for all of them, which is the third class this ladder found and is not a rate at all.

So the honest statement has two clauses and the second is not decoration. Almost every tail reads nothing in front of it; two tails read one bit, and the bit is whether the prefix is worth a whole move. A figure showing only the two exceptional tails would leave a reader thinking prefixes generally matter, which is the reverse of what four rungs of this ladder established.

It is also the reason the exception is interesting rather than embarrassing. A law with no exceptions and a law with exceptions everywhere are equally uninformative; a law that holds on seventy-eight tails out of eighty and fails on two, for a stated reason, shows what the law was made of.

The suffix law, restated to survive this page, reads: the rate is a function of the tail, except that two tails also read whether the prefix is worth a whole move to the player their first coin belongs to. That is longer than what read from the back forwards hoped for and it is still an invariant — the prefix enters through one bit, and every other property of it, its length, its shape, its number of gaps and its number of coins, is ignored by every tail including these two.

What the two tails have that the others do not #

Nothing here explains why LRRL and RLLR are the two, and it is worth saying so plainly rather than gesturing at it.

What can be said is what they are not. They are not distinguished by their own rate, since sixteen of the eighteen prefixes give them the same quarter that LLR gives all eighteen. They are not distinguished by length, since every tail to four squares was swept and only these two behave this way. They are not distinguished by their last run, which is the thing the rung below showed sets the rate: LRRL ends in a single L exactly as RL, RRL and LRL do, and all three of those ignore their prefix completely. And they are not distinguished by being colour-palindromes, which they are — LRL and RLR are palindromes too and are ordinary.

So this page has no shape description of the two, and offering one would be repeating the mistake it exists to correct. What it has is a description of the prefixes, which is exact, tested out of sample, and says nothing about why these two tails are the ones listening. What separates them is that the value of the whole strip has a second regime available to it, reachable only when there is a whole move waiting on the far side of the gap, and the census says which prefixes have one without saying what the second regime is made of.

That is a smaller claim than a mechanism and it is the claim the sweep supports. The rung below made a larger claim from a smaller sample and it did not survive; making the same mistake one rung up, with a better number in it, would be worse rather than better.

Where the ladder goes next #

The push anchor has eight rungs: the game and its cliff, the reading and where it fails, the criterion that decides the failures, how wrong the reading is when it is wrong, the numeral in the empty squares, the cut that invents a cliff, which run sets the rate, what the rate is a function of, and now what the two exceptions are reading.

The rung above is the mechanism. The criterion is exact on four hundred and eighty-six prefixes and it is still a criterion — it says which prefixes change the rate and not what the change consists of. What would settle that is the values themselves rather than the ratios of their differences: take one prefix from each side of the threshold, L.L and LL, put both in front of the same widening gap and the same LRRL, and read the two sequences of values against each other rather than reading each one’s rate. If the whole-move prefix is contributing a constant that the fractional one is not, it will be visible in the values as a shift, and the rate will turn out to be an artefact of measuring differences rather than the primary thing. If it is not visible there, the second regime is not a shift and the question is harder than this page makes it look.

Two neighbours are worth the trip. A numeral in the empty squares is the reading of a Push strip that turned out to be over the gaps rather than over the coins, and it is the other place on this ladder where the answer was a different object from the one everybody was looking at. And the criterion that cannot exist is worth reading beside a criterion that plainly does, because the reason this one exists and that one does not is a fact about what a value can be a function of.

Normal play throughout, as on every rung here. Under misère play none of these positions is a number, and a ratio of successive differences between things that are not numbers is not a quantity at all.