Two strips that end the same way

Read from the back forwards widened each of a three-run Push strip’s three gaps in turn and found the same answer every time: the value converges at $2^{-k_3}$, where $k_3$ is the length of the strip’s rearmost run, whichever gap is being widened. It closed by naming what that would have to mean if it were a law rather than a pattern:

Everything here says a strip’s sensitivity is governed by its last run and nothing else, which is a statement about a suffix — so the sharp version is that two strips agreeing in their last $j$ squares have the same rate, and the natural experiment is to hold a suffix fixed and vary everything in front of it.

The experiment is affordable and it has been run. The grouping is single-valued on 78 of the 80 tails, which is the ladder’s invariant arriving; and it is not single-valued on two of them, which is the more interesting half.

The experiment #

A family here is three things written down in order: a prefix $W$, then $i$ empty squares, then a tail $T$. The gap in the middle is the one being widened, $i$ running from one to eight, and the rate is the limit of the ratio of successive differences of the strip’s value — the quantity the rung below measured, computed the same way.

Two conventions matter and both are there to stop a gap being counted twice. A tail ends in a coin, since a trailing empty square is part of the widening gap rather than part of the tail. A prefix begins and ends in a coin for the same reason at the other end. That leaves 80 tails of up to four squares and 18 prefixes of up to three, and 1,440 families.

The question is exactly the rung below’s: does the rate depend on $T$ alone?

The ratio of successive differences is the right instrument and it is worth saying why, because a cruder one would have answered a different question. A strip’s value is a rational number, and comparing the values themselves across two prefixes says nothing — the prefixes are different positions and of course they are worth different amounts. What is being compared is how the value responds to one more empty square, and the second difference divided by the first strips out both the level and the scale. Two families whose values differ by a hundred and whose responses differ by a factor of ten can still have the same ratio, and that ratio is the only thing the rung below’s law was ever about. It is the same move the two numbers at the top made on the temperature ladder: to find out what a quantity depends on, hold everything else and vary one input, and look at the derivative rather than the reading.

For 1,440 families the answer comes back in four shapes. The ratio locks on a power of two, agreeing with its predecessor to the last bit — 972 families. It is still approaching one when the sweep stops, which is the interior-gap behaviour the rung below already reported and which is a different statement from having arrived — 360. The differences become exactly nought — 108, and that class is new. Or the sweep ends with the ratio still moving and going nowhere identifiable, which happens on none of them.

What the tails do #

The single-valued tails fall into five classes, and two things about the list are worth stopping on.

The front of the tail is ignored, and so is its geometry. LR, ..LR and LRLR all converge at a half. LLR, L.LR, LL.R and RRL all converge at a quarter. The empty squares inside a tail make no difference to the rate, and neither do coins in front of the tail’s rearmost run — which is the rung below’s reading holding up on a population it was never measured on, since these tails are arbitrary words and its were three tidy runs.

And there is a class the rung below did not have. On six tails — L.R, R.L and their padded forms — the successive differences do not shrink geometrically. They become exactly nought. Widen the gap past three squares and the strip’s value stops changing at all: the gap has ceased to matter, rather than mattering less and less.

The largest class is the one with no decay at all: 54 of the 78 tails, every one of them a tail whose last two coins are the same colour or whose last coin has no coin of the other colour anywhere behind it. On those the strip has a net push in one direction that a wider gap simply lengthens, so the value marches off and the differences settle on a constant. It is the uninteresting answer, and it is more than two thirds of the population — which is worth saying plainly, because a law demonstrated on the interesting quarter of a population and quietly not mentioned on the rest is a law measured on a sample of its own choosing. Here the reading covers all of it: no decay is what the reading predicts on every one of those 54.

That class is not a rounding artefact and it is not a slow convergence misread. The differences are computed on exact rationals — every strip here is a number, and the values come out of the ordinary simplicity rule machinery rather than out of floating point — so a difference of nought is a difference of nought. What produces it is the tail being one coin, one empty square and one coin of the other colour, and nothing else in the eighty produces it.

Wrong only in one direction #

Reading the rate off the rearmost run is the rung below’s rule, and it can now be scored on a population fifty times larger than the one it was found on. The result is one-sided in a way worth naming.

On 62 tails the reading names the rate and is right. On 10 it names a rate — a half, in every case — where the values do not settle at all, because the strip is unbalanced enough that widening the gap keeps changing it by roughly a constant. On the 6 above it names a half where the differences vanish outright. And on none does it name a rate different from the one the family actually converges at.

So the rule over-predicts decay in two different ways and never mis-describes decay that exists. That makes it a filter rather than an answer: read the run and learn what the rate would be, and find out separately whether there is one. It is the same shape wrong in one direction only found for a very different test in a very different corner of this site, and it is worth the same treatment — a one-sided rule is not a failed two-sided one, it is a cheaper object with a narrower claim.

The 62 also include every tail whose run is nought — a tail whose last two coins are the same colour, where the reading predicts no decay and there is none — so the count is not flattered by easy cases. Forty-four of the seventy-eight are of that kind, and they are as much a prediction as the others.

Where a prefix still speaks #

Two tails break the grouping, and they are mirror images: LRRL and RLLR.

Behind sixteen of the eighteen prefixes, LRRL converges at a quarter, which is what its rearmost run says. Behind L.L and R.L it converges at a half. RLLR does the same behind L.R and R.R.

The four exceptional prefixes have one thing in common and it is a small thing: each is a prefix whose own last coin is isolated by an empty square. LL and LR and RL and RR all behave; L.L and R.L do not. A single empty square, sitting in front of a gap that is being widened without bound, changes what happens at the far end of the strip.

The four are also not a special corner of the tail space. LRRL and RLLR are the only tails in the eighty on which any prefix disagrees, and they are ordinary-looking words: two coins of one colour between two of the other, which is the shape nothing worth fighting over built Shove’s first example out of. Nothing about them is marked out in advance, and the sweep found them rather than being pointed at them.

The obvious objection is that the sweep has not gone far enough and the ratio is still moving. It has, and it is not. Taken out to eleven empty squares the ratios lock on the fourth term and stay there: 0.5 nine times running behind L.L, and 0.25 nine times running behind LL. Two families, the same tail, an unboundedly large gap between the prefix and the tail, and two different answers to the bit.

That is the single result on this page that the rung below’s sentence forbids. Governed by its last run and nothing else is right on 78 tails and wrong on two, and the two are wrong because of a coin arbitrarily far away from the run in question, separated from it by an arbitrarily wide stretch of nothing.

Why an empty square can carry that far #

The mechanism is not mysterious once the moves are looked at, though it is easy to miss.

Push moves a coin one square towards the cliff, pushing everything in its path. A coin with an empty square in front of it therefore has a move that changes only its own position; a coin backed up against another coin has a move that changes both. So a prefix ending L.L offers Left a move that is local, and a prefix ending LL does not. The widening gap does not insulate the tail from that difference, because the difference is not about distance — it is about which moves exist, and a move that exists at one gap width exists at every gap width.

What the widening gap does insulate the tail from is everything quantitative about the prefix. Sixteen prefixes of quite different values all give the same rate; the exceptions are not larger or closer, they are structurally different. So the suffix law is not false so much as stated at the wrong grain: the rate is a function of the tail together with one bit of the prefix, and that bit is whether the prefix’s last coin can move on its own.

This page does not have that as a theorem. It has it as a description of four exceptions among 1,440 families, which is a hypothesis with a small sample and is labelled as one. What would settle it is a prefix census: sweep every prefix to five or six squares against these two tails and check that the rate is a half exactly on the prefixes whose last coin is isolated. That is one sweep and it was not affordable here, because the strips reach sixteen squares and the recursion is what it is.

The control widens the law rather than breaking it #

The other way to move a row is Shove, which is Push with one clause changed: the moved coin goes to the end of the row rather than one square along. Its sequences are arithmetic rather than geometric — a constant increment per extra empty square, so the value never converges — and the rung below used it as the control that compounds.

Asked the suffix question, Shove gives the same answer as Push. The increment is a function of the tail on all 26 tails tested, with no prefix changing it. That was not the expected result, and it is the better one.

What separates the two games is how much of the tail each reads. Push gives LR and LRLR the same rate, a half, because it reads only the tail’s rearmost run and the two tails have the same one. Shove gives them different increments — $-1/2$ and $-5/8$ — because it reads the whole tail. Push’s answer at LLR is a quarter and Shove’s is $-1/4$; at RLLR Push says a quarter again and Shove says $-3/8$.

So the suffix law is not a fact about Push’s geometry. It survives the clause change that destroys the geometric convergence, the numeral reading and everything else on this ladder, which suggests it is a fact about widening gaps in this family of games rather than about either game in particular. What the clause decides is the resolution: one run of the tail, or all of it.

What this cannot say #

It is a statement about rates and not about values. Where a family ends up is emphatically the prefix’s business — the sixteen prefixes behind LRRL give sixteen different limits — and nothing here touches that. The clean division is that the prefix chooses the destination and the tail chooses the speed, and only the second half is a law.

The tails are short. Four squares is enough to hold a rearmost run of three and a coin in front of it, which is what the reading needs, and it is not enough to ask whether a tail’s own tail behaves like a tail. That recursion is the obvious next thing and it is not here.

It says nothing about strips that are not numbers. Every family here was checked to be a number at every gap width before its ratios were taken, and the ones that are not were dropped. On this ladder that is a small exclusion, because a Push strip with a cliff is nearly always a number — but nearly always is the kind of qualifier what a value leaves out exists to warn about, and a rate is a statement about numbers converging and has no meaning for a position that is not one.

And the two exceptions are a sample of four families. Four families out of 1,440 is enough to refute nothing in front matters and is not enough to establish what replaces it. The description offered above — that the deciding bit is whether the prefix’s last coin can move alone — fits all four and has not been tested against a fifth.

Normal play throughout, as on every rung of this anchor. Under misère play none of these positions is a number and the ratio of successive differences is not a quantity that exists.

Where the ladder goes next #

The push anchor has seven 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, and now what the rate is a function of.

The rung above is the prefix census. This page has a hypothesis about four families and the sweep that would settle it is completely specified: every prefix to five squares against LRRL and RLLR, with the rate recorded and compared against whether the prefix’s last coin is isolated. If the split is exact the law gets its correct statement — the rate is a function of the tail and one bit of the prefix — and if it is not, the four exceptions are something else and the description offered here is a coincidence with a sample of four. Either answer is worth having, and unusually for this ladder the experiment is fully written down before it is run.

Two neighbours are worth the trip. The cliff a cut invents is where the geometric convergence arrived and where the rate first became a quantity, and it is the page this whole line of questions descends from. And 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, which is worth reading beside a result about a gap that grows until it stops saying anything.