Read from the back forwards

The cliff a cut invents widened the gap between two Push runs and found the strip’s value falling geometrically, at a ratio of exactly $2^{-k}$ where $k$ is the length of the back run. It closed on the question a third run raises:

The rung above is the strip with three runs. If a two-run strip is a single numeral whose rate is set by the last run, the obvious question is what a third does — whether the rate is still $2^{-k}$ for the rearmost run alone, which would say the strip is read from the back forwards, or whether the rates compound, which would say something else entirely.

It is the rearmost run alone, and the sweep says so more strongly than the question asked for.

The last gap first #

Take the obvious version of the experiment: three runs, and widen the gap in front of the last one. A strip $.LR\,.LR\,.^gLR$ has its value falling by half each time $g$ grows, and $.LR\,.LR\,.^gLLR$ falls by a quarter each time, and $.LR\,.LR\,.^gLLLR$ by an eighth. The rate is $2^{-k_3}$, with $k_3$ the length of the run the gap sits in front of.

That is what the two-run result predicts and it holds exactly — to the last bit of a dyadic rational, not to a tolerance — on 40 of the 41 families a sweep to fifteen squares can measure. It is not, on its own, evidence for either of the readings the rung below offered. The gap being widened is immediately in front of the rearmost run, so the rearmost run and the run behind the gap are the same run, and the family cannot tell them apart.

The gap in the middle #

Widen the middle gap instead, and the two readings separate. Now the gap has one whole run behind it and another in front, so the run immediately behind the gap is $k_2$ and the rearmost run is $k_3$, and they can be given different lengths.

The rate is $k_3$’s. On $.LR\,.^gLLR\,.LR$ the ratios run $0.7083, 0.4853, 0.4924, 0.4962$ — heading for $\tfrac12$, which is $2^{-k_3}$ with $k_3 = 1$ — while $2^{-k_2}$ is a quarter and never comes near. On $.LR\,.^gLR\,.LLR$ the ratio is a quarter exactly, which is $2^{-k_3}$ with $k_3 = 2$, while the run behind the gap would say a half.

Widening the front gap, two runs away from the end, gives the same answer again. So the quantity that decides how fast a gap stops mattering is not a property of the gap’s neighbourhood at all. It is a property of the far end of the strip.

That is what read from the back forwards means, and it is worth saying plainly: a Push strip’s value responds to a change anywhere in it at a rate set by its last run.

The two ways the rate arrives #

There is one difference between the cases, and it is a difference in how rather than in what.

Widening the last gap gives the ratio exactly, after a transient of at most two terms. Widening an interior gap gives it in the limit: the ratios climb toward it — $0.4853, 0.4924, 0.4962$ — and do not reach it inside a sweep that can afford fifteen squares. The further forward the gap, the slower the climb: the last gap is exact on 40 of 41 families, the middle gap on 24, the front gap on 20.

The sweep is short enough that a tolerance test would be measuring the length cap rather than the game, which is why the census is a vote instead. For each family, which of the four candidate rates does the measured ratio lie closest to? The rearmost run’s rate wins all 123 families, and the compound rate $2^{-(k_2+k_3)}$ — the alternative the rung below named — wins none. All 123 families have runs of differing lengths, so the four candidates are four different numbers and the vote is between real alternatives.

Why the tail, and not the neighbourhood #

The mechanism is visible once the move rule is read carefully, and it is the same clause the rung below blamed for the cut failing.

A push moves the chosen coin and the run in front of it. So a coin near the front of the strip cannot be pushed without disturbing everything between it and the cliff, and a coin at the very back can be pushed with no consequence for anything ahead of it at all. The strip’s moves are therefore not symmetric in the two directions: the back of the strip is where the cheap moves are, and the front is where every move is expensive.

Widening a gap adds a move to whichever coins sit behind it. What that move is worth is decided by how much slack the position still has, and the slack that survives longest is at the back — because the rearmost run’s pushes are the ones nothing else has to pay for. A run of $k$ coins at the back has its own moves halving in value $k$ at a time, which is the $2^{-k}$, and every gap in front of it inherits that scale because the position’s remaining freedom is measured against the same reserve.

The criterion that cannot exist established the general form of this asymmetry — that no local reading of a Push strip can be right, because the moves available at a square depend on the whole prefix in front of it. This page is the other half of the same fact, seen from the value side rather than the reading side: what a change is worth depends on the suffix behind it. A Push strip is directional twice over, and the two directions carry different information.

Nothing worth fighting over is why any of this is a number at all. Every Push position is a number, so there is no temperature to confuse the reading and no fight to interrupt the fall; a sequence of values here is a sequence of dyadic rationals and nothing else, which is what makes a ratio a meaningful thing to measure. On a ladder where the positions were hot, the same experiment would be measuring two quantities at once.

Where the rate breaks, and what breaks it #

One family is worth printing in full, because it is the only place on this ladder where the simplicity rule is visible in a rate.

The strip $..LLR\,..LR\,.^gLR$ falls by $\tfrac1{16}$, then $\tfrac1{32}$, then $\tfrac1{64}$ — halving exactly — and arrives at $-2$. Then the next step is $\tfrac1{32}$, four times what the halving predicts, and after it the halving resumes from the new value.

$-2$ is the only whole number the sequence passes through. Every other term is a dyadic rational with a denominator of sixteen or more, and the value of a Push position is whatever the simplicity rule allows: the simplest number strictly between the best Left option and the best Right option. An integer is the simplest number there is, so a position whose confusion interval happens to contain one takes it, and the term the geometric fall would have produced is not the term the rule produces.

The rate resumes immediately afterwards. So the geometric law is a law about the shape of the sequence and not about every term of it, and the term it fails on is one the simplicity rule reached first. That is the ordinary relationship on this site between a smooth reading and the rule underneath it, and it is the same shape as the exceptions a second level of stops found on a different ladder — a formula that is exact except where a simpler description got there first.

Where the strip separates #

The rung below found the two-run limit failing to be the sum except when the back run is a single coin, and the law extends with nothing changed.

Extrapolate each family’s limit from its last difference on the measured ratio — an extrapolation, since no strip of infinite length is evaluated anywhere on this site — and compare it with the front two runs’ value plus the rearmost run’s own limiting value $-1/(2^k - 1)$. It agrees on all 26 families whose rearmost run is one coin, and on none of the 36 where it is longer.

A single coin is the case where the run has nothing behind its own front, so pushing it does not hand anything to a neighbour; anything longer keeps a piece of the interaction at every separation. That was the two-run reading and it survives a third run intact, which is a small piece of evidence that the object here is a tail rather than a pair.

The game where the rates do compound #

The alternative the rung below named is not idle. It is what the neighbouring game does.

Shove is Push with one clause changed: a shove moves the pushed coin and everything behind it, rather than the coin and the run in front. Run the identical sweep on Shove and both halves of the finding invert.

Nothing converges. Every one of the 33 Shove families is arithmetic — a constant increment per extra square, forever — so a Shove strip’s value diverges as a gap grows rather than settling. There is no rate to attribute to any run, because there is no convergence to have a rate.

And the increment compounds. For the last gap it is a clean $2^{-k_3}$ on all 11 families. For an interior gap it is $\tfrac58$, $\tfrac9{16}$, $\tfrac5{16}$, $\tfrac{21}{64}$, $\tfrac{37}{64}$ — dyadic rationals that are not powers of two at all, and that depend on every run behind the gap.

So the two candidate behaviours the rung below set against each other both exist, in two games one clause apart. That is the strongest form of control this site gets: not a null result but a positive result in the game the finding is being distinguished from, which says the measurement can tell the two apart and did.

What has not changed #

The parts are still not independent. Sixty-four three-run strips fit inside fifteen squares and three of them are worth the sum of their three runs — the same failure the rung below measured on ninety-three two-run strips, undiminished by having a third run to spread it over. The worst of them, ..LR..LR..LLR, is worth about $-0.436$ against a sum of $-1.8125$: a discrepancy larger than the value.

Knowing the rate does not repair that and was never going to. What it says is that the dependence has one parameter, and which parameter it is: the length of the last run. A correction term built from anything else — the gap, the neighbouring run, the number of runs — is looking in the wrong place, and this is the second rung running to say so.

The sum is the object is the thread this ladder keeps failing to join, and the failure is now specific rather than general. A Push strip is not a sum of its runs; it is a single object whose sensitivity to everything in it is graded by its own tail.

A control one clause away #

The finding here rests on a comparison with Shove, and the comparison is what makes it a finding rather than an observation. It is worth saying why that particular control is the right one.

A control has to differ in one thing. Shove and Push have the same board, the same pieces, the same wall, the same players and the same ending condition. One clause differs: a coin slides alone rather than shoving the run in front of it. So any difference in behaviour is attributable to that clause and to nothing else — not to the geometry, not to the sweep, not to how the values were computed.

And it has to be capable of behaving differently. A control that gave the same answer would say nothing; Shove compounds where Push does not, on the same strips and by the same measurement, so the property being tested is one this pair of games actually distinguishes.

Together those make the non-compounding a fact about the shoving clause. Without the control it would be a fact about Push, compatible with being a fact about strips, about walls, about how far apart runs were placed, or about the sweep’s range.

That is worth more than a wider sweep of Push alone. More strips would have confirmed the same statement about the same game; one clause changed tests where the statement comes from — and in a family where every game differs from its neighbour by a clause, that is the experiment always available and rarely run.

What this does not say #

It is a rate, not a value. Nothing here computes what a three-run strip is worth. The recursion does that and remains the only thing that does; what the rate gives is how much a change matters, which is a derivative rather than a value.

Fifteen squares is a short strip. Runs of one, two and three coins and gaps of one and two squares are all the sweep affords, and a three-run strip with a gap of six is already at the cap. Every claim here is about the first few terms of sequences that are infinite in principle.

The limits are extrapolated. They are computed from the last measured difference on the assumption that the ratio holds beyond the sweep, and a family whose ratio has not locked on the rate is not extrapolated at all. That is a decision recorded in the code rather than a detail: extrapolating from a term still inside a transient reports the length cap and calls it a limit.

And the picture cannot show a limit. Every figure here is a table of ratios, because the thing being argued about is a sequence and a sequence of dyadic rationals falling by halves is a picture with nothing in it after the third term. What a drawing could show — the strips themselves — is drawn in the rung below, and repeating it here would be showing the position rather than the finding. The short side only says how many makes the same concession on a different game and for the same reason: where the position is arithmetic, the honest figure is the arithmetic.

The convention, named #

Normal play throughout: a player who cannot move loses.

Push is played on a strip of squares carrying blue and red coins with a cliff at the left end. A player pushes one of their own coins one square left; the coins immediately in front of it — the contiguous run between it and the first empty square — move with it, and the move is illegal if that run is against the cliff. Shove is the same board with the other clause: a shove moves the chosen coin and everything behind it, and a coin at the cliff edge falls off.

A run here is the family the ladder has used throughout: $g$ empty squares, then $k$ blue coins, then one red one, written $.^gL^kR$. A three-run strip is three of those concatenated, and its gaps are the three runs’ own leading empty squares — front, middle and last.

The rate is the ratio of successive differences as one gap is widened a square at a time. A sequence whose ratio is a constant $r$ with $|r| < 1$ converges; one whose differences are constant does not.

The limit is an extrapolation, not an evaluation: the last measured value plus the last difference times $r/(1-r)$. No infinite strip is evaluated here or anywhere on this site.

Where the ladder goes next #

The push anchor has six rungs: 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, and now which run sets the rate.

The rung above is the tail as an object. 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. That is a grouping test rather than a fit, of exactly the kind the two numbers at the top used to settle whether a formula’s stated inputs were its real ones, and it is affordable on strips this size. If the grouping is single-valued, the rate is a function of the suffix and the ladder has an invariant rather than a pattern.

Two neighbours are worth the trip. A numeral in the empty squares is where the geometric behaviour comes from, and it is the one-run case this page keeps reducing to. And the other way to move a row is Shove given its own page, which is worth reading beside the control here: the game that compounds is not a foil invented for the comparison but a game with its own ladder and its own reasons.