The shortest hop was a coin flip

The first account anybody reaches for is that the rule keeps its shortest hop. It has everything a good guess needs: the rule places each organ at the minimum of a sum of inverse powers of distance, so it is a rule about being far from near things, and the family made of the shortest steps should be the one most strongly held.

It was refuted at twelve of twenty-nine across the census — right less than half the time, which reads as a decisive answer. This essay re-scores it along a single rung and finds sixteen of thirty-one, which reads as no answer at all. Both numbers are correct. The difference between them is the sampling, and the difference is the point.

The ordering reverses inside a rung #

Rank the lags of a settled stem by the length of their step across the surface — the distance the placement rule itself uses — and the two contact families come first and second. Which of them is first is not a property of the counted pair.

Across the 5/8 rung, from a rise of 0.018 down to 0.007, the shorter step belongs to the five at the coarse end and to the eight from 0.015 down. At 0.016 the two are within 1.3 per cent of each other.

So a census that takes one rise per rung is not sampling the reading’s own variable. It is sampling a variable the reading depends on, at one arbitrary point per rung, and reporting the result as though it were a property of the rung.

The reversal itself deserves a sentence, because it is not obvious that it should happen at all. The two contact steps are the two shortest distances across the surface between an organ and its neighbours, and their lengths depend on the settled divergence and the rise together. As the rise falls the stem stretches vertically and the divergence slides, and the two effects do not act equally on the two lags: one step is dominated by its azimuthal component and the other by its height. There is therefore a rise at which they must cross, and the only question is whether it falls inside a rung or between two. Here it falls inside — which is what makes the counted pair unable to report it.

Sixteen of thirty-one #

Re-scored on every wrecked offset at every rise of that rung — thirty-one rows, one counted pair throughout — the survivor is the shorter step at sixteen and the second-shortest at fifteen.

That is a coin flip, and a coin flip is a different result from a refutation. A reading that is right at twelve of twenty-nine has been tested and found wanting. A reading that is right at sixteen of thirty-one has been tested and found to carry no information about the answer — which is worse for it, and better for knowing what to do next.

The distinction is not pedantry. Twelve of twenty-nine invites the obvious follow-up: if the shortest step loses more often than it wins, perhaps the longest wins, and that reading was duly scored and found to hold at seventeen — also not a rule. Sixteen of thirty-one kills both at once. There is no reading of the form “the survivor is the one whose step is longer/shorter” that can do better than chance on rows where the ordering has been decorrelated from everything else, because on those rows the ordering is not carrying the signal. Scoring a statement and its negation on the same rows is the cheapest way to find that out, and it is the shape this collection uses throughout.

Why the two numbers differ #

The census’s twelve of twenty-nine is not wrong and it is not a smaller sample of the same thing. It is a sample in which the step ordering is correlated with the counted pair, because each pair appears at one rise and that rise fixes the ordering.

There is a second reason the two numbers differ in kind rather than in size, and it matters more than the arithmetic does.

A reading that scores twelve of twenty-nine is not merely unsupported. It is supported in reverse: on a binary choice between two families, being right at twelve rows means the opposite reading — the longer of the two steps survives — is right at seventeen. Seventeen of twenty-nine is not much, but it is on the right side of half, and it is exactly the sort of residue that gets picked up later as a hint worth chasing. A refutation that leaves its own inverse looking mildly promising has not finished the job.

Along the rung it does finish. Sixteen of thirty-one for the shorter step is fifteen of thirty-one for the longer, and neither number is anything. The two are not one right reading and one wrong one; they are a claim and its negation, both indistinguishable from a coin, on a sample built to vary the quantity they disagree about.

That is also the case for re-scoring a reading somebody had already answered, which is usually waste. The case is narrow and stateable. The original score was computed on a sample in which the quantity under test was pinned to the quantity used to select the sample, so the number it produced could not be read as being about that quantity in either direction — not as support and not as refutation. Re-scoring on a sample that breaks the pinning does not overturn the conclusion. It replaces a conclusion that happened to be right with the same conclusion held for a reason, and it shuts the door the first score had left ajar behind it.

Both numbers are quoted here for that reason rather than the later one alone. Sixteen of thirty-one on its own tells a reader nothing about whether the sampling was ever a problem, and reporting it without the twelve would repeat the original mistake pointing the other way.

Along a rung the correlation is broken by construction: the pair is held and the ordering moves. What survives the breaking is the information the reading actually carries, and it is none.

It is worth being precise about which correlation is doing the damage, because the census is not badly designed. Each lattice in it appears at one rise, and that rise was chosen to produce the pair. So within the census, “which pair” and “which step is shorter” move together — not because they are physically linked but because one rise was picked per pair. A reading about step lengths scored on that census is therefore partly scored on which pairs happened to be included, and changing the pair list would change its score without anything about the reading or the physics changing at all.

This is a general hazard and it is worth naming in the abstract, because this collection will meet it again. A sample assembled to vary one quantity holds others fixed by construction. Any reading later scored on that sample inherits the constancy, and its score is a statement about the sample’s design as much as about the reading. The only fix is to vary the quantity the reading is about, on purpose, which is what a rung sweep is.

What the reading was competing with #

The reading that replaced it is the offset — smaller family if the cut lands no further back than the smaller number, larger beyond — and it scores twenty-five of thirty on the same census. Re-scored along this rung it does better than a coin flip and worse than its census figure, for reasons that are about the front rather than about the rule.

Three readings, then, in order of what they cost to state and what they buy: the shortest hop, which is free and carries nothing; the offset, which is arithmetic and carries most of the census; and the family that lost a member, which is mechanism-shaped, is right on every row it applies to, and applies to nine.

Three controls #

The lengths are computed, not estimated. A step’s length is derived from the settled divergence and the rise, not measured off a drawing, so the ordering at 0.016 — where the two differ by 1.3 per cent — is exact arithmetic rather than a close call.

The result does not rest on the crossing. If the three rises nearest the crossing are dropped, the score moves by less than the difference between sixteen and fifteen. The coin flip is a property of the whole rung, not of the rises where the two steps are nearly tied. That check matters because the obvious objection to a null result is that it was manufactured by including rows where the quantity under test barely varies — rows near a crossing, where “which is shorter” is decided in the third decimal place. Dropping them makes the reading no better, which is the direction the objection needed.

And thirty-one rows is the whole rung, not a selection. Every offset out to two organs past the front is cut at every rise; the rows scored here are every one that failed to repair, and the ones that repaired are reported as such rather than dropped. The thirty-one are also not thirty-one independent experiments, and the essay does not claim they are: rows at neighbouring rises share most of their geometry, so the effective number is smaller than the count. That cuts against reading anything into a sixteen-fifteen split, which is exactly the reading being avoided — the claim is that the split is indistinguishable from chance, and correlated rows make it harder, not easier, to say anything stronger than that.

What it does not license #

It does not license re-scoring every earlier result along a rung and expecting them all to dissolve. Two of them cannot dissolve this way, and the reason is worth stating.

That the survivor is a contact family at all is a statement about which lags are available, and the available lags are what the counted pair names. Holding the pair and moving the rise cannot disturb it, and the sweep here confirms it: every one of the thirty-one rows keeps one of the two.

That a wrecked stem keeps exactly one hop rigid is a statement about the motif it settles into, measured against a control that shares its history, and it is unaffected by which hop that is.

What is fragile is anything scored over the census as a set of comparable rows. The census was built to answer which pair, and it answers that. Rows drawn from it are not thirty independent tests of a rule about step lengths.

The same caution applies to this essay’s own thirty-one rows, in the other direction. They are one rung of one branch, and the reading could carry information on a rung where the ordering does not reverse — where the shorter step belongs to the same family throughout, and the census’s correlation is a fact rather than an artefact. Whether such a rung exists is a question about the arithmetic of the ladder, and answering it needs the same sweep run on the other branch, where the divergences and the step lengths are different numbers.

Where this leaves it #

A refutation that turns into a coin flip is not a weaker result. It is a different one, and it changes what the next measurement should be.

Against a refutation, the move is to find the reading that beats it. Against an absence of information, the move is to stop scoring readings over this census altogether and to sweep the quantity each one is actually about — which for the shortest-hop reading is the step ratio, and for the offset rule is the depth of the front.

The reading is not resurrected by any of this. Sixteen of thirty-one is not evidence for it. What the number says is that the census figure of twelve of twenty-nine was never the measurement it looked like, and that a reading can be disposed of for the wrong reason and stay disposed of for the right one.

That is a small thing to be careful about and it is the whole difference between a collection that accumulates results and one that accumulates numbers. The earlier essay reached the right conclusion; its arithmetic simply was not doing the work the prose gave it credit for. Recording that here, against the same reading, with the better sample, costs one sweep and leaves the conclusion where it was — which is the only outcome that would have been worth the sweep either way.