The side the census sat on

The ablation census is twelve lattices. Ten of them produce at least one stem that never repairs after an organ is removed, and those ten contribute thirty wrecked cuts between them. It is the table on which nearly every reading in this thread has been scored, and it was built to span the counted pairs and both branches, which it does: five pairs, two branches, rises from 0.032 down to 0.005.

Once each row carries the fraction of its own rung it was grown at, a property of the table appears that was invisible while the rise stood alone. Eight of the ten wrecking lattices sit past their rung’s handover. One sits before it. And one sits so close to a handover that its two contact steps differ by two and a half parts in a thousand, which is a lattice where the words shorter and longer mean nothing at all.

Counted by cuts rather than by lattices, because a lattice with five wrecking offsets contributes five rows to any score: twenty-five of the thirty wrecked cuts were made past the handover.

What that does to a reading #

The reading in question is that a wrecked stem keeps its shortest hop. It came out of the mechanism, which minimises a sum of inverse powers of distance and so ought to hold its nearest neighbours hardest, and it was scored at twelve of thirty and refused.

The refusal stands. What the position column changes is what the twelve was evidence of.

Past a handover the shorter step belongs to the larger of the two counted families. So on twenty-five of the thirty rows, “the shortest hop survives” and “the larger family survives” are the same prediction. The census scores them at twelve and eleven, and those two numbers are close because on five-sixths of the table they are one number.

That is not a defect in the arithmetic of the score. It is a statement about what a score on this table can distinguish. Two readings that agree on twenty-five of thirty rows cannot be separated by more than five rows, whatever they say about the mechanism, and five rows is not a lot of evidence to hang a distinction on.

How much a five-row difference can carry #

It is worth doing the arithmetic rather than gesturing at it, because “only five rows” is the kind of phrase that can be used to dismiss anything.

Thirty cuts, of which twenty-five agree between the two readings and five disagree. On those five, the shortest-hop reading is right once and the larger-family reading is right none of the times — a difference of one row. A census of this size cannot resolve a difference of one row: with thirty binary outcomes, two readings differing by one are indistinguishable by any standard this collection would accept, and by most it would not.

So the honest statement about the shortest-hop reading, after the correction, has two halves. It is wrong — twelve of thirty is far from thirty of thirty, and the rows it fails on are not close calls. And it is wrong for reasons this table cannot fully attribute, because on most of its rows it is not making an independent prediction at all.

That distinction is not pedantry. A reading that fails because the mechanism does not work that way is a dead end; a reading that fails because it was tested against its own restatement is an untested reading. Only the first is a result, and the census on its own cannot say which of the two happened.

Why the census landed there, and why any census would #

The obvious question is whether somebody chose badly, and the answer is that nobody chose at all — and that a person choosing sensibly would land in the same place.

A census row starts with a pair. Wanting a stem counted at five and eight, a person picks a rise that reliably gives five and eight, which means a rise comfortably inside the rung rather than near either boundary, because near a boundary the two candidate pairs have nearly equal steps and the counter’s answer is a decision about a near-tie. So rises get picked near the middle of their rungs.

And a handover is always in the coarse half — six of six on this ladder, at six to forty per cent. A rise picked near the middle of its rung is therefore past the handover as a matter of course. The sampling is not a mistake anybody made; it is what the sensible procedure produces, given a fact about the ladder that nobody had measured.

That is worth separating from the more usual kind of sampling complaint. The census is not unrepresentative of the lattices somebody cared about; it is unrepresentative of a quantity nobody knew was a quantity. The correction is not “choose better rises” but “record the side, and if the reading is about the ordering, straddle a handover on purpose”.

The one row on the other side, and the one row with no side #

Two rows are worth naming individually, because with only ten lattices each of them carries a tenth of whatever the table can say.

The lattice at a rise of 0.016, counted 5/8, sits at twelve per cent of its rung — just above the handover at fifteen. It is the only wrecking lattice on the coarse side, where the shorter step belongs to the smaller family. It wrecks at exactly one offset, and the family it keeps is the five.

That single row is the whole of the census’s evidence about the coarse side. One cut. It is consistent with the shortest-hop reading and with the smaller-family reading and with the offset rule, all three, and it distinguishes none of them.

The lattice on the Lucas branch at 0.008, counted 7/11, sits at thirty-three per cent of its rung — which is, to three decimal places, where that rung’s handover is. Its two contact steps differ by two and a half parts in a thousand. Any statement about which of them is shorter is a statement about the fourth significant figure of a quantity measured on a grid of 1,536 azimuth steps, and this collection’s own tolerance for calling two steps ordered is one per cent.

That row contributes four of the thirty cuts. It has been scored in every table here as though its ordering meant something, and it does not. Marking it as unordered rather than dropping it is deliberate: dropping rows that inconvenience a reading is how a score gets made true, and the four cuts are perfectly good evidence about everything except the ordering.

What survives the correction #

Most of the thread, and it is worth being specific about which parts and why.

That the survivor is one of the two contact families is untouched. That reading is about which lags exist, and the counted pair fixes those; the ordering between them is not in the claim.

That a wrecked stem keeps exactly one hop rigid is untouched for the same reason, and more strongly: it is measured against each stem’s own control, so it makes no comparison across lattices at all.

The offset rule — the smaller count while the cut lands no further back than it, the larger beyond — is scored at twenty-five of thirty, and the position column does not move it. Its five failures sit at eighty-nine, eighty-nine, nineteen, sixty-six and thirty-three per cent, which is a spread rather than a cluster: two of them are the single lattice at the far fine end of the 8/13 rung, one is the unordered Lucas row, and two are the pair of runs that closed the class.

What does not survive is the strength of the case against the shortest-hop reading. Twelve of thirty on a table that is nearly one-sided is weaker evidence than twelve of thirty on a balanced one, and this collection said the first while sounding like the second.

The experiment the correction implies #

If a census cannot distinguish the ordering from the family, then the way to test the ordering is to hold everything else and move only that — which is what a handover is for. Grow a band of rises around one, where the pair is constant and the divergence is held to a twentieth of a degree, and cut at every offset on both sides.

Done on two bands, on two branches, at fifty-five wrecked cuts: the survivor does not change while the ordering flips underneath it. That is the test the census could not run, and it answers the question the census could only gesture at.

It is also a much cheaper experiment than the census, which is the part worth remembering. Thirty-seven stems and a hundred and fifty cuts, all at one rung each, against a census spanning the whole ladder. A design that varies one thing does not need to be large.

The general shape #

This is the second time in two rounds that a result here has turned out to be partly about how the tables were built rather than about what they measured. The first was that a census taking one rise per rung samples everything the rise moves; this is the specific version of that with a name attached to the quantity.

Both have the same fix and it is not a methodological principle so much as a habit: before scoring a reading over a quantity, ask what the range of that quantity was in the rows being scored. Here the answer was a range of nearly zero, and it took one column to see.

The uncomfortable version is that the census was published, quoted and built on for three rounds before anybody asked. Nothing about the question required new machinery — the hop ranking was already in every row of the table — and what was missing was the idea that a ranking has a place as well as a value.

What the correction is not #

It is not a retraction. Everything published on this table is still there and most of it is unaffected, which is the ordinary case when a sampling problem is found: the tables were built for a question about the counted pair, they answered it, and a later question turned out to need a coordinate they did not carry.

It is not an argument for larger censuses. A census of thirty lattices instead of ten, built the same way, would sit past its handovers on twenty-five of thirty rows exactly as this one does, and would carry three times as much of the same one-sidedness. Size does not fix a design; the band experiment fixes it, and the band is smaller.

And it is not a claim that the position is doing anything causal. Nothing here says a stem grown at eighty per cent of its rung behaves differently from one grown at twenty because of where it sits. The position is a coordinate that happens to correlate with a quantity a reading was stated over, which is exactly what makes it a confound and exactly what stops it being a mechanism.

The distinction matters for what gets recorded. A confound is fixed by a design; a mechanism would be fixed by an explanation, and this collection has neither the explanation nor a reason to think one is owed. What the tables need is the extra column and the occasional band, both of which are cheap. What they do not need is a story about why eighty per cent of a rung is a different place to be, because the experiment that would have supported one found nothing.

What is still open #

Whether a census straddling handovers on purpose would find anything the bands do not. The bands are narrow — eighteen and nineteen rises, a few per cent of a rung each — and they hold the divergence by construction. A census that sampled each rung at, say, five per cent, twenty, forty, seventy and ninety would vary the position and the ordering and the divergence together, which is the confound this thread started with, but it would at least have rows on both sides of every handover.

And whether the same one-sidedness is in the other tables. The front-depth work, the jugacy work and the noise work all grow stems at rises chosen the same way, by the same procedure, for the same reason — so the presumption is that they sit past their handovers too, and the presumption has not been checked row by row. That is an afternoon’s arithmetic on tables already published, which is precisely the kind of debt this essay exists to point at rather than to leave implied. The reason to name it here rather than to quietly do it is that the answer is only interesting if it is written down before the checking starts: the presumption is that they sit past their handovers too, and a presumption recorded in advance is a prediction rather than a summary.