The family that lost a member

The best account of which family a wrecked stem keeps is the offset: take the smaller counted number when the removal landed no further back than it, the larger when it landed beyond. Twenty-five of thirty, and the arithmetic form of it explains nothing.

Restated in the arrangement rather than in the sequence it stops being arithmetic. An organ five places back from the tip of a stem counted at 5 and 8 spirals lies on the tip’s own five-chain; an organ eight places back lies on the eight-chain. So the rule reads as a statement about whose neighbour was taken, and it comes with a prediction anybody would make: the chain that lost a member is the chain that breaks.

It is refuted, and it is refuted backwards. That is a better outcome than a confirmation would have been, and this essay is mostly about why.

A prediction that fails in the direction nobody proposed is worth more than one that succeeds, because success is compatible with the prediction being a restatement of the data it was drawn from. This one was drawn from an intuition about chains, scored on rows the intuition had never seen, and came back inverted at every row — which means the intuition was about the wrong object and now names the right one. The census that raised it could not have found this, because the column it needed was never in the table.

The column the census never carried #

The census records, for each stem that never repaired, the lattice, the offset and the surviving family. It does not record which family the removed organ belonged to, and adding it is one line of arithmetic: an organ k places back lies on the tip’s p-chain exactly when p divides k.

Of the thirty wrecked offsets in the census, nine remove an organ that lies on exactly one of the two chains. Seven of those are members of the smaller family and two of the larger. Twenty-one remove an organ that lies on neither. None removes an organ on both, which is not an accident: the two counted numbers of a contact pair are coprime, and no offset inside the front is a multiple of both.

The coprimality is worth a sentence because it is what makes the reading testable at all. If the two counted numbers shared a factor there would be offsets belonging to both chains, and every such row would be silent for a different reason — not because the organ was nobody’s neighbour but because it was everybody’s. A pair whose numbers share a factor is exactly what a wrecked stem can be counted at when it lands somewhere the lattice did not come from, so the clean split here is a property of the lattices this census is drawn from rather than of contact pairs in general.

Nine of nine, the other way #

On all nine, the family that lost a member is the family left standing.

• A stem counted at 5 and 8 spirals, cut five places back, keeps the five.
• The same lattice cut eight places back keeps the eight.
• A Lucas stem counted at 4 and 7 spirals keeps the four when cut four back and the seven when cut seven back.
• The 7/11 lattice cut seven back keeps the seven; the 8/13 lattice cut eight back keeps the eight.

There is no scatter to argue about and no threshold to tune. The reading was stated before the column was computed, it is a binary prediction on nine independent rows, and it lost all nine.

Nine rows is not many, and it is worth being explicit about what nine buys. If the two outcomes were equally likely on each row, nine of nine in one direction happens by chance about twice in a thousand tries. That is not a p-value anybody should lean on — the rows are not draws from an urn, and two of them share a lattice — but it is enough to say that the result is not the shape a coin makes, and the direction was fixed in advance by a prediction that had already been written down.

Why backwards is the right way round #

The prediction had an intuition behind it — take a link out of a chain and the chain is what breaks — and the intuition is about the wrong object.

A contact family is not a physical thread. It is a set of organs whose angular spacing happens to be a particular lag, and the organs above the hole are still spaced at that lag after the removal, because the rule that placed them is still minimising the same sum of inverse powers. Removing one member does not unpick the others.

What the removal actually costs is the other chain. Every organ near the tip was positioned against a neighbourhood that included the one now missing, and the members of the family the removed organ did not belong to are exactly the ones whose spacing was set with it in view.

So the correct reading is the opposite of the stated one and it is mechanism-shaped in the same way: the chain that keeps its spacing is the chain whose member was taken, because that chain’s remaining members are still mutually consistent while the other chain’s are not.

That has a consequence worth stating even though this census cannot test it. If the surviving chain is the one left mutually consistent, then what survives is not really a family at all — it is whatever subset of the arrangement the removal did not put out of register. The family language is a convenience that fits because the two contact chains are the only subsets a settled lattice has at that scale. On an arrangement with three comparable lags, or one where a removal falls between chains, the same mechanism would predict something the vocabulary of pairs cannot express, and the pair a counter returns would stop being the right way to ask.

What it does not cover #

Twenty-one offsets of thirty remove an organ on neither chain, and on those the reading has nothing whatever to say.

That is the honest limit and it is a large one. A rule that is perfect on three tenths of a census and silent on the other seven is not the account; it is a clue about what the account will look like. Quoting the nine without the twenty-one would be reporting a subset dressed as a result, which is the failure this collection has already caught itself making with “the shortest hop survives” — right at twelve of twenty-nine and stated as though it were the rule.

Two of those facts fit together, and the fit decides what the reading is actually worth.

On all nine answerable rows the chain reading and the offset rule return the same answer. They have to. An organ five back on a 5/8 stem is on the five-chain, and is also a removal landing no further back than the smaller counted number, so both say five; an organ eight back is on the eight-chain and lands beyond the smaller number, so both say eight. The same holds at 4/7, at 7/11 and at 8/13. The two accounts agree on every row either of them can be checked against.

So the chain reading does not win a single row the offset rule was losing. What it supplies is not accuracy but the boundary — it says why the rule’s cut falls at the smaller counted number rather than anywhere else, which is the one thing the arithmetic form could never say about itself.

It also forces a conclusion about the failures. The offset rule is right at twenty-five of thirty, so five rows defeat it; and since it agrees with the chain reading on all nine rows where the organ taken was somebody’s direct neighbour, and the chain reading takes all nine, every one of those five failures lies inside the silent twenty-one. The rule breaks exactly where the mechanism has nothing to say, and nowhere else.

That is encouraging and damning at once. Encouraging, because two accounts that fail in the same place is what one expects when one of them is the mechanism behind the other. Damning, because it means the nine rows carry no information about the disagreement between them: the census that could separate a mechanism from an arithmetic coincidence is exactly the census of offsets that are nobody’s neighbour, and that census is the one this collection has not run.

The twenty-one are not a ragged remainder either. They are the offsets strictly between the two counted numbers and below the smaller one, and the offset rule covers them by arithmetic without saying what is physically true of them. Any account that finishes this thread has to say what a removal does when the organ taken was nobody’s direct neighbour.

There is a reading available for those rows and this essay deliberately does not adopt it. An organ four places back on a 5/8 stem is not on either chain through the tip, but it is on the five-chain through the organ one place above the tip, and on the eight-chain through some organ further down. Chains are everywhere; what makes the tip’s own two special is that the tip is where the next organ is about to be placed. Extending the reading to the twenty-one would mean choosing which organ’s chains count, and that choice is exactly the sort of free parameter this collection has been caught by before. It is a hypothesis to test, not a gap to fill in with a definition.

Three controls #

The column is arithmetic, not judgement. Whether an organ lies on a chain is decided by divisibility, computed from the counted pair the figure itself reports. Nothing about the surviving family enters the classification, so the score is not circular.

The rows are independent. Nine offsets across five lattices on two branches, each a separately grown stem with its own control. They are not nine readings of one run.

And the reading is not the offset rule in disguise. On these nine rows the two agree, which is why the offset rule scores well: an offset equal to the smaller number is at most the smaller number, and an offset equal to the larger is beyond it. The reading here is narrower and says something the offset rule does not — why those rows come out as they do — and it is falsifiable in a way the offset rule is not, because a single row where the lost chain broke would have ended it.

What would settle it #

The reading covers nine rows because only nine offsets in this census are multiples of a contact number. That is a property of the census, not of the question, and it can be changed.

A stem at a finer rise has a deeper front and larger counted numbers, so more of its offsets are multiples of one of them: at 8/13 the answerable offsets are 8 and 13, at 13/21 they are 13 and 21. Sweeping the offset at two or three finer rungs would take the reading from nine rows to twenty or thirty without changing what is being asked, which is the cheapest strengthening available.

The second thing is a genuine test rather than more of the same. If the chain that keeps its spacing is the one whose member was taken, then removing two organs from the same chain should leave that chain standing more surely than removing one from each — and the two-organ census already exists.

Where this leaves it #

The thread has narrowed the question four times: to one of two families, to not-the-shorter, to the offset at twenty-five of thirty, and to a rule that needs the rise attached. This is the first narrowing that says anything about why, and it says it in the only way this collection accepts — by making a prediction, scoring it on rows chosen before the answer was known, and reporting that it came out backwards.

A refutation that hands back a cleaner statement than the one it refuted is the best thing a test can do. What it does not hand back is coverage, and the next move is not another argument. It is more rows.