One offset, two answers

Two essays have narrowed the question. A stem that never repairs after a single removal keeps one lattice hop rigid; the hop is one of the two contact families at twenty-nine of thirty offsets; and which of the two is not decided by which is shorter. What is left is a binary choice with the length explanation removed, and one obvious remaining candidate: where the cut landed.

This essay scores every rule that can be written over the offset and the two counted numbers, reports the best one, and then reports the pair of runs that says no rule of that form can be the whole account.

Five rules, scored on the same thirty offsets #

A rule here is a function that takes the counted pair and the offset and returns one of the two numbers. Five of them are worth stating, and all five are scored on the same census rather than on the rows that suggested them.

The rules, and what each gets right:

The winner is worth reading aloud, because it is almost embarrassingly simple. Take the smaller of the two counted numbers if the removal landed no further back than that number; take the larger if it landed beyond it. At a 5/8 lattice that means an organ removed four or five places back leaves the five-family standing, and one removed six, seven or eight places back leaves the eight.

That the two nearest rivals are so much worse is worth a line. “Always the smaller” is the same rule with the second half removed and it drops seven offsets; “always the shortest step” — the account refuted one rung down — is twelve. So the offset is carrying real information rather than being a proxy for something already known.

What twenty-five of thirty is worth #

A score on a binary choice needs a null, and the null here is easy to state: thirty offsets, two answers each, decided by nothing.

A coin gets fifteen. The two constant rules get eighteen and eleven, which is what a coin looks like from either side — about one census in five would produce eighteen or better by chance alone, so “always the smaller count” is not a finding and is only in the table to make that visible. Twenty-five is a different object. Under the same null, twenty-five or more of thirty comes up about once in six thousand, and five candidate rules were scored, so even charging the search its full price leaves about once in twelve hundred.

Two things stop that being a stronger claim than it is, and both are worth stating. The offsets are not thirty independent trials: four of them come from one lattice, and a lattice whose behaviour is consistent contributes correlated rows, which makes the true tail fatter than the arithmetic above. And the score is not the interesting quantity anyway — a rule that got twenty-nine of thirty would still be refuted by the contradiction below, and a rule that got twenty of thirty would still be worth reporting if its failures had a shape.

What the number does establish is the thing it was computed for: the offset is not being read into noise. A quantity that decided nothing would have produced a score near fifteen for every rule stated over it, and the spread from eleven to twenty-five is too wide for that. Whatever the missing ingredient is, it is a correction to an offset rule rather than a replacement for one.

Where the rule fails #

Five offsets refuse it, and they are worth naming rather than counting, because a rule with a hit rate is a rule whose failures ought to be described.

• At 8/13 with a rise of 0.005, an organ six places back should leave the eight standing and leaves the four — which is the one survivor in the whole census that is not a contact family at all, so it is a failure of a different kind.
• At the same lattice, an organ nine places back should leave the thirteen and leaves the eight.
• At Lucas 4/7 with a rise of 0.020, five places back should leave the seven and leaves the four.
• At Lucas 4/7 with a rise of 0.013, three places back should leave the four and leaves the seven.
• At Lucas 7/11 with a rise of 0.008, eight places back should leave the eleven and leaves the seven.

Four of the five have the same shape: the removal landed beyond the smaller count, the rule therefore says the larger, and the smaller survived anyway. So the rule’s second clause is the weak one — the first clause, which says that a cut inside the smaller count leaves the smaller family standing, is right at every offset it applies to except one.

The pair of runs that ends it #

A hit rate of twenty-five in thirty is consistent with a better rule over the same two ingredients existing and not yet having been found. What is not consistent with it is two runs that agree on both ingredients and disagree on the answer.

The census contains fourteen places where the same counted pair is cut at the same offset on two different lattices. Thirteen of them agree. One does not.

At a Lucas 4/7 lattice with a rise of 0.020, an organ removed five places back leaves the four-family standing. At a Lucas 4/7 lattice with a rise of 0.013, an organ removed five places back leaves the seven.

Same rule. Same branch. Same counted pair — a counter shown either stem returns 4/7 and has nothing else to say. Same offset. Different surviving family.

So no function of the counted pair and the offset can produce both rows, and that is a stronger statement than any score. It does not say the rule above is a poor summary; it says that the class of rules the search was over does not contain the answer, and that a better member of that class cannot exist.

Why the rules were scored on the whole census #

A rule that is proposed after looking at some rows and then reported on those rows is not a measurement, so it is worth saying how the five above were arrived at and what would have happened otherwise.

Each of them is a rule somebody would state out loud without having seen the table. “The shortest step survives” comes from the mechanism — the rule minimises a sum of inverse powers of distance, so it should hold what is nearest. “Always the smaller” and “always the larger” are the two constant answers, and a census on which neither of those scored well is a census where something is varying. “The nearer of the two to the offset” is the obvious continuous version. The winner is the obvious discrete one.

All five are scored on all thirty offsets. That is not a formality: the winning rule scores 25 of 30 over the whole census and would score 12 of 12 over the 5/8 rows alone. Restricted to the lattice that suggested it, it is a law; over the census it is a summary with five exceptions, and the difference between those two descriptions is the whole reason for sweeping more than one lattice.

What the rise is doing there #

The two rows differ in one thing: the rise. Both are 4/7 lattices, but one sits near the coarse end of the 4/7 rung and one nearer its middle, and the divergence they settle at differs — 138.86° against 137.35°.

That is exactly the quantity a counter cannot see. A rung is a range of rises over which the same pair is returned, and within it everything geometric changes: the divergence, the two step lengths, the ratio between them, how close the lattice is to the boundary where the pair changes.

There is a measured effect on the same rung that points the same way. The displacement a removal causes at the offset equal to the larger count grows steadily down a rung — on the golden branch at 5/8 it runs 2.58°, 4.92°, 8.91°, 11.95° from the coarse end to the fine — so the larger family gets more responsive as the stem goes down its rung. A choice between two families that is sensitive to how strongly each responds would be expected to change hands somewhere on the rung, and to do so at a place the counted pair says nothing about.

What the offset is a proxy for #

The winning rule uses the offset as a raw number of organs, and it is worth asking what that number stands for, because “the cut landed inside the smaller count” does not obviously mean anything on its own.

An offset of k organs is an offset of k places in the sequence and something quite different in the arrangement. The organ five places back from the tip of a 5/8 stem is a member of the five-family through the tip; the organ eight places back is a member of the eight-family. So the offsets the rule’s first clause covers — one through five — are exactly the offsets at which the removed organ is not yet a five-family neighbour of the growing tip, and the offsets past it are where it is.

Read that way the rule is not about counting places at all. It says: the family whose member was removed is the one that does not survive. Take out a five-family neighbour and the eight survives; take out an eight-family neighbour and the five does. That is a mechanism-shaped statement rather than an arithmetic one, and it would explain why the offset matters and why the length of the step does not — the shape a mechanism claim has to take if it is to be worth more than a fit.

It also predicts the failures should be at offsets that belong to both families or to neither, and three of the five are: nine at an 8/13 lattice, eight at 7/11, three at 4/7. Whether that survives being made precise is not settled here, and saying so is the point of stating it as a reading rather than as the result.

A control: within a rung, most rows do agree #

The disagreement is one row of fourteen, and the other thirteen are worth reporting, because a census in which every repeated cell disagreed would be saying something about reproducibility rather than about the rise.

The 5/8 pair appears at four rises — 0.016, 0.013, 0.010 and 0.008, a factor of two in the rise. All four are cut four places back and all four keep the five. The two that also wreck at six and seven places back both keep the eight at both. So across a factor of two in the rise, with the divergence sliding and both step lengths changing, the survivor at a given offset is the same at every rise but one — and the one is on the other branch.

That is the right shape for a result that is nearly a rule. The offset does most of the work, the rise does a little, and the little it does is enough to make the rule false rather than approximate.

The rise is a small term, and that is a constraint on what it can be #

The contradiction closes the class of rules, and it is easy to read that as saying the rise matters as much as the offset. The census says the opposite, and the opposite is the more useful reading.

Fourteen cells in the census repeat a counted pair and an offset on two different lattices. If the rise were a co-equal ingredient — if the survivor were as much a function of where a stem sits on its rung as of where the cut landed — those fourteen would agree about half the time. They agree thirteen times. The 5/8 pair is cut four places back at four rises spanning a factor of two, with the divergence sliding and both step lengths moving, and keeps the five at all four.

So the missing quantity is nearly constant along a rung and changes hands somewhere inside one. That is a real constraint and it rules things out. It is not the divergence, which slides continuously and by more than a degree across the disagreeing pair. It is not the ratio of the two step lengths, which moves smoothly from one rung boundary to the other and would flip the answer at a predictable place on every rung rather than on one. Anything that varies smoothly and appreciably along a rung would have shown up as disagreement in far more than one of fourteen cells.

What fits is a quantity with a threshold in it: something that stays on one side of a comparison over most of a rung and crosses somewhere near one end. The two crossing response curves proposed below are exactly that shape, which is why they are the candidates rather than a general appeal to geometry — and it is why the sweep that would settle it is several rises inside one rung, where a threshold would announce itself as a single hand-over, rather than more rungs, which would add rows without adding resolution.

What would settle it #

Worth stating, because “the rise is in it somewhere” is not a hypothesis and the census points at two things that are.

The first is the position on the rung. A rung is a range of rises over which the counted pair is constant, and a stem’s position inside it can be measured without reference to the pair: the settled divergence, or the distance to the rise where the pair changes. The one disagreement in the census is between a lattice near the coarse end of 4/7 and one nearer its middle, so a sweep of the offset at five or six rises inside a single rung — rather than at one rise per rung, which is what this census does — would say directly whether the survivor changes hands at a stated place on every rung.

The second is the relative strength of the two families’ responses, which is already measurable and has already been measured on one branch. The displacement a removal causes at the offset equal to the larger count grows monotonically down a rung. The same quantity at the offset equal to the smaller count has not been swept. If the survivor is the family whose response is weaker at that offset, the two curves should cross where the survivor changes hands, and that is a prediction with a shape rather than a hit rate.

Neither of those is a large experiment. Both are the same sweep this essay reports, run along a different axis, and the reason they are not here is that the axis was only identified by the row that refused the rule.

Where this leaves the question #

Better off than it was, and short of an answer, which is worth saying plainly.

The offset accounts for five sixths of the census by the simplest rule available. The five it misses are named. And the class of accounts that use only what a counter can see has been closed off, not by a poor score but by a contradiction inside it — which means the remaining work is to find the quantity that varies along a rung and decides the choice, rather than to keep trying arrangements of the same two numbers.

That is a smaller residue than the question started with. What a wrecked stem is, what it can settle at, how far it slips and over which family — all of that is settled. What is not settled is a choice between two, at five sixths accuracy, with the missing ingredient named and located.