One rung, two answers

A stem that never repairs after organs are removed keeps one lattice hop rigid, and the hop it keeps is one of the two contact families at twenty-nine of thirty offsets. Which of the two is not decided by which is shorter. The best account left standing is the offset — take the smaller counted number if the removal landed no further back than it, the larger if it landed beyond — and it is right at twenty-five of thirty.

It is refuted by exactly one thing, and the refutation is worth restating precisely because it is so small. Two runs agree on the counted pair, agree on the offset, and disagree on which family survives. They differ in the rise.

A single disagreeing pair is the kind of result that can be a mistake, a tolerance, or a seed. This essay makes it reproducible by sweeping the quantity the two runs differed in — and it turns out not to be an anomaly at all but the ordinary behaviour of a rung, invisible until now because every census this collection has run sampled one rise per rung.

A rung, swept at a thousandth #

A rung is a range of rises over which a counter returns the same pair. Sampling one rise from each is the natural thing to do when the question is which pair, and exactly the wrong thing when the question is what else varies.

Swept from 0.018 down to 0.007 in steps of a thousandth, twelve rises return the same pair. Nothing a counter can measure separates them.

The choice of a thousandth is not arbitrary and it is the only free parameter in the sweep, so it is worth stating what it buys. Coarser than that and the rung holds four or five samples, which is what every earlier census here effectively used and is too few to see an ordering reverse inside it. Finer and each stem still settles, but the twelve rises become twenty-four that say the same thing at twice the cost, because what is being resolved is a monotone slide rather than a feature with a width. A thousandth is the resolution at which the rung stops being a point and has not yet become a continuum.

Two quantities move across those twelve rises, and a counter is blind to both.

That blindness is not a defect of the instrument. A spiral count is deliberately a topological reading: it follows chains of near neighbours and reports how many run in each direction, and it returns the same answer for every arrangement in which those chains connect the same way. A whole rung is, by construction, the set of geometries about which it says one thing. So asking a counter to distinguish two stems inside a rung is asking it to do the opposite of its job, and any account of ablation stated purely in counted numbers inherits that limit whether or not it declares it.

The divergence slides, and the two steps change places #

The settled divergence runs from 136.5469° at the coarse end to 137.8672° at the fine one — a slide of one and a third degrees, monotonic apart from a flat pair at the top, and entirely inside one rung.

The second is sharper, and it is the one this collection has an argument riding on. Rank the lags 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 changes hands inside the rung.

At the coarse end the shorter step belongs to the smaller family. At 0.016 the two are within 1.3 per cent of each other. From 0.015 down the shorter step belongs to the larger family. The whole reading that “the rule keeps its shortest hop” was refuted against is a statement about an ordering that reverses under a parameter the counter cannot read.

The survivor changes hands with it #

The offsets that never repair grow from one at the coarse end to five near the fine one. That is the front deepening: the run of recent organs at which a removal is felt at all is the larger of the two counted numbers, and as the rise falls there are simply more places a cut can land and fail to heal.

The deepening is worth separating from the survivor question, because it is the part that needs no new mechanism. A front three organs deep offers three places to cut; a front eight organs deep offers eight. If the rule for which family survives were written down correctly and never changed, the number of wrecked offsets would still grow down a rung simply because there are more offsets. What the grid adds is that the offsets which appear at the fine end are not a random extension of the ones already there: they are the offsets past the smaller counted number, and they are exactly the ones that keep the larger family.

Read the grid along any row and the counted pair is constant. Read down the row at an offset of four:

• from 0.018 to 0.008 it keeps the five;
• at 0.007 it keeps the eight.

Same lattice. Same pair. Same offset. Different survivor. The disagreement that appeared once in the census appears here on demand, and the thing that produces it is the position on the rung.

What this does to the rule #

The offset rule is not overturned by this. It is relocated.

Its first clause — a cut inside the smaller count leaves the smaller family standing — holds at every offset it applies to, at every rise of the rung, with one exception at the very fine end. Its second clause is the one that needs a rise attached: the offsets beyond the smaller count only exist once the front is deep enough to have them, and they are the ones that keep the larger family.

It is worth putting a number on how much of the rung each answer owns, because the number is uncomfortable.

Offset four is the offset the two disagreeing runs shared. Across the twelve rises of this rung it keeps the five at eleven of them and the eight at one; the crossing sits at the fine end, between 0.008 and 0.007, and only a single rise inside the rung lies below it. The column is not evenly divided. It is eleven to one.

That ratio is this essay’s own result read backwards, and it says something awkward about how the result was found. A census that samples one rise per rung draws one cell from that column. Eleven times in twelve it draws a five, agrees with the rule stated over the offset, and records a row that supports it. Once in twelve it draws an eight, disagrees, and records the anomaly that sent this sweep out in the first place.

Before calling that unlucky it is worth doing the arithmetic on the census as a whole, because the answer is not what the phrase suggests. If every lattice in the census carried a column like this one — a crossing near one end, eleven rises agreeing with the offset rule and one not — then each lattice sampled at a random rise has one chance in twelve of landing past its own crossing. Twelve lattices were sampled. The probability that none of them landed past a crossing is (11/12)¹², which is 0.35, so a census of this size had about a two-in-three chance of turning up at least one disagreement.

That reframes the discovery. It was not a fluke that the census contained a contradiction; it was very nearly the expected outcome of sweeping twelve lattices, and a census of four or five would more likely than not have contained none. The luck was in the size of the sweep rather than in the draw, which is a better thing for a result to depend on and a worse thing to rely on twice — a census assembled the same way with half as many lattices would have reported the offset rule as unrefuted and would have had no way of knowing.

So the refutation was available because the sweep was large enough, and the column it came from is eleven to one against showing it. The history in which the census happened to sample 0.009 rather than 0.007 is a history in which the offset rule scores twenty-six of thirty rather than twenty-five, nothing looks wrong anywhere, and the missing ingredient stays invisible for exactly as long as nobody sweeps a rung.

The general form of that applies to every row this collection has published from a one-rise census. A rule that is right at eleven of twelve rises and wrong at the twelfth will be scored as right, with a small residue of anomalies, by any procedure that samples one rise per rung. The residue is not noise, and it does not shrink as more stems are measured: it is a fixed fraction of the rung’s width, so measuring more stems one rise at a time estimates that fraction more precisely while doing nothing whatever to explain it.

So the rule stated over the offset alone is a rule with the rise left out, and quoting it without the rise quotes half the argument. The honest form is conditional: given a front of this depth, the offset decides. The depth is a function of the rise, and the rise is exactly what a counter throws away.

Stated that way the rule stops being a fit and starts being a claim with a domain, which is the difference this collection keeps insisting on. A rule that scores twenty-five of thirty over a census assembled from one rise per rung is being scored on a sample that never varies the thing it depends on. The same rule scored along a rung — where the offset is held and the rise moved — is either right at every rise or it is not, and here it is not.

Three controls #

The pair really is constant. The claim is worth nothing if the sweep quietly crosses a rung boundary, so the counted pair is measured at every rise and the generator refuses to draw a grid whose columns are not all the same pair. Twelve of twelve return the same two numbers.

The steps are not tied. At 0.016 the two contact steps differ by 1.3 per cent, which is small enough to ask whether the ordering there is a measurement rather than a fact. It is a fact about the geometry — the lengths are computed from the settled divergence and the rise, not estimated from a drawing — but the essay does not rest on that rise. The survivor changes hands at 0.008 to 0.007, five rises away from the crossing, where the two steps differ by seven per cent.

And the sweep is not an artefact of its own resolution. A thousandth is fine enough to put twelve rises inside this rung and coarse enough that each is a settled lattice by the same test every other run on this site uses.

The ordering reverses in the wrong place #

One of the two moving quantities can be separated from the survivor by this sweep alone, and it is worth doing before appealing to a lattice the collection does not have.

The ordering of the two contact steps reverses at 0.015. The survivor changes hands between 0.008 and 0.007, five rises further down the rung. Those are not the same place and they are not close: between them lie five settled lattices at which the shorter step belongs to the larger family and the family left standing is still the smaller one.

So the reading the rule keeps whichever step is currently shorter is refused here for a second time, and refused more sharply than a hit rate refuses it. It is not that the ordering wins twelve of thirty across a mixed census; it is that on one rung, with everything else held, the ordering flips at one rise and the outcome flips at another, and the two are separated by half the rung.

What survives is weaker and more specific. The step ratio slides monotonically across the rung and could still be the acting quantity if what matters is its value crossing some threshold rather than its passing one — a ratio of about 1.07 with the larger family shorter, say, rather than a ratio of exactly one. The divergence slides monotonically over the same rises and fits equally well. Those two are what a rung with a non-monotone geometry would separate, and they are the honest remaining pair.

The discrete reading is finished, though, and finishing it costs nothing beyond reading two numbers off the same grid.

What is left over #

Two things this does not settle, and both are worth naming rather than implying.

Which of the two moving quantities is the one that acts. The divergence slides and the step ordering reverses across the same rises, so this sweep cannot separate them: any rule written over the position on the rung fits, and so does any rule written over the step ratio. Separating them needs a lattice where the two come apart — a branch on which the divergence slides one way and the ordering the other — and whether such a rung exists is a question about the arithmetic of the ladder rather than about ablation.

Whether the crossing sits at the same place on every rung. Offset four hands over between 0.008 and 0.007 on this rung, which is near its fine end. Nothing says the fraction of a rung the smaller family owns is the same on the next one down, and there is a reason to expect it is not: the front deepens as the rise falls, so a rung whose front reaches thirteen has offsets that no rung with a front of eight can offer, and the offsets past the smaller count are the ones the hand-over is about. A sweep of the 8/13 rung at the same resolution would say whether eleven-to-one is a number about this rung or a number about rungs.

Whether the same thing happens on the other branch. Everything here is one rung of the golden branch. The Lucas branch has rungs of its own and its own ordering of contact steps, and the pair of runs that started this was a Lucas pair.

Where this leaves the question #

The question this thread has been asking since the first removal is what decides which hop the rule keeps, and the answer has been narrowed three times: to one of two families, then to not-the-shorter, then to the offset at twenty-five of thirty. This narrows it a fourth time, and it is the first of the four that adds an ingredient rather than removing one.

What a counter reports is a pair. What decides the survivor is a pair and a position on the rung the pair does not name. That is not a failure of counting; it is a statement about what counting is for. A spiral count is a robust, scale-free description of an arrangement, and its robustness is exactly its blindness: it is the same number over a range of geometries, which is what makes it worth measuring on a real plant and what makes it insufficient here.

The next thing to do is not another sweep of this rung. It is to ask whether the family that survives is the one that did not lose a member — a reading the offset rule already suggests and the census cannot currently test, because it does not record which family the removed organ belonged to. That is one column, and it is the difference between a rule that fits and a mechanism that explains.