The angles name the branch

A counting instrument that only ever returns Fibonacci numbers is not counting. It is a very compact way of writing down the opinion that plants are Fibonacci, and this site has spent four rounds of work showing that the opinion is a statement about a branch rather than about plants.

So the first thing to do with a new counter is take it somewhere the answer is not Fibonacci.

The Lucas branch, briefly #

The ladder of parastichy pairs is not a single sequence. The placement rule’s forks — the rises at which the counted pair changes — form a tree, and the Fibonacci pairs are one path through it. Keep the larger count at every fork and the divergence converges on 137.508°; make one different choice at the first fork and it converges on 99.502°, whose ladder runs 1/3, 3/4, 4/7, 7/11, 11/18. Those are the Lucas numbers, and plants on that branch exist — they are uncommon and they are not rare.

The reason the branch matters for a counting instrument is that its numbers are different in every way that a counter could accidentally rely on. Lucas pairs are not consecutive Fibonacci numbers; their ratio still tends to φ but the numbers themselves — 4, 7, 11, 18 — share no small factor structure with 5, 8, 13, 21; and the divergence angle is nowhere near 137.5°.

The readout on Lucas stems #

Seed the rule at 99.502° and hold the rise fixed, exactly as the golden-angle stems were held. At a rise of 0.013 the position counter says the pair is 4 and 7. At 0.008 it says 7 and 11.

The angle readout, given the divergence sequence and told nothing else, returns 4 and 7 on four stems in five at the first rise, and 7 and 11 on five in five at the second.

The arithmetic that produces them is the arithmetic of the previous essays with different integers in it. At the second rise the significant lags sort into 0 mod 7 and 4 mod 7; the spacing is seven and the offset is four; the pair is 7 and 7 + 4 = 11. Nothing in pairFromAngles knows what a Lucas number is, and nothing in it knows what a Fibonacci number is either — the only integers it has are the lag indices it was handed.

What a failure would have looked like #

A control is only worth running if its failure is legible, so it is worth saying what the Lucas stems would have shown had the readout been Fibonacci in disguise.

There are two shapes and both are visible without a judgement call. A readout with the integers baked in would refuse everywhere on the branch: the significant lags sort into 0 mod 7 and 4 mod 7, and an instrument looking for 0 mod 8 and 5 mod 8 finds neither, so its clearance tests fail and it returns nothing. Or — worse and equally legible — it would return a Fibonacci pair anyway, at which point the position counter on the same stem says 7 and 11 and the two instruments disagree on a head where one of them is known to be right.

Neither is a subtle signal, which is the point of choosing a branch whose integers share no structure with the ones under suspicion. The 1-in-1,000-scale pricing that this collection reaches for when a pattern might be luck is not needed here: a counter cannot accidentally produce 7 and 11 while looking for 8 and 13.

Where the one stem in five belongs #

At the coarsest of the three rises the readout returns 4 and 7 on four stems in five, and the fifth is a refusal rather than a wrong pair. That number is the same on the Fibonacci branch at the corresponding rise, which is what makes it reportable.

The reading is that the error budget belongs to the rise, not to the branch. A coarse rise puts fewer organs in the window the lag statistic is computed over, the significant lags thin out, and the clearance test that separates a real comb from noise stops clearing — the same mechanism that empties a band in every other instrument here. A branch-specific failure would have shown as a different rate on the Lucas stems, and it does not.

What the branch does not rule out #

The control is strong against one shortcut and weak against another, and the distinction is worth stating because it decides whether a third control is needed.

Lucas numbers are not Fibonacci numbers, so the test rules out an instrument with the integers written into it. But their ratio still tends to φ, so it does not rule out an instrument reasoning from the ratio — a counter that assumed consecutive families stand in the golden ratio would pass on both branches and would be just as wrong.

What rules that out is the whorled head. At 144° the divergence is rational, the ladder terminates rather than continuing, and the family list is the three numbers 2, 3 and 5 with nothing after them. There is no ratio to tend to. The readout returns the pair on that head as well, and it is the only one of the three controls that could have caught a ratio assumption.

So the instrument is tested against three heads chosen for what each one breaks: Fibonacci for the ordinary case, Lucas against hardcoded integers, whorled against a hardcoded ratio. Passing all three is a much weaker claim than passing any one of them looks, and stating which shortcut each control kills is the only way that arithmetic is available to a reader.

Why this is the test that matters #

The founding essay records the failure this is guarding against, and it is instructive because the picture was no help.

The first spiral counter returned the two smallest index offsets rather than the two shortest hops, which on a nine-hundred-point head gives 21 and 34 where the real neighbours are 34 and 55. Twenty-one spirals were drawn and there were twenty-one of them, so every assertion in the figure passed. What caught it was the angle recovery refusing: no divergence angle makes that pair the closest at that radius. The counter had been returning Fibonacci numbers, correctly-formed ones, from the wrong part of the head.

A counter that returns consecutive Fibonacci numbers on Fibonacci stems is therefore consistent with two states of the world: it counts, or it has a habit. The Lucas stems separate them, and they separate them by returning numbers that a habit would have to be very specifically wrong to produce.

The classification, and what it costs #

Once a pair is in hand the branch follows from it by arithmetic that needs no data: classifyPair asks whether the two numbers are consecutive in the Fibonacci sequence, consecutive in the Lucas sequence, share a common factor — which makes the arrangement whorled or multijugate — or none of those.

So the chain from a botanist’s notebook to a classification is:

1. list the divergence angles up one stem, in order;
2. autocorrelate the differences;
3. sort the significant lags into residue classes to get the pair;
4. look the pair up.

with no coordinate measured, no photograph, and no divergence angle needed beyond the mean of the list. The cost is the cost of step one, which the next essay prices: about two hundred and fifty internodes and a quarter of a degree per organ.

That is worth setting against what the site’s survey specification asks for by the other route. A reported pair pins the divergence angle to about 221°/mn — so a count of 7 and 11 leaves a band nearly three degrees wide, and a count of 34 and 55 leaves a band of a ninth of a degree. Counting the spirals in a photograph and reading the angles off a stem are two different measurements with different costs, and until now only the first returned a pair.

Where the two branches differ in difficulty #

They are not equally easy to read, and the difference is arithmetic rather than biological.

The window the readout uses is thirty lags, and it needs three multiples of the smaller parastichy number inside it while staying short enough that the larger number is not itself a candidate spacing. On the Fibonacci branch the ratio of the pair is close to 1.618, so 8 and 13 sit comfortably inside that condition. On the Lucas branch the low rungs are 3/4 and 4/7, whose ratios are 1.33 and 1.75 — one much closer to a whole number and one much further from it than φ.

At 4/7 the difference of the pair is three, so the second comb sits at 3 mod 4: lags 3, 7, 11, 15, 19, 23, 27. That is a dense class and the reading is easy. At 3/4 the difference is one, so the second comb would sit at 1 mod 3 — lags 1, 4, 7, 10 — and lag one is excluded from every search on this site because it carries a measurement artefact. A pair whose members differ by one cannot be read this way at all. The readout refuses at the coarse Lucas rung, and that is one of the reasons why.

What a whorl would do to it #

There is a third kind of pair and the readout has never been shown one, so it is worth working out what it would do rather than finding out later.

A bijugate plant puts two primordia down at a time, a trijugate three, and the site’s own counter for those reads the jugacy off the rotational symmetry rather than off the counts. The consequence for the pair is arithmetic: a k-jugate arrangement’s parastichy numbers are k times an ordinary pair, so a bijugate plant on the 5/8 rung counts 10 and 16.

Feed that to the residue-class reading and the spacing comes out as ten, the offset as six, and the pair as 10 and 16 — correct. The arithmetic does not care about the common factor, because n < 2m still holds and the lags are still two progressions. What it does not do is report the jugacy: it returns 10 and 16 where a person wants to hear “bijugate, on the 5/8 rung”.

That last step is one line of arithmetic — take the greatest common divisor — and the site already has it in jugacyOfPair. So in principle the sequence gives the jugacy too. In practice it is untested, because a divergence sequence for a bijugate plant is not what the model produces: the two organs at each whorl arrive at the same node, so the “sequence of divergences” a botanist writes down is a sequence of whorl rotations and there is a decision to make about what the readout is being handed. That decision has not been made and the essay says so rather than asserting the extension works.

The branch and the attractor #

There is a reason the two branches exist at all and it is worth restating, because it is what makes reading the branch off a plant interesting rather than merely tidy.

The placement rule contains no reference to Fibonacci, to spirals or to the golden angle. What it does is minimise a repulsion, and the divergence it settles on depends on where it started and how fast the rise falls. The earlier work measured the threshold: below about ninety nodes per rung a shoot keeps the branch it started on, and above it finds the Fibonacci one. So a plant’s branch is a statement about its own history — what its apex was doing early, and how quickly its pattern developed — rather than about a preference for particular integers.

Which means an instrument that reads the branch off a finished stem is reading something about the plant’s past. That is a modest version of the same claim the mechanism essay makes for the comb itself, and it comes for free once the pair is available: the branch is a lookup, and the lookup’s input is the pair.

The census this cannot replace #

It is tempting to read “the angles name the branch” as the beginning of an answer to how often plants are Lucas, and it is worth blocking that reading.

The site’s census machinery measures how often the geometry produces a given pair, over all divergence angles at a given rise, and the answer is the frequency essay’s: at a rise of 0.12, 67.5% of divergences give consecutive Fibonacci counts; at 0.008, 14.6%, with whorled at 35% and unnamed pairs at 48%. That is a statement about the space of angles, not about plants, and the site has been careful to say so.

An instrument that reads a branch off one stem does not change that. It changes what a survey of plants could measure, from “the pairs visible in photographs of heads” to “the pairs readable from angle sequences up stems”, and those are different samples of plants with different biases — the first favours species with countable heads, the second species with long stems. Neither is a random sample of anything, and a frequency computed from either is a frequency of a sample.

What it does not settle #

It does not settle how often plants are on the Lucas branch. That is a question about a population and the site’s answer to it is a specification rather than a number: the census of what a survey would have to be is four rounds of work old and still unpaid.

What it changes about that specification is which measurements can contribute. Until now the only instrument that returned a branch was a spiral count off a photograph, which needs the head to be visible, developed and countable — and which on a stem means either a cleared shoot or a cross-section. An angle sequence needs a ruler, a protractor good to a quarter of a degree, and patience over two hundred and fifty internodes, and it can be taken on material where the spirals cannot be traced at all.

The two are also independent in a useful way. A photograph of a head and a list of angles up its stem are different data with different failure modes, and the site’s habit is to trust a number more when two routes with unrelated failures agree on it. On the model’s stems they do agree, at every rise where both work, on both branches.

The honest reading of five stems #

Five seeded stems at two rises is a small experiment, and the claim it supports is correspondingly small: that the readout’s arithmetic is not tuned to Fibonacci numbers.

It is not a claim that the readout works equally well everywhere. It works at two rungs on each branch, refuses at the coarse rung on both, and cannot go finer than 8/13 for the window reason set out in the previous essay. Four rungs on two branches is the whole of its demonstrated range, and the honest way to describe it is as an instrument with a stated operating window rather than as a general method.

What makes the four rungs worth having is that they were not chosen after the fact. The rises were fixed by the earlier work’s ladder, the clearance tests were set from white sequences rather than from these ones, and the Lucas branch was run with the same defaults as the Fibonacci one — no re-tuning, no separate threshold, no exception. A method that needed adjusting to reach the second branch would be a method that had learned the first.

What five stems on two branches is worth #

Ten runs is a small experiment and the conclusion it supports is a narrow one: the readout’s arithmetic contains nothing Fibonacci-specific, demonstrated by running it unchanged on a branch whose numbers are not Fibonacci.

That narrowness is deliberate. A broader claim — that the readout works on any lattice — would need the whorled and multijugate cases, which are untested here for the reason given above, and would need rungs finer than 8/13, which the thirty-lag window cannot reach. What is established is that the two branches the placement rule actually settles into are both readable, at the rungs where the instrument works at all, with no adjustment between them.