The blur was at the centre

The empty interval is the rings located where on a seed head the two meanings of “neighbour” part company. A three-family contact cut — the three shortest index lags a spiral count keeps, taken forward and back — names a cell’s Voronoi walls exactly everywhere except inside a band about two thirds of a wall spacing either side of each flip ring, the radius at which the two counted families are perpendicular and the lattice cannot decide which diagonal carries the wall. With that band set aside, one hop-ratio cut-off serves every head.

That result was stated on nine heads and qualified on three of them. On a golden head a cell’s distance from its ring decided whether it was disputed: the nearest cell read exactly and the farthest cell read wrongly were within a hundredth of a spacing of each other. On a Lucas head the distance only bounded it. Exact cells sat as close as 0.23 of a spacing to a ring, and disputed six-sided cells reached out to 0.59 where the golden heads’ stopped at 0.43. The essay said plainly that why the two divergence sequences differ inside the band had not been measured. This is the measurement.

Two readings of the blur #

There were two ways the Lucas band could differ, and they predict different things.

The Lucas angle is the other branch the placement rule settles on, and its counts run 3, 4, 7, 11, 18 where the golden angle’s run 3, 5, 8, 13. If the difference is in the lattice, then the Lucas angle’s local geometry near a flip ring is not the golden angle’s — the third and fourth families approach their tie along a different curve, or tie at a different offset — and the blur should be on every Lucas ring, the outer ones as much as the inner. If the difference is in where the organs happen to sit, it should be patchy, and it should be concentrated wherever the Lucas head’s organs sit unlike a lattice.

Neither reading can be tested on a whole head, because both whole-head numbers are extremes. The nearest exact cell and the farthest disputed hexagon are each one cell, and a whole head reports whichever ring that cell happens to be on. So the head is read again a ring at a time: each cell is assigned to its nearest flip ring, and the census of disputed hexagons, fives, sevens and exact cells is taken separately for every ring.

Read over its resolved rings, the Lucas edge is sharp #

A ring is resolved when both of its neighbouring rings are more than two wall spacings away, so that its band is not crowded by another’s. Restricted to the resolved rings, every head’s edge is sharp, and the Lucas heads’ are as sharp as the golden heads’. On the 4,000-organ Lucas head the nearest exact cell is 0.638 spacings from its ring and the farthest disputed cell 0.642. On the 2,400-organ Lucas head the two numbers are 0.645 and 0.649; on the 900-organ Lucas head, 0.644 and 0.648. The golden heads read 0.634 against 0.636, 0.634 against 0.637, and 0.649 against 0.652.

So the inner edge is not blurred on any resolved Lucas ring. Over the whole head it is, and the 0.23 comes back, because the whole head includes rings that are not resolved. The first measurement already rules out half of the lattice reading: whatever made the Lucas band look soft, it is not on the rings where the band has room to be itself.

One band, on every ring of both heads #

Drawn a ring at a time, the band has an anatomy, and the anatomy is the same on every resolved ring of both heads. The six-sided cells whose walls the three-family cut gets wrong lie along the ring’s centre line, never more than 0.154 to 0.163 of a spacing from it. The five- and seven-sided cells lie on its two flanks, from about 0.14 out to between 0.631 and 0.652. The exact cells begin where the flanks end, between 0.634 and 0.655.

Those ranges hold on the golden rings keeping 21, 34, 55, 89 and 144, and on the Lucas rings keeping 29, 47 and 76, on every head size read. There is no Lucas ring whose hexagons reach further, no Lucas ring whose exact cells come nearer, and nothing about the outer Lucas rings that separates them from the outer golden rings by more than the hundredths the golden rings differ among themselves.

That disposes of the lattice reading altogether. The Lucas angle’s lattice near a flip ring does what the golden angle’s does, at the same offsets, and the band it produces cannot be told apart from the golden band by any of the four numbers that describe it.

A ring holds its own number, and the number before it #

Counting rather than measuring gives the same answer more sharply. Every five- and seven-sided cell lies on a ring, and each resolved ring holds exactly its own family number of each — 21 fives and 21 sevens on the ring keeping 21, 144 and 144 on the ring keeping 144, and on the Lucas rings 29, 47 and 76 of each. Each of those fives is paired with a seven along the ring.

The disputed hexagons turn out to be counted by the ring too. The ring keeping 21 holds 13 of them; 34 holds 21; 55 holds 34; 89 holds 55; 144 holds 89. On the Lucas head the ring keeping 29 holds 18, 47 holds 29, and 76 holds 47. Every whole resolved ring on every head carries exactly the family number before its own in disputed hexagons, with no exception among the sixteen ring censuses taken across the six heads.

Why the number before is the number to expect #

A flip ring of the pair $(m, n)$ is where the $m$ and $n$ families are perpendicular, and there the two diagonals of the cell they span — the sum, $m + n$, and the difference, $n - m$ — have the same length. Those are the third and fourth families, and the tie between them is the whole reason no cut-off can separate the two relations on a ring.

On a Fibonacci or a Lucas ring the difference $n - m$ is the number before $m$. So the ring holds as many disputed hexagons as there are spirals in the fourth family, the family the third ties with. That the count matches is measured on sixteen censuses; why each spiral of the fourth family contributes exactly one disputed hexagon at the ring is suggested by the tie and is not derived here. It is the kind of equality a proof should be able to reach from the closed form for the ring, and until one does it is a measured fact about eight rings on two angles.

The tie is the same curve on a Lucas head #

The same reading taken off the lattice directly agrees. On the 4,000-organ Lucas head a fifth of a ring’s radius inside it, the median third and fourth hop ratios are 1.24 and 1.75; at the ring they close to 1.390 and 1.410. On the 9,000-organ golden head the same reading gave 1.25 and 1.78 a fifth inside, 1.399 and 1.412 at the ring.

That is the reading the earlier essay proposed as the way to decide the question — whether the third and fourth families meet at one offset or over a spread — and it answers it: one offset, the ring’s own, on both angles. A spread would have put the blur in the lattice. There is no spread to find.

Where the rings crowd together #

If the difference is not on the resolved rings it has to be on the others, and the others are all at the centre. A head is a family of cylinders at every rise at once, and its flip rings — halfway between the transitions, not at them — sit a factor of about $\varphi$ apart in radius, which puts the inner ones only fractions of a spacing apart. On the 4,000-organ Lucas head the rings keeping 4, 7 and 11 sit at radii 2.25, 3.69 and 5.94 — 0.75 and then 1.18 wall spacings apart. The ring keeping 18 is 1.94 spacings beyond 11, and only from 18 to 29, 3.12 spacings, is there a gap wider than two.

A band reaches 0.66 of a spacing either side of its ring, so two bands overlap whenever their rings are less than 1.32 apart. On the Lucas head the rings of 4, 7 and 11 all overlap, and the first pair of consecutive rings whose bands separate is 11 and 18. On the golden head the rings keeping 5 and 8 are 0.87 apart and overlap, and the first pair to separate is 8 and 13, 1.42 apart — one ring earlier, and at a radius of 4.3 rather than 5.9.

The reason is arithmetic. A ring keeping $m$ sits at a radius of about $0.54\,m$, so the gap from it to the next ring is about 0.54 times the family number before $m$ — about 0.28 of it in wall spacings. Two bands clear each other once that gap passes 1.32, which needs the number before $m$ to be five or more. The Fibonacci numbers run 1, 2, 3, 5, 8, so the ring of 8 is the first with five before it. The Lucas numbers run 1, 3, 4, 7, 11 and go from 4 straight to 7, so the first clear gap waits for the ring of 11, one ring further out.

The two old numbers are half a gap each #

That explains the two numbers the whole-head reading reported, and explains them exactly.

The farthest disputed hexagon on a Lucas head was 0.59 spacings from its ring. It belongs to the ring keeping 7, and the ring keeping 11 is 1.18 spacings further out — so a cell midway between them is 0.59 from each, and inside both bands at once. The farthest disputed hexagon on a golden head was 0.43 from its ring. It belongs to the ring keeping 5, and the ring keeping 8 is 0.87 beyond it. Half of 0.87 is 0.435.

So neither number measured how far a disputed cell can sit from a flip ring. Each measured how far apart the two most widely spaced overlapping rings are on that angle. “Distance from the nearest ring” is a good coordinate wherever rings are more than a band’s width apart, and a meaningless one where they are not, because a cell between two overlapping bands can be far from its nearest ring and still well inside the next one’s. The Lucas number was larger because the Lucas centre’s widest overlapping gap is larger.

The centre of a Lucas head, drawn #

Drawn, the crowding is plain. Out to the ring of 11 the shaded annuli round the rings of 1, 3, 4, 7 and 11 run into one another and cover the centre without a break, so every cell there is inside some band, and most of them are disputed. Between the ring of 11 and the ring of 18 the first clear ground appears, and beyond 18 each annulus stands alone with undisputed cells between it and the next.

One cell in the crowded centre is not disputed although it sits only 0.23 of a spacing from the ring keeping 7, and it is ringed in the drawing. It is the 0.23 of the original report, and it has an explanation that has nothing to do with rings.

The centre of a golden head, for comparison #

The golden centre crowds too, and its disputed hexagons reach 0.43 for the reason given above, but its overlapping annuli stop sooner, and no cell well inside any band — nearer its ring than 0.6 of a spacing — is read exactly. The exact cells nearest a golden ring are all at the band’s outer edge, between 0.63 and 0.66, where exact and disputed cells interleave on every head of either kind. That interleaving is the edge’s own width, about three hundredths of a spacing, and it is the same on both angles.

Organ seventeen #

The exceptional cell is organ 17, at a radius of 4.12. Its shortest index lags, by hop ratio, are 11 forward, 7 back, 11 back, 18 forward and 7 forward, and those five are exactly its five Voronoi walls. A three-family cut keeps the first three distinct lags — 11, 7 and 18 — in both directions, which on any organ with partners on both sides names six. Organ 17 has no partner 18 behind it: that would be organ −1, and the head begins at organ 0.

So the cut names five partners for a five-sided cell and gets all five right. It is not reading the lattice correctly inside the band. It is reading a cell whose count of candidate partners is short by one in exactly the family whose second member would have been wrong. On every Lucas head read, 900, 2,400 and 4,000 organs, it is the same organ at the same radius, and on none of them is any other cell nearer its ring than 0.6 of a spacing read exactly.

Why no golden organ does the same #

The same accident is available to any organ younger than the third family it keeps. It happens on the Lucas head because the Lucas rings are numbered from 3 and 4, so a small organ near the centre can keep a third family of 18 while its own index is 17. On a golden head the families an organ near the radius of four keeps are 5, 8 and 13, and every organ at that radius is older than 13. The accident needs an organ whose index is just under a family number it is also close enough to the centre to keep, and the Lucas numbers provide one at 17 and 18.

It is worth being exact about what this is. It is an edge effect in the index, not in space: the head has no organ before organ 0, so the youngest few organs have one-sided index neighbourhoods. It lives at the same centre as the crowded rings, which is why the two were easy to mistake for one blur.

What the whole-head numbers were measuring #

Put together, the two numbers that separated the Lucas band from the golden one were both about the centre and neither about the band.

The 0.59 was half the widest gap between two overlapping Lucas rings, 7 and 11. The 0.23 was one organ at the start of the index, reading five walls correctly because it had only five candidates. Neither says anything about the band round a resolved ring, and on resolved rings the two angles’ bands are the same band to within the hundredths that separate one golden ring from another. The finding that the contact cut and the tessellation are the same relation off the rings therefore needs no Lucas qualification. What does need one is the whole-head summary, which was averaging over two regimes — rings that stand apart, and rings that do not.

What changes in the published claim #

The earlier essay said that on a Lucas head distance from a ring bounds the disagreement without deciding it cell by cell. That sentence was true of the whole head as read, and it located the cause in the wrong place. Distance from a ring decides the disagreement cell by cell on every resolved ring of both angles. It fails to decide it at the centre of both, over a wider radius on the Lucas head because the Lucas numbers skip five.

This corrects a qualification rather than a result. The third family is still the family a stem needs to make the two relations one; the band is still the lattice’s tie, located; and the second moment of a cell’s side count still counts the same cells the band does. What goes is the suggestion that the Lucas angle had a band of its own. The same ring-by-ring reading is owed to the finding that a count names two thirds of a cell’s walls, which is also an average over both regimes at once.

What this does not establish #

It does not say that a real Lucas plant has a centre like this one. A capitulum’s first organs are the youngest and the least regular, and the crowded centre is exactly where a real head departs most from Vogel’s rule, so the drawn centre is a statement about the model.

It does not derive the hexagon count. That each resolved ring holds the family number before its own of disputed hexagons is measured on sixteen censuses and explained only as far as naming the family it matches.

And it does not say what the band does to a head whose organs are not exactly where the rule put them. Every head here is ordered.

What would withdraw it #

A resolved ring on either angle, at any size read, with a disputed hexagon more than 0.2 of a spacing from it, or an exact cell nearer than 0.6.

A Lucas cell well inside a band, read exactly, that is neither in the overlap of two bands nor short of a partner in the index.

A whole resolved ring whose disputed hexagons are not the family number before its own. Sixteen are counted and none is.

Still open: whether a displaced head’s band widens ring by ring #

Every head here and in the essay before it is ordered, and the one number that does distinguish rings from each other — the family number — is exactly what should decide how they respond to disorder. The gap between the third and fourth hop ratios a spacing from a ring is what a displacement has to overcome before a cell’s walls change, and that gap should be smaller on a ring of 144 than on a ring of 21 at the same distance. If it is, a displaced head does not widen its band uniformly. It fails from the rim inward, and the single cut-off between the rings fails before any of it. A hundredth of a spacing takes that measurement.