The empty interval is the rings

No cut-off on distance turns a head’s counted contacts into its cell walls. The largest hop ratio of any wall and the smallest of any contact that is not a wall overlap on every whole head measured, so the interval a threshold would have to lie in is empty. On a stem the two relations are the same set of edges, exactly, once the contact cut keeps three families instead of two — and on a head the same three-family cut still gets 7.57 per cent of a 900-organ head’s relations wrong.

Those two results sit uneasily together. A stem is a lattice at one rise, and a head is a family of cylinders at every rise at once, so whatever a head gets wrong that a stem gets right has to come from the rise changing with radius. The question is where on the head the 7.57 per cent is.

What the stem result predicts, before any head is read #

On a stem the tessellation changes which third family it carries at a single rise, the rise at which the two counted families are perpendicular, and that rise can be written in closed form. On a head a rise is a radius, so the prediction is a set of radii: the flip rings, each a factor of φ outside the one before, each halfway between two consecutive transitions of the counted pair.

If a head is locally a stem, a three-family cut should be exact wherever the local rise is well away from a flip, and every disagreement should sit near one of those radii. If the disagreement is spread over the head, or sits at the transitions, or in the untidy middle, the head is doing something a stem cannot.

The prediction has a second half. Every five- and seven-sided cell on a golden head lies on those same rings, within a cell of a radius computed from the angle alone. So if the disagreement is on the rings, the two relations and the exceptional cells are one structure, and the empty interval is a statement about that structure rather than about the head.

The instrument, and what it is not told #

Each interior cell of a head gets one number: its distance from the nearest flip ring, measured in the head’s own wall spacing, which is the median length of a wall between two interior cells — 1.92 on a golden head of 2,400 organs. The rings come from the divergence angle and the rise law. Nothing about where the disputed cells are is used to place them.

Each cell also gets the two relations, recomputed from the same points: its Voronoi walls, and the three shortest index families a count ranks, taken in both directions. A cell is disputed when the two lists differ in any member.

The heads are golden at 900, 1,500, 2,400, 4,000, 6,000 and 9,000 organs and Lucas at 900, 2,400 and 4,000, all cut at 0.86 of the radius as every earlier comparison of the two relations was. How far the scan over index lags reaches is derived from where the rings sit rather than chosen, and it was checked by reading two of the heads again with the scan half as long again. No count moved.

Every disputed cell is within two thirds of a spacing of a ring #

On the 2,400-organ golden head 353 of the 1,680 interior cells are disputed, and the figure above sorts every one of the 1,680 by its distance from the nearest ring. Out to 0.6 of a spacing every group is disputed without exception. The group from 0.60 to 0.65 is disputed in 19 cells of 27. From 0.65 outward there is not one disputed cell, in any group, all the way to the rim.

The farthest disputed cell is 0.638 spacings from its ring, and the 1,292 cells at 0.66 spacings or more have walls a three-family cut names exactly. On the other five golden heads the farthest disputed cell is 0.637, 0.657, 0.656, 0.637 and 0.653 spacings out, and on the three Lucas heads 0.648, 0.649 and 0.642. Beyond 0.66 nothing is disputed on any of the nine.

That is the result in one sentence, and the rest is what follows from it. The contact relation and the wall relation are the same relation on a head everywhere except a shell about one and a third spacings thick around each flip ring.

Drawn on the head, the band is the rings #

Two kinds of cell make up the band. All 264 of the head’s five- and seven-sided cells are disputed, which is no surprise: a cell with five walls, or seven, cannot be named by three families taken in both directions, which always name six. The other 89 are hexagons, and they sit closer in. No disputed hexagon on any golden head is more than 0.43 spacings from its ring, while the exceptions reach 0.66.

So the band has a structure across it. Along its centre line are six-sided cells whose six walls are the wrong six, and on its two flanks are the fives and sevens that pair up along the ring. Between the rings there is no disputed cell of either kind, on this head or any other read.

The bands the comparison was always read in #

The measurement that found no cut-off read the head’s disagreement band by band at its best threshold and found it running 13.4, 2.8, 2.3, 0.1, 2.5 and 0.0 per cent from the centre outwards, with no reason given for the zeros. On the 2,400-organ head a three-family cut disputes 9.78 per cent of the relations between 0.2 and 0.3 of the radius, 11.41 between 0.3 and 0.4, nothing between 0.4 and 0.5, 5.49 and 5.64 in the next two bands, and nothing in the outer two.

The zeros are the bands with no cell near a ring. The band from 0.4 to 0.5 of the radius lies between the ring keeping 34, at a radius of 18.2, and the ring keeping 55, at 29.5; the outer two lie beyond the last ring the head has. The band from 0.5 to 0.6 holds no ring of its own, but 66 of its cells are within 0.66 spacings of the ring just outside it, and those are its disputed relations.

So the variation from band to band that looked like a head being untidy in some places and tidy in others is the rings falling into some bands and not others. A band reading is an annulus reading, and an annulus either contains a ring or it does not.

Why the whole-head number falls as the head grows #

The same fact decides what a head’s overall disagreement is. The three-family cut gets 7.57 per cent of a 900-organ golden head’s relations wrong, 7.53 per cent of a 1,500-organ head’s, 4.70 of 2,400, 4.62 of 4,000, 3.08 of 6,000 and 3.35 of 9,000.

That sequence is not a slow convergence. It falls when the head grows without taking in a ring — from 1,500 to 2,400 organs, and from 4,000 to 6,000 — and it stalls or rises when a ring enters the cut: 900 to 1,500 takes in the ring keeping 55, 2,400 to 4,000 the ring keeping 89, and 6,000 to 9,000 the ring keeping 144. A disputed share is, near enough, ring cells over all cells, and the number of ring cells grows with the head’s radius while all cells grow with its area.

The 7.57 per cent the stem comparison reported for a head is therefore a property of a 900-organ head rather than the price of a disc. The mean squared departure of a cell’s side count from six obeys the same law for the same reason, because it counts the same cells.

Setting the ring cells aside opens the interval #

Now return to the question the empty interval asked. For a cut-off to exist on a set of cells, the largest hop ratio of any wall among them has to be smaller than the smallest hop ratio of any contact that is not a wall. Over the whole 2,400-organ head those two numbers are 2.236 and 1.446, and the interval between them runs the wrong way.

Set aside the cells within a quarter of a spacing of a ring and the interval is still shut on all six golden heads. Set aside a little more and it opens: past 0.68 spacings on the 900-organ head, 0.74 on 2,400, 0.90 on 9,000. Beyond that it only widens, because setting more cells aside can only remove the cells that decide either end.

Those widths are a little larger than the 0.66 at which disputes stop, and the difference is worth a paragraph. A three-family cut is a cut at a count, and it is exact from 0.66 outward. A cut on distance asks more of the cells, because between 0.66 and 0.9 of a spacing they already carry the right three families while their third and fourth hop ratios are still close enough that no single number lies between them across the whole head. Counting families recovers the tessellation closer to a ring than any distance does.

One cut-off for every head #

With one spacing set aside, every head’s interval is open, and they overlap. On the golden heads the intervals run from 1.4258–1.5330 at 900 organs down to 1.4297–1.4437 at 9,000, and every one of them contains 1.4302 to 1.4437. The three Lucas intervals, 1.4229–1.5020, 1.4298–1.4697 and 1.4291–1.4708, contain it too.

So a single threshold anywhere between 1.4302 and 1.4437 shortest lags turns counted contacts into cell walls on all nine heads, on both divergence sequences, at every size from 900 organs to 9,000, everywhere except the band around each ring. That is a value for a free parameter found by locating the cells it cannot serve, rather than by minimising a residual across them.

It is not the 1.47 that minimised the whole head’s disagreement. That value was a compromise between the ring cells and the rest; this is the threshold for the rest alone, and 1.47 lies above it. On the 9,000-organ head a cut at 1.47 admits contacts with a hop ratio of 1.444 that are not walls.

The upper end of each interval needs a caution. On every golden head it is set by a cell just past the exclusion edge on the head’s outermost resolved ring — 1.005 spacings out on the 900-organ head, 1.000 on the 9,000 — so the narrowing from 1.533 to 1.444 records where the edge fell on the newest ring. It is not a trend the interval would follow to zero, and it is not evidence that it would not.

Why the rings are where the lattice cannot decide #

The reason is in the closed form. A flip ring is where the two counted families are perpendicular, and when two vectors are perpendicular their sum and their difference have the same length: the squared lengths of a + b and of a − b differ by four times the dot product of a and b, which is zero there. The sum is the third family and the difference is the fourth.

So at a flip ring the third and fourth families tie, the four cells they join sit on one circle, and nothing in the lattice decides which pair shares the wall. A fifth of a ring’s radius inside it, the two hop ratios are 1.25 and 1.78 in the median on the 9,000-organ head; at the ring they are 1.399 and 1.412. The common cut-off, 1.430 to 1.444, lies just above the place where they meet.

That is why no count and no distance can separate the two relations on a ring. Both instruments rank lags by length, and at the ring the lattice offers two lags of the same length and a wall that belongs to one of them. The empty interval is not a defect of either definition of neighbour. It is the lattice’s own tie, located.

The lattice’s own vectors make the same errors #

The check that settles whose tie it is uses no counting at all. Replace the three families a count ranks with the lattice’s three shortest vectors at each cell’s local rise, computed from the divergence and the radius alone, and compare those with the walls.

They dispute 440 relations on the 2,400-organ golden head, against 484 for the counted families, and 1,194 on the 9,000-organ head, against 1,322. On every one of the nine heads they dispute nothing beyond 0.66 spacings of a ring. An ideal lattice makes the same errors in the same place as the counting instrument, so the band belongs to the lattice and not to how the families were counted. That also retires the remaining suspicion about a count’s ranking of its families here: an ideal ranking fails on the same cells.

On Lucas heads the band has no sharp inner edge #

The golden band is a shell with a sharp edge. On every golden head the nearest exactly read cell is within a hundredth of a spacing of the farthest disputed one — 0.633 against 0.637 at 900 organs, 0.649 against 0.653 at 9,000 — so a cell’s distance from its ring is enough to say whether it is disputed.

The Lucas heads share the outer bound, 0.642 to 0.649, and nothing else about the edge. Exactly read cells sit as close as 0.23 spacings to a Lucas ring, disputed hexagons reach 0.59 where golden ones stop at 0.43, and one five- or seven-sided cell of the 154 on a 900-organ Lucas head is read exactly. On a Lucas head, distance from a ring bounds the disagreement without deciding it cell by cell. Why the two divergence sequences differ inside the band is not measured here.

What the negative result was about #

No cut-off makes them one is still true as it was stated. On no whole head, at any size, does a single threshold separate the two relations, and nothing above changes that.

What changes is what the sentence is about. On a 9,000-organ golden head it is about 978 of the 6,464 interior cells, the ones within two thirds of a spacing of a flip ring, and about nothing else on the head. Off those cells the answer is not “no cut-off” but one cut-off, the same on nine heads and two divergence sequences. On them the answer is not a badly chosen threshold but a tie in the lattice that no threshold could break.

That is a better use of a negative result than repeating it, and it was only available because the stem comparison said where to look. The finding that a count names two thirds of a cell’s walls is the same kind of statement — a disagreement averaged over a head — and it would be worth reading against the rings in the same way.

What this does not establish #

That a real capitulum’s cells are the Voronoi cells of its seed centres, or that its organs sit close enough to the divergence to put the rings where the closed form says. A flip-ring band is about one and a third spacings wide, and resolving it needs organs located to a small fraction of a spacing, which is finer than a photograph usually gives.

Nor does it show that the common interval stays open past 9,000 organs. Its upper end is set by the outermost ring’s flank on every head read, so a larger head could narrow it again when a new ring enters. That is untested.

And every head here is ordered. Positional disorder moves the whole-head best value from 1.47 to 1.81 at a sixth of a spacing, and whether it widens the band, or moves the off-ring cut-off, was not measured.

What would withdraw it #

A disputed cell more than 0.66 spacings from every flip ring, on any golden or Lucas head. The nine heads hold 22,022 interior cells between them and not one is.

A golden head whose interval stays shut once the cells within one spacing of a ring are set aside, or whose interval does not contain 1.4302 to 1.4437.

A scan over index lags lengthened by half that changes any count on the heads it was checked on, 900 and 2,400 organs.

Still open: why a Lucas ring blurs its inner edge #

On a golden head a cell’s distance from its ring decides whether it is disputed; on a Lucas head it only bounds it. The measurement that would say why is the reading drawn above for a golden head, repeated across a Lucas ring: the third and fourth hop ratios cell by cell, asking whether they meet at a single offset from the ring or over a spread of offsets. A spread would put the blurred edge in the lattice; a single meeting point would put it in how the Lucas head’s organs sit around the ring.

The same reading on a displaced head would answer the other open question — whether the band widens under disorder at the rate the rings themselves dissolve, or faster.