No cut-off makes them one

Two things on this site are called a neighbour and they are not the same relation.

The first is what a count works from. For a cell, every index lag is measured, the lags are ranked by the length of the hop they make, and the shortest few are kept. The second is what a tessellation gives: two cells are neighbours when their Voronoi regions share a wall. One of them has a knob in it and the other has none, and the collection has used the two words interchangeably — a wall labelled 34 and a hop of 34 read as one fact about one pair of organs.

The question here is whether a single number could make them one relation, and the answer is no. Not a poorly chosen number, and not a number that needs more care: no number.

Two relations, and one word #

The contact relation is a ranking with a cut in it. The counting instrument measures the hop at every index lag out to two hundred, sorts them, and keeps the shortest few — two whole families, which is four partners, because a lag runs both ways.

The tessellation relation is a partition with no cut in it. A cell’s walls are decided by the point set alone, and the six sides Euler forces come out of the construction rather than out of a setting.

So the two differ in kind before they differ in any number. One is a threshold applied to a list; the other is a fact about a partition of the plane. The interesting question is whether the first can be made to reproduce the second, which is a question about the threshold.

What a single cut-off would have to do #

State the cut in length rather than in count, and per cell, in units of that cell’s own shortest lag — which removes the scale, so the same value can be asked of a cell near the centre and a cell near the rim.

Call it θ. For θ to reproduce the tessellation exactly, every wall of every cell would have to be at a distance below θ, and every partner of every cell that is not a wall would have to be at or beyond it. Two bounds, and they are read off the data rather than chosen: θ must exceed the furthest wall anywhere, and θ must not exceed the nearest non-wall anywhere.

That is the whole of the test. It has no tolerance in it and nothing to tune.

One cell at a time, it nearly always exists #

Asked of a single cell, the test almost always passes. Of the 608 interior cells of a nine-hundred-point golden head, 606 have some cut-off that separates their own walls from their own non-walls.

That is the number that made the confusion reasonable. A relation that is locally reconcilable in 99.7 per cent of cases looks like a relation that is reconcilable, and nothing about a single cell hints otherwise.

The per-cell margins are not even tight. The ratio of a cell’s nearest non-wall to its furthest wall has a median of 1.202 and reaches 1.403 at best, so a typical cell has a fifth of a spacing of room either side of wherever its own threshold is put.

The two cells that have no cut-off at all #

Two cells fail the test on their own terms, and where they sit is worth recording: index 2 at 4.7 per cent of the radius, and index 6 at 8.2 per cent.

Both are in the innermost handful of organs, where the √i radius law has not yet produced anything resembling a lattice and a cell’s own shortest lag is a poor unit for anything. The worst per-cell gap ratio anywhere is 0.898, which is one of those two.

They are not the reason the whole head fails. That is the part worth being careful about, and the next three sections are about it.

The head, and the interval that is empty #

Ask the same question of all 608 cells at once and the two bounds cross.

A single cut-off would have to exceed 2.236 shortest lags, because that is the furthest wall anywhere in the head, and it would have to not exceed 1.441, because that is the nearest non-wall anywhere. There is no such number. The interval is empty by a factor of 1.55.

That is a stronger statement than the reconciliation is imperfect. It is not that some choice of θ leaves a residue; it is that the constraint the reconciliation would have to satisfy has no solution, and the two bounds are the wrong way round by more than half again.

The two extremes, located #

Neither bound comes from a cell in trouble, which is what makes the emptiness structural rather than an artefact of the middle.

The furthest wall, at 2.236 shortest lags, is √5 exactly, and it belongs to the centre point — the first organ placed, whose own shortest lag is short and whose walls reach out to a ring that is not yet close to it. The nearest non-wall, at 1.441, is a lag of 21 at about 60 per cent of the radius, in the tidiest part of the head, where a twenty-one-hop is short enough to be a plausible neighbour and the tessellation declines it anyway.

So the two ends of the empty interval are contributed by two different regimes, and neither is a rim effect. The rim cut at 86 per cent of the radius is already made before any of this is read.

Cutting the untidy middle away #

The obvious repair is to throw away the part of the head that is not a lattice, and it does not work. Cutting the inner 5 per cent leaves the interval running from 1.637 down to 1.441; cutting the inner 10 per cent leaves 1.518 down to 1.441; cutting the inner 30 per cent still leaves 1.490 down to 1.441.

The lower bound moves and the upper bound does not, because the upper bound was never in the middle. Removing a third of the head buys a factor of 1.03 on an interval that needs a factor of 1.55, and there is nothing further in to remove.

Every head size, and the same emptiness #

Five golden heads were read the same way, from 300 points to 1,500. The interval is empty on all five: 2.236 against 1.457 at 300 points, 2.236 against 1.441 at 600, 900 and 1,200, and 2.236 against 1.431 at 1,500.

The mean side count rises across those sizes — 5.386, 5.535, 5.697, 5.775, 5.715 — as the rim cut leaves proportionally less boundary behind, which is the head being honest about its own edge. The emptiness does not move with it.

And a head with no Fibonacci number in it #

A negative result about a golden-angle head invites the suspicion that something about the golden angle produced it. So the same reading was made on a Lucas head of nine hundred points, where the families are 29, 47 and 76 rather than Fibonacci.

The interval there runs from 2.646 down to 1.421, which is emptier still. Whatever this is, it is not a fact about Fibonacci numbers, and it is not a fact about one divergence angle.

The best available value #

If no cut-off works, the next question is which one comes closest, and it has an answer: θ = 1.47 shortest lags, which leaves 1.80 per cent of the edges in dispute — 63 of 3,669 directed edges on the nine-hundred-point head.

The curve of disagreement against θ has a genuine minimum rather than a shoulder. At 1.20 it is 38.71 per cent, at 1.40 it is 6.75, at 1.47 it is 1.80, at 1.55 it is 6.44 and at 2.00 it is 37.94. So the minimum is inside the swept range and is not a boundary effect.

What the residue is made of #

Sixty-three edges is a small number and it is worth knowing what kind of edges they are, because 1.80 per cent invites the reading that the two relations agree apart from noise.

The disputed edges are systematically longer than the agreed ones: 1.504 shortest lags against 1.153. They sit systematically further in: mean radius 0.379 of R against 0.551. And they are tilted further from the local radial direction, 54.1 degrees against 44.6.

None of that is noise. It is a population with its own statistics, concentrated where the lattice is coarsest, and a threshold cannot separate a population from the one it overlaps.

Where the disputed edges are #

Band by band from the centre outwards, the dispute rate at 1.47 runs 13.4 per cent, 2.8, 2.3, 0.1, 2.5 and 0.0. The outermost band of the interior has no disputed edge at all.

By lag, the disputes are led by 21, with 24 of the 63, then 55 with eight and 13 with seven; the remainder are 5, 34, 2, 3 and 8.

So the best available cut-off fails almost entirely in the inner third, and its worst single family is a lag that is a genuine contact family at some radii and not at others — which is the transition structure of the head arriving in a place nobody was looking for it.

The parameter is sharp #

A value that came close and was flat around its minimum would be a mild problem. This one is neither.

A fifth of a shortest lag either side of 1.47 multiplies the disagreement by between five and ten. At 1.30 the disputed share is 19.41 per cent; at 1.70 it is 15.07 per cent. Both of those are values a reasonable person could have chosen, and both are an order of magnitude worse than the best.

A free parameter the collection has never stated #

Put those together and the position is uncomfortable in a specific way. Every statement here that treats a counted contact and a shared wall as the same object has been made at some implicit θ. That θ has never been written down, it is not 1 and it is not 2, and moving it a fifth of a lag changes the answer fivefold.

This is the same defect a recency window in the placement rule turned out to have, arriving from the other side. There the fix was to make the neighbourhood a stated hypothesis with a width; here there is nothing to state, because the value that would have to be stated does not exist.

The best that can be said is that 1.47 is stable across head sizes — 1.49, 1.47, 1.47, 1.46, 1.45, a spread of 0.04 — and 1.47 on the Lucas control. A parameter that is stable and unstated is the worst combination available: it is exactly the kind that gets away with never being named.

What disorder costs #

A cut-off that has to be refitted on every specimen is not a constant of the pattern, so the head was displaced and read again. Gaussian noise in both coordinates, σ stated as a fraction of the mean shortest lag, three seeds at each of eight levels, and both relations recomputed from the displaced points.

At σ = 0.16 of a spacing the best cut-off has drifted from 1.47 to 1.81, a move of 23 per cent, and holding the ordered head’s own value costs 1.64 times the refitted residual — 40.98 per cent against 24.93.

The exponent, and what it means for a specimen #

The excess disagreement above the ordered head’s floor grows as the 2.39 power of the displacement, fitted in the logarithm of both over seven levels. Doubling σ multiplies the excess by about 2.5 at every step above σ = 0.005.

An exponent above one is the part that matters. A head twice as untidy is more than twice as ambiguous about what a neighbour is, so the ambiguity is not a fixed tax on measurement — it is a quantity that runs away from the disorder that causes it.

The number of cells with any separating cut-off of their own falls too, from 606 at σ = 0 to 430 at σ = 0.16. So even the local reconciliation that made the confusion reasonable stops holding on a specimen that is not a model.

A lattice where the cut-off does exist #

The machinery is capable of reporting a non-empty interval, and it does so on the case where one should exist.

On a sheared triangular lattice of 18 by 18, indexed by row and column, every wall is at 1.000000000 shortest lags and every non-wall at or beyond 1.732050808, which is √3. All 320 interior cells separate; the interval that works for the whole lattice is (1, √3], width 0.732; and at the midpoint of it the symmetric difference is exactly zero over 1,176 agreed walls.

That is the positive control, and it is what makes the empty interval on a head a measurement rather than a property of the test. A perfect lattice has one cut-off; a head has none.

Why the interval is empty rather than narrow #

The reason is that a head is not one lattice. It is a sequence of them, and the counted pair changes with radius in a way that is computable and sharp.

A cut-off stated in units of a cell’s own shortest lag corrects for scale but not for rung. At the coarse inner bands the ratio between the second contact family and the third is one thing; at the fine outer bands it is another; and a single θ is being asked to sit between them at every radius at once.

The regular lattice has one rung and one ratio, which is why its interval has width 0.732. A head has six or seven bands of rung, and the interval it offers is the intersection of theirs.

What this does not rule out #

It does not rule out a cut-off per band. Nothing here was measured with θ allowed to vary with radius, and the per-cell figures suggest a banded threshold would do far better than a global one — the dispute rate at 1.47 is already 0.0 per cent in the outermost band.

Such a rule would be a different object, with a stated banding and a stated θ in each band, and it would have to be shown to reject. It is not a repair to the claim that the two relations are one; it is the admission that they are two, written as a table.

It also does not rule out reconciliation on a stem, where there is one rung and no radius. That is where the two do coincide, and it is a positive result rather than this one.

Neither instrument is wrong #

Every statement this site has made about wall counts, mean side number, Lewis’s law or the Aboav relation is a statement about the tessellation and is untouched by any of this. So is the charge arithmetic of the interior, which counts sides and never asks about a lag.

Every statement about hop lengths, transitions and recovered divergence angles is a statement about the contact ranking and is equally untouched.

What is wrong is the joint statement — that a cell’s walls are its contact families — made without saying at what threshold. Each instrument is correct about its own object, and the object is not the same object.

What would refute it #

A point set on which the interval is non-empty and which is a head rather than a lattice. The test costs one triangulation and one lag scan, it is the same code that reports (1, √3] on the regular control, and it would take a single counter-example.

Or a demonstration that the furthest wall on a head is not a real wall — that the centre point’s √5 contact is an artefact of the construction rather than a fact about the partition. That is the one bound doing most of the work, and if it went the interval would still be empty at 1.490 against 1.441 with the inner third cut away, which is why the inner cuts were run.

The measurement that would make it worse is a head large enough for a seventh band. Every new rung adds another ratio to the intersection, and the interval can only shrink.

The triangulation, measured rather than caveated #

There was a known defect in the triangulation these readings sit on: its enclosing triangle was built too small, so a handful of fans against the convex hull came out incomplete.

The honest thing is to measure what repairing it would do rather than to note it. Simulated by padding the point set far enough out that no padding point can take an edge from a real one, the repair changes the neighbours of zero cells at every head size read here. One head of the six is damaged at all — the six-hundred-point one, at 13 points sitting between 0.972 and 0.982 of the radius — and every reading in this essay is cut at 0.86 or below.

So the caveat was a number and the number was nothing. That is a better position than a warning, and it took one simulation to reach.

The repair itself has since been made — the enclosing triangle is built a hundred times the point set’s radius rather than four, and the triangulation now holds exactly the 2n − 2 − h triangles Euler’s formula requires on every head tested — and it changed no reading here, which is what the simulation said it would. A prediction about a repair is worth as much as the repair only if somebody afterwards checks that it held.

What the comparison refuses to answer #

Two refusals are worth recording, because a comparison that answers everything is answering from somewhere other than its input.

Nine points on a circle is refused rather than compared: every one of them is on the convex hull, so no cell has a complete neighbourhood, and there is nothing to read. A point set the library does not hold is refused by name rather than answered about the golden head, which is the failure mode where a default quietly becomes the result.

Neither refusal is decoration. The first is the case where a reader would most expect a number and where a number would be meaningless.

Where this leaves the word #

Neighbour on this site now needs a qualifier, and there are only two of them. A contact neighbour is one of the four partners the shortest two lags supply. An adjacent neighbour shares a wall. They agree on about 98 per cent of edges at the best threshold anybody could have picked, on 66 per cent at the cut the counting instrument actually makes, and on no threshold exactly.

The size of the second gap is the subject of the essay that follows, and it is a third rather than a rounding error. The ordering the two relations induce on the families is a separate disagreement with its own numbers.

What is settled here is only the negative, and it is settled at the level of the definition rather than the tolerance: the two relations cannot be made one by choosing a number, because the number would have to be greater than 2.236 and no greater than 1.441.