The width carries the denominator

The dip in disorder at a rational divergence angle narrows as the head grows, and the previous phase measured how fast: the half-width falls as the square of the number of organs, not as the first power. That was a prediction that came out wrong and was reported wrong — the argument written down first gave 1/n, from the total twist across the head, and the measurement gave 1/n², because the twist has to stay inside a radial gap that is itself shrinking while the arc it makes grows.

What the phase could not say was whether the coefficient of that law depends on the denominator. It had three: 3/8, 5/13 and 8/21, with coefficients running 1,400 to 3,300, and it recorded plainly that three denominators do not support a q-dependence.

Six do, and the answer is that they do depend on it, by a factor of twelve.

The law within a denominator, which holds #

Before the coefficient can carry anything the exponent has to be checked at the new denominators, because a law measured on 8, 13 and 21 might simply stop working at 55.

It does not. Within a denominator, above a head-size floor described below, n²·w is flat to a few per cent. Taking the two larger heads at each denominator: 3/8 gives 1,166 and 1,210, a spread of 1.04; 5/13 gives 1,308 and 1,261, also 1.04; 8/21 gives 2,711 and 3,286, a spread of 1.21; 13/34 gives 6,914 and 7,097, a spread of 1.03; 21/55 gives 9,069 and 9,180, a spread of 1.01; and 34/89 gives 13,909 and 14,685, a spread of 1.06.

That is the previous phase’s exponent, confirmed on three denominators it could not reach, and confirmed harder than it was originally measured: 21/55 and 34/89 give the two flattest columns in the table.

The floor, which is a real effect and not a nuisance #

Each denominator is measured at three head sizes: the smallest at which its rows have organs in them, and two multiples of it. The smallest is defined as fifteen organs per row — max(300, 15q) — and at every denominator from 21 upward its coefficient comes out below the two larger heads’, by a factor of two at q = 21 and q = 34.

That is not noise and it is not excluded quietly. It is the head-size floor showing itself: at fifteen organs per row the rows are only just rows, the cells along one are still being cut by neighbours from the next, and the arrangement at the exact rational has not reached the order it will have. The dip is shallower and its shoulders are closer in, so the width measured at a quarter of the way up comes out short.

The floor is therefore a statement about heads: a rational angle does not look ordered until its rows are populated, and how many organs that takes is proportional to the denominator. A sunflower head of 900 organs at 34/89 has ten organs per row, which is below the floor, and would not show the dip that the same angle shows on a head of 1,335.

Dividing by q #

Take the six coefficients and divide each by its denominator. They come out 149 at q = 8, 99 at 13, 143 at 21, 206 at 34, 166 at 55 and 161 at 89.

The undivided quantity spans a factor of 12.0. Divided, it spans 2.1. So most of what was left after the 1/n² law is accounted for by a single factor of q, and what remains is a factor of two with no trend in it.

So the law is w ≈ 150·q/n², to within a factor of two, and the factor of two is stated as a factor of two rather than absorbed into a fitted constant. This site has a standing objection to fitting: a prediction with nothing free in it that comes out within a factor of two is worth more than a one-parameter curve through six points, and the previous phase made the same choice about the reading-error law for exactly the same reason.

Why the old numbers were what they were #

The previous phase reported coefficients of 1,400 to 3,300 across 3/8, 5/13 and 8/21 and read them as one number with scatter. With the corrected background the same three come out 1,188, 1,285 and 2,999 — the same range, the same ordering, and the same conclusion available from them, which is that three points spanning a factor of two and a half do not establish a trend.

What changed the picture is not a correction to those three. It is the three that were added, and the reason they could not be added before is the subject of the previous essay: at 13/34 and above the background was contaminated, the dips read as shallower than they are or as inversions, and the widths that came out of them were not measurements of anything.

So the sequence is worth stating in order, because it is the ordinary shape of a result and it is easy to present as though the answer had been visible all along. The question was asked. The obvious extension of the measurement was attempted. It produced a number that was wrong in a way nothing flagged. Finding that took the phase, and once it was fixed the answer took an afternoon.

What the coefficient is not #

Two readings of “the coefficient carries the denominator” are available and only one of them is supported.

Supported: the half-width, at fixed head size, is larger at a larger denominator. A head of 1,980 organs at 21/55 tolerates about eleven times the angular error that the same head at 3/8 does before its dip is a quarter filled in.

Not supported: that a larger denominator is somehow more robust, or that rational angles with large denominators are more likely to be found in nature because they are easier to hit. That inference does not follow, and the reason is the floor. A larger denominator needs a proportionally larger head before it has a dip at all, so the tolerance is wider on a scale where the head is also bigger. Measured per organ per row — the natural unit — the two effects run in opposite directions and the site has not measured which wins.

That distinction matters because the disorder thread’s whole point is that μ₂ at a given angle is a statement about a head of a stated size, and a q-dependence in the width is another instance of the same thing rather than a new fact about angles.

What a factor of q means #

The previous phase’s argument for the exponent went: the pattern stays ordered while the total twist across the head stays inside the radial gap between neighbouring rows; the twist grows as n·δ and the arc it makes grows as n^1.5 while the gap shrinks as 1/√n; so δ goes as 1/n².

Nothing in that argument mentions the denominator, which is why it gave no q-dependence. Putting one in requires asking what the rows are like, and the rows are what q counts.

At denominator q the head has q rows, so each row holds n/q organs and the rows are 360/q apart in azimuth. A wider azimuthal separation between rows is more room for a twist to develop before organs from one row start landing between organs of the next — which is a direct argument for the tolerance rising with the spacing between rows, and the spacing between rows goes as 1/q. That gives the wrong sign.

The right sign comes from the other direction: what has to stay small is the twist relative to a row’s own internal spacing, and a row with n/q organs in it has them q/n apart along the row. Making the denominator larger makes the rows sparser, so a given twist displaces an organ by a smaller fraction of its row’s spacing, and more twist is tolerated. w ∝ q/n² is what that says.

That is a reconstruction, not a derivation. It gets the sign and the power right and it was written after the measurement rather than before it, which on this site is the difference between an explanation and a story. The prediction that would make it a derivation is available and is not made here: the same argument says the coefficient should be independent of which rational of a given denominator is used, so 21/55 and 34/89 — with denominators of 55 and 89 — should not be the only cases; 12/55 or 23/55 should give the same 166. Nobody has looked.

The two methods, side by side #

The machinery still carries the previous phase’s background — one sample at two tenths of a degree — alongside this one. That is deliberate rather than an oversight, and the reason is worth recording since a site with two methods for one quantity is normally a site with a bug in it.

The old method is what three essays and one refutation-index entry were measured with, and every one of those is at a denominator where the two agree: at 3/8 and 5/13 the fixed offset sits in a flat stretch, and the numbers those essays quote are the numbers either method gives. Replacing the method under them would change published figures by a few per cent for no gain in correctness at those denominators, and would leave four pages quoting numbers that no longer match what they draw.

What is not acceptable is leaving the old method undocumented, so its own file now says plainly that it is superseded above q = 21 and why. The rule this site works to is that a fix belongs where the bug is — and the bug here is not in the arithmetic of dipWidth, which is correct, but in the assumption that one offset stands for a neighbourhood, which stops being true at a denominator no essay written with it goes near.

The line to hold is that no new measurement uses the old background, and that any future extension of the dip results to a large denominator uses the new module. That is a rule about what comes next rather than a rewrite of what exists, and it is the smallest thing that keeps both halves honest.

What this closes, and what it opens #

Closed: the previous phase’s leaving. The exponent survives to q = 89, the coefficient does not, and the coefficient’s dependence is a single factor of the denominator to within a factor of two.

Opened, and smaller: whether the coefficient depends on the rational or only on its denominator. Every fraction measured here is a Fibonacci convergent, which is a very particular set — they are the best rational approximations to the golden angle, so their neighbourhoods are as empty as neighbourhoods of that denominator ever get. A non-convergent with the same denominator sits in a more crowded stretch and might not be measurable at all, which would be an answer of a different kind.

And a caution carried forward. Every claim in this essay depends on a background measured over the offsets that are not on somebody else’s dip, and that method has its own ceiling: it works while a dip’s half-width is much smaller than the distance to the nearest other rational, and at 34/89 that ratio is down to about ten. The law extends further than the measurement can follow it.