The memory was the rise

The previous phase opened its best thread with a number: 0.54, the correlation between one divergence angle and the next, measured with no noise in the model at all. It was read as the sequence’s own memory, and it was explained — plausibly, and in a way that felt like it needed no checking — as the rule correcting itself. A lattice is an arrangement in which each element sits between its neighbours, so a node pushed to one side leaves a gap the next falls into, and the pattern spends its whole length pulling itself straight.

That explanation is about a lattice. It should therefore hold of any lattice the rule produces, including one that is not going anywhere.

It does not.

The test #

Hold the rise fixed. Grow four hundred and sixty nodes at a constant rise instead of a declining one, with no noise, and look at the divergences.

There are none to look at. Every divergence is the same number, to the resolution of the sample grid. The variance is zero. The autocorrelation function refuses the sequence, correctly, because there is no sequence — there is one angle, repeated three hundred and twenty times.

That is the first half of the answer and it is nearly the whole of it. A rule at equilibrium has no sequence. It converges, it stays, and a converged rule produces a constant. The memory the previous phase measured cannot be a property of the lattice, because a lattice sitting still has nothing to remember.

The second half, which fixes the sign #

The obvious objection is that a constant sequence is a degenerate case and proves nothing about a disturbed one. So disturb it.

At a fixed rise with a jostle of three tenths of a degree — a real pattern, four tenths of a degree of scatter, intact by every measure the site has — the lag-one correlation is −0.68.

Not zero. Negative, and strongly, against a sampling band of ±0.11.

So the sign is wrong as well as the attribution. If the self-correction story were the whole explanation, a disturbed lattice at a fixed rise should show the positive correlation most clearly, because that is the case in which the rule has nothing to do except correct. What it shows is a large negative one.

What the negative correlation is #

It is the same arithmetic the previous phase already worked out for placement noise, arriving from a different direction.

A disturbance that displaces a node enters two consecutive recorded divergences with opposite signs — into $d_i$ and $d_{i+1}$ — which is a moving-average term whose autocorrelation at lag one is exactly $-\tfrac{1}{2}$. Under a jostle the node itself is not displaced, but the choice it induces in its successors is, and the effect on the recorded sequence has the same alternating shape: a node that lands a little to one side makes one gap wide and the next narrow.

At a fixed rise there is nothing else in the sequence, so the alternation is all there is, and the correlation goes to the negative value that alternation implies.

On a rising stem there is something else, and that something else is the finding.

The thing that was there instead #

A rising stem’s equilibrium is moving. The rise falls, the ladder’s preferred pair changes, and the divergence the rule wants drifts — slowly most of the time, and quickly through a transition.

The pattern does not track that instantly. It lags, it overshoots, it settles. And because the drift is smooth, the deviations it produces are shared between consecutive nodes: two angles measured a node apart are at nearly the same point on the same drift, so they deviate from the local mean in the same direction.

That is a positive correlation at lag one, and it is what +0.54 is.

The detrending in the previous phase’s analysis was supposed to remove exactly this. It removes a centred moving average over twenty-one nodes, which takes out the slow climb and leaves the fast structure. It works: the drift over twenty nodes is gone. What it cannot remove is the pattern’s response to the drift, which lives at exactly the timescale the statistic reads, and which is not a trend but a chase.

The measurement that settles it #

The clean way to see which of the two it is, is to change the rate rather than the noise. If the correlation is the pattern chasing a moving target, it should depend on how fast the target moves.

Swept, it is a threshold:

• 38 nodes per rung → −0.30
• 67 nodes per rung → +0.54
• 115 nodes per rung → +0.58
• 192 nodes per rung → +0.58
• 308 nodes per rung → +0.55

Below about sixty nodes per rung there is no positive memory at all. Above it, there is one, and it is flat across a fivefold change in rate.

A property of a lattice would not do that. A property of a chase does exactly that: too fast and the pattern cannot follow at all, so the sequence is dominated by the alternation; slow enough to follow and the response saturates, because the pattern is now tracking the equilibrium closely and the residual is set by the rule’s own correction time rather than by the rate.

The evidence that was already in hand #

It is worth asking whether the previous phase could have caught this with what it had, because the answer is yes and the reason it did not is instructive.

That phase ran a robustness check on the one free choice in the analysis: the detrending window. It varied it from eleven nodes to forty-one and confirmed that the separation between the noise kinds survived at every setting. That is a good check and it was the right check for the question being asked, which was whether the filter was manufacturing the correlation.

But it varies the analysis and holds the experiment fixed. Every one of those three runs was a stem at the same rate, climbing the same ladder over the same range of rise. A parameter of the analysis was swept and a parameter of the model was not, and the quantity in question turns out to depend on the second and not the first.

There is a second piece of evidence that was in the record and unread. The phase’s own figures plot the autocorrelation to six lags, and on the coarse rungs the curve dips negative around lag five before recovering. That dip was noticed and set aside as a curiosity. It is the trough of an alternation — the negative lag-one behaviour this essay measures, showing through a positive drift term — and the shape of the whole curve was on the page.

Neither omission is careless. Both are the same thing: a number with an explanation attached stops being interrogated, and the explanation offered was good enough that nobody asked where else it should hold.

The transient, and what it is made of #

The runs in this essay drop their first hundred and forty nodes. That is more than twice what the previous phase dropped, and the difference is not caution.

A run begins from a seeded stretch of ideal lattice at a stated divergence, which puts the pattern on a branch. It is not the pattern the rule would have made, so the rule spends its first several dozen nodes moving away from it — and that motion is itself a chase, with its own strong positive correlation, towards an equilibrium that the seed was not at.

On a rising stem the drift removal takes most of it out, because the settling is slow and a moving average is exactly what removes slow things. At a fixed rise there is no drift removal, because there is no drift to remove — so the transient sits in the sequence undiminished, and it looks precisely like the memory this essay is arguing does not exist.

Dropping a hundred and forty nodes is what makes the fixed-rise measurement a measurement of the equilibrium rather than of the approach to it. That the two look alike is the point: the settling from a seed and the chase down the ladder are the same phenomenon, and it is the phenomenon the correlation reads.

What changes in the previous phase’s conclusions #

This is the part worth being careful about, because a correction that is announced loudly and then turns out to change nothing is worse than no correction.

Nothing in the noise-discrimination result changes. That result compares three kinds of disturbance on rising stems at matched scatter, and finds that placement noise takes the correlation to zero while the other two leave it standing. Every one of those measurements was made on a rising stem and every one of them still holds. The separation is real, it is five sampling bands wide, and the arithmetic explaining it — placement noise contributes a term correlated at $-\tfrac{1}{2}$ — is untouched.

What changes is what the baseline is. The previous phase describes +0.54 as “the sequence’s own memory”, a property the pattern brings with it, which the noise then either erases or leaves alone. The right description is that it is the correlation of a stem in motion, and the noise either erases that or leaves it alone. The mechanism of the discrimination is the same; the thing being disturbed is not what it was said to be.

And one thing becomes a requirement that was not stated. If the correlation is a property of a growing stem, then a stem that is not growing — or one growing too fast — has none, and the test is unavailable on it. The eligibility gate the previous phase specified has one clause: measure the scatter first, and stop if it is above about nine tenths of a degree. It needs a second, and the next essay is that.

The general shape of the mistake #

It is worth naming, because this collection has made it before in a different form.

The previous phase measured a quantity at one value of a parameter and described the result as a constant of the model. The parameter was the rate; it was set to the value everything else in the thread used; and there was no reason at the time to think it mattered, because the explanation offered — a lattice corrects itself — does not mention the rate at all.

The site’s own notes record the same error with maxWindow, where raising a parameter from 120 to 480 changed nothing and read as a converged answer, when in fact the parameter had never been binding. That one was caught by asking what the parameter actually controlled. This one is caught by asking what the explanation predicts and testing it somewhere the explanation says it should hold.

A constant measured at one parameter value is a function evaluated once. The defence is not to sweep everything, which is unaffordable; it is to notice when a story is being told about a quantity, and to check the story where it makes its strongest claim. “The lattice corrects itself” is a claim about lattices, and the cheapest lattice to test it on is one that is not going anywhere.

What a stationary plant would look like #

The consequence for fieldwork is concrete and it is not comfortable.

A great many shoots are, over the stretch anybody would measure, at something close to steady state. The plastochron ratio settles, the rise stops falling appreciably, and the pattern holds one parastichy pair for the whole length of the specimen. That is the easy case for every other measurement here: a steady pattern is what makes a count reliable and a scatter meaningful.

For this statistic it is the empty case. A stem at equilibrium has a divergence sequence whose only structure is the alternation that disturbance puts there — the correlation is negative or zero, and it is negative or zero whichever kind of disturbance the plant has. The instrument that separates the three kinds needs the pattern to be moving.

So the specimen the previous phase asked for is more specific than it said. Not merely a quiet stem and a long one, but a stem visibly climbing — one where the counted pair changes along its length, which is the same feature that makes the rising-phyllotaxis thread interesting and is uncommon enough that it is usually remarked on when it occurs. A shoot that transitions once over sixty internodes is the target; a shoot at 8/13 from top to bottom will report nothing whatever kind of noise it carries.

There is a compensation, and it is the one this phase’s other statistic supplies. The readout that names the parastichy number wants the opposite specimen — a steady rung, held for sixty internodes, and enough disturbance to excite it. The two instruments between them cover the two kinds of stem, and neither covers both.

What the number is worth now #

Less as a description of a lattice and more as a measurement of a shoot.

+0.54 is not a constant of the placement rule. It is what a stem climbing the ladder at between sixty and three hundred nodes per rung produces, and it saturates across that range, which makes it a usable feature rather than a fragile one. A stem showing it is a stem in that regime; a stem showing nothing is either too fast, at equilibrium, or carrying placement-like error — and telling those apart needs the scatter as well, which is the mixture problem this thread keeps arriving at.

The claim that survives, stated so it can be tested again: a divergence sequence’s lag-one correlation measures how a pattern responds to a changing equilibrium, and a pattern at equilibrium has none.