The drift goes the other way

The prediction is a filter and it is worth writing out before it fails, because the reasoning is good and the failure is instructive.

A placement rule puts each organ where the repulsion from the organs already there is least. If every organ in that neighbourhood is displaced by the same amount, the whole profile shifts and the minimum shifts with it, so the rule places the next organ in the displaced position and the error passes through untouched. If the organs are displaced by different amounts, the profile changes shape, the minimum moves less than the disturbance did, and the error is partly corrected.

A disturbance correlated over few organs looks like the second case to any neighbourhood deeper than its correlation length. A disturbance correlated over many organs looks like the first case to any neighbourhood shallower than it. So the rule should be a high-pass filter whose corner sits where the correlation length meets the depth of the neighbourhood, and the prediction is a crossover that tracks the depth.

The depth is measurable — the number of organs carrying nine tenths of the profile the rule minimises — and it can be varied by a factor of sixty by changing the exponent of the falloff. The correlation length is a parameter of the disturbance. Nothing had varied them together.

How the sweep is built #

Two things have to be held still for the sweep to mean anything, and both are worth stating because getting either wrong would produce the observed result without any physics in it.

The disturbance is the same size at every correlation length. A correlated stream built by the obvious recursion has a larger variance than an uncorrelated one, so comparing at equal nominal amplitude would compare a disturbance with one nearly twice its size and report the difference as an effect of correlation. Every stream here is divided by its own measured standard deviation, so a jostle of a quarter degree is a quarter degree of displacement per organ at every setting.

The stems are the same pattern. Every run in the sweep is counted at 8/13 by a blind counter shown its positions, at every depth and every correlation length. A depth at which the rule grew a different lattice would be comparing two objects rather than one, and the exponents at the ends of this sweep are chosen inside the range where the lattice holds.

Three seeds are averaged at every cell, and the spread between them is carried through the figures as a band rather than reported once. At the white control that spread reaches a factor of four, which is why the companion null in this thread is stated against it rather than against a constant.

No crossover, at any depth #

The first thing to say about the figure is what is missing from it.

Every line rises smoothly with the correlation length. None of them has a knee, a plateau, a step or a turn. There is no correlation length at which any of the five rules stops suppressing and starts passing, and no ordering among the lines that changes anywhere in the sweep.

That absence is checked rather than eyeballed: at every depth, the wander must be strictly larger at each longer correlation length than at the one before, and the sweep must contain correlation lengths shorter than the neighbourhood, where a filter would have suppressed them. At a depth of a hundred and eighty-two organs, every correlation length in the sweep — 2.8, 9.5 and 32.8 organs — is comfortably inside the neighbourhood, and all three pass.

What the numbers are, line by line #

The figure compresses five sequences and they are worth reading out, because the regularity in them is stronger than the headline.

At an exponent of 1.5 — a hundred and eighty-two organs — the wander runs 0.98, 10.03, 30.59, 82.26 across independent errors and correlation lengths of 2.8, 9.5 and 32.8 organs. At an exponent of 2, a hundred and twenty organs: 0.82, 7.24, 27.61, 68.51. At 3, thirty organs: 0.91, 5.05, 19.08, 46.47. At 4, eight organs: 0.69, 4.80, 15.06, 35.70. At 6, three organs: 0.60, 3.42, 10.94, 22.37.

Two things fall out of the columns rather than the rows.

The white column is close to one at every depth, which is the calibration working: independent errors give one by construction of the statistic, and a column that drifted with depth would mean the normalisation was itself depth-dependent and nothing else in the table could be read. It runs 0.98, 0.82, 0.91, 0.69, 0.60 — sagging slightly at the shallow end, by a factor of 1.6 across a depth range of sixty, which is inside the seed spread.

The ordering is preserved down every column. It is not that the deep rule wins at the longest drift and the ordering shuffles elsewhere; the deep rule is highest at 2.8 organs, at 9.5, at 32.8, and the shallow rule is lowest at all three. A filter would produce crossings between the lines somewhere in that range, since each line’s corner would sit at its own depth, and the depths differ by a factor of sixty.

And the direction is reversed #

The second thing is worse for the prediction than the first.

At a correlation length of 32.8 organs, the wander runs 82.3, 68.5, 46.5, 35.7, 22.4 from the deepest rule to the shallowest. The rule that looks at a hundred and eighty-two organs passes 3.7 times as much as the one that looks at three.

The filter reading says the opposite. A shallow rule should be the leaky one: a disturbance correlated over thirty-three organs is a uniform translation to a neighbourhood of three, so a three-organ rule should pass all of it, and a hundred-and-eighty-two-organ rule should see the variation within the drift and correct most of it.

The measured ordering is not marginal and it is not a seed. It is monotone across all five depths, at every correlation length in the sweep including the shortest, and the ratio between the ends is far outside the seed-to-seed spread.

The prediction was about the wrong quantity #

The account of what went wrong is the subject of the next rung and is worth stating here in outline, because otherwise the result looks like a mystery rather than an error.

The wander is a ratio. Its numerator is how much the block means of the divergences move, which is what “how much drift got through” ought to mean. Its denominator is the variance of the divergences themselves — the scatter between neighbours.

A deeper rule suppresses the relative part of a disturbance better than a shallow one, because it has more neighbours to compare against. That is the denominator falling. If the numerator barely moves, the ratio rises — and it rises for a reason that has nothing to do with how much drift the rule let through.

So the sweep did test something, and what it tested was not what it was built to test. The filter reading is a claim about the numerator and the statistic is dominated by the denominator.

Why the prediction was worth making anyway #

It is tempting, having found the error, to say the experiment was ill-posed. It was not, and the distinction matters for what gets built next.

The filter reading came out of a measurement rather than out of nowhere: a rule driven by a correlated jostle leaves more wander than a kinematic lattice driven by the same jostle, while cutting the neighbour-to-neighbour scatter by more than half. Both halves of that were measured. The reading assembled them into a mechanism — the rule corrects the relative part and passes the shared part — and the mechanism is, as far as this sweep can tell, correct.

What the reading got wrong is the step from a mechanism to a prediction about a particular number. It assumed that if the rule corrects the relative part, then a statistic that is supposed to measure the shared part will show the correction turning on and off with the depth. That step needed the statistic to be a measurement of the shared part alone, and it is not.

The general form is worth keeping: a mechanism can be right and every prediction derived from it through a ratio can be wrong, because the ratio’s denominator is also a function of the mechanism. That is the same hazard the companion sweep of the loop bound ran into from the other side, where a parameter that did nothing produced a figure indistinguishable from a parameter the answer did not depend on.

What would have shown the crossover #

If the filter is real, the place to look is the numerator on its own — how many degrees of slow drift reach the divergences, without dividing by anything.

That measurement is available from the same runs and it is the next rung’s subject. The short version is that it is nearly flat across the whole sweep: a neighbourhood sixty times deeper passes about a fifth more drift, where the ratio moves by a factor of 3.7. So the filter, measured directly, is very weak rather than sharply tuned, and there is no corner in it either.

That is a second negative and it is the honest place to leave the filter reading: the rule does correct what is relative and does pass what is shared, and the amount it passes is almost independent of how deep it looks — which is not what a filter with a corner does.

The comparison against no rule at all #

There is one more control worth putting beside the sweep, because it answers a question the sweep alone leaves open: is the rule doing anything to the drift, or is the whole table simply the disturbance showing through?

The comparison is a kinematic lattice — organs placed at a fixed divergence plus the same disturbance, with no placement rule anywhere in it. That is what “all of it gets through” looks like, and at a correlation length of 32.8 organs it gives a wander of about 36, over three seeds whose individual answers run from 22 to 58.

The rule at an exponent of three gives 46.5, which is above the no-rule control. The rule at an exponent of six gives 22.4, which is below it. So the rule is not uniformly a filter or uniformly an amplifier of this statistic; where it sits relative to no rule at all depends on how deep it looks.

That is a strange thing for a filter to do and an entirely ordinary thing for a ratio to do. A rule that suppresses relative scatter shrinks the denominator below the kinematic lattice’s, and a deep rule shrinks it enough to push the ratio above one even though the numerator is not larger. The control does not rescue the filter reading; it makes the case for looking at the numerator alone, which is what the next rung does.

It is also worth saying that the control is noisy — a factor of two and a half between its own seeds at three seeds — so it is used here to bracket rather than to measure. Sharpening it would mean many more runs of a lattice nobody is otherwise interested in.

What this does not say #

It does not say the rule does not correct anything. It corrects a great deal: the scatter between neighbouring divergences falls from 0.32° to 0.21° across the depth sweep, which is the correction happening and being measured.

That number also lets the reversal be checked arithmetically rather than taken on trust. The variance ratio between the two ends of the scatter is (0.32 ÷ 0.21)² = 2.3, and the drift reaching the divergences moves by about 1.2 across the same sweep, so the two together predict a ratio of about 2.8 in the wander. The measured ratio is 3.7. The denominator therefore accounts for most of the reversal, the numerator for a little of it in the same direction, and the residue of about a third sits inside the seed-to-seed spread, which reaches a factor of four on the white control. The decomposition closes, which is what says the reversal is arithmetic rather than a phenomenon — and undoing the normalisation altogether is the same statement made without a ratio in it.

It does not say there is no crossover anywhere. It says there is none in the ratio across correlation lengths from 2.8 to 32.8 organs at neighbourhoods from 3 to 182; read as degrees rather than as a ratio there is one, and it does not track the depth. A disturbance correlated over thousands of organs, or a neighbourhood of one, are not reached.

It does not say the exponent only changes the depth. It changes how sharp the profile’s minimum is as well, and the companion sweep of the loop bound — which changes the number of organs and nothing else, and does nothing — is the control for that rather than a proof.

And it does not say a plant’s inhibition falls as any of these exponents. The exponent here is being used as a handle on depth, not proposed as a measurement. What a real inhibitor’s range is remains this collection’s most testable open quantity and is untouched by this.

What is left of the filter #

Something survives, and it is worth separating from what does not.

The mechanism survives: a rule places against neighbours, so it responds to what differs between them and is blind to what they share. That is not a hypothesis about plants, it is a property of taking a minimum over a sum, and it is why a drift passes through where a jitter does not.

The tuning does not survive. The filter reading implied that the rule has a characteristic scale — its own depth — separating what it corrects from what it does not, and that moving the scale moves the separation. Across a factor of sixty in the depth, no separation appears at any correlation length in the sweep.

The reason is visible once the question is asked in the right units. The depth of the neighbourhood is a number of organs, and the correlation length of the disturbance is a number of organs, but the rule does not compare organs at equal weight. Thirty organs carry nine tenths of the profile at an exponent of three, and the nearest half dozen of those carry most of the thirty. So the scale that would set a corner is not the depth as measured; it is something much shorter and much less sensitive to the exponent, which is exactly what a nearly-flat sweep looks like.

That is a claim this thread does not test and states rather than proves. It is written down because a negative result with no account of why the prediction failed is an invitation to make the same prediction again with a different parameter.

The check that would refuse it #

Three assertions, and they are arranged so that the negative cannot be satisfied by a sweep that failed to sweep.

The first is that the neighbourhood really varies: the number of organs carrying the profile must fall at every step of the exponent sweep, and the range must span at least a factor of ten. It spans sixty-one.

The second is the absence of a crossover, stated as monotonicity: at every depth, the wander must be strictly larger at each longer correlation length. A crossover would show as a violation somewhere. The same assertion requires that the sweep include at least one correlation length shorter than the neighbourhood at every depth, since a filter can only be refuted where it predicts suppression.

The third is the reversal: at the longest correlation length, the wander must fall at every step as the rule is made shallower, and the ratio between the ends must be at least two. That is the assertion that turns “the prediction is unsupported” into “the prediction is backwards”, and it is the one that would fail first if the effect were noise.