What the sharing costs a lattice

The disturbance that forges this site’s observables also damages the arrangements it is applied to, and the two facts have always travelled together. A stem jostled by errors inherited from its contact neighbours loses its lattice at half the displacement that independent errors need, and its comb ratio moves a fifth of the way towards the value a forgery gives.

The control that separates the disturbance’s structure from its history says the two effects have different causes, and that the more interesting of them belongs to the less interesting half.

The experiment #

Three disturbances driven through the placement rule itself — not into a kinematic lattice, but into stems whose every organ is placed where the repulsion from the existing ones is least. Independent errors, errors shared once with the contact neighbours, and errors inherited from them. Same amplitude per organ, each stream divided by its own measured spread so that a quarter of a degree means a quarter of a degree in every row.

The damage is the sharing #

Take the amplitude at which each disturbance still leaves every stem with a lattice — a divergence scatter under two degrees, on five stems.

The shared disturbance is exactly as harmful as the inherited one. Both cost a lattice half of what it withstands from independent noise of the same size.

That locates the harm where the earlier control put it. The two contact organs dominate the minimum the rule is taking; correlating their displacements makes them push the same way instead of partly cancelling, and the next organ is displaced further. Whether those correlated displacements were themselves inherited makes no difference to that — the rule sees the neighbours’ positions, not their provenance.

Why a correlated disturbance is harsher at all #

The result that shared and inherited disturbances are equally harmful is easier to hold on to with the mechanism in view, and the mechanism is short.

The rule places an organ at the minimum of the repulsion from the organs already there, and at any settled rise two families dominate that sum: the organs m and n places back, whose hops are the two shortest. The minimum sits between them.

If those two organs are displaced independently, their displacements partly cancel in the position of the minimum — one pushes it one way, the other the other, and the expected shift is smaller than either. If they are displaced together, they push the same way and the minimum moves by the full amount. The next organ inherits that as a genuine displacement, and its own displacement is added to it.

So a correlation between the two contact neighbours converts partial cancellation into addition. Nothing in that argument involves where the correlation came from — only that it is there when the rule takes its minimum — which is exactly why the control and the original are equally harmful.

The earlier off-contact control is the other half of the same argument, from the other direction: correlate a disturbance at offsets the rule is not dominated by — seven and eleven at this rise, neither of them a short hop — and the damage disappears, leaving a stem as intact as white noise leaves it. Together the two controls say the harm is neither correlation as such nor inheritance as such, but correlation at the offsets that hold the minimum up.

Two effects, two properties, and they come apart #

Put this beside the forgery result and the pair of them is tidier than either alone.

The transported disturbance has two properties: it is correlated at the contact offsets, and it is re-transmitted, so an error that enters it is still in it hundreds of organs later. Two effects have been measured on it: it damages a lattice, and it forges a comb.

The damage belongs to the first property. Shared once and inherited are equally harmful, so the harm needs the correlation and does not need the history.

The forgery belongs to the second. Shared once forges nothing at any coupling; only the inherited disturbance reproduces the comb.

One property each, cleanly, with a control for each — which is an unusually tidy outcome for two effects of one object and is the reason both controls were worth building.

Which leaves the comb an escape #

That separation has a consequence for the retraction, and it is the largest thing in this essay.

The comb stopped being evidence of a placement rule because a transported disturbance forges it. But the forgery needs re-transmission, and a plant could in principle have correlation without it — errors shared between touching organs because they are made under shared conditions, rather than passed from one to the next.

On such a plant the comb would be evidence again, and the plant would still be at the harsher end of the tolerance table, because the damage does not need the history either.

So the retraction is conditional on a property of a real plant’s disturbance that nobody has measured — and it is measurable on the same sequence, since the two shapes differ in whether the autocorrelation carries anything at the combinations of the two contact offsets.

That is a better position than the retraction left, and it is not a reprieve: the physically obvious transmission is the re-transmitting one, so the escape needs a plant whose disturbance is shared without being passed on. It is worth stating because it is the one arrangement in which the whole thread’s headline observable comes back.

And the boundary is in the quantity a botanist has #

One practical note, because the halving of the tolerance sounds worse than it is for a field measurement.

The displacement a lattice survives halves with a correlated disturbance. The recorded scatter at which it fails does not — it is about two degrees whichever disturbance produced it, because a correlated disturbance produces more scatter per unit of displacement and the two effects cancel in the one quantity anybody can read.

So a botanist looking at a shoot’s divergence scatter is reading how close it is to losing its lattice, and the reading means the same thing whatever the disturbance’s colour. The halving matters to somebody modelling the disturbance and not to somebody measuring the plant.

And the scatter follows the same line #

At a fixed displacement of a quarter of a degree per organ, the recorded scatter of the surviving stems is 0.57° for independent errors, 0.71° for shared ones and 0.97° for inherited ones.

The shared disturbance sits between the two, closer to white. So the damage is not entirely the structure — the inheritance adds to it — but the structure accounts for the step that matters, since 0.71° is already most of the way to the harsh end in survival terms and the survival boundary falls between 0.25° and 0.5° for both.

The amplitudes in between #

The survival table is coarse — it reports the largest amplitude at which every stem of a colour still has a lattice, from a list that doubles — so it is worth saying what the boundary looks like between its steps.

At a quarter of a degree, every stem of every colour survives: five of five standing for white, shared and inherited alike, with scatters of 0.57°, 0.71° and 0.97°. At half a degree, white still has five of five and the two correlated disturbances do not. Past that the failures are not marginal — a stem that loses its lattice does not scatter by three degrees, it scatters by tens, because the pattern has stopped being a lattice rather than become a worse one.

So the boundary is between 0.25° and 0.5° for both correlated disturbances and above 0.5° for white, and nothing in the tables sits near a threshold. That matters for the claim being made: “the same survival” is not two numbers that happen to round together, it is two disturbances failing in the same interval while a third does not.

It also sets what a finer sweep would buy. Halving the step would put the boundary within a fifth of a degree for each colour and might separate the shared from the inherited one — the scatters at a fixed amplitude do differ, so the survival boundaries probably differ a little too. That is five times the runs for a refinement of a number that is already doing its job, which is why the coarse table is what is here.

The ratio is mostly the history #

The comb ratio — the second comb divided by the main one — was this site’s last discriminator between a rule and a forgery, and it was retired when it turned out to follow the disturbance’s colour on arrangements that all contain the same rule. The control says which half of the colour it follows.

The shared disturbance moves the ratio by 0.11 and the inherited one by 0.22. But those three rows are at three different recorded scatters, and the ratio is known to move with scatter — so the comparison a protractor could make is against a white run at the same scatter.

Drive white noise at 0.35° per organ and it records 0.77° of scatter, against the shared disturbance’s 0.71°, and gives a ratio of 0.86.

So of the shared disturbance’s 0.11, about 0.06 is what any disturbance of that recorded size would produce, and 0.05 is the structure. Of the inherited one’s 0.22, the same accounting leaves most of it unexplained by scatter — a white run at its scatter would be beyond the amplitudes at which stems survive at all.

Which is a small rehabilitation and not a rescue. The ratio still moves with the disturbance and still cannot be quoted as a property of the mechanism. What the control adds is that the movement is mostly driven by the property that also forges the comb — the inheritance — rather than by the structure a plant is most likely to have.

An accounting note about equal sizes #

Every row above is at “the same size”, and that phrase has caused trouble in this thread once already, so it is worth spelling out what it means here.

Each disturbance is a stream of deviates multiplied by an amplitude in degrees. The streams are not automatically the same size: the inherited one’s recursion draws from two lagged terms and normalises as though it drew from one, so at a coupling of 0.7 its deviates have a standard deviation of 2.01 rather than one. An earlier comparison in this thread was made without correcting for that, and the result was a control that was described as being at one scatter and was at another.

So every stream here is divided by its own measured standard deviation, over twenty thousand draws, before the amplitude is applied. A displacement of a quarter of a degree is a quarter of a degree of root-mean-square displacement per organ in all three rows.

Recorded scatter is then a result rather than an input, which is the right way round: the whole point of the comparison is that disturbances of equal size leave different scatters, because what a protractor records is relative displacement.

Two effects, two causes, and they are not the same #

Set the two results beside each other, because the pattern is the point.

A disturbance shared between touching organs is dangerous to a pattern and useless for forging evidence about it. A disturbance that is handed on and on is dangerous and forges everything.

That asymmetry has a consequence for how a plant’s own noise should be thought about. The kind of coupling everybody would grant — touching organs push each other — is the kind that costs a lattice most of what it costs and manufactures none of what was being read as evidence. The kind that manufactures evidence is a stronger hypothesis about the tissue, and it is also the kind that would destroy the pattern fastest.

What a botanist would see, and what they would not #

The three rows of this comparison are three different disturbances producing three recorded scatters, and the recorded scatter is the only one of the three quantities a photograph carries.

A stem jostled by independent errors at a quarter of a degree records 0.57°. One jostled by shared errors of the same size records 0.71°. One jostled by inherited errors records 0.97°. So a botanist measuring the scatter of a real stem’s divergences and finding, say, 0.7° cannot tell whether they are looking at a plant with a large independent noise or a smaller correlated one — the two are the same number.

That is the same difficulty the whole thread has been circling, and it is why the observables here are sequence statistics rather than spreads. A spread is one number and the accounts differ by two: what the disturbance is, and how it is shared.

The consequence for an experiment is a design note rather than a result. Reporting a scatter is nearly free and settles nothing; reporting the autocorrelation of the sequence at a handful of lags costs the same measurements and separates accounts that a scatter cannot. Any survey of divergence sequences should therefore publish the sequence, not a summary of it — a point this site has now arrived at from four different directions.

The awkward corollary #

Put those two together and the transported account has an internal tension worth stating.

To forge the comb, the inheritance has to be strong: at a coupling of 0.5 it manages it, and below about 0.4 there is no comb at all. But a coupling that strong is also what makes the disturbance harsh, and the harshness is what destroys a lattice at half the amplitude white noise needs. A plant whose errors propagate strongly enough to produce the observable would be a plant whose lattice is fragile.

The numbers do not quite close that argument — a stem at a quarter of a degree survives at either coupling — but they narrow the window. The account needs an inheritance strong enough to populate the whole residue class of lags and weak enough not to accumulate the pattern away, and the two requirements pull in opposite directions.

What this does not say #

It does not say the shared disturbance is what a plant has. It is a control, and an unphysical one: no process passes a displacement between two organs without passing the displacement they themselves received.

It does not restore the comb ratio. The ratio moved with the colour; it still does; what has changed is the account of which property of the colour moves it.

And it does not settle the discrimination. After all of this the position is that a comb needs a propagating disturbance, a propagating disturbance is harsher than a shared one, and a placement rule leaves a signature in the lag-one correlation that neither has. Three statements, none of which is a clean observable, and the clean observable of the previous work is the intervention rather than any reading.

The check #

Two assertions carry the essay and they are separately falsifiable.

The first requires the shared and inherited disturbances to break a lattice at the same amplitude, and both to break it before white noise does. A control that was merely gentler would fail the first half; one that was harmless would fail the second.

The second requires the ratios to be ordered — white below shared below inherited — and requires the matched white run to account for more of the shared disturbance’s shift than of the inherited one’s. That is the claim that the ratio belongs mostly to the history, stated so that a measurement in which the scatter explained everything, or nothing, would stop the build.

The same comparison at long scales #

The damage and the comb ratio in this essay are both short-scale quantities. Measured at long scales — block means over tens or hundreds of organs — the inherited disturbance and the shared one look almost identical, and both look like independent errors.

That is not a failure of the comparison; it is where the difference between them is not. The inheritance shows itself at the contact offsets and at combinations of them, which is a set of lags rather than a decaying tail.