Phyllotaxis
the spirals are counted, not admired
What a plant might be doing

What the ratio was hiding

The statistic that says a rule sharpens a drift rises by a factor of nearly four across a sweep of the rule's depth. Undo the normalisation and ask instead how many degrees of drift actually reach the divergences, and the answer changes by a fifth. Nearly all of the effect was in the denominator, and the denominator is the thing the rule is good at.
24 min read 16 figures Two routes, one numberState the claim, state the testCounted, not admired

Worth reading first: A disturbance with a memory · The sequence has a memory · Two degrees of scatter.

The wander of a stem’s divergences is a ratio. Cut the divergences into consecutive blocks, take the variance of the block means, multiply by the square of the block size, and divide by the variance of the divergences themselves. The division is what makes it dimensionless, comparable between disturbances of different sizes, and equal to one for independent errors.

It is also what makes it a poor instrument for the question it was being asked.

Across a sweep of the rule’s neighbourhood — from three organs to a hundred and eighty-two, a factor of sixty-one — the wander rises from 22.4 to 82.3 at a disturbance correlated over thirty-three organs. That was read as the deep rule passing 3.7 times as much drift as the shallow one.

The wander climbs because its denominator fallsThree quantities across the same sweep of the rule's depth, each divided by its own value for the shallowest rule so that they share an axis. The wander rises by a factor of 3.7 as the neighbourhood goes from 3 organs to 182. The scatter between neighbouring divergences falls by 1.57, because that is the part of a disturbance a placement rule corrects and a deeper rule corrects it better. What is left — how many degrees of slow drift actually reach the divergences — changes by 1.20, from 1.52 to 1.82 degrees. The rule barely filters a drift at any depth.012342.262.081.480.9030.477organs in the neighbourhoodagainst the shallowest rulewanderscatterdrift througha drift correlated over 33 organs · 3 seeds a pointdrift through: 1.52° to 1.82°
Fig. 1 The same sweep with the ratio taken apart. Three quantities, each divided by its own value for the shallowest rule so that they share an axis: the wander, the scatter it divides by, and the degrees of drift that actually reach the divergences.
What the sequence sees that the scatter cannotEach point is an ensemble at one amplitude, placed at the scatter it produces. A stem at three quarters of a degree of scatter has a lag-one correlation near zero if its noise arrived after the primordium was placed, and near 0.7 if it arrived before — and no measurement of the scatter can tell those apart. The separation closes above about a degree, because what the other two kinds preserve is the correlation of a lattice.-0.20000.2000.4000.6000.5000.75011.251.50divergence scatter, in degrees — the one quantity a plant offerscorrelation between one divergence and the nextplacement noisejostle noisefield noise4 runs per point · band ±0.13every point is a lattice
Fig. 2 The denominator, on its own terms: how much of a sequence’s spread sits between adjacent terms. It is the part a placement rule corrects and therefore the part that moves when the rule changes.

The quantity without the division

The numerator of the wander, before it is normalised, is how far the block means move. Multiply by the block size and take a square root and it becomes a quantity in degrees: how much slow drift reaches the divergences, per organ, in the units a protractor would report.

Across the same sweep it runs 1.82°, 1.74°, 1.65°, 1.61°, 1.52° from the deepest rule to the shallowest.

That is a change of a fifth. A neighbourhood sixty times deeper passes twenty per cent more drift.

Meanwhile the denominator — the scatter between neighbouring divergences, which is the quantity a rule is supposed to be good at suppressing — runs 0.205°, 0.212°, 0.246°, 0.269°, 0.321°. A change of 1.57 in the other direction.

Multiply the two effects and the ratio moves by 3.7, which is what the wander does. Nearly all of it is the denominator.

A deeper rule passes more of a drift, not lessThe wander left in a stem's divergences, against how many organs its disturbance stays correlated over, for rules whose neighbourhoods run from 3 organs to 182. The prediction under test said a rule should pass a drift once the drift outlasts its neighbourhood, so the shallow rules should be the leaky ones and each line should turn where its own depth is crossed. Every line rises smoothly and the deepest rule is the highest of them at every correlation length — 82 against 22 at the longest drift. There is no crossover anywhere in the sweep.02550750102030how many organs the disturbance stays correlated overwander at a block of 64182 organs120 organs30 organs8 organs3 organsneighbourhood measured as the organs carrying 90% of the profileno crossover at any depth
Fig. 3 The ratio as it was read before this decomposition, for comparison. Nothing in that figure is wrong; what is wrong is the sentence that was attached to it.
Through the rule, the drift survives and the inheritance still does notHow much of a divergence sequence's variance survives being averaged over blocks, on stems the rule grew. The vertical quantity is B² times the variance of the block means divided by the variance of the sequence, which is one at every block size for independent errors — the arithmetic is normalised for a *differenced* stream, since a divergence is the difference of two organs' errors. A line that climbs is a sequence with power at frequencies below one per block. The disturbances that remember the last error climb to 46 at a block of 64. The ones inherited between touching organs do not climb at all — 1.51 and 1.90 at the same block — although their own deviates carry ×— and ×— an independent stream's variance in exactly this statistic. What they do instead is dig a hole: at block sizes of 8 and 13, which are the offsets they couple at, the statistic falls to 0.06 and 0.12.10.11024813163264block size, in organsvariance of the block means, against independent errorsindependenta memory, ρ = 0.9a memory, ρ = 0.97inherited, a = 0.5inherited, a = 0.7900 organs · jostled at 0.25° · 3 stems eachgenerated from a stated rule, not drawn to look right
Fig. 4 The statistic on the rule’s own stems, before it is taken apart. Everything this essay does is undo one division in it.

Getting from the ratio to degrees

The step from a dimensionless number to a quantity in degrees is one line of algebra and it is worth writing out, because everything here depends on it being the right line.

The wander at block size B is the variance of the block means, times B squared, divided by the variance of the divergences. So the variance of the block means is the wander times the divergences’ variance, divided by B squared. Take a square root: the standard deviation of the block means is the square root of the wander, times the standard deviation of the divergences, divided by B.

Multiply by B and what is left is the square root of the wander times the scatter — a quantity in degrees, and the natural one: it is how far a block of B organs is displaced from its neighbours’ blocks, scaled up by the block size, which is the size of the slow excursion the disturbance is producing.

The memory of a divergence sequence, at 0.75° of scatterWith no noise at all the lag-one correlation is 0.54: the rule corrects itself, so a lattice arrives with a memory in it. Matched at the same recorded scatter, placement noise leaves -0.10, jostle noise leaves 0.66, field noise leaves 0.47. The band is ±0.13, which is what an uncorrelated sequence of this length gives.-0.25000.2500.500123456lag, in nodescorrelation between a divergence and the one that many nodes laterno noiseplacement noisejostle noisefield noisesampling band4 runs each · 243 divergences per runmatched at 0.75° of scatter
Fig. 5 The other end of the same arithmetic: a correlation read at a lag rather than at a block size. The two are one function evaluated in two places, so a quantity in degrees at one is a quantity in degrees at the other.

Nothing in that derivation is specific to this thread, and its consequence is general: the square root of a normalised second moment, times the scale it was normalised by, is the unnormalised quantity. The reason it is worth spelling out is that it is available at no cost from every measurement this collection has already made with the statistic, and it was never taken.

Which is a result about the rule, just not the one it looked like

Untangling the two does not leave nothing. It leaves two statements, both worth having and neither the one that was claimed.

A deeper rule corrects relative errors better. The scatter between neighbours falls by a factor of 1.57 as the neighbourhood grows from three organs to a hundred and eighty-two. That is exactly what a rule placing against neighbours should do: more neighbours in the comparison means a better estimate of where the next organ belongs, and less of the disturbance survives as scatter.

And a drift passes almost regardless. The degrees of slow drift reaching the divergences change by a fifth across the same range. The rule is not filtering the drift at any depth; it is passing nearly all of it at every depth, and what changes with depth is how clean the rest of the signal is.

Two stems at 0.75° of scatter, one angle at a timeThe divergence of each node, with the slow climb of the ladder removed. Both stems scatter by about 52.26° and a botanist would report them identically. The jostled one wanders in runs — its lag-one correlation is 0.70 — and the one with placement noise alternates about the mean at 0.21, because a displacement of one node enters two consecutive divergences with opposite signs.jostle noise — correlation 0.70placement noise — correlation 0.2160 nodes each, both at 52.26° of scattercorrelations 0.70 and 0.21
Fig. 6 The distinction in one picture: two stems can agree on how much their divergences scatter and disagree entirely about how that scatter is arranged, which is why a single number about spread is not a description of a disturbance.

Put together: the rule is a good corrector of the relative part and an indifferent filter of the shared part, at every depth tried. The wander rises with depth because it is measuring the corrector, not the filter.

A memory manufactures nothingThe largest comb mean found in a kinematic lattice whose azimuth errors are an AR(1) process, against the coefficient of that process, over eight seeds at each point. The dashed line is where the rule's own stems sit, at 0.64; the shaded strip is three sampling bands. Every point is inside the strip — 0.026, 0.022, 0.014, 0.015 at ρ = 0.5, 0.7, 0.9, 0.97 — and the readout returns nothing on 32 runs out of 32. A correlated error is not a periodic one.00.2000.4000.6000.5000.7000.9000.970how strongly each error remembers the last, ρthe largest comb mean anywhere in the thirty lagsthe rule's own stems: 0.64three sampling bands0.0260.0220.0140.015kinematic lattice · AR(1) errorgenerated from a stated rule, not drawn to look right
Fig. 7 The disturbances at the coefficients this essay uses, read at lags. A drift correlated over tens of organs is nearly smooth between neighbours, which is why it contributes so little to the quantity below.

How much of the denominator is the drift itself

There is one honest complication and it limits how cleanly the two parts separate.

The denominator is the variance of the divergences, and the divergences contain everything: the relative scatter the rule failed to correct, and whatever neighbour-to-neighbour difference the drift itself produces. A drift correlated over thirty-three organs is not perfectly smooth from one organ to the next, so it contributes to the denominator as well as to the numerator.

That means the decomposition is not an identity. It is three measured numbers, and the statement “the ratio moved because the denominator fell” is a statement about their observed sizes rather than an algebraic fact.

What the sequence sees that the scatter cannotEach point is an ensemble at one amplitude, placed at the scatter it produces. A stem at three quarters of a degree of scatter has a lag-one correlation near zero if its noise arrived after the primordium was placed, and near 0.7 if it arrived before — and no measurement of the scatter can tell those apart. The separation closes above about a degree, because what the other two kinds preserve is the correlation of a lattice.-0.20000.2000.4000.6000.5000.75011.251.50divergence scatter, in degrees — the one quantity a plant offerscorrelation between one divergence and the nextplacement noisejostle noisefield noise4 runs per point · band ±0.13every point is a lattice
Fig. 8 The quantity the complication is about: how much of a sequence’s spread sits between adjacent terms rather than across the run. A disturbance that is smooth on the scale of a few organs puts almost nothing there, and one that is not, does.

How much does it matter here? The drift is correlated over thirty-three organs and the scatter is measured between adjacent ones, so the drift’s contribution to the denominator is small — it changes by about three per cent per organ over the correlation length, against a scatter of a fifth of a degree. That is why the three numbers multiply out as closely as they do: 1.20 times 1.57 is 1.87 against the square root of the wander’s own ratio, which is 1.92.

Agreement to within three per cent is the best available evidence that the decomposition is close to exact at this correlation length. At a much shorter one it would not be, and no claim is made there.

Where each kind's lattice gives wayThe largest amplitude at which every run still has a lattice, and the scatter it produces there. The amplitudes are incomparable — field 0.015 (fraction of the barrier), jostle 1 (degrees of azimuth), placement 0.8 (degrees of azimuth) — and the scatters agree to 19%. The boundary belongs to the pattern rather than to the disturbance: a lattice fails at about a degree and a half of scatter, and which of three mechanisms produced it does not move where.field1.64°intact to 0.015, broken by 0.02jostle1.72°intact to 1, broken by 1.4placement1.42°intact to 0.8, broken by 13 runs per amplitudescatters 19% apart
Fig. 9 Why an unnormalised spread is not usable on its own: what a stem tolerates depends on the kind of disturbance as well as its size, so a number in degrees carries the experiment as well as the subject.

Why the normalisation was there in the first place

It would be wrong to conclude that the division was a mistake. It was put in for a reason and the reason still holds.

Without it, the statistic has units of degrees and depends on the size of the disturbance driving the stem. Comparing a stem jostled at a quarter of a degree with one jostled at a half would then report the second as having twice the wander, whatever the structure of either disturbance — and the whole point of the statistic is to separate structure from amplitude.

A lattice or a wreck, with nothing in betweenEvery run in the sweep, at both kinds of noise. The line at 6° is where a run stops being counted as a lattice, and nothing lands near it: the intact runs reach 1.97° and the destroyed ones start at 8.06°, a factor of 4.1 away.00.50011.5001234567amplitude, by stepscatter, log₁₀ degrees6°intactno latticeplacementfield48 runs · both kindsan empty factor of 4.1 at the cut
Fig. 10 Why an unnormalised measure of spread is not usable on its own: it is a function of how hard the stem was disturbed, and this thread’s questions are all about disturbances of equal size and different shape.

The normalisation does that job correctly. What it cannot do is survive a sweep of a parameter that changes the denominator, and that is precisely what the depth sweep is. Every earlier use of the statistic in this collection compared disturbances at a fixed rule; this was the first comparison of rules, and it is the first place the denominator was free to move.

That is a general shape and this collection has met it before. A ratio is safe across a sweep of anything its denominator does not depend on, and misleading across a sweep of anything its denominator does depend on — and which of those a new sweep is cannot be read off the statistic’s definition without thinking about the mechanism.

Two combs, at a rise of 0.005The autocorrelation of 760 divergence angles from one stem held at a rise of 0.005. The filled teeth are the lags at multiples of 8; the open teeth are the second comb, at the same spacing offset by 5. Reading the spacing off the first and the offset off the second gives the pair 8 and 13, which is what the position counter reports for the same stem — from angles alone, with no coordinate anywhere in the calculation.-0.50000.500125810131621242629lag, in internodescorrelation between a divergence and the one that many internodes later816245132129spacing 8 · offset 5pair 8/13 — counter says 8/13the shaded strip is the sampling bandone stem · 760 divergences · disturbance 0.25generated from a stated rule, not drawn to look right
Fig. 11 The sibling statistic, which has the same structure and the same hazard: a ratio whose denominator is a property of the rule. Reporting the scale beside the ratio is the same repair in both.

The repair

The repair is not to abandon the ratio. It is to report the two parts beside it, which costs nothing since both are computed on the way to it.

So the measurement is now three numbers per cell: the wander, the scatter, and the drift in degrees. The first is comparable with everything this collection has already measured; the second says what the rule did to the relative part; the third says what reached the sequence. Any claim of the form the rule passes more or less of a drift has to be made in the third.

The disturbance with the largest wander leaves none in the sequenceHow much of a divergence sequence's variance survives being averaged over blocks, on kinematic lattices. The vertical quantity is B² times the variance of the block means divided by the variance of the sequence, which is one at every block size for independent errors — the arithmetic is normalised for a *differenced* stream, since a divergence is the difference of two organs' errors. A line that climbs is a sequence with power at frequencies below one per block. The disturbances that remember the last error climb to 32 at a block of 128. The ones inherited between touching organs do not climb at all — 1.06 and 0.83 at the same block — although their own deviates carry ×5 and ×49 an independent stream's variance in exactly this statistic. What they do instead is dig a hole: at block sizes of 8 and 13, which are the offsets they couple at, the statistic falls to 0.22 and 0.21.10.11024813163264128block size, in organsvariance of the block means, against independent errorsindependenta memory, ρ = 0.5a memory, ρ = 0.9a memory, ρ = 0.97inherited, a = 0.5inherited, a = 0.7shared once, a = 0.76000 organs · 6 runs eachgenerated from a stated rule, not drawn to look right
Fig. 12 The statistic on a lattice with no rule in it, where the denominator is fixed by construction and the ratio is safe. Every earlier result in this thread was made in this regime, which is why they stand.

The same repair has been made twice before here in different clothes. A period was reported without the resolution of the grid it was computed on, and it turned out to be the grid; the fix was to return the span of the motif beside the period. A dip width was reported without the window it was integrated over, and it turned out to be the window; the fix was to report the width at several windows. The rule of thumb they add up to is short: a summary statistic returned without the scale of what it summarises can be read as anything.

Three disturbances, three places to get inThe rule reads its neighbours, builds a profile of the energy at every azimuth, takes the least of it, and records a position. field noise enters at the profile; jostle noise enters at the neighbours. Two of the three are upstream of the choice and can change which minimum is taken; the third is downstream and never can.upstream of the choicethe neighboursalready placedthe profileenergy by azimuththe choicethe least of itthe recordwhat a ruler readsfield noisejostle noiseone rule, three entry pointsthe order is the argument
Fig. 13 The entry points a disturbance has into a placement rule. Every measurement in this thread uses one of them, so a difference between cells is a difference in the rule rather than in the disturbance.

The third number, and why it is reported per cell

Adding a number to a table is cheap and adding the wrong one is not, so it is worth saying why the drift in degrees is reported at every cell rather than once.

A single headline — the drift through changes by a fifth across the sweep — would hide the thing that makes it convincing, which is that it changes monotonically and by small steps: 1.52°, 1.61°, 1.65°, 1.74°, 1.82°. Five values in order, none of them out of sequence, over three seeds each. A quantity that wandered between those values would be reporting seed noise; one that stepped cleanly is reporting something small and real.

It also matters that the wander and the scatter are reported at the same cells rather than summarised. The claim is a relation between three quantities and relations between summaries are how a decomposition goes wrong: two effects that each look monotone in a mean can be uncorrelated cell by cell.

The ratio follows the disturbance, not the ruleThe ratio of the second comb to the main comb on stems grown by the placement rule and jostled by seven different disturbances, all at 0.25° of displacement per organ and all on the same rule. Independent errors and errors with a memory return 0.76–0.81, which is the value this site measured for the rule. A periodicity at the smaller parastichy number takes it down to 0.45; errors inherited from the contact neighbours take it up to 1.09, most of the way to the 1.24 a transported disturbance gives with no rule in it at all. So the quantity separates arrangements by how their errors are related, not by whether anything computed the positions.second comb ÷ main comb, at 0.25° of displacementthe rule, 0.79no rule at all, 1.24independent0.80a memory, ρ = 0.50.78a memory, ρ = 0.90.76a memory, ρ = 0.970.81repeating every 80.45inherited, a = 0.51.02inherited, a = 0.71.095 stems a row · rise 0.005generated from a stated rule, not drawn to look right
Fig. 14 The habit generalised, on a different statistic in the same thread: a ratio reported per arrangement rather than pooled, so that a difference between arrangements cannot be manufactured by averaging.

The cost of all this is three numbers where there was one, and the benefit is that the next sweep of anything in this thread will show immediately whether it is moving the numerator, the denominator or both. That is the whole of the repair, and it is smaller than the mistake it is fixing.

What this changes about the earlier claim

The claim this thread started from is that a placement rule sharpens a drift rather than correcting it, and it was made from a single pair of numbers: a rule’s stems giving a wander of 46.5 against a kinematic lattice’s 30.0 at the same disturbance, while the scatter fell from 0.553° to 0.246°.

Both halves of that measurement stand. What changes is which of them carries the claim.

The wander is in the disturbance and not in what a plant lets you measureEach disturbance measured twice, in the same statistic. On the left, the variance of the block means of the disturbance's own deviates, over blocks of 100, as a multiple of what independent draws would give; on the right, the same quantity for the divergence sequence those deviates produce, over blocks of 128. The left column is what this site measured when it proposed a slow wander as a second observable. The right column is what a botanist would have: a divergence is the difference of two organs' errors, and differencing is exactly the operation that removes power at low frequencies. The disturbance inherited between touching organs goes from ×49.1 — the largest here — to 0.83, which is what independent errors give. The one with a memory in time keeps most of its own.in the disturbanceblocks of 100in the divergencesblocks of 128the horizontal rule is what independent errors giveindependent×0.930.96a memory, ρ = 0.9×16.2410.25inherited, a = 0.7×49.090.83shared once, a = 0.7×2.550.956000 organs · 6 runs eachgenerated from a stated rule, not drawn to look right
Fig. 15 The measurement the claim came from, in its original form: what is in a disturbance against what survives into the divergences. Nothing in it is withdrawn.

The scatter falling by more than half is the rule correcting the relative part, and it is the larger of the two effects. The wander rising is what a ratio does when its denominator halves. So sharpens is the right word for what the rule does to the visibility of a drift, and the wrong word for what it does to the drift: it does not amplify anything, it cleans up everything else.

That is a smaller claim and a more useful one, because it is a claim about an instrument a botanist would use. A plant whose placement rule corrects well is a plant in which a slow disturbance is easier to see, not one in which a slow disturbance is larger.

What a botanist would do with this

The practical form of the correction is worth stating, because it changes the advice this thread gives about a real measurement.

A slow drift in a plant’s divergences — a wander over tens of organs rather than a jitter between neighbours — is a distinct kind of disturbance and the collection has argued that it is worth looking for, because it is the one signature of a disturbance with a memory in time rather than in space. The question a botanist faces is whether a placement rule would hide it.

The answer, corrected, is no and for a slightly surprising reason. The drift comes through nearly intact at any depth, so what reaches the sequence of divergences is essentially what the disturbance produced. What the rule changes is the background the drift has to be seen against: a rule that corrects relative errors well leaves a quieter sequence, and the same drift is then a larger fraction of what is left.

What the experiment costs, in internodesThe combined sampling band of two autocorrelations falls as one over the root of the sequence length. The difference to be resolved is 0.76 — between noise that arrives before the primordium is placed and noise that arrives after — so the count needed is 56 internodes on a single stem. Every other open question in this collection is priced in tens of specimens.00.2500.5000.750100200300internodes counted on one stemsmallest difference in correlation the count can resolvethe difference to resolve — 0.7656 internodesmatched at 0.75° of scatterone stem, counted once
Fig. 16 What the measurement costs in organs, which is the question this makes easier rather than harder. A cleaner background means a shorter run suffices for the same confidence.

So the practical claim is that a well-corrected plant is a better subject for this measurement than a poorly corrected one, and the earlier wording — that the rule sharpens the drift — was right about the observation and wrong about the mechanism. Nothing about how many organs a botanist has to count changes; what changes is that the number now follows from a quantity in degrees rather than from a ratio whose parts had not been separated.

What this does not say

It does not say the wander should not be used. It is the right statistic for comparing disturbances at a fixed rule, which is what it was built for and what every earlier measurement here does with it.

It does not say the twenty per cent is nothing. The drift through does rise with depth, monotonically, across the whole sweep. It is a real and small effect and this essay does not have the seeds to say whether its shape is interesting.

It does not say the scatter is the whole denominator story. The variance of the divergences includes whatever the drift itself contributes to neighbour-to-neighbour differences, so the two parts are not perfectly separable. The decomposition here is a good approximation rather than an identity, and it is reported as three measured numbers rather than as an algebraic split.

And it does not settle what a real inhibition’s range is. The exponent is a handle on depth in a model. What it corresponds to in tissue is the open question this collection has repeatedly named as its most testable one.

The check that would refuse it

Two assertions, and their relation is the content.

The first is that the drift reaching the divergences changes by less than a factor of 1.5 across the whole depth sweep. That is the claim that the numerator is nearly flat, and it is the one that would fail if the filter reading were right after all — a rule with a corner in it would show a large change in the numerator somewhere in a sweep spanning a factor of sixty in depth.

The second is that the wander changes by more than twice as much as the drift through does. That is what makes the essay’s title a measurement rather than an opinion: it says the ratio moves for a reason its numerator does not supply, and it fixes how much of the movement has to be unaccounted for before the claim is allowed. Measured, the wander moves by 3.7 and the drift through by 1.20, so the margin is comfortable and the assertion has room to fail.

Both are computed from the same three seeds per cell as the figures, and the spread between those seeds is carried rather than hidden — because a decomposition of an effect into two smaller effects is exactly the kind of claim that a noisy measurement can be talked into.

Where this goes next

02550750102030how many organs the disturbance stays correlated overwander at a block of 64182 organs120 organs30 organs8 organs3 organsneighbourhood measured as the organs carrying 90% of the profileno crossover at any depth The drift goes the other way A rule that corrects what its neighbourhood shares should let through any disturbance slower than its own reach, and should suppress anything faster — a crossover, tracking the depth. Swept over a neighbourhood that changes by a factor of sixty, there is no crossover anywhere, and the deep rule passes nearly four times as much as the shallow one. The prediction is not weakly supported; it is backwards. 10.11024813163264block size, in organsvariance of the block means, against independent errorsindependenta memory, ρ = 0.9a memory, ρ = 0.97inherited, a = 0.5inherited, a = 0.7900 organs · jostled at 0.25° · 3 stems eachgenerated from a stated rule, not drawn to look right What the rule does to a drift A placement rule was supposed to leave no slow wander in a divergence sequence, because its errors are corrections rather than inheritances. Driven by a disturbance that drifts, it leaves a larger one than a lattice with no rule in it at all — while cutting the per-organ scatter by more than half. The rule removes what is relative between neighbours, and a drift is not. 10.11024813163264128block size, in organsvariance of the block means, against independent errorsindependenta memory, ρ = 0.5a memory, ρ = 0.9a memory, ρ = 0.97inherited, a = 0.5inherited, a = 0.7shared once, a = 0.76000 organs · 6 runs eachgenerated from a stated rule, not drawn to look right The second statistic was the first The experiment this collection has been specifying was priced as two readings off one sequence, the second of them free. The two readings turn out to be one function looked at twice, so the specification loses a statistic — and gains a cheaper one, a warning about how observables get priced, and a question it could not previously ask.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

  • The window was not the neighbourhood — both name artefact, autocorrelation, falsifiability, honest limits, measurement, negative result, neighbourhood depth, noise, the placement rule, repulsion, self correction
  • A disturbance that is not passed on — both name artefact, autocorrelation, discrimination, falsifiability, honest limits, measurement, noise, summary statistic
  • The disturbance that travels — both name autocorrelation, discrimination, honest limits, measurement, noise, the placement rule, repulsion, self correction
  • A comb is evidence of a rule — both name autocorrelation, discrimination, falsifiability, measurement, noise, the placement rule, self correction
  • A difference forgets a drift — both name artefact, autocorrelation, honest limits, measurement, negative result, noise, summary statistic
  • A disturbance the organs share — both name artefact, autocorrelation, honest limits, measurement, noise, the placement rule, self correction

Named objects

A flat tag is an object no other essay names yet.

ArtefactAutocorrelationDiscriminationDriftFalsifiabilityHonest limitsMeasurementNegative resultNeighbourhood depthNoiseThe placement ruleReproducibilityRepulsionSelf correctionSummary statistic