The grid was in the number

The placement rule this collection runs on is continuous in its statement and discrete in its implementation. Each organ goes where the repulsion from the organs already present is least, and least is found by evaluating the repulsion at a fixed number of azimuths around the circumference and taking the smallest.

Every flat-rise run on this site uses 384 of them. That is a step of 0.94°.

The disturbance those same runs carry is a quarter of a degree.

The arithmetic that should have been done long ago #

A quantisation to a grid of step s is, to a good approximation, an additive error uniform on ±s/2, which has a standard deviation of s/√12. At 384 azimuths that is 0.27°.

The disturbance the stems are given is 0.25°.

So the larger of the two displacements each organ carries is the one nobody put there, and it has a property the intended one does not: it is white. It is independent from organ to organ by construction, because it is a rounding of one organ’s answer that has nothing to do with the rounding of the next one’s.

A white disturbance dilutes a correlation, and it dilutes the two combs differently, because they have different numbers of members and sit at different lags. Which means the ratio of the two — the quantity that separates a placement rule from a transported disturbance, and the only quantity that survived the retraction of the comb — has a grid in it.

The measurement #

The whole sweep, at a rise of 0.005, five stems at each grid, everything else identical:

The correction is a fifth of the quantity, in the direction that matters, and it comes almost entirely from the second comb: the main comb barely moves while the second gains a sixth of its value between the coarsest grid and the finest.

That asymmetry is what the arithmetic predicts. The main comb here has three members and the second has four, at longer lags, and a white contamination costs a longer-lag correlation more.

What “converged” is being claimed here #

The last three grids give 0.79, 0.79 and 0.82, which is not the same number three times, and it is worth being exact about what is and is not being claimed.

Each row is five stems, so each ratio has a sampling error of its own. The five stems at 1,152 azimuths give individual ratios spanning about a tenth either side of their mean, so a standard error on the mean of roughly four hundredths — which makes 0.79 and 0.82 a difference of well under two standard errors, and 0.62 against 0.79 a difference of four.

The claim is therefore: the ratio changes by an amount the sampling cannot explain between 384 and 768, and by an amount it can explain between 1,152 and 2,304. Read that way the study says the grid has stopped mattering by 1,152, and it does not say the converged value is 0.79 rather than 0.82 — the honest figure is 0.79 ± 0.03, which is the number this thread now carries and quotes to two figures.

That distinction is not pedantry. A convergence study that reports its last two points as identical when they differ by three per cent has hidden the size of its own residual, and the residual here is a fifth of the correction it was measuring.

The scatter, which is the check #

The recorded divergence scatter falls from 0.70° to 0.51° as the grid is refined, and that is not an incidental observation — it is the confirmation that the extra correlation loss was quantisation and not something else.

A divergence is a difference of two azimuths, so a quantisation error of 0.27° on each contributes 0.27 × √2 = 0.38° to the scatter, in quadrature with whatever else is there. Taking the converged 0.51° as the stem’s own scatter, the predicted coarse-grid figure is √(0.51² + 0.38²) = 0.64°, against the measured 0.70°.

Within a tenth of a degree, on a prediction with nothing fitted in it. The grid is doing what a disturbance of 0.38° would do, because that is what it is.

The prediction across the whole sweep, and where it misses #

The quadrature check is stronger if it is run at every grid rather than only at the coarsest, and running it that way turns a confirmation into a second finding.

Take 0.51° as the stem’s own scatter and add each grid’s quantisation in quadrature. At a step of 0.47° the prediction is 0.55° against a measured 0.61°; at 0.31° it is 0.53° against 0.53°; at 0.23° it is 0.52° against 0.51°; at 0.16° it is 0.51° against 0.51°. The three fine grids agree to a hundredth of a degree, which is better than the measurement deserves and is the sense in which the model is right.

The two coarse grids do not. Both sit about six hundredths of a degree above the prediction, in the same direction and by nearly the same amount — 0.70° against 0.64°, and 0.61° against 0.55°. A model that is exact at three settings and short by a constant at two is not a model with noise in it; it is a model missing a term that only switches on at coarse grids.

The candidate is not far to seek and it is the other thing a grid does: at a coarse enough step the rule occasionally picks the wrong minimum outright rather than rounding the right one. A wrong minimum is a displacement of degrees rather than of tenths, so a small number of them per run adds scatter without adding much to the rounding statistics — and it should appear exactly where the ratio was still moving, which is the two coarsest grids and nowhere else.

So the scatter column carries the same boundary the ratio column does, found by a different route. That is worth more than either column alone. Two quantities with unrelated sensitivities agreeing on where 384 azimuths stops being adequate is the kind of agreement a single convergence study cannot manufacture.

The disturbance budget, restated #

The consequence for every flat-rise run already published is one addition, and it is worth doing explicitly rather than leaving as an implication.

A run declared to carry a quarter of a degree of disturbance was carrying that and 0.27° of quantisation, which in quadrature is 0.37° — half again as much displacement as its specification says, with the majority of it from a source not in the specification at all.

The direction of the resulting bias is knowable and it is not symmetric. An undeclared white admixture dilutes a correlation, so a result claiming that a stem’s disturbance is correlated — transported through the neighbours rather than applied independently — was measured against a background working against it, and the true effect is larger than reported. Those results are conservative and none needs re-reading.

The results at risk are the opposite kind: any claim that a stem behaves as though its disturbance were white, or that some quantity is insensitive to the character of the disturbance. Those were made on stems whose disturbance was already substantially white without anybody saying so, and a white result on a contaminated stem is the one reading the contamination could have produced by itself.

Why nobody caught it #

Three reasons, and none of them is carelessness, which is the reason to write them down.

The grid was chosen against a different question. 384 azimuths were fixed when this site’s flat runs were introduced, and the question then was whether the divergence sequence has a memory — a lag-one correlation of −0.6, which is large and which a tenth of a degree of extra scatter does not touch. A parameter adequate for one measurement was inherited by every measurement after it.

A coarse grid does not look like a coarse grid. It produces a lattice, at the right divergence, with the right counted pair, and with a scatter that is plausible for a plant. Nothing about the output says some of this scatter is mine.

And the site’s own convergence habit was applied to the wrong axis. The essays before these did a careful study of how many organs a reading needs and how long a window has to be, and they did name the sample grid as an untested axis — a convergence study on it was one of four things recorded as asked for and not delivered.

What it costs to fix #

Refining the grid is not free, and the arithmetic of what it costs is worth having, because the temptation with a finding like this is to refine everything.

The rule’s inner loop evaluates the repulsion at every sampled azimuth against every organ in the neighbourhood, so the cost of a stem is proportional to the number of azimuths. Going from 384 to 1,152 triples every flat run on the site. The comb thread’s ensembles, the mixture thread’s grids of rate against disturbance, the window thread’s overlapping readings — all of them triple.

That is affordable for a sweep of fifteen rises and five stems, which is what this thread needed. It is not obviously affordable for the grids of a hundred and more stems the mixture and window threads use, and those threads’ results are verdicts rather than continuous numbers, which is exactly the category the measurement above says is safe.

So the practical answer is not refine everything: it is refine what is quoted. A number that appears in a conclusion gets a convergence study; a verdict that clears its threshold by a factor of three does not, and the reason it does not is now measured rather than assumed.

The other symptom, which is worse #

While checking whether the grid decides the ablation experiment’s answer, the same 384 azimuths produced something less subtle. Run the rule at this rise with no noise at all on that grid, and the stem does not settle on the golden angle: it settles into a repeating cycle of divergences at a mean of 185.8°.

That is the arrangement a mid-front ablation produces — a block of eight organs precessing slowly, which a counter reads as a two-jugate lattice — reached with no ablation and no noise, purely by rounding.

Every flat run on this site is a noisy one, and the noise is what has been keeping them off it. That is a fortunate accident rather than a design, and it has held all this time without anybody knowing it was holding.

So the grid is not merely adding scatter. At this rise, on this grid, it puts a second attractor within reach of the arithmetic — and the reason the site’s results are not contaminated by it is a property of the disturbance amplitudes that happened to be chosen.

What else on this site runs on 384 azimuths #

The honest answer is: everything with a flat rise in it. The constant-rise stems that the comb, the mixture, the colour and the window threads are all built out of use the same setting.

So the question is which of those results are of a kind a fifth of a dilution could move, and it has a short answer: verdicts are safe and continuous numbers are not.

Every claim of the form this stem reads 8/13, this arrangement has a comb and that one does not, these two windows agree, this readout refuses is a threshold crossing with a wide margin, and the grid moves the quantity by far less than the margin. Those are most of the results this thread has produced, and they are unaffected.

The exposed ones are the numbers quoted to two figures: the comb ratio, which this essay corrects; the lag-one correlation of −0.6, which is a correlation and so is diluted in the same direction; and the recorded scatters, which now have a known 0.38° of grid in them wherever they were measured at 384. Of the three the first is load-bearing and has been redone, the second is used qualitatively and is not, and the third has been quoted as a plant-like amplitude in several places where it is a little too large.

A working rule falls out of the arithmetic and is worth stating for whoever runs the next flat stem: refine the grid until the quantisation error is under a third of the disturbance the stems are meant to carry. At a disturbance of a quarter of a degree that is a step of 0.29°, which is 1,240 azimuths — the same place the measurement above says it has converged, arrived at from the other direction.

The general shape of the mistake #

This collection has made this mistake before, in a form that was recorded and then not generalised.

The essays on the cut-off found that the neighbourhood the rule sums over was a property of the program rather than of the model: the code took the most recent so-many organs, and that recency window turned out to manufacture patterns — an inverse-first-power rule with no lattice at all acquires a clean 8/13 once the loop is cut. The repair was to make the neighbourhood a stated cut-off with a width, so that it became a hypothesis about how far inhibition reaches rather than an artefact nobody had meant to state.

The azimuth grid is the same category of object and it had not been through the same repair. It is not a parameter of the placement rule; nothing in the rule says the circle has 384 positions on it. It is a parameter of the program, and like the recency window it turns out to do physical work — adding a disturbance, diluting a correlation, and at one rise putting a second attractor within reach.

The difference is that a cut-off can be made into a hypothesis and a grid cannot. There is no plant-side quantity that the number of sampled azimuths corresponds to. So the only correct treatment is convergence: refine it until the answer stops moving, and report the grid at which it stopped.

What has to be re-read, and what does not #

The comb ratio is 0.79 and not 0.65. Every use of that number is affected: the discriminator’s margin against a distance-weighted transported disturbance narrows from a factor of 2.0 to a factor of 1.6. It is still a separation and it is a weaker one.

The forgery results are not affected. The kinematic lattices used to forge a comb are built arithmetically — organ i at i times the divergence plus an inherited error — with no minimisation and therefore no grid. Their numbers are exact and stand as reported.

The comb’s presence or absence is not affected. A dilution of a fifth does not turn a comb of 0.6 into no comb, and this thread’s qualitative results — that a rule makes a comb, that a memory does not, that a transport does — survive at every grid tried.

And the pairs are not affected. Every reading of 8/13 in this thread is 8/13 at every grid. The instrument’s verdicts are robust; it is the one continuous number it produces that was not.

There is one more thing to say, and it is about how the defect was found rather than about what it costs. Nobody went looking for it. The sweep across a rung was set up to answer a different question — whether the ratio is a constant — and the grid study was run only because the answer to that question was a curve and a curve invites the question is the curve real. The ablation thread found the second symptom independently, for the same reason: a displacement measured in degrees on a grid whose step is a degree is an obvious thing to check.

Two threads, run for unrelated reasons at the same time, both landed on the same parameter. That is the argument for doing the convergence study on a quantity before it becomes load-bearing rather than after: the ratio has been this site’s whole discriminator since it was published, and for all that time it was a fifth away from its own value.