The corner moves with the rise

A rule with a deep neighbourhood was supposed to correct a slow drift and let a fast one through. It does not: the deeper rule passes more at every correlation length, which is the prediction backwards. What the sweep found instead was a corner — a short correlation, one to three organs, past which the deeper rule stops losing — and two readings fitted it equally well.

Either the corner is the contact scale, the distance over which an organ’s own neighbours sit, so that a disturbance outlasting it is one the lattice cannot treat as local. Or it is any memory at all, two organs being simply the shortest correlation distinguishable from none at these run lengths.

Every sweep in this thread has varied the exponent, the amplitude and the colour. None of them has ever varied the rise — which is the one knob that moves the contact numbers while leaving the rule, the amplitude and the run length exactly where they are. This essay varies it.

What the sweep can reach, and what it cannot #

Less than the question wants, and the shortfall is worth stating before the result rather than after it.

The contact numbers grow without bound as the rise falls, so in principle the sweep could run from 2/3 to 21/34 and beyond. In practice a stem at a rise below about four thousandths does not settle into a clean lattice at any run length this collection can afford: the counter comes back with pairs like 2/13 and 13/24, which are not contact families at all but the arithmetic of an arrangement that has not finished sorting itself out. Lengthening the stem does not fix it — 2,600 organs at a rise of 0.0012 still returns 13/15.

So the honest range is 3/5 to 8/13 — a factor of about four in the contact scale rather than the factor of eleven the question was posed with. The conclusion is bounded accordingly, and a factor of four turns out to be enough.

It is worth saying why it is enough, because a shortfall against a stated plan usually is not. The two readings make different qualitative predictions, not different slopes: one says the corner sits at a fixed correlation regardless of the lattice, the other says it moves with the contacts. Distinguishing a constant from a non-constant needs only that the quantity change at all, and a factor of four in the contact scale changes it by more than the sweep’s own resolution in correlation. What the missing factor of eleven would have bought is the form of the dependence — whether the corner scales with the contacts linearly, or with their logarithm, or not smoothly at all — and that is exactly the question this essay ends up unable to answer.

Three rises, three shapes #

The rule, the amplitude, the run length and the seeds are identical in all three panels. Only the rise differs — and the rise is the parameter the whole depth thread has held fixed while sweeping everything else.

At the coarsest, the deeper rule wins at white noise, loses across the middle of the correlation range, and wins again at the longest correlation — a dip rather than a corner. At the middle rise it wins essentially everywhere, and there is no corner to locate. At the finest it loses at white noise and wins from about seven tenths upward, which is a clean single crossing.

Three rises, three different curves. Not one curve translated along the axis; three shapes.

The seed counts underneath are worth reading rather than taking on trust. At the coarsest rise the deeper rule takes five of six seeds at white noise, two of six across the middle, and five of six at the longest correlation; at the finest it takes one of six at white noise and rises monotonically to six of six. Six seeds is not many, and a two-of-six against a four-of-six is not a difference anybody should lean on in isolation. What carries the result is that the pattern across the row differs — monotone in one panel, flat in another, U-shaped in the third — and a pattern across six correlations is a stronger thing than any one cell in it. The seed-to-seed spread within a cell is larger than the difference between two rules, which is precisely why the comparison is made seed by seed rather than between two means.

What that settles #

It refutes the “any memory at all” reading, and it refutes it cleanly.

That reading requires the corner to sit at the same short correlation wherever it is looked for, because its whole content is that two organs is the shortest memory the instrument can distinguish from none. An instrumental floor does not move when the rise moves; the run length and the amplitude are what set it, and neither changed. A corner that is at seven tenths on one lattice, absent on another, and at nine tenths on a third is not a floor.

The refutation is worth being careful with, because a floor could in principle be disguised by the correlation axis rather than by the rise. A correlation coefficient is a fraction, and the number of organs it corresponds to depends on nothing but itself — the correlation length is defined in organs and converted, not read off the stem — so a given coefficient is the same memory at every rise. That is what makes the three panels comparable at all, and it is why a corner moving between them cannot be explained by the axis meaning different things in different panels.

There is a second refutation in the same table, and it does not involve the corner at all.

Read the white-noise column on its own — the leftmost cell of each panel, where the disturbance has no memory whatever. At the coarsest lattice the deeper rule takes five of six seeds there. At the finest it takes one of six. The same two rules, the same amplitude, the same run length, the same seeds, no correlation anywhere in the comparison, and the answer is reversed.

Whatever that is, it cannot be a fact about the shortest memory an instrument can tell apart from none, because there is no memory in that column to be told apart. The floor reading needs the white-noise cells as the baseline a corner is found against, and a baseline that changes sign between two panels is not a baseline.

That makes the refutation stronger rather than merely doubling it. The corner-moves argument depends on locating a crossing, and locating a crossing from six sampled correlations at six seeds each is the part of this measurement most open to objection: the crossing at nine and a half tenths is a midpoint between two sampled values, and the middle panel’s absence is an absence rather than a number. The white-noise column needs none of that. It is two cells read directly, five of six against one of six, at the one place on the axis where the two readings under test say the same thing as each other.

It also moves the question. If the lattice already decides the comparison with no disturbance memory in play at all, then what a longer memory does is modify an effect that was there first — and the corner is a feature of that modification rather than a scale being discovered. That is a smaller claim than the thread set out to test, and a considerably more secure one.

What it does not settle #

It does not establish the contact-scale reading, and the temptation to say it does should be resisted.

If the corner tracked the contact scale, the expectation would be a corner in every panel, sliding smoothly as the contact numbers grow. What is there instead is a corner at 3/5, no corner at 5/8, and a corner at 8/13 — with the two corners on either side of the panel that has none. A quantity that vanishes in the middle of its own range is not sliding.

There is a weaker claim the data does support, and it is worth separating from the strong one so that neither gets quoted as the other. Something about the lattice reaches the comparison. Whatever sets whether a corner exists is not the rule, not the amplitude and not the run length, because all three are held; it is the only thing that changed. That is a real finding and it is much less than “the corner is the contact scale”, which would require the corner to be a function of the contacts rather than merely to notice them.

The distinction matters because the thread’s original question was whether the neighbourhood has a scale that the disturbance can be compared against. Finding that the comparison depends on the lattice is consistent with such a scale existing and with several other things — the settled divergence, the step lengths, the front depth — all of which move with the rise too. Separating quantities that move together needs a lattice where they come apart, and this sweep does not have one.

There is a reading that would explain the middle panel and it is not tested here. The comparison is between two exponents, and what the panels report is which of those two passes more; if the middle rise is one where both rules happen to sit on the same side of some threshold, the comparison would flatten without anything about the corner changing. Distinguishing that from a genuine absence needs a third exponent at the same rise, which is a sweep and not an argument.

Four controls #

The rule is unchanged. The same two exponents are compared at every rise, so nothing about the neighbourhood’s depth in organs is being varied along with the contact scale.

The amplitude is normalised. The disturbance is divided by its own measured standard deviation before it is applied, so a change of colour or of rise does not change how much disturbance there is.

The seeds are paired. The same seed drives both rules in every cell, and what is counted is how many seeds agree about which rule passed more — not a difference of two means that overlap.

And each rise is a settled lattice. Every one of the three returns a genuine contact pair, which is the criterion the sweep’s own generator refuses to draw without. That check is not a formality here: the whole reason the sweep stops at eight thousandths is that below it the criterion fails, and a panel drawn from an unsettled arrangement would have contact numbers that mean nothing while looking exactly like the others. The generator asserts the pair rather than printing it, so a rise that stopped settling would stop the build rather than quietly widen the claimed range.

Where this leaves the thread #

The depth thread has now produced three negative results in a row, and they are not the same negative result.

The prediction that a deep neighbourhood filters a slow drift is refuted: the deep rule passes more, not less. The corner that appeared instead is not an artefact of the ratio the sweep was read in. And it is not a fixed number of organs either, because it does not survive a change of rise.

What is left is a comparison whose shape depends on the lattice it is run on, in a thread whose whole purpose was to find a quantity that does not. The neighbourhood itself has already been shown to be four different numbers depending on how it is summarised, and the one summary that barely moves — the weighted mean lag — is the one whose top five terms are the same five organs at every exponent. Read alongside that, this result is less surprising than it first looks: a comparison between two rules whose neighbourhoods are both dominated by the same handful of lattice neighbours will notice the lattice, because the lattice is most of what the rules are reading. That is not a failure of the sweep. It is the sweep doing the only thing that was ever going to distinguish an instrumental floor from a physical scale, and reporting that the answer is neither of the two things on offer.

The next measurement is not a better statistic. It is the third exponent at the middle rise, which is the one thing that would say whether the flat panel is a fact about the lattice or a fact about the pair being compared.

After that, the sweep worth running is the one this essay could not: the same comparison at a rise fine enough to carry 13/21 contacts, on a stem long enough to settle there. That is a computation rather than an idea — the obstacle is that settling at those rises takes stems several times longer than anything here has grown, and the sweep needs six seeds of two rules at six correlations for each one. It is the difference between a factor of four in the contact scale and a factor of ten, and a factor of ten is what would turn “the comparison notices the lattice” into a statement about which property of the lattice it notices.