A maximum in the gap

The falloff exponent is the power in the sum each new organ is placed at the minimum of, and four values of it have been carried through this collection for a long time without any measurement telling them apart. The share of starting angles that still settle onto a lattice does not separate them. Neither does the rise at which half of them stop settling, even after the sweep that located it.

Here is one that does, and it is not a threshold on a curve. It is the shape of the curve.

The gap nobody could see into #

The published list of rises stepped from 0.02 straight to 0.013, a factor of 1.54, and every reading of the settling table for four samplings was taken on it. Between those two entries there was nothing, and a curve is a straight line between two points whether or not it is one.

The refinement cut four rises into the two empty stretches, two of them into that one: 0.01732 and 0.01501. Seven of the nine rises now sit between 0.02 and 0.008, where there had been two, and the step across the interesting region falls from 1.538 and 1.625 to 1.155 and 1.176.

The nine rises contain the published five, and the eighty starting angles contain the published forty, which is the containment rule every doubling in this thread has followed. So the old readings are not a rival design to be reconciled with this one. They are this table with rows removed.

What sits inside it #

At exponent 5 the settling share goes 0.550 at 0.02, 0.725 at 0.01732, 0.550 at 0.01501 and 0.450 at 0.013.

The share rises by 0.175, which is close to a third of the range that column covers from its highest value to its lowest, then falls back to exactly where it started, then keeps falling. On the published list only the two outer values are visible, and one step of that list carries the column from 0.550 down to 0.450.

So a stretch that had been read as a single downward step contains an excursion larger than the step itself, in the other direction.

Three point one three standard errors #

The peak sits 3.13 standard errors above the higher of the two ends of the stretch.

The standard error on a share at eighty starting angles is 0.0559, and it is the most precisely known quantity in this table — a count of settled runs over a count of runs, with no fitting, no interpolation and no level in it. Three standard errors is not a marginal excursion on a quantity of that kind.

It is worth saying which comparison that is. The peak is not being tested against zero, or against a fitted line; it is being tested against the coarser of the two rises that bracket it, which is the value the published reading would have assigned to that stretch.

Why a share is the right thing to be reading #

Every other number this thread has argued about is derived. A wall is an interpolation between two rises at a level somebody chose; a bracket is a pair of entries from a list; an ordering is the sign of a difference between two of those.

A settling share is none of that. It is a count of stems that stopped moving over a count of stems grown, at one rise and one exponent, with eighty angles behind each figure and 1,200 organs behind each stem. It is the quantity the whole table exists to hold, and it is the only one in the thread whose uncertainty is known exactly rather than argued about.

So a claim made directly about the shares is made on the strongest footing this table has.

Present in both halves, independently #

The eighty starting angles are the published forty and forty added between them, so the column can be read three times: on all eighty, on the published forty alone, and on the added forty alone.

The peak is at 0.01732 in all three. On the published forty it reaches 0.650 against 0.500 at the coarse end of the stretch; on the added forty it reaches 0.800 against 0.600.

Two disjoint sets of starting angles, each of which has never seen the other, place the maximum at the same rise and give it the same sign. That is the whole difference between a measurement and a bump.

The added forty are the stronger half #

They give the larger excursion — 0.800 against 0.600 — which is the opposite of what a selection story would predict. The published forty are the angles this collection has been growing stems from since the settling table existed, so if the peak were an artefact of a particular set of starting angles it would live in that set.

It does not. It lives more strongly in the forty that were added for the first time to reach this rise at all.

Exponent 4 does the same thing #

Exponent 4 goes 0.563 at 0.02, 0.662 at 0.01732, 0.475 at 0.01501 and 0.450 at 0.013. The peak is 1.79 standard errors above the higher end, and it too is present in all three readings — 0.650 on the published forty, 0.675 on the added forty.

At 1.79 it is not on its own evidence of much. Its interest is that it agrees with exponent 5 in every particular: the same rise, the same sign, and the same independence across the two halves.

Two of four, and which two #

The two exponents that turn are the two steep ones. The two that do not are the shallow ones.

That is what makes this a statement about the exponent rather than about the sweep. A feature of the machinery, or of the rise list, or of the settling criterion would appear in all four columns of the same table, read at the same rises, over the same angles. This appears in two, and the two are adjacent in the parameter.

Exponent 2 falls straight through #

At exponent 2 the share goes 0.650, 0.525, 0.500, 0.425 across the same four rises. It falls at every step.

Its value at 0.01732 is 2.24 standard errors below the higher end of the stretch. The identical comparison at exponent 5 gives 3.13 above, so the two exponents differ by more than five standard errors in the one number this essay is about. There is no reading of exponent 2, on any of the three sets of angles, in which anything turns there.

Exponent 3 is flat rather than turning #

Exponent 3 reads 0.575 at 0.02, 0.575 at 0.01732 and 0.575 at 0.01501 before dropping to 0.450 at 0.013. Its peak is 0.00 standard errors above the end, because the peak is the end.

On the published forty alone it does turn, weakly. On the added forty it does not. One of three readings, at a rise the other two put nothing at, is exactly the pattern that separates this exponent from the two that turn — and it is the control the whole claim needs, because a column with no real feature in it is expected to produce one reading in three that looks like one.

The first quantity that tells them apart #

That sentence has to be read carefully, because it is a claim about a thread that has produced nothing but negatives.

The settling share does not order the four exponents. The wall does not either, at any level, and after the sweep that located it, none of the six pairs separates at a half. What does differ is how long a settled run takes to settle and which destinations a steep rule reaches — both of which are properties of settled runs rather than of whether a run settles.

The turn is the first quantity read off the settling share itself that puts the four exponents into two groups.

A shape survives what a threshold cannot #

A threshold on a curve inherits every weakness of the curve. Its location depends on the level chosen, on how steeply the curve passes through that level, and on which two entries of the rise list bracket the crossing — and each of those turned out to be doing the work that the exponent was supposed to be doing.

A maximum has none of that. It has no level in it; the peak is where it is whatever share is called interesting. It does not need the curve to be steep. And it is not read off two entries of a list, but off three.

The second gap has nothing in it #

The other empty stretch of the published list runs from 0.013 to 0.008, and the refinement cut 0.01106 and 0.00941 into it.

Nothing turns there. At exponent 5 the share goes 0.450, 0.313, 0.362, 0.300 across those four rises: it wobbles, and the wobble sits well below the coarse end rather than above it. At no exponent does the eighty-angle reading show a maximum in that stretch.

Which is what a negative control looks like #

The second gap is the same treatment applied to the same column by the same design at the same cost. If cutting rises into an empty stretch manufactured maxima — by chance, by the sampling, by anything about the procedure — it would have manufactured one here too.

It did not, at any of the four exponents. So the feature in the first gap is a feature of that stretch of the rise, and the four added rises are not a machine for producing bumps.

The one wobble anywhere in the second gap that has the right sign is at exponent 2, in the published forty alone, and the added forty contradict it. That is what a fluctuation looks like when it is caught: one reading of three, at a rise the other two put nothing at, and no agreement between two halves that have never seen each other.

Why two brackets had no coarse end #

This explains something the earlier reading recorded and could not account for. Exponents 4 and 5 both had a bracket on the wall that was open at the coarse end: no published rise had a settling share confidently above a half, so the wall might have sat above the coarsest rise ever grown.

The two exponents whose bracket was open are the two whose share turns. The published rises sat on the far side of a maximum they could not see: the share was confidently above the level at 0.01732, and 0.01732 was not in the list.

A crossing read on the wrong side of a turn #

A wall is defined as the rise at which the share crosses a level, and that definition assumes the share falls as the rise falls. Where the curve turns, the assumption fails, and everything built on it fails quietly rather than loudly.

Three of the thirty-six cells of the finer level scan have a share that crosses its level more than once. A quantity defined as the crossing is choosing, in those cells, and nothing in the reading says which.

The rule returns an interval that runs backwards #

The instrument shows the same thing from the other side, and this is the sharpest form of it. The shared rule that computes a bracket takes the finest rise whose share is confidently above the level down to the coarsest whose share is confidently below it.

Read at a level of three fifths, at exponents 4 and 5, those two ends cross over. The share is confidently below three fifths at 0.03 and confidently above it at 0.01732, so the rule returns an interval whose upper end is beneath its lower end — a ratio of 0.58, where every honest bracket is a ratio above one.

Which exponents, and at which level #

Only the two that turn, and only at a level high enough to sit between the peak and the ends of the stretch. At three fifths, exponent 2 returns 1.73 times the right way round and exponent 3 returns 2.31. Exponents 4 and 5 return nothing at all.

Three fifths is the level that became readable when the sampling doubled: five of the nine levels resolved at forty starting angles and six do at eighty, the sixth being this one. So the defect appears at exactly the level the refinement made available, which is the ordinary way an improved instrument exposes a rule nothing had ever stressed.

Two empty intervals cannot overlap #

Here is why the defect matters rather than merely being untidy. A pair of exponents is declared separated when their brackets do not overlap.

An interval that runs backwards contains no rises at all. Two such intervals therefore do not overlap, and a separation test that asks only whether two ranges intersect will report the pair as cleanly separated — on the strength of two readings that failed.

That is a false positive produced by a failure, which is the worst kind, because the failure and the finding are the same event.

What the collection does about it #

The reading refuses an inverted bracket rather than reporting its ratio, so no separation is ever claimed from one. The shared rule itself is left as it stands, and that decision is worth stating rather than burying: the published rises cannot produce the case, because they never resolve the rise where the share turns, and the rule is read by a great many other files whose answers would move.

So the defect is real, bounded to a case only the finer rises can reach, and recorded rather than patched. A reader who finds a backwards bracket elsewhere in this collection has found either a curve that turns or a rule that was applied where it does not hold.

What this does not establish #

It does not say why. Nothing here explains what is special about a rise near 0.01732 for a steep placement rule, and no mechanism in this collection predicted a maximum there. The finding is that the share is not monotone in the rise at two of four exponents, and that is all of it.

It does not order the four exponents. Two turn and two do not, which sorts them into a pair and a pair; it says nothing about exponent 4 against exponent 5, whose peaks differ by less than either one’s uncertainty.

Nor does it rescue the wall #

The wall is still located to a factor of about one and a half at every exponent, the four still sit inside a factor of 1.111 of one another, and no pair separates at a half. Finding a feature elsewhere in the column does not make a threshold on it sharper.

If anything the turn makes the wall worse. A crossing is well defined only where the curve is monotone, and the curve is now known not to be, at two of four exponents, at a rise inside the region every wall sits in.

Nor does it touch what the wall was originally asked to settle. Stems do stop settling at the fine end and longer runs do not reach past it, and that remains the finding the whole thread rests on. A column that turns once on its way down is still a column that falls by a factor of four to six from end to end.

What would refute it #

A third set of forty starting angles, disjoint from both, in which the share at 0.01732 does not exceed the share at 0.02 at exponents 4 and 5. That is about half an hour of machine time and it is the check this finding most deserves, because two halves agreeing is two.

Or a rise cut between 0.02 and 0.01732 at which the share is already above 0.725, which would make the peak an edge of something wider rather than a maximum. The current list cannot tell a narrow maximum at 0.01732 from a broad plateau whose fine end happens to fall there, and a plateau read as a peak is a mistake this collection has made in the other direction.

What it costs to look #

Nothing that has not already been paid. Both checks are stems grown at rises this design already knows how to grow, at exponents already in the table, and both are decidable in an afternoon.

That is the argument for writing the finding down at 3.13 standard errors rather than waiting. A coarse list is not merely imprecise about a feature narrower than its step; it reports the feature’s absence with the same confidence it reports everything else, and the absence is what four samplings of this table recorded.

What is claimed #

That at eighty starting angles on nine rises, the settling share at exponent 5 reaches 0.725 at a rise of 0.01732 against 0.550 at 0.02 and 0.550 at 0.01501 — a maximum 3.13 standard errors above the higher end of a stretch the published rise list crossed in one step.

That it is present independently in the forty starting angles the published table was grown from and in the forty added, at the same rise in both; that exponent 4 shows the same maximum at 1.79 standard errors, also in both halves; and that exponents 2 and 3 show none, with exponent 2’s value at that rise 2.24 standard errors below the coarse end of the stretch.

That the second gap, cut with two rises by the same design, produces no maximum at any exponent, which is the negative control.

And that the two exponents whose share turns are exactly the two whose bracket on the wall was open at the coarse end, and exactly the two for which the shared bracket rule, read at three fifths, returns an interval running backwards at a ratio of 0.58 — so a separation test that asks only whether two ranges intersect would report those two exponents as separated on the strength of two failed readings.