The window that was not a plateau

Two windows of this fit have been swept and each found a plateau. The near end — where the fit is allowed to start — moves the exponent by at most half its effect and is flat from six bonds outward. The far end — where it is allowed to stop — moves it by at most 5.3 per cent inside each profile’s reach and is flat by construction beyond it.

Two plateaus in a row is the kind of run that produces a habit, and the habit is to expect the third. The third is easy to name and its shape easy to predict: the local decay rate is read from the first quarter of the chain, the rule looks conservative, and sweeping it should let the stiff cases reach further and move their exponents a little.

The prediction is right about the direction and badly wrong about the size, and it is wrong about which end of the series has the problem.

What the third window is #

The quantity being fitted is the rate at which the Peierls distortion’s excess over bulk decays with distance from the chain’s end. That rate is read bond by bond, and the reading stops at the first quarter of the chain.

The quarter is not arbitrary in kind. Half is a hard ceiling: past the midpoint of a chain with two ends, the healing being measured belongs to the other end, and the reading would be nonsense. A quarter is half of that, chosen as safely inside.

But “safely inside” the ceiling is not the same as “not binding”, and there are two quite different ways a profile can end:

• on the floor — the excess over bulk falls below the relaxation’s own convergence, the local-rate reading stops producing rates, and the reach is a property of the physics. The window never touched it.
• on the cap — the reach is exactly the quarter. The profile has more to say and is not being asked.

Nothing in the earlier sweeps distinguished these, and telling them apart needs no new physics at all: it is a comparison of the reach against the cap, at each fraction.

The series splits at a stiffness of two #

The answer is cleaner than a sweep usually gives. Five of the ten stiffnesses end on the floor inside a quarter of the chain; five sit exactly at the cap. The split is at K = 2 and there is nothing ambiguous in between.

The soft chains — K from 1.1 to 1.8 — heal over 23 to 74 bonds and then their excess drops below the noise floor. The cap at a quarter is 77 bonds, so those profiles were never being cut. Opening the window to the midpoint changes their reach by nothing and their exponents by nothing: the agreement is exact to a part in a billion, which is the check that the sweep is doing what it claims rather than perturbing everything slightly.

The stiff chains do not heal so fast, and all five of them run into the cap. K = 2 reads 77 bonds when 77 is the limit and 101 when it is not. K = 2.2 reads 77 and then 135.

What it costs #

For the five affected cases, opening the window moves the exponent by 0.96, 2.4, 5.3, 9.9 and 13.3 per cent — rising with stiffness, and every one of them downward.

That is larger than either window already swept, and it is on the cases quoted at the top of the range. The published exponent at K = 2.8 is 0.6586; with the window opened it is 0.5932. At K = 3.1 it is 0.6498 and becomes 0.5564.

The direction is not luck. A window that truncates a decay whose local rate is still falling reads a rate that is systematically too fast — the points that would have pulled it down are the ones outside the window. So a cut profile overestimates the exponent, always, and the size of the overestimate grows with how much profile is missing. That is exactly the pattern: no move where nothing is missing, a small move where a little is, and a seventh where most of it is.

The floor argument, where the exponent turned out to be a floor, found the published exponents rising with stiffness and then turning over at the top. With the window opened, the turnover is much stronger and starts earlier. Whether the turnover is real at all is a question this sweep cannot close, for the reason in the next section.

And two of them are not fixed #

Opening the window resolves two of the five. K = 2 and K = 2.2 reach the floor before the midpoint and their opened exponents are measurements.

K = 2.5, 2.8 and 3.1 do not — by the ruler the next section builds, they have run 9.2, 6.5 and 5.2 of their own decay lengths against the 13.26 a completed profile runs. Their profiles are still being cut, and there is no wider window available: half is the ceiling, not a convention, and reading past it means reading the far end’s healing.

So those two exponents are not mis-measured. They are unmeasured. Whatever number the fit returns for them is a statement about how long the chain is, and it will keep changing as the window opens until it stops — which on a 320-site chain it never does.

The natural expectation is that a longer chain is needed at the soft end, on the reasoning that the softest case had a single point past its reach. That is the right instinct pointed the wrong way. The soft cases heal quickly and fit inside anything; it is the stiff ones, whose healing runs on for hundreds of bonds, that a 320-site chain cannot hold.

The ruler that says which reaches are real #

Comparing a reach against the cap answers the question crudely — a reach three bonds short of a cap of 157 is at the cap in every sense that matters, and an off-by-one test says it is not. There is a better ruler, and the sweep produces it for free.

A profile ends when its excess over bulk drops below the noise floor. The floor is the same for every case and so is the surface amplitude the decay starts from, so the number of decay lengths a profile runs before it disappears should be the same for all of them — it is the logarithm of one over the other, and neither depends on the stiffness.

It is. Across the five cases where the quarter rule was never binding, the reach divided by the fitted decay length is 13.42, 13.32, 13.34, 13.20 and 13.02 — a mean of 13.26 and a spread of three per cent over decay lengths that differ by a factor of three and a half.

That constant is a ruler. Any case whose reach falls short of thirteen of its own decay lengths did not end on its own; it was cut. And by that measure the sweep resolves fewer cases than the cap comparison suggested:

K = 2 runs 12.9 lengths and K = 2.2 runs 12.5, both within the spread — those two are genuinely resolved by opening the window. K = 2.5 runs 9.2, K = 2.8 runs 6.5 and K = 3.1 runs 5.2. Three cases, not two, are still being cut at the hard ceiling, and the stiffest of them has seen barely a third of its own profile.

This is worth more than the correction it makes. A test built from the data’s own internal consistency is not sensitive to a tolerance somebody chose, and it would keep working on a longer chain, at a different floor, or on a different model — anywhere the same two quantities set where a profile ends.

What can still be said about the exponent #

The honest requote is a narrower range over fewer cases. Seven of the ten stiffnesses have exponents that are measurements on this chain; the range across those is what can be claimed, and it is narrower than the published one because the published one’s upper end was made of window.

Setting three cases aside is not the same as dropping outliers, and the distinction is the reason for saying it this way. An outlier is a measurement one distrusts. These are not measurements at all, and quoting them inside a range would put the chain’s length into a number that is supposed to describe the physics.

What was computed, and how #

Nothing was recomputed. The relaxation profiles are the same ones used in the chain distorts hardest where it stops, and the entire sweep is arithmetic over them: for each stiffness and each of nine fractions from a tenth to a half, take the local rates up to that fraction of the chain, note the last bond that produced one, compare it against the largest bond the fraction could admit, and fit the exponent over the standard window.

The classification of each reach is the load-bearing part and it is a comparison rather than a judgement: a reach within one bond of the cap is window-bound and anything else is floor-bound. That test is what makes the split at K = 2 a measurement rather than an impression, and it is what lets the good news and the bad news be stated as separate claims — that the rule was free on the soft half, and that it was deciding the answer on the stiff half.

Seven checks. Two say the rule was not binding below K = 2 and that those exponents are unchanged to a part in a billion. Two say it was binding above it and that opening it moves an exponent by more than a tenth. One says every affected exponent moves downward, which is the claim the truncation argument predicts and the one that would be most embarrassing to have wrong. Two more build and use the ruler: that a profile ending on its own runs a fixed number of its own decay lengths, to within five per cent, and that by that measure the stiffest cases are still capped at the midpoint and are the stiff ones rather than the soft ones — because that is the half of the natural prediction that came out backwards, and a check that did not name the direction would pass either way.

Where the model stops #

The chain is 320 sites, the model is the same tight-binding one with a fixed elastic constant standard for this chain, and the excess is measured against a bulk value taken from the chain’s middle, which is bulk because the distortion is decided at the ends. That last point matters more here than usual: the bulk reference is itself read from the region the window is being opened towards, so at fractions near a half the reference and the reading are drawn from overlapping stretches of chain. The measurements at 0.5 should be read as the edge of what the method allows rather than as comfortably inside it.

The noise floor is set at 10⁻⁸ and it is what makes a profile end for the soft cases. A lower floor would extend those reaches and might make the quarter rule bind on cases where it currently does not — so “the rule was free on the soft half” is a statement about this floor, not about the chains.

And the exponent is fitted over a window of six to a hundred and twenty bonds throughout, so the two swept windows are held at their plateau values while the third moves. That is the right way to sweep a third choice and it assumes the plateaus found for the first two survive at the longer reaches opened here, which was not rechecked.

The generalisation #

The transferable point is that a run of negative results is not evidence about the next one.

Two windows in this fit were swept and both were plateaus, and the reasonable expectation after that is a third plateau. The expectation is reasonable and it was wrong, and what made it wrong is not bad luck: the first two windows are choices about which part of an available profile to fit, and the third is a choice about how much profile exists. Those are different kinds of parameter and there was never a reason for them to behave alike.

So the useful question about an unswept choice is not “have the similar ones mattered” but “what would it be truncating”. A parameter that selects from what is there tends to have a plateau, because the thing selected from does not change. A parameter that decides what is there does not, because outside its value the data simply are not present, and a fit cannot know what it was not given.

The second point is smaller and is about how the good news should be reported. Half this series was genuinely unaffected, to a part in a billion. Writing that the quarter rule “matters” would be true and would misdescribe five of the ten cases; writing that it “does not matter” would be worse. The sweep’s value is that it says which cases, and the boundary — a stiffness of two — is a fact about the series that nobody could have guessed.

What it does to the floor argument #

One earlier result deserves rechecking in the light of this, and the answer is that it survives and is now better supported than it was.

The exponent was the floor argued that the fitted tail exponent tracks the noise floor rather than the physics — that what the fit measures is where the profile stops being visible. The ruler is that argument made quantitative from the other side: a profile runs 13.26 of its own decay lengths and then vanishes, which is precisely a statement that the reach is set by the floor and the decay length together, with nothing else in it.

And the exponents moving downward when more profile is admitted is the same finding again. If the exponent were a property of the tail, adding tail would leave it alone; it moves, and it moves in the direction that having less profile predicts.

So the three sweeps together say something no one of them could. The two window choices that select from an available profile are free, the one choice that decides how much profile exists is not, and the reason is that the exponent was never a property of the decay in the first place. A quantity that is really a property of where the data run out will be insensitive to how they are fitted and sensitive to how many there are — which is exactly the pattern across these three sweeps.

Who found it, and when #

The Peierls distortion and the healing of a defect into a bulk are old; the local-rate reading — the observation that the decay keeps slowing down rather than settling — its window, and all three of the sweeps are new arithmetic. The quarter rule is the kind of default that is easy to leave adjustable and never adjust, and testing it costs nothing but arithmetic on profiles that already exist.

That is worth noticing as a small piece of luck. A truncation hard-coded rather than defaulted would have made this sweep a code change on a module whose cache holds eleven hundred seconds of stored relaxations, and the sweep would have been proposed and not run.

Still open: a longer chain, and the other two plateaus #

The obvious open question is a longer chain, and the ruler says exactly how long and for which cases. A profile needs 13.26 of its own decay lengths and K = 3.1’s length is 29.9 bonds, so it needs about 397 bonds of profile inside half the chain — a chain of some eight hundred sites for that case alone, and rather more to have room to spare. That is a real cost rather than an arithmetic one, and it buys two exponents and the answer to whether the turnover at the stiff end is physics.

The nearer question is whether the two swept plateaus survive. Both were established on profiles truncated at a quarter, so the near-end and far-end sweeps were run over reaches that are now shown to be too short for half the series. Re-running them on the opened profiles is arithmetic over the same stored relaxations, it costs nothing, and until it is done there are three sweeps of which only one was run under the correct conditions.