The other window was a plateau too

The rule of thumb was on the flat part swept the near end of the window the tail exponent is fitted over and found a plateau: from six bonds outward the exponent moves by half a per cent to nine, against a sixty per cent range across the series, so the rule of thumb that set the near end was doing no harm.

It closed by naming the other window. Sixty bonds was chosen for the same kind of reason six was, and the same sweep run on the far end would say whether it too sits on a plateau — with the difference that the far end runs into the noise floor rather than into a profile that has not become asymptotic, so its failure mode is the one the exponent was the floor characterises in detail, and its plateau, if there is one, should end abruptly rather than gradually.

It does end abruptly, and the abruptness is not what the prediction meant.

Flat because the window is not honoured #

Every curve is exactly flat past a stiffness-dependent point, and “exactly” is the word. The softest chain returns 0.4115194… at a far end of thirty bonds, at forty, at fifty, and at every value out to a hundred and twenty — to ten decimal places, the same number.

That is not the choice failing to matter. It is the choice not being made: the profile gives local decay rates only out to twenty-three bonds, so a window ending at forty and one ending at a hundred and twenty contain the same points and are the same fit.

That distinction matters because a flat sweep is the evidence a plateau is claimed on, and a sweep that is flat because nothing is varying is evidence of nothing. The fit records how many points it used, which is how the two cases are told apart, and every number quoted below is from a window that was actually honoured.

Inside the reach, a plateau #

Restricting to the far ends inside each profile’s own reach, the exponent moves by 0.00, 0.12, 0.23, 0.72, 0.87, 0.50, 1.24, 2.89, 3.52 and 3.02 per cent of its value at sixty bonds. The worst is 5.3 per cent when measured against the smallest published exponent rather than each case’s own.

So the far end is a plateau as well, and a slightly better one than the near end: 5.3 per cent against 9.6.

The ordering across cases is also the same in both sweeps: the softest chains move least and the stiffest most, at both ends of the window. That is one mechanism rather than two — the stiff cases have long tails, so both the non-asymptotic region at the near end and the noise-limited region at the far end are a larger part of what is being fitted, and both windows bite harder there. It is also, again, the cases already set aside for the noise floor that carry the largest numbers.

Put together, the exponent’s dependence on where the fit begins and where it ends is bounded by about a tenth of itself from both directions, against a quantity that varies by sixty per cent across the series. The two arbitrary choices in the fit are both doing no harm, which is the strongest statement available about the procedure and it took two cheap sweeps to make.

The softest case’s zero per cent is not a good number, though, and it is worth flagging: it has exactly one far end inside its reach, so a curve that is perfectly flat is a single fit repeated eleven times. The stiffest cases have six ends inside, which is where the larger movements are and where the sweep is doing work.

What the abrupt end was predicted to be #

The near-end sweep predicted that the far-end plateau would end abruptly, and reasoned that it would do so because the noise floor arrives suddenly rather than gradually. That prediction is right about the shape and wrong about the mechanism, and the difference is worth recording because it is the kind of thing a prediction can be right about for no good reason.

The plateau does end abruptly. But it ends because the sweep stops having an effect, not because the fit starts going wrong — past the reach the points are already gone and the fit is unchanged, whereas a noise-floor failure would show as the exponent moving to a wrong value once corrupted points were admitted. Nothing here admits corrupted points, because the local decay rate drops them before the fit ever sees them.

So the abruptness is a property of the dropping rather than of the noise, and a version of this calculation that fitted the raw excess instead would show the other behaviour — a curve that stays flat and then goes wrong. This one cannot.

Two different things end a profile #

The interesting finding is not the plateau. It is what stops it.

The five softest chains stop at 23, 30, 39, 54 and 74 bonds, and each stops where the excess over the bulk falls below the relaxation’s own convergence — the noise floor, which is the mechanism the exponent was the floor identifies as having produced a spurious exponent in four cases.

The five stiffest all stop at exactly 77. That is not the floor. It is a quarter of the 320-site chain: the reading rule takes the profile only over its first quarter, on the reasoning that the middle of the chain is bulk and a rate computed there is a difference of two numbers that are both nearly the bulk value. Seventy-seven is where that rule bites, and it bites at the same place for every case that reaches it.

So the far end is limited by two entirely different things, and which one is limiting depends on the stiffness. For the soft half it is physics — the profile really has decayed into the noise. For the stiff half it is a convention. And the two halves are not distinguished anywhere in the reported exponents, which is a version of the same defect found for the noise floor: a fixed window that is a different fraction of the available data for every case.

It is worth asking whether the quarter is even the right rule to have. Its justification is that the middle of a long chain is bulk, so a rate computed there is a difference of two numbers that are both nearly the bulk value and the subtraction loses its precision. That is a real concern and it is the same concern the noise floor addresses — so the two rules are guarding against the same thing by two different criteria, one geometric and one numerical, and only whichever bites first has any effect.

For the soft half the numerical rule bites first, which is the sensible order: it responds to the actual size of the excess rather than to a fraction of the chain. For the stiff half the geometric rule bites first, which means those cases are being cut by a criterion that does not look at their data at all.

What was computed, and how #

Nothing is recomputed. The relaxed profiles are the ones already computed, the local decay rate is the same three-point logarithmic derivative, and the exponent is the same least-squares slope of that rate against 1/b. Only the far end of the fitting range is varied, over eleven values from twenty bonds to a hundred and twenty, with the near end held at six.

The check requires five things: that enough cases have a measurable exponent; that each returns its published value at the original far end, so this is the same fit; that the reaches differ by more than a factor of two, so a fixed far end is a different window for each; that every case saturates inside the sweep, so the flatness past the reach is exhibited rather than assumed; and that inside the reach the exponent moves by under a tenth of itself, which is the plateau.

What the two sweeps together license #

Three arguments have quoted exponents from 0.41 to 0.66, and until the near-end sweep there was no statement at all about how much of that was the fit’s windows. There are now two, and they compose.

The exponent at a stated pair of windows moves by under ten per cent — which is the kind of statement the noise-floor analysis could not make about the four cases it excluded, and which is now available for the six it did not. when the near end is varied over its plateau and by under six when the far end is, and the two effects are of similar size and the same sign pattern. Taken together, a fair error bar on any one exponent from window choice alone is of order ten per cent — which is small against the sixty per cent trend across the series and large against the three decimal places the numbers were first reported to.

That is the practical output of both sweeps and it is a number not available before. It is also a floor rather than a full budget: the noise floor and the quarter-chain rule are two further choices, and the first of them is worth tens of per cent on four cases.

Where the model stops #

The quarter-chain rule is not swept here, and it is now the thing worth sweeping. It is the only remaining unexamined choice in the exponent’s definition, and unlike the two windows it is not obviously harmless: it caps five of the ten cases, so it decides how much profile those five are read over.

The noise floor is not swept either, and moving it by three decades moves four of the ten exponents by tens. That result stands and nothing here touches it — these are the six cases left as measurable, and their behaviour under a moved floor is a separate question.

And the whole exercise is a study of a procedure applied to a model. The relaxed dimerised chain is a classical lattice with a tight-binding electronic structure, and the exponent is a property of how the profiles are read. That has been the standing caveat throughout and nothing here changes it.

What a reader should take from three window sweeps #

Three of the four choices in the exponent have now been examined, and the pattern of results is worth stating as a whole because it is not the pattern anybody would have guessed.

The noise floor matters enormously, on four of ten cases, and moving it by three decades moves those exponents from 0.21 to 0.58 and from −196 to −36.9. That is the noise-floor finding and it is what led to those four being excluded.

The near end matters by half a per cent to nine, on every case, and its choice was made by a rule of thumb that turns out to sit on the flat part.

The far end matters by nothing to 3.5 per cent inside each reach, and by nothing at all outside it.

So one choice out of three dominates, and it is not either of the two that look like windows. A study that had swept the two obvious parameters and stopped would have concluded the quantity was robust; the parameter that was not obviously a parameter is the one that decided four of the ten answers. That is a general hazard and it is the reason all three were swept rather than only the obvious two.

The generalisation #

The transferable point is that a flat region in a robustness sweep has to be interrogated before it is believed, and the question is whether the parameter is doing anything.

Here half the flatness is genuine — the exponent really is insensitive to the far end inside the reach — and half is an artefact of a window that cannot be honoured. The two look identical on a plot and are distinguished by one number, the count of points in each fit, which cost nothing to record and would have been easy not to.

This is the same shape as a decay that keeps slowing down read from the other side: there a quantity looked constant because the window was moving through a region where it happened not to change, and here a quantity looks constant because the window has stopped moving at all.

The failure mode has a name worth remembering: a saturating parameter, one whose effect stops because the data run out rather than because the answer stops moving. Any sweep of a range endpoint has it, and so does any sweep of a resolution, a cutoff or a basis size that the underlying calculation silently truncates. The check is always the same — report how much data each setting actually used.

The second point is smaller and is about reporting. The reaches here span a factor of 3.35 and are capped by two different mechanisms in two halves of the series, and none of that appears in any exponent as usually quoted. A quantity fitted over “bonds six to sixty” sounds like one procedure and is ten different ones.

The repair is cheap. Quote each exponent with the number of bonds its fit actually used and the reason the fit stopped there — the noise floor or the rule — so that a reader comparing two stiffnesses can see at once whether the two numbers were read off the same kind of window, and discount the comparison when they were not.

One more consequence follows for how the exponents should be quoted. An exponent of 0.5168 written to four decimal places implies a precision the fit does not have; with a window budget of a tenth of the value, the honest form is 0.52 ± 0.03, and the digits beyond that are the arithmetic’s rather than the physics’. Four decimals were quoted throughout, which was defensible while nothing was known about the windows and is not any longer.

Who found it, and when #

The Peierls instability, the coherence length and the asymptotic form of a gapped chain’s response are standard. The relaxation, the profiles, the exponent and both window sweeps are new arithmetic, and this sweep is a loop over one parameter of a fit that already existed, reusing profiles already computed — the same arrangement the near-end sweep used, and the reason both were cheap enough to be worth doing.

Still open: the quarter-chain rule, and a longer soft chain #

The obvious open question is the quarter-chain rule, which is now the only unexamined choice left. Sweeping the fraction from a tenth to a half would say whether the five capped cases have more usable profile than they are being given — and the middle of a 320-site chain is already known to be bulk to 10⁻⁸ at every stiffness here, which suggests the rule is conservative and that the stiff cases are being cut short for no reason. If so, their reaches would rise and their exponents would move, and the headline range would need requoting.

The nearer question is the softest case’s single point. One far end inside a reach means the plateau claim rests on nothing for that stiffness, and the fix is not a finer sweep — it is a longer chain, which would push the noise floor further out in bonds without changing the physics. A chain of six hundred and forty at K = 1.1 would give perhaps forty usable bonds instead of twenty-three, and would say whether that case’s exponent is as stable as the others or merely unmeasured — and the chain distorts hardest where it stops, so a longer chain changes nothing about the physics being measured, only how much of it is visible.