The rate that turned back

The last of the three windows on this fit was the fraction of the chain the local decay rate is read from, and it was not a plateau. Opening it from a quarter of the chain to a half left the soft chains exactly where they were and moved the stiff ones’ exponents down by up to thirteen per cent. The two earlier windows — where the fit starts and where it stops — had each been swept on profiles cut at a quarter, and the obvious next step was to sweep them again on the opened ones.

That step was taken, and the first thing it produced was not an answer about the plateaus. It was a curve that should not exist.

The rate that should keep falling #

The quantity underneath every number in this argument is the local decay rate: at each bond, how fast the Peierls distortion’s excess over its bulk value is shrinking, read as a centred difference of its logarithm. The whole reason the tail is interesting is that this rate is not constant. It falls with distance from the end, roughly as a constant plus a term in one over the distance, and that falling is what makes the excess a power law times an exponential rather than a plain exponential.

A rate that falls towards a constant should fall monotonically, and a chain that cannot stay even has a constant to fall towards: the bulk alternation. There is nothing in the physics of one end healing into a bulk that would make it turn around.

On the soft chains it does not. At a stiffness of 1.8 the rate falls from 0.25 per bond near the end to 0.184 at bond 74, where the excess meets the relaxation’s noise floor and the rates stop. Every rate it produces is smaller than the one before.

The stiff chains do something else. At K = 2.2 the rate falls to 0.0993 per bond and then, from bond 99 onward, rises — slowly at first and then steeply, to 0.118 by bond 135 where the rates finally stop. At K = 2.8 the minimum is at bond 75 and the rise past it is violent: by bond 150 the rate is nearly five times its minimum.

Every bond past a minimum like that is being read as tail, and none of it is. The opened window was taken to admit more profile, and on the four stiffest chains it admitted mostly something else.

The middle is not bulk #

The explanation is in how the excess is defined, and it was stated as a caution when the window was opened: the bulk value the excess is measured against is read from the chain’s middle.

A chain of 320 sites has two ends and each of them heals towards the bulk — the distortion is decided at the ends, and the middle inherits it. At the midpoint both healings have run 160 bonds, and on a soft chain that is a hundred of its decay lengths and the middle is bulk to every digit the relaxation carries. On a stiff chain it is not. At K = 2.2, whose decay length is about eleven bonds, the midpoint sits fifteen lengths from each end, and the excess each end leaves there is small but not zero.

Subtracting that value from every bond does two things. It makes the measured excess exactly zero at the midpoint, whatever the true one is — so the logarithm heads to minus infinity and the rate read from it heads to plus infinity. And far from the midpoint it removes a sliver so small that nothing notices. Between those two regimes there has to be a bond where the sliver starts to matter, and that is the turn.

The claim can be tested without looking at the turn at all. Take the tail the quarter-chain window fitted — a power of the distance times an exponential, with its two parameters read from bonds six to sixty, which no opened profile changes. Put one copy at each end of the chain. Subtract the sum’s value at the midpoint. Take the local rate of the result and find its minimum.

For K = 2.2 the model puts the minimum at bond 100 and the measurement is 99. For K = 2.5 it is 87 against 85, for K = 2.8 76 against 75, and for K = 3.1 69 against 69. Nothing was fitted to the turn. Two numbers from the first sixty bonds of each profile, and the chain’s length, place a feature a hundred bonds away to within two.

The model also says why the soft chains have no turn. For K = 1.8 its minimum is at bond 118, and the profile stops at 74 on the floor — so the floor arrives first and the turn is never reached. At K = 2 the two are closest: a model turn at 109 against a floor at 101. The line of model turns falls with stiffness, the line of reaches rises, and they cross between K = 2 and K = 2.2, which is the split the earlier sweep found by a different test.

Both ends are needed. A model that subtracts the midpoint reference but keeps only one end puts the minima at 106, 95, 86 and 80 — seven to eleven bonds too far out on every stiff case. The far end’s own healing, arriving across the whole chain, is a real part of what turns the rate.

Why a sliver is enough #

It is worth being clear about how small the contamination is, because the size is the surprising part.

At the turn on K = 2.2 the reference is about half a per cent of the excess it is subtracted from — 0.58 per cent, in the two-ended model. A one-per-cent change in a quantity does not usually turn anything around. It does here because of what the rate is doing by then: after ninety bonds it has almost stopped falling. A rate going as a constant plus $p/b$ falls by about $p/b^2$ per bond, and at $b = 99$ with $p = 0.62$ that is six parts in a hundred thousand of a unit per bond. The reference’s pull on the rate grows as its share of the excess grows, and it only has to match that tiny fall to cancel it.

The slower the rate falls, the less contamination it takes to turn it. That is why the turn moves inward as the chain stiffens. A stiffer chain has a longer decay length, so its excess at the midpoint is larger — but it also has a larger $p$ and, at any given bond, a rate that has settled less. Both push the turn towards the end, and the model says by how much.

This is the connection worth carrying out of the calculation. The quantity the tail exponent is read from is the curvature of the rate: its slope against one over the distance. The feature that ends a stiff profile is also the curvature of the rate: the place where it goes to zero. The same small number decides what is measured and where the measurement stops being possible.

The ruler, read to the turn #

The previous sweep built a ruler out of this model’s own internal consistency. A profile that ends on the floor runs a fixed number of its own decay lengths before it disappears, because the floor and the surface amplitude are the same for every stiffness. So any case that has run fewer was cut. By that ruler, reaching to the last bond with a rate, K = 2 and K = 2.2 were resolved by opening the window and the three stiffest were not.

The ruler was right and the reach was the wrong thing to hold it against. Counted to the last rate, K = 2.2 ran 12.54 of its decay lengths, inside the spread of the finished profiles. Counted to its turn — the last bond that is still tail — it ran 9.13. Thirty-six of its 135 bonds were the reference.

The finished profiles themselves are untouched. None of the six stiffnesses from 1.1 to 2 has a turn, so both counts are the same for them, and they average 13.19 lengths with a spread of three per cent. K = 2.2 has run about seven tenths of that.

Six of the ten stiffnesses have exponents that are measurements on this chain, not seven. The three stiffest, which were already known to be unmeasured, turn out to be much further from finished than their reaches suggested: 4.87, 2.80 and 1.85 lengths to their turns, against 9.18, 6.52 and 5.25 to their last rates. K = 3.1 has seen less than a seventh of the profile a finished case shows, and more than half of what the opened window gave it lies past the turn.

Where the seventh came from #

The opened window moved the five stiffest exponents down by 0.96, 2.4, 5.3, 9.9 and 13.3 per cent, and the explanation given was truncation: a window that stops while the rate is still falling reads a rate that is too fast, so admitting the missing points must pull the exponent down, and the size of the pull must grow with how much is missing. It fitted the pattern exactly.

It was the wrong mechanism. Fitted only to each profile’s turn, with the same window of six to a hundred and twenty bonds cut short there, the four stiff exponents come back to 0.6167, 0.6488, 0.6563 and 0.6468. Against the quarter-chain values of 0.6201, 0.6505, 0.6555 and 0.6418 the moves are −0.5, −0.3, +0.1 and +0.8 per cent. Two of them are upward.

So the missing points between the quarter’s cap and the turn — twenty-two bonds for K = 2.2, eight for K = 2.5 — change nothing that matters. What pulled the exponents down was the bonds past the turn, where the rate is rising. The fit is a straight line in the rate against one over the distance, and a cluster of high rates at the far end, where one over the distance is smallest, drags the line’s slope towards zero. The more rates past the turn, the harder the drag: none for K = 2, thirty-six for K = 2.2, eighty-eight for K = 3.1. That is the same monotone pattern the truncation argument predicted, produced by something else entirely.

A mechanism that predicts the right direction and the right ordering can still be the wrong mechanism. The truncation argument was not careless; it made a specific prediction — every move downward, growing with stiffness — and the prediction held. It held because the contamination has the same sign and the same ordering, which no amount of agreement with the argument could have revealed. Only reading the rates themselves could.

The two plateaus, re-run #

The question the opened window was meant to answer has two answers, and one of them is that it was never a question.

The near-end sweep was never exposed to the quarter rule. It moved the start of the fit from two bonds to thirty with the far end held at sixty, and sixty is inside every stiffness’s quarter-chain reach of 77 or fewer bonds. So the sweep reads exactly the same rates whether the chain is read to a quarter or to a half, and its exponents come back bit for bit. Its plateau stands, and it did not need re-establishing.

What the opened profiles do show is that this plateau belongs to a far end held short. Move the far end to a hundred and twenty and K = 2, which has no turn, barely notices: its near-end sweep moves by 8.0 per cent across the plateau instead of 7.2. K = 2.2’s moves by 18.8 per cent, and with the far end at its last rate by 53. Held at its turn instead, it moves by 9.3 per cent — larger than at sixty but a plateau of the same kind.

The far-end sweep was run with the start at six and the far end opened from twenty bonds to a hundred and twenty, and it was flat past each profile’s reach by construction, because no further rates exist to add. For the stiff cases that flatness began at 77 bonds, which was the quarter’s cap and not the floor. On the opened profiles the far end moves the exponent by 1.5 per cent for K = 2 and 2.0 for K = 2.2 up to the turn, both under the 5.3 per cent published as its worst — so inside the turn the far-end plateau survives. Past the turn the sweep is no longer choosing how much tail to fit. It is choosing how much of the reference to fit, and the exponent falls without limit.

One plane, not two lines #

Each window was swept with the other held fixed — the near end with the far end at sixty, the far end with the near end at six — and each found a plateau along its own line. Two plateaus along two lines through a plane say nothing about the rest of the plane, and on an opened profile the rest of the plane is available.

The flat region is large and its edge is not where the turn alone would put it. A window starting anywhere from four to twelve bonds stays within five per cent of the standard value out to far ends of 120 to 130 — well past the turn at 99. A window starting at fifteen stays flat to 110, one starting at twenty to 95, one starting at thirty only to 65.

The shape follows from what the fit is. A window that starts near the end contains dozens of rates from the steep, well-measured part of the profile, and they pin the line; a handful of contaminated rates at the far end cannot move it far. A window that starts deep has already given up those anchoring points, so the same contamination is a larger share of what remains. The edge of the plateau is a diagonal because the harm done by the reference depends on how much genuine tail is in the same fit. Neither one-dimensional sweep could have seen that, because each held one coordinate where the plane happens to be most forgiving.

What was computed and how #

Nothing new was relaxed. The profiles are the same stored relaxations of a 320-site chain at ten stiffnesses that every sweep in this argument has used — the ones in which the chain distorts hardest where it stops — with the same tight-binding band and elastic constant set up for this chain, the same noise floor of 10⁻⁸ and the same bulk value read from the middle.

What was added is a reading. The turn is the bond of the smallest local rate on the profile opened to half the chain. The model turn is the minimum of the local rate of $f(b) + f(N - b) - 2f(N/2)$, with $f(b) = b^{-p} e^{-b/\lambda}$ and $p$ and $\lambda$ from the standard window of six to sixty on the quarter-read profile; a one-ended version drops $f(N - b)$. Lengths are the reach, or the turn if it comes first, divided by the decay length fitted up to that bond. The window plane is 239 fits for K = 2.2, one for each near end from two to thirty and each far end from forty to the last rate in steps of five.

The table is the whole calculation in one place, and its last column is the finding: six rows say yes, and the row that changed is K = 2.2. The checks that go with it state each finding in a form that could fail: that the near-end sweep with its far end at sixty returns identical exponents under both fractions for every stiffness; that every profile with no turn is at K ≤ 2 and every one with a turn is stiffer; that the model’s turn is within three bonds of the measured one on each stiff case and past the reach on each soft one; that K = 2.2 counted to its turn has run under eighty per cent of a finished profile’s lengths; that each stiff exponent fitted to its turn is within two per cent of its quarter-chain value. The refusal is K = 1.8, which must have no bond past its turn.

What the picture cannot show #

A turn is found, not proved. The minimum of a noisy sequence is where it is found, and on K = 2.2, whose rate is very flat near the bottom, the minimum could move by a few bonds under a different floor or a relaxation converged a little further. The model’s agreement to within two bonds on four cases is what makes the turn believable, and it is agreement with a model whose tail is only the power-law-times-exponential form the fit assumes. A tail with a third term would move the model’s turn, and nothing here tests that.

The six measured exponents are measurements of this model on this chain. The chain is 320 sites. The ruler says a finished profile runs about thirteen of its decay lengths, and the turn says that on a chain this long a stiff profile cannot run that far before the midpoint’s reference intrudes. Both are statements about 320. The previous essay estimated eight hundred sites for K = 3.1, and the turn makes that estimate optimistic: the chain has to be long enough that the reference at its middle is smaller than the floor, not merely long enough to hold the profile.

And the reference is only one way to define the bulk. Reading it from the middle is natural and it is what produced the turn. A bulk taken from a separate periodic ring at the same stiffness would carry no healing at all, and would remove the turn by construction. It would also change every excess this argument has measured, by a small amount at the ends and a large one near the middle, and that is a different calculation.

Still open: a bulk with no ends, and the chain the turn asks for #

The obvious open question is the reference itself. A periodic ring of the same length relaxes to an alternation with no end in it, and its alternation is a bulk value that owes nothing to either end of the chain. Measuring every excess against that value would remove the turn on the stiff cases and would say whether what lies past it is tail after all — and it would move the soft cases not at all, which is the check that the new reference does only what it claims.

The nearer question is the chain length the turn implies. The model says where the turn falls for any chain length and any stiffness, from two numbers each stiffness already has, and so it says how long a chain has to be before K = 2.2’s turn lies past its floor. That is one line of arithmetic per stiffness, and it replaces the previous estimate — the length a profile needs — with the one that matters: the length at which the middle of the chain is bulk.