The exponent was the floor

A decay that keeps slowing down established that the alternation profile of a relaxed chain is not an exponential, and gave the honest replacement: measure the decay rate at a bond rather than over a window, then fit those local rates against the reciprocal distance from the end. The intercept of that fit is a rate with no window in it, and its reciprocal is a coherence length.

The slope is a second number and it is not a nuisance. A rate going as $1/\lambda + p/b$ is an excess going as $b^{-p}\exp(-b/\lambda)$, so $p$ says the profile is a power law times an exponential rather than an exponential. Across ten stiffnesses it runs 0.41, 0.45, 0.48, 0.52, 0.56, 0.59, 0.62, 0.65, 0.66, 0.65 — rising, then flattening.

That shape is what a quantity approaching a limit looks like, and it was read that way: $p$ “appears to be settling near two thirds”. The natural question is what it is obliged to be.

It is obliged to be nothing, because over the last four of those ten cases it is not being measured.

Why there is a power there at all #

Before asking what the power is, it is worth saying why anybody expects one. A chain that has been relaxed to a fixed point is not settling towards its bulk alternation by a single exponential — a decay that keeps slowing down established exactly that, and it is why the local rate has to be measured at a bond rather than fitted over a window.

The reason is the gap. The gap is not the band width separated the two energies a distorted chain has, and it is the gap that sets how far a disturbance at the end reaches: a gapped one-dimensional system responds to a local perturbation over a length set by the gap, and the response of a band edge is never a clean exponential because the density of states at the edge is singular. A square-root singularity gives an inverse-square-root prefactor, which is where a reader’s expectation of one half comes from.

The measurements below do not reach one half and do not stop there, so that expectation is worth holding loosely. What matters here is only that a power is the right shape to look for.

A rate with two numbers in it #

The local rate is a centred difference of the logarithm of the excess — the alternation at a bond minus the bulk value. Two things bound where it can be taken.

At the near end the profile has not settled into any asymptotic form, so the first few bonds are excluded — and the near end is where the interesting physics is, which the chain distorts hardest where it stops is about. Everything measured here is deliberately far from it. At the far end the excess has decayed, and the relaxation that produced it converged to a fixed point with a tolerance. Below that tolerance the “excess” is not a small number, it is the difference between two numbers that are equal, and its logarithm is noise.

So the fit drops points whose excess falls under a floor. That floor is a property of the arithmetic and it is the same absolute number at every stiffness. The excess it is compared against is not: a stiffer spring gives a bigger alternation and a bigger enhancement at the end, so the tail starts higher and has further to fall before it disappears.

Two repairs that do not work #

The obvious suspicion is the window. The exponent was the windows established that a fitted exponent can be a property of the stretch it was fitted over rather than of the system, and the decay analysis says as much itself — that a fit over a fixed number of bonds “gets shallower the longer the true length is”.

Refitting over a window scaled to each case’s own coherence length gives 0.47, 0.51, 0.54, 0.57, 0.61, 0.67, 0.68. That is the same drift. Whatever is happening does not care where the window sits in bonds.

The second suspicion is a missing term. A two-parameter fit will absorb anything the truth has that the model does not, so add the next one and fit $1/\lambda + p/b + c/b^2$. That makes it worse: $p$ then runs from 0.42 to 0.75, and $c$ changes sign halfway through the range, which is what a coefficient does when it is fitting nothing in particular.

Both repairs assume the fit is being asked a question it can answer. It is not.

There is a third suspicion worth killing explicitly, because it is the one most people would try first: that a chain of 320 sites is simply too short, and the tail is running into the far end. It is not. The centre of the chain is bulk to better than $10^{-8}$ at every stiffness measured — the excess forty bonds before the centre is $8 \times 10^{-5}$ at the softest spring and the centre itself is $-3 \times 10^{-8}$, which is a sign-flipped nothing. The two ends’ healing regions do not meet. Doubling the chain would change none of what follows.

What the floor does #

Count the tail in its own units. The excess at the far end is roughly the enhancement at the near end times $\exp(-b/\lambda)$, so the number of e-foldings visible before it reaches the floor is $\ln(\text{excess}_0/\text{floor})$ — and both the numerator’s parts move the wrong way as the gap closes. The coherence length grows, so each e-folding costs more bonds; and the enhancement shrinks, so there are fewer e-foldings to spend.

Measured, the reduced reach falls from 6.4 at the stiffest spring to 3.8 at the softest. A power law multiplying an exponential only shows itself once the exponential has been divided out over a decent stretch, and under four e-foldings is not a decent stretch. So the fit at the soft end is being asked for the shape of a curve over a piece of it where the shape is not yet expressed.

The test that separates the two is the floor itself. Raise it from $10^{-8}$ to $10^{-5}$ — throwing away the least trustworthy points — and the six stiffest cases move by about 0.02, which is what a measurement does when its worst data are trimmed. The case at $K = 2.2$ moves from 0.21 to 0.58, which is a rescue: its $10^{-8}$ tail was noise, and removing it recovers a sensible number. The three softest move from $-196$ to $-36.9$, from $-37.8$ to $-6.0$ and from $-8.2$ to $0.30$, which is not a rescue and not a measurement.

The asymmetry between the two directions is worth noticing. Trimming noise raises a fitted power towards a sensible value where a real signal is present, because the discarded points were dragging the rate estimate around at random. Where no signal is present it moves the answer by an arbitrary amount in an arbitrary direction, which is what $-196 \to -36.9$ is. Both behaviours are visible in one table, at neighbouring stiffnesses, which is the cleanest evidence available that the boundary between them is real and is where it has been drawn.

The cases that survive both tests #

A case is measurable here if it has at least six e-foldings of visible tail and its answer does not move by more than 0.05 across three decades of floor. Six of the ten pass.

Across those six the power rises monotonically — 0.410, 0.450, 0.479, 0.514, 0.542, 0.567 — with the coherence length going from 1.7 bonds to 7.9. It passes through a half without pausing and shows no flattening anywhere. The flattening in the published column begins at $K = 2.2$, which is the first case that fails.

That is the whole finding. The apparent limit is assembled entirely from the cases that have no range to measure in, and the cases that do have a range say the quantity is still moving.

It is worth being careful about what this does not overturn. The coherence length itself is unaffected: it comes from the intercept of the same fit, and an intercept is far more robust to a short range than a slope is — the fitted $\lambda$ agrees with the published value to better than a per cent in every measurable case, and the whole picture of a length that grows as the gap closes stands exactly as it was. What fails is the second number, and it fails because a slope needs range in a way an intercept does not.

The curve that looked like it was settling #

Drawn together the two are hard to unsee. The published curve is smooth, monotone and flattening, which is the signature every reader is trained to accept as convergence — and it is smooth because the failure is smooth. The fitted power degrades gradually as the reach shrinks, so nothing in the sequence looks like a break.

That is what makes this failure mode worth an essay of its own rather than a footnote. A fit that fails abruptly announces itself. A fit that fails gradually produces exactly the shape a converging quantity produces, and the two are told apart only by asking how much curve each point was fitted over — which is not a number the fit reports.

Where the measurable cases point #

The six points lie close to a straight line in $1/\lambda$, and that line meets the axis at $0.602$. If the relation stays straight, an infinitely long coherence length gives a power near six tenths.

It should be quoted as a direction and not as a value, and the reason is visible in the picture: every measured point is to the right of $1/\lambda = 0.12$, and the intercept is at zero. The extrapolation reaches further than the whole measured range, on six points, with nothing checking that the relation is linear rather than merely locally straight. It is worth more than the two thirds it replaces because it is built only from cases that were measured, and that is all it is worth.

What was computed, and how #

Ten chains of 320 sites at stiffnesses from 1.1 to 3.1, each relaxed bond by bond to a fixed point, giving an alternation profile and a bulk value. The local rate at each bond is a centred difference of the logarithm of the excess; the fits are least squares of that rate against $1/b$ over the stated range, at each of three floors.

Nothing here needed a new relaxation. Every number is arithmetic over profiles already computed, taken at different floors and over different stretches — which is why the whole of it needed no new diagonalisations.

The refusal is the softest spring. Asked for a power over 3.8 e-foldings whose last are numerical noise, the fit returns $-196$. An instrument that failed quietly there — returning 0.7, say — would be far more dangerous than one that returns an absurdity, and the check requires the absurdity: if that fit ever comes back plausible, the guard this essay is built on has stopped working.

The same distinction decides how much of the related work is touched. The distortion the ends decide and a distortion needs two states are about which pattern a chain picks and why it distorts at all, and neither depends on the shape of the tail. This essay is narrow on purpose: it is about one slope in one fit.

Where the model stops #

The floor is the relaxation’s own convergence tolerance and it is not fundamental. Relaxing harder, in higher precision, would push it down and buy more e-foldings — a decade of floor is one more e-folding, so reaching six at $K = 3.1$ needs about two decades. Whether the fixed-point iteration is stable that far down is a separate question this has not asked.

And a longer chain would not help, which is worth stating because it is the first thing anybody would try. The centre of a 320-site chain is already bulk to better than $10^{-8}$ at every stiffness here: the excess forty bonds before the centre is $8 \times 10^{-5}$ at the softest spring and the centre itself is $-3 \times 10^{-8}$. The tail is not being cut off by the far end. It is being cut off by the arithmetic, and more sites do not change that.

Ten stiffnesses on one chain length is also a thin basis for the claim that the power keeps rising. What would settle it is the same measurement at a floor two decades lower, where the four excluded cases would become measurable and could be checked against the line the six draw.

The generalisation #

The habit this argues for is small and it is nearly free: report the range a fitted exponent was measured over, in the units the exponent is about.

A power law fitted over a decade is a different object from one fitted over four, and a decay exponent fitted over two e-foldings is barely an exponent at all. Neither fact is visible in the fitted value, its uncertainty, or its goodness of fit — a two-e-folding fit can have a beautiful $R^2$, because over two e-foldings almost anything is a straight line on a log plot.

The failure this produces is specific and it is the one above: a sequence of such fits, taken across systems whose accessible range varies systematically, produces a smooth trend that is a picture of the range rather than of the physics. And it will look like convergence, because a shrinking range and a converging quantity both produce a curve that flattens.

The check is one number per point and it is already computed. It just has to be printed beside the answer.

There is a reason this particular collection keeps arriving at that conclusion. Every quantity here is computed rather than quoted, so the range a fit had available is always in hand — and the whole argument of a chain cannot stay even, that a one-dimensional chain distorts because the electrons pay for it, was settled by counting rather than by fitting anything. A number that comes out of a fit is a different kind of object from a number that comes out of a sum, and the difference is that the fit has a window and the sum does not. Where both are available the sum should be preferred; where only the fit is, the window belongs in the report.

Who found it, and when #

That a power-law prefactor multiplying an exponential requires many e-foldings to extract is standard practice in every field that fits decays, and is written down in the numerical-analysis literature rather than the chemical one. The specific arrangement here — where the accessible range shrinks systematically with the parameter being scanned, so the artefact is a smooth trend rather than scatter — is the part worth carrying, and it is a property of the model rather than a discovery about it.

What is new here is the measurement: which of the ten cases have a range, what they say, and that the published limit is drawn from the four that do not.

Still open: the floor, and the near-end cut #

The obvious open question is the floor. Two more decades of convergence would make the four excluded cases measurable, and the prediction is already on the page: they should fall on the line the six draw, near 0.60 at the far end rather than near 0.66. That is a prediction made here that a longer convergence can check, which is the strongest form available here.

The nearer question is the near end. Everything above discards the first few bonds because the profile has not reached its asymptotic form there, and how many is set by a rule of thumb — three coherence lengths — rather than by anything measured. The local rates are computed at every bond and the fit could be run from every possible starting point, so the honest version is a plot of the fitted power against where the fit was allowed to start. If it has a plateau, the rule of thumb is doing no harm; if it does not, then the near-end cut is a second window with a second exponent hiding in it, and the quantity has two arbitrary choices in it rather than one.