Three points, and they all go down

Matching the rings found that the five cases of the collapse used in the exponent turned out to be the windows were not five different temperatures of one system: their product n·δ∞ — the ring size times the bulk amplitude, which is the number of coherence lengths the ring holds — ran from 1.36 to 9.77, a factor of seven. Choosing sizes so that every case sits at the same product tightened the collapse to 1.62 per cent.

It also leaves a floor. A dimerised ring needs an even number of sites, so the matching is approximate: the sizes 40.2, 56.5, 78.6, 108.7 and 149.7 the target asked for become 40, 56, 78, 108 and 150, and that rounding leaves 1.09 per cent of the product un-matched. Against a residual of 1.62 that is not obviously smaller, so one target cannot say whether what remains is a real departure from a single scaling function or the arithmetic of even numbers.

It asked for two more targets, because three points on a curve of residual against target would say which.

Falling, and not flattening #

At a target of five the residual is 2.515 per cent; at seven, 2.070; at 9.6, 1.621. Monotone, and still falling at the largest target that could be afforded.

That is what a finite-size effect that keeps shrinking looks like, and it is not what a departure from a single scaling function looks like — a genuine departure would settle on a value as the rings grew and the scaling regime was entered. Three points cannot distinguish a slow approach to zero from a slow approach to something small, so what is established is a direction rather than a limit. The direction is downward.

It is worth being explicit about how weak three points is. The residual falls by a factor of 1.55 across a factor of 1.92 in the target, which is consistent with a power a little under one half, with a slow logarithm, and with an approach to any floor below about 1.4 per cent. Every one of those is a different physical statement and the sweep does not choose between them. What it does rule out is the one reading that prompted the sweep — a residual that is the rounding — because the rounding does not behave like any of them.

And the rounding is not what is being measured #

The sharper part of the answer is at the small end, and it is a comparison that cannot be made with one target.

At a target of five the sizes are 20, 30, 40, 56 and 78, and the rounding leaves 6.48 per cent of the product un-matched — a much worse mismatch than at 9.6, because the rings are smaller and a change of two sites is a larger fraction of them. The collapse residual there is 2.515 per cent.

The collapse is a third as wide as the mismatch it is supposed to be limited by.

That cannot happen if the rounding translates one-for-one into spread, and it settles the question one target left open. Whatever the residual is, it is not the rounding — the rounding is a loose bound at the small end and, by the largest target, the two have crossed and the residual is the larger of the pair. Neither end supports reading the residual as the arithmetic of even numbers.

The reason is not mysterious once stated. The rounding’s spread is a spread in the product, and the collapse’s residual is a spread in the scaled curves, and the curves are not linearly sensitive to the product. Near the matched value they are flat in it — that is what a collapse means — so a mismatch of six per cent in the product produces much less than six per cent of spread in the curves. The supposed floor was a bound and never a floor.

A rough estimate shows how large the gap between the two can be. If the scaled curves depend on the product only through a smooth function whose slope vanishes at the matched value, a mismatch of a fraction ε in the product moves them by something of order ε². A six per cent mismatch then contributes spread of the order of a few tenths of a per cent rather than six, and the better the collapse, the more the rounding overstates its own effect.

What the sizes mean #

It is worth pausing on what n·δ∞ is, because the whole exercise turns on it and it is not an obvious quantity.

δ∞ is the bulk dimerisation amplitude at a given stiffness — how strongly an infinite chain distorts — and 1/δ∞ is, up to a constant, the coherence length in bonds. So n·δ∞ is the number of coherence lengths that fit round the ring, and it is the natural finite-size variable: a ring of forty at a soft stiffness and a ring of a hundred and fifty at a stiff one are the same system in this variable, and a ring of forty at three different stiffnesses is three different systems.

That is the whole matching argument in one sentence, and it is why the published five cases — three of them rings of forty — were never a test of one scaling function. Matching them is not a refinement of the collapse; it is the first time the collapse has been run on comparable systems. The amplitude the collapse left behind is where δ∞ acquires a closed form, which is what makes choosing the sizes arithmetic rather than a search.

What matching buys, target by target #

The unmatched cases are the same five rings whatever target is being aimed at, so their 3.41 per cent is a constant and the comparison is one number against three. Matching buys a factor of 1.36 at a target of five, 1.65 at seven and 2.10 at 9.6.

So the single-target 2.10 is the best of the three rather than a typical one, and the improvement is still growing at the largest target computed. That is worth recording because a single factor quoted from a single target invites being read as what matching buys, and what matching buys depends on where the match is made.

It also has a straightforward reading: at a small target the rings are small, the rounding is coarse, and the matched cases are not very well matched — so matching helps less. The two effects one target was trying to separate are correlated through the ring size, which is why one target could not separate them.

One more thing follows from the crossing and it is a small correction to the single-target presentation. At that target the residual (1.62 per cent) is above the rounding (1.09), and the difference — 0.53 per cent — was reported as what is left once the rounding is taken off. On the evidence here that subtraction is not the right operation: the rounding does not contribute its full size to the residual, so taking it off wholesale removes too much. The residual at 9.6 is 1.62 per cent and the honest statement is that the rounding accounts for some unknown fraction of it, bounded above by a response that has yet to be measured.

The rings each target asks for #

The sizes rise together with the target, because the softest stiffness has the largest bulk amplitude and needs the smallest ring to hold a given number of coherence lengths. At a target of 9.6 the largest ring is a hundred and fifty; at seven, a hundred and ten; at five, seventy-eight.

The natural targets to add are five and twenty. Five is here. Twenty is not, and the reason is cost rather than principle: it needs a ring of three hundred and twelve at the stiffest setting, whose bulk amplitude is smallest and whose ring must therefore be largest, and a single point on a single curve at that size is an hour of golden sections over the ring free energy. The three targets here are all at the low end, which weakens the sweep — a longer lever would say more about whether the fall continues — and the honest report is the three that were affordable rather than a gap where a fourth should be.

It also says something about the choice of 9.6. That target was picked so that the largest ring came out at a hundred and fifty, which was what the budget allowed; the sweep shows it is also the target at which matching does most among the three, so the headline factor was computed at the most favourable affordable point. That is not a criticism — the affordable point was the only one available — but it is the kind of coincidence worth recording, because a number chosen for cost and reported as a result invites being read as neither.

A target the test would flatter itself at #

There is a lower limit to the sweep and it is not cost. At a target of one half the sizes are 2, 4, 4, 6 and 8: two of the five stiffnesses round onto the same ring of four. Two identical rings agree with each other perfectly, so a collapse computed across them would report a spread narrowed by a coincidence of rounding rather than by anything about scaling.

The sweep detects that and reports it rather than running. That refusal is the reason the sweep stops at five rather than going lower where the rings are cheap — the same discipline a bracket that has to be tested before it is bisected needs.

What was computed, and how #

For each target, the size that puts each of five stiffnesses at that value of n·δ∞, rounded to the nearest even number. For each of those fifteen rings, the distortion amplitude against temperature by golden-section minimisation of the ring free energy, scaled by the ring’s own cold amplitude and plotted against reduced temperature. The residual is the worst relative spread across the five scaled curves at any reduced temperature.

The unmatched comparison is the original five cases, recomputed identically at every target so that the ratio is against a fixed thing. Ten of the fifteen rings are new; the five at 9.6 were already relaxed, which is why the sweep was affordable at all.

The test requires eight things: that the published unmatched and matched residuals both come back at the original target, so this extends that calculation rather than replacing it; that there are three points; that matching tightens the collapse at every one of them; that the residual falls with the target; that it is below the rounding at the smallest target, so it is not the rounding; that the two cross; and the refusal, that a small enough target produces a collision and is reported rather than run.

Where the model stops #

Three targets spanning a factor of two in n·δ∞ is a short lever. Twenty is the natural target because a longer lever says more, and the answer here is about the low end of the range. If the residual has a floor it would show as a flattening, and a flattening between five and 9.6 would be hard to distinguish from the curvature of a slow decay.

The model is the same classical-lattice tight-binding chain as the one whose exponent turned out to be a window, at the mean-field level, with no quantum lattice fluctuations — the caricature behind the amplitude’s closed form. A collapse that failed in a real material would fail for reasons this model has no term for.

The scaled curves are also read at a fixed grid of reduced temperatures, and the residual is the worst spread over that grid rather than an integral of it. A worst-case statistic is sensitive to one bad point in a way a mean is not, and it was chosen deliberately — but it means the three residuals here are three worst cases and not three averages, and a collapse whose curves crossed would report better than it deserves.

And “the same n·δ∞” is one choice of what to hold fixed. It is the right one if the scaling variable is the number of coherence lengths in the ring, which is the standard argument; a different scaling variable would give different sizes and a different residual, and nothing here tests the choice itself.

The generalisation #

The transferable finding is about floors that are not floors.

The un-matched spread in the product plainly limits how well the matching can be done, and it is natural to treat it as a floor under the residual. It is not, because the residual is a spread in something else, and the map from one to the other is not the identity. The way to find out is to move to a regime where the supposed floor is large and see whether the residual follows it up. Here it does not: the floor quadruples and the residual rises by half.

That test is cheap and general, and it is the same discipline a rule of thumb on a plateau needs: an unexamined bound is a liability until somebody varies the thing it bounds. Any time an error budget is assembled by identifying a source and quoting its size, the question is whether the response to that source has been measured or assumed — and the assumption is usually that the response is one-for-one, which is what a flat direction in the fit specifically denies. A collapse is a flat direction by construction, so it is exactly the case where the assumption is worst.

The second, smaller lesson is that a ratio quoted from one condition is a ratio at that condition. Matching buys 2.10 at one target and 1.36 at another, and the difference is not noise.

One caution about the three points themselves. Three is enough to see a direction and not enough to see a shape: a residual that rises by half while its supposed floor quadruples is clearly not tracking the floor, but three points cannot distinguish a shallow power law from a saturation from a straight line with a small slope. The finding as stated — that the bound is not a floor — needs only the direction, and is safe. Any stronger reading, such as an exponent relating the two, would need intermediate cases not run here, because each one is a full relaxation and the three chosen are the ones that bracket the regime.

Who found it, and when #

The Peierls instability and the finite-size scaling of a dimerised chain are standard. The collapse, the matching, the targets and every number above are new arithmetic, and what is added here is two more targets and the observation that the supposed floor is not one.

Still open: a longer lever, and the response coefficient #

The obvious open question is the long lever the cost prevented here. A target of twenty is expensive because of one case: it puts K = 1.2 at a ring of eighty-four and K = 2.0 at three hundred and twelve, and the cost of a curve grows fast with the size. Dropping the stiffest setting from the set and running the remaining four at twenty caps the sweep at a ring of two hundred and twenty-five, which is under half the work — and four matched cases still test a collapse. Whether four cases at a long lever say more than five at a short one is a judgement to make deliberately rather than by default.

The nearer question is the response the last section named. The sweep shows that a mismatch in the product does not produce a proportional spread in the curves, and the coefficient is measurable: deliberately mis-matching the five rings by a stated amount and reading the resulting residual would give the derivative of one against the other, and that number turns the rounding from a bound into a correction. It is five collapses at sizes already computed, which is the cheapest calculation left.