The lever that was supposed to be smaller

Three defensible ways of counting frozen rotors, swept across all three measured accelerations, give a clean two-part result: every verdict survives every convention, and every margin moves by up to a factor of eleven. That result came with a prediction about the other quoted input.

The supply argument above suggests the answer for the five-membered ring is that nothing is at risk — a factor of eleven in the ceiling did not reach it, and a plausible range in one energy is a smaller lever than that.

The prediction is wrong, and it is wrong in a way that is worth more than the sweep it was attached to. A factor of eleven that moves nothing and a factor of nine that moves two verdicts are not comparable quantities, because what decides whether a lever matters is not how far it moves the ceiling but how near the ceiling already is to a measurement. That is the same reading a published answer holding over a window seven tenths of a per cent wide forces on the counterpoise correction.

What the range is, and what it is not #

Two different objects are in circulation and both are used here.

The standard compilation value for butane’s gauche energy is 3.8 ± 0.4 kilojoules a mole, which is 0.9 ± 0.1 kilocalories, and it is the sort of quoted input the conformer statistics themselves are otherwise free of, and it is what the comparison with error bars on both sides of a verdict already carried. That is an error bar on one determination.

The spread among determinations is wider. Electron diffraction, Raman intensities, thermodynamic fits and computed values do not agree to four tenths, and the figures a reader of the literature would find run from about 2.5 to about 4.5 — 0.6 to 1.1 kilocalories in the units they were reported in. That is not an error bar on anything; it is the range of numbers somebody might have used.

A sweep over the first says whether a result is stable against measurement error. A sweep over the second says whether it is stable against which paper was read. Both are run here, and it turns out the first is enough.

Eight point seven, against eleven #

Across 2.5 to 4.5 kilojoules a mole the ceiling moves by up to 8.70-fold — from 5.62 to 16.58 at two rotors, from 13.32 to 67.49 at three, and from 31.59 to 274.77 at four.

The rotor conventions moved it by 11.0-fold and changed nothing. This moves it by 8.7 and changes two verdicts out of nine. So the natural comparison — bigger factor, bigger threat — is the wrong one, and the right one is geometric rather than arithmetic: what matters is whether a ceiling’s range crosses a measurement, and a range that starts nearer one needs to be smaller to do it.

That is visible in the figure without any arithmetic. The three measurements are three horizontal lines. Six of the nine ceiling curves stay entirely on one side of the line they are being compared against for the whole sweep. Two cross.

Where they cross #

The six-membered ring’s survival goes first, at 3.630 kilojoules a mole, under the convention that gives it two rotors — which is the convention behind the decided verdict. Below that energy the ceiling is under ten and the account is refuted; above it, unrefuted.

3.630 is 0.42 standard deviations below the compilation value of 3.8. It is not at the edge of the quoted error bar; it is comfortably inside it, nearer the centre than one sigma. So the six-membered verdict is decided by a quantity whose own stated uncertainty spans the decision, which is precisely the arrangement a band on the verdict warned about — and that calculation was looking at the same energy from the other end, asking how far it would have to move rather than whether it already had.

The five-membered ring’s refutation goes second, at 4.422, again under the convention the decided verdict used, which gives a five-membered closure four rotors. There the ceiling reaches 274.77 and the measured 250-fold acceleration stops being above it.

That one is outside the compilation’s error bar and inside the literature’s range. It is the headline result — the one the rotamer ceiling established and every calculation since has quoted — the rotamer account cannot produce a 250-fold acceleration on a five-membered closure — and it depends on a number reported over a range that contains the point where it stops being true.

Only the 11,000-fold case is safe. It is refuted at every energy in the range under every convention, by a margin of at least forty, and it is the one result in the comparison that nothing has yet moved.

The two unsettled choices interact #

Neither crossing happens under all three conventions, and that is not an accident of the numbers.

The six-membered case crosses under the convention that gives it two rotors and not under the ones giving it three or four, because at three and four the ceiling never comes down as far as ten anywhere in the range. The five-membered 250-fold case crosses under the convention that gives it four and not under three or two, because at three and two the ceiling never gets up as far as 250.

So the two unsettled quantities do not add their uncertainties. A case is at risk from the gauche energy exactly when the rotor convention has already placed its ceiling near the measurement, and the convention that does that is a different one for each case — the one that gives the six-membered closure the fewest rotors and the five-membered closure the most. Under the single convention that is internally consistent about direction, n − 3, neither case moves.

That is a result about the conventions rather than about chemistry, and it points somewhere useful: the convention the decided verdict used is the one under which both crossings happen. It was already known to be the odd one out — it is the only one of the three that gives a larger ring fewer frozen rotations — and it now turns out to be the only one under which the gauche energy is dangerous.

The argument that depended on strictly less #

The convention sweep had a second argument, and it is the stronger one. Rather than comparing a measurement against a ceiling and quoting a margin, it computes the least rotor count that would reach each measurement and compares that against the number of bonds in the chain — which is not a convention at all.

That is the refutation without a margin in it. The account does not fall a little short on the five-membered ring; it would need more internal rotations than the molecule contains.

The requirement is computed from the ceiling, and the ceiling contains the gauche energy. At 2.5 kilojoules a mole the 250-fold acceleration needs seven rotors; at 3.8 it needs five; at 4.5 it needs four. A five-atom chain has four bonds.

So the argument holds across most of the range and fails at the top of it. It is still the better argument — it depends on one quantity rather than two, its failure needs a gauche energy at the extreme of what is reported, and it never came close on the 11,000-fold case, which the explanation with the wrong sign established as the extreme of the measured set, which needs eleven rotors at the bottom of the range and seven at the top. What it is not is an argument that depends on strictly less. It depends on strictly less of one kind of choice.

That distinction is worth keeping. A refutation with no convention in it can still have a measurement in it, and the two are different vulnerabilities that get described with the same word. An estimate that can be wrong by two is the standing case of a bound that had to be counted rather than assumed.

Why a small lever moved more than a large one #

The arithmetic of what happened is short and worth stating, because the prediction failed on a reasonable-sounding principle.

A verdict changes when a ceiling crosses a measurement. Whether a sweep of the ceiling crosses depends on two things: how far the ceiling moves, and where it started relative to the line. The rotor sweep moved the ceiling by eleven-fold and started every case a long way from its line — the tightest was the six-membered case at 11.0 against 10, and eleven-fold in that direction took it to 121, further from the line rather than across it. The gauche sweep moves each ceiling through its own starting point in both directions, so a case sitting close to its line crosses whichever way it is nudged.

So the two sweeps are not two samples of one kind of uncertainty. A convention sweep asks which of these discrete choices was made, and the answers are scattered; an input sweep asks what value did this continuous quantity have, and the answers are a connected interval around where every calculation so far has stood. An interval around the working point is far more dangerous to a marginal verdict than a scatter away from it, whatever the widths are.

What can now be said about the three cases #

Three measurements have been tested against the rotamer account from the start, and it is worth setting out where each stands after four sweeps.

The 11,000-fold acceleration is refuted, and nothing has moved it. Three rotor conventions, a temperature range, an error bar and a literature range have all been applied, and under every combination of them the account falls short by at least a factor of forty. It needs seven internal rotations at the most favourable energy in the range and the chain has four bonds. This is the one secure result.

The 250-fold acceleration is refuted under two conventions and at every energy inside the compilation’s error bar, and not at the top of the literature range under the third. That is a weaker statement than the one made until now and it is still a refutation for any reader who accepts either of the two conventions that order ring sizes the same way.

The 10-fold acceleration is undecided, and now visibly so. Its verdict changes with the rotor convention’s margin, with the temperature, and with the gauche energy inside a single determination’s error bar. A comparison that three separate levers can each move is not a verdict, and the honest report of it is a bound in both directions.

What was computed, and how #

The ceiling is the closed form used throughout — $((1 + 2x)/2x)^r$ with $x = \exp(-g/RT)$ — evaluated at forty-one energies across the range, for each of the three rotor conventions, against each of the three measured accelerations. Nine combinations, forty-one energies, and the whole thing is 369 evaluations of an expression.

The crossings are located by bisecting on the verdict rather than on the ceiling. That is the right thing to bisect because the verdict is what is being reported, and it means the number quoted is the energy at which the reported conclusion changes rather than the energy at which some proxy for it does.

The supply figures come from the same closed form as the convention sweep. Enumerating every one of the three-to-the-rotors conformations explicitly is an independent check on that closed form, and it is made once rather than at each of the forty-one energies, where at eleven rotors it would mean 177,147 conformations per point.

Six results are checked numerically. The 11,000-fold case is refuted at every energy under every convention, which is the one safe result and has to be established before the unsafe ones mean anything. The 250-fold case is not. The six-membered case is not. The two flip under some conventions and not others, so the two unsettled choices interact rather than adding. The margins spread by more than a factor of two.

And two checks that point in opposite directions. The quoted error bar of ±0.4 must flip something, or the finding would be about which paper was read rather than about the measurement — it does, at 3.630. A band of ±0.02 must flip nothing, or the sweep would be reporting instability by construction rather than measuring it — it does not.

Where the model stops #

The range is quoted and its width is a judgement. 2.5 to 4.5 is what a reader of the literature would find, stated as such, and somebody with a narrower reading of the literature would get a narrower answer. The finding that does not depend on that judgement is the six-membered one, which moves inside a stated error bar rather than inside an assembled range.

The ceiling is still a ceiling. Everything above concerns whether a measurement is above or below the largest ratio the rotamer account can produce. A case that moves from refuted to unrefuted has not become explained; it has become uninformative, which is a worse outcome than a refutation rather than a better one.

And the temperature is held at 298 K throughout, which is the same choice the temperature sweep swept. Whether the two sweeps are independent is not asked here, and the answer turns out to matter.

The generalisation #

The transferable point is about which quantity to sweep when there is more than one candidate, and the rule that comes out of it is not the obvious one.

The obvious rule is to sweep the quantity with the widest range, since that is where the largest movement is. That is the natural reasoning, and it is why the gauche energy looked harmless: an eleven-fold lever had already been pulled without effect, and this one looked smaller.

The better rule is to sweep the quantity that moves the answer through its current position. A discrete choice among conventions moves a result to a scatter of other places, most of which are further from any threshold. A continuous input moves it along a line through where it is, so the nearest threshold in either direction is reachable at half the range. On a marginal comparison the second is the more dangerous by a large factor, whatever the ranges are.

There is a diagnostic in that which costs nothing: before sweeping anything, look at the distance from each verdict to its own threshold in the units the sweep will move it in. Two of the nine combinations here sat within a factor of two of their line and seven did not, and the two that sat close are exactly the two that moved. That could have been read off the convention sweep’s own table.

The second half is about what a strong argument is strong against. The supply argument is genuinely better than the margin argument, and it is right to prefer it — but “depends on strictly less” was doing more work in that sentence than it could carry. It removed a convention and kept a measurement, and the two failure modes are unrelated. Naming which kind of dependence has been removed is worth the extra clause, and the same correction is needed about a control that ruled out one candidate and not another.

Who found it, and when #

The rotamer account and its ceiling are standard; butane’s gauche energy is a measured quantity with a long literature; the three accelerations are measured. The sweep, the two crossings and the observation that the supply argument moves with them are new here.

A prediction written down before the calculation is what makes this a refutation instead of an extension — and a prediction that turns out wrong is worth more than four that turn out right, when it names the reasoning rather than only the conclusion.

Still open: whether temperature and gauche energy are two levers #

The obvious open question is now sharper than it was. The temperature and the gauche energy have been swept separately and reported as two independent checks on the same verdict. The ceiling depends on both, and it is worth asking whether it depends on them separately — because if it does not, then one of the two sweeps explored ground the other had already covered, and one instrument has been counted as two.

The nearer question is the six-membered case, which has now been found marginal by three different levers: a rotor convention that gives it a margin between 9.9 per cent and a factor of twelve, an error bar on one energy that spans its verdict, and a temperature at which it changes. Three levers all reaching one comparison is not three problems; it is one comparison that should never have been quoted as a verdict. Restating it as a bound — the rotamer account is not refuted by the six-membered measurement at any defensible setting, and is not supported by it at any either — would say what is actually known, and would cost nothing that was ever real.