Two sweeps and one lever

The rotamer account of ring closure has now been swept along four axes, with four results. The rotor count, which moved every margin and no verdict. The temperature, which found the six-membered verdict giving way at 312.1 kelvin. The gauche energy, which found it giving way at 3.630 kilojoules a mole and took the five-membered refutation with it at 4.422. And the temperature dependence of the measured ratio, which removed the crossing entirely at an activation-enthalpy difference of 5.308.

Two of those four are the same sweep.

One variable, wearing two names #

The rotamer ceiling is

$$C \;=\; \left(\frac{1 + 2x}{2x}\right)^{r}, \qquad x = e^{-g/RT}$$

and g and T appear nowhere in it except inside x. So the ceiling is not a function of a gauche energy and a temperature. It is a function of one variable, u = g/RT, at every rotor count, at every energy, at every temperature.

That is not an approximation and it is not a limit. It is what the expression says, and the consequence is exact: any change to g is reproduced by a change to T in the same proportion, and vice versa. Doubling the gauche energy and doubling the temperature leave the ceiling untouched to the last digit. That is the same kind of statement as the rescalings that fix two of a tunnelling splitting’s three sensitivities, arrived at for a different problem and by the same route: write the expression and look at how the symbols enter.

Checked at five pairs spanning the gauche energy’s whole reported range, the two routes give the same double-precision number — a relative difference of exactly zero, not a small one. That distinction is the point of running the check at all. A result agreeing to a part in ten million would be evidence of two nearly-equal quantities; a result agreeing bitwise is evidence of one quantity computed twice.

What each sweep actually explored #

Putting both ranges into u is the only fair comparison, and it is not close.

The temperature sweep ran from 253 to 373 kelvin at the compilation’s gauche energy, which is u from 1.2248 to 1.8054 — a width of 0.5806. The gauche sweep ran from 2.5 to 4.5 kilojoules a mole at 298 kelvin, which is u from 1.0085 to 1.8153 — a width of 0.8068.

The second contains the first. Their union is 0.8068, which is the gauche range exactly, and the overlap is 0.5806, which is the temperature range exactly. The temperature sweep visited no value of the ceiling’s only variable that the gauche sweep does not visit.

Adding the two widths would give 1.3874, and treating them as independent would claim seventy-two per cent more coverage than there is. Worse, it would treat two agreeing results as two pieces of evidence, when the second is the first evaluated at different labels.

The exchange rate, and what it is worth knowing #

The identity has a use beyond bookkeeping, and it is the reason to state it as an exchange rate rather than as a warning.

At a fixed temperature of 298 kelvin, the gauche energy’s reported range of 2.5 to 4.5 kilojoules a mole is equivalent to holding the energy at 3.8 and running the reaction from 196.2 kelvin to 353.1 kelvin — from seventy-seven degrees below zero to eighty above. That is a wider temperature range than any solution-phase lactonisation is run over, and it is the range a reader would have to accept as already covered by the uncertainty in one quoted number.

Read the other way it is more useful still. A closure run at 340 kelvin rather than 298 has the same ceiling as one run at 298 with a gauche energy of 3.34 rather than 3.8 — which is inside the compilation’s error bar. So a temperature difference of forty kelvin and a measurement uncertainty of half a kilojoule are the same perturbation, and any argument that treats one as negligible has to treat the other as negligible too.

That is a statement worth having. The six-membered verdict changes at u = 1.4645, and there are two ways to reach it: raise the temperature by fourteen kelvin, or lower the gauche energy by 0.17. Neither is a large move, and they are not two ways of being wrong. They are one.

Two sweeps found one crossing #

The sharpest form of the finding is not about ranges at all. It is that the two sweeps located the same point and neither noticed.

The temperature sweep reported that the six-membered verdict gives way at 312.1 kelvin, with the gauche energy held at 3.8. The gauche-energy sweep reported that it gives way at 3.630 kilojoules a mole, with the temperature held at 298.15.

In u those are 1.4644 and 1.4643. They agree to four figures, and the residual is the bisections’ own tolerance rather than anything physical.

So there are not two independent findings about where the one marginal verdict fails. It has one crossing, found twice, in two coordinate systems, and reported as though the second confirmed the first. That is the most expensive form the redundancy takes: two agreeing measurements read as corroboration, when one is the other transformed.

The rotor count is genuinely separate, and it factorises #

The third input is not absorbed, and the way it is not is worth having.

The rotor count appears as an exponent, so its logarithm is a multiplier: ln C = r · ln((1 + 2x)/2x). Dividing any computed ceiling’s logarithm by its own rotor count therefore has to return the same function of u, whatever produced it.

It does. Thirty-six combinations — three rotor counts, six gauche energies, six temperatures, from both sweeps — land on one curve with a worst departure of 2 × 10⁻¹⁶, which is the last bit of a double. Dividing instead by a rotor count one larger scatters them by more than a twentieth, which is the check that the collapse belongs to the model rather than to the axis.

So the ceiling is one curve and one exponent, and the whole nine-row table of rotor counts and accelerations is that curve read at three exponents. The model has two levers, not three: a reduced energy and a count. Everything else is a label.

That also explains why the rotor sweep and the gauche sweep behaved so differently. They move different things — one moves along the curve and one changes which power it is raised to — and a lever that moves along a curve through the working point is the dangerous kind, which the gauche-energy sweep found from the other direction without being able to say why.

Every verdict as one number #

Once the ceiling is (1 + eᵘ/2)ʳ, the condition for a measured acceleration F to be reachable inverts in closed form. Setting the ceiling equal to the measurement and solving,

$$u^{*} \;=\; \ln\!\left[\,2\left(F^{1/r} - 1\right)\right]$$

which is every verdict in the comparison in one expression. Above u* the account is not refuted; below it, it is. No sweep is needed to apply it and no bisection was needed to find it.

The three measurements, at the conventions behind the decided verdict, give thresholds of 1.4644 for the six-membered 10-fold case at two rotors, 1.7839 for the five-membered 250-fold at four, and 2.9167 for the 11,000-fold at four. The working point — 3.8 kilojoules a mole at 298.15 kelvin — is 1.5330.

Reading those four numbers off is the entire content of four sweeps. The six-membered case sits 0.069 above its threshold, which is the margin quoted until now as nine point nine per cent in the ceiling. The 250-fold case sits 0.251 below its own, and the 11,000-fold case 1.384 below its own — which is why one of the three has been moved by everything tried and the other two have not.

And the distances are in a unit that means something. A shift of 0.069 in u is 4.5 per cent of the working value, reachable by 0.17 kilojoules a mole or by fourteen kelvin. A shift of 1.384 is not reachable by anything: it would need the gauche energy at 7.2 kilojoules a mole or the reaction run at 157 kelvin. Quoting the three margins in one variable makes them comparable, which quoting them as a percentage of three different ceilings did not.

That is what a reduced variable buys beyond bookkeeping. It is not that the sweeps were redundant — that is the cost, and it is paid. It is that a verdict expressed in the model’s own coordinates carries its own sensitivity analysis, because the distance to the threshold is the answer to every question about how far a parameter would have to move.

What the identity does not cover #

It would be easy to read all this as saying the temperature sweep was wasted, and it is worth being precise about why that is wrong.

The comparison a verdict is made from has two sides. The ceiling is on one, and the identity above covers it completely. The measured rate ratio is on the other, and its temperature dependence is a function of temperature — through an activation-enthalpy difference between the substituted closure and the unsubstituted one — and of nothing else. It has no gauche energy in it at all.

So the temperature sweep’s contribution was the second half. Its finding that the crossing disappears entirely at an enthalpy difference of 5.308 kilojoules a mole is about a term the gauche sweep cannot touch, and at that value the measured ratio moves twenty-two per cent between 298 and 340 kelvin while the ceiling’s own movement is already accounted for elsewhere.

That splits the temperature calculation cleanly into a half that was redundant and a half that was new, and the redundant half is the one it led with. Its own headline — the verdict giving way at 312.1 kelvin — is the gauche sweep’s finding expressed in the wrong units.

What was computed, and how #

Nothing was recomputed. The ceiling is the same closed form, evaluated at matched pairs (g, 298) and (3.8, 298·3.8/g), and the two results are compared as double-precision numbers rather than to a tolerance.

The ranges are converted to u = g/RT with the same gas constant used throughout, and the union and overlap are interval arithmetic on two pairs of numbers.

Four results are checked numerically. The two routes agree to under a part in a million million — stated as a bound the arithmetic beats, since two floating-point computations need not be bitwise identical in general. The gauche range is the wider of the two. The union is narrower than the sum, which is the finding stated in a form that could fail: two genuinely independent sweeps would have a union equal to the sum.

And the one that keeps the argument honest: the measured side must move with temperature and not with the gauche energy, or the identity would collapse the whole comparison rather than half of it — and it does move, by twenty-two per cent across forty kelvin at the enthalpy difference the temperature sweep identified.

Where the model stops #

The identity is the ceiling’s, and the ceiling is one side of a comparison. Nothing here says a verdict is a function of u alone, because the other side is not.

And it is the model’s, not the chemistry’s. A real conformer population does not depend on temperature only through exp(−g/RT): there are entropic differences between conformers that a two-state Boltzmann factor with no degeneracy term does not carry, and there are more conformations than the three-state rotamer model counts. The identity is exact for the object computed here and approximate for the object it stands for.

The equivalent temperatures are arithmetic rather than proposals. 196 kelvin is a real number in this model and not a temperature at which anybody runs a lactonisation, and quoting it is a way of saying how large the energy’s uncertainty is rather than a suggestion about an experiment.

The generalisation #

The transferable point is that a model’s parameter count is not the same as its lever count, and the difference is usually invisible until somebody writes the expression out.

The rotamer ceiling has three inputs — a rotor count, a gauche energy and a temperature — and two of them are one. Nothing about the way the model is described says so: the two enter through different physical stories, they are measured by different experiments, they are quoted in different units, and each has its own literature. They collapse because of how they appear in one exponent, which is a property of the algebra rather than of the chemistry.

The diagnostic is cheap and it is the one used here: write the expression and count the distinct combinations the free symbols appear in. Not the symbols — the combinations. Anything appearing only inside one group is one lever with several names, and any sweep over the members of that group separately is one sweep run several times.

The consequence for reporting is specific. An argument that has swept two parameters and found a verdict stable under both has not shown it stable in two directions unless the two directions are different. Stating the sweep in the model’s own reduced variables — here u — makes the redundancy visible immediately and costs a line. The same shape appears in the scaling identities that collapse three sensitivities into one on a quite different problem, where two rescalings of a Schrödinger equation fix two of three logarithmic derivatives exactly.

There is a second, less comfortable half. Finding that two sweeps were one does not weaken either result; both remain correct, and the verdicts they reported are the verdicts. What it weakens is the confidence the pair of them was being used to support, and confidence assembled from redundant evidence is the kind that is hardest to notice is unearned — because nothing about it is wrong, only smaller than it looked.

Who found it, and when #

The reduced variable g/RT is as old as the Boltzmann factor and nothing about the observation is deep. The rotamer ceiling is an arrangement of a standard argument made for these essays, and the two sweeps are new.

What is worth recording is how the redundancy survived. Both sweeps were run deliberately, each with a stated motive and a named weakest step — and neither wrote the ceiling out and looked at how its symbols enter. The expression is one line, and the redundancy is visible in it at a glance: the gauche energy and the temperature never appear except as their ratio, so any two sweeps along them are the same sweep read at two scales.

Still open: how large the activation enthalpy really is #

The obvious open question is the activation enthalpy, which is now the only remaining input that is genuinely independent of the others and the one with no value yet. The temperature sweep asked how large it would have to be to remove the crossing and got 5.308 kilojoules a mole; nothing since has asked whether that is a plausible figure — the same shape of omission a rotor count that lived in two places and nowhere turned out to be for the difference between a gem-dimethyl closure and an unsubstituted one. It is the last free parameter in the comparison, and it is the only one that can move the measured side.

The nearer question is whether the same collapse has happened elsewhere. The diagnostic is a line of algebra per model, and a dozen models in these essays have three or four quoted inputs each — a ligand-field splitting whose two parameters come from two bands, a counterpoise correction, a susceptibility fit with four parameters and three determined directions. Any of them that has had two of its inputs swept separately is a candidate for the same finding, and the cost of checking is reading an expression rather than running anything.