A denominator needs three currencies

Three essays have now deferred the same question, each time in a sentence of about the same shape. The slope floor’s denominator is left untouched here; the run is computed by the predictor rather than measured, so it carries no experimental error and every price prices only the numerator; pricing it needs a different currency from the one used here, and inventing it is the work.

Inventing it produces three, which is why the deferral kept happening. The three predictors’ runs are three different kinds of object and only two of them can be priced at all.

What a floor is made of #

A slope floor comes from a near tie. Take two systems whose predicted values are close and whose measured values are not; any model of the form measurement = f(predictor) must have a derivative somewhere of at least the measured difference over the predicted one. The floor is that ratio. It is a bound on every model of the form, it involves no fitting, and it is the collection’s only instrument that returns a number rather than a verdict.

The rise is measured and every price so far has been a price on it: the standard error that would halve the floor by halving its rise, expressed as a fraction of the set’s own measured range so that three predictors in three units can be compared.

The run is computed. It carries no experimental error — which is the observation that made the deferral reasonable — and it does not follow that it carries no uncertainty. What it carries depends on what the computation is.

Three runs #

The spin-only moment’s run is arithmetic on a count. The predictor is $\sqrt{n(n+2)}$ where $n$ is a number of unpaired electrons, so every value it can take is one of five numbers and every difference is a difference of two of them. There is no parameter, no basis, no convention, no model form. The steepest pair is vanadium(III) against cobalt(II), whose predicted moments are 2.8284 and 3.8730, and that run is 1.0446 because $\sqrt{8}$ and $\sqrt{15}$ are what they are.

The Hückel eigenvalue’s run is exact given the connectivity and not given the form. The predictor is the highest occupied eigenvalue of a molecule’s adjacency matrix, which is a graph invariant: no parameter in Hückel theory moves it, since α and β are an affine map on the energy and the eigenvalue is dimensionless. But Hückel theory makes a structural approximation — it sets every overlap integral to zero — and restoring the overlap changes the eigenvalue. So the run has no parametric uncertainty and a real formal one.

The angle strain’s run is computed through choices. An angle strain is a force constant times a squared departure from a reference angle, and both the constant and the reference are decisions already found to matter here. Its steepest pair is cyclobutane against cyclohexane, a run of 32.5 kilojoules a mole out of a predicted range of 280.

So the deferred question has three answers of three kinds, and the reason nobody wrote them is that the second and third need sweeps of quite different things while the first needs no sweep at all.

The price that was quoted, and the one that was meant #

Before pricing anything there is a correction to make, and it is inside the very quantity this essay is about.

An earlier essay reported the largest floor here as a rise of 0.880 electronvolts over a run of 0.0308 — one per cent of that predictor’s range. One per cent is right for a different denominator. The predictor’s range is 0.4142 to 1.0000, a span of 0.5858, and 0.0308 is 5.26 per cent of it. One per cent is what the run is as a fraction of the measured range of 3.10 electronvolts, which is a perfectly sensible number to compute and is not what it was called.

The slip is a factor of five and it is in the denominator’s denominator. It matters because the sentence it appeared in was an argument: a run that is one per cent of its predictor’s range makes the floor look as though a tiny change in the model would move it enormously, which is the conclusion that sentence was reaching for. At 5.26 per cent the conclusion is weaker and still holds — which is the right outcome for a correction to have, and is why it is worth stating rather than quietly fixing.

The model price #

With the ranges straight, the second currency is definable. Halving a floor means doubling its run, so the model price is the change in the run required — expressed, like the measurement price, as a fraction of the predictor’s own range.

The Hückel floor needs 5.26 per cent of the predictor’s range against a measurement price of 10.04 per cent of the measured range. The angle-strain floor needs 11.61 per cent against 33.82 per cent.

Both are cheaper to move through their model than through their measurement, by factors of 1.91 and 2.91.

That is the finding and it needs a caveat in the same breath. The two prices are in different currencies and their ratio is not a physical quantity — a fraction of a measured range and a fraction of a predicted range are commensurable only because both are fractions. What the ratio says is narrow and it is enough: a change in the computed predictor of a size that nobody would call large moves the floor further than a measurement error of a size nobody would call small. A floor advertised as a bound on every model of a form, established without fitting, turns out to be more sensitive to the computation that produced its own abscissa than to the experiment that produced its ordinate.

Why the floors are so unequal to begin with #

The two prices are worth reading beside the floors themselves, because the floors differ by an order of magnitude and the prices do not.

The Hückel floor is 28.5 electronvolts per unit of eigenvalue, the angle strain’s is 3.38 kilojoules a mole per kilojoule a mole, and the spin-only moment’s is 1.96 Bohr magnetons per Bohr magneton. The last two are dimensionless and near one; the first is large and carries a unit. Nothing follows from the comparison, which is the first thing to say — a floor’s size is set by whatever units its two axes happen to be in, and only a floor divided by the predictor’s overall slope is comparable between predictors at all.

Divided that way the three are 8.04, 14.04 and 1.93. So a Hückel model must be eight times steeper somewhere than the set is overall, an angle-strain model fourteen times, and a spin-only model twice. The angle strain — the control, with no exact ties — demands the most extreme non-linearity of the three, which is the statement the continuous instrument was built to make and is where its value lies.

And that ratio is where the denominator’s arbitrariness lands hardest. Both the numerator and the denominator of floor divided by overall slope contain the predictor’s own differences: the floor divides by one run and the slope is fitted across all of them. So a change in how the predictor is computed moves both, and whether it moves them together is not something inspection can answer. It is the question still to be asked of a family of models rather than of a single change, and the two prices above cannot settle it because each of them varies one run while holding the rest fixed — which is exactly what a change of model form does not do.

The one that cannot be moved #

The third floor has no model price, and reporting that correctly is the refusal this essay turns on.

A tempting way to write it is that the spin-only run is infinitely expensive to move. That is a different claim and it is false. An infinite price says the quantity could move if somebody paid enough; this one says there is nothing to pay for. $\sqrt{8}$ and $\sqrt{15}$ are not the outputs of a calculation with assumptions in it. The predictor reads an integer and returns a square root, and the only way to change its run is to change the integer — which means claiming a different number of unpaired electrons, which is not a reparameterisation but a different chemical assignment.

So the spin-only floor is the one floor here that was priced correctly by accident: its whole uncertainty is in its measurement, because its run is arithmetic.

It is worth noticing how narrow that exemption is. The predictor is exact because it reads an integer, and it reads an integer because somebody decided how many unpaired electrons each ion has — which for four of these nine ions is the assignment the measured moment then contradicts. The exactness is real and it sits one step downstream of a judgement, so nothing moves this run is true of the arithmetic and not of the chain of reasoning the arithmetic sits in. A predictor can have an exact denominator and an arguable input, and this one does. And it is the floor whose measurement price is the largest of the three, at 17.38 per cent, which makes it the most robust of the three floors on both readings at once. The best-behaved predictor in the collection is again the one about which the least can be said.

What was computed, and how #

The rises, runs and floors are the earlier essays’, recomputed from the same quoted measurements: nine magnetic moments, six ionisation energies, six ring strains. The measurement price is unchanged from those essays — a one-sigma statement, a measured difference destroyed at about $\Delta/\sqrt{2}$, as a fraction of the measured range. The model price is the run divided by the predictor’s own range, since doubling a run is a change of one run.

Nine things are checked: that every numeric predictor has a floor to price; that the three runs are three distinct kinds; that exactly one of them is unsweepable and that it is the integer case, named with the reason; that the Hückel run is one per cent of the measured range, which is the number that was reported; that it is 5.26 per cent of the predictor’s range, which is what it called it; that the two readings differ by more than a factor of four, so the correction is a correction rather than a rounding; that the angle-strain floor is several times cheaper to move through its model; and that the spin-only floor has no model price at all, reported as absent.

Recording both readings of the run is deliberate. A correction that replaced one number with another would leave a reader unable to tell which of the two had been computed; stating both says that it computed a true thing and named it wrongly, which is a different and more useful report.

Where this stops #

The model price is a crude instrument and the crudeness is in one place. It asks what change in the run halves the floor and normalises by the predictor’s range, which treats every part of that range as equally available. For the angle strain that is roughly fair; for the Hückel eigenvalue it is not, because the eigenvalue is a graph invariant and no change of parameter moves it. Its 5.26 per cent is therefore a statement about what would have to happen rather than about a knob anybody can turn, and what is still owed is a statement of which family of models supplies the turning.

The two currencies are not converted and cannot be. An experimental standard error and a change in a computed quantity are not the same kind of thing, and a ratio of two fractions is a comparison of two normalisations rather than of two uncertainties. Anybody wanting a single number would have to say how much they believe the model, which is exactly the judgement the whole tie-based instrument was built to avoid needing.

And the control is the predictor most affected. The angle strain was included as a control — a predictor with no exact ties, so that the exact instrument would have something to be silent about — and it turns out to be the one whose floor is most sensitive to its own computation. A control chosen to have no ties was not chosen to have a robust denominator, and nothing had asked whether it did.

The generalisation #

The habit is to ask what a quantity is made of before asking how uncertain it is.

Every deferral of this question rested on a true observation — a computed number carries no experimental error — and treated it as though it settled the matter. It does not, and the reason is that computed covers at least three situations that behave differently: a quantity fixed by arithmetic, a quantity fixed by a structure with a modelling choice sitting beside it, and a quantity produced by a calculation with conventions in it. The first has no uncertainty of any kind. The third has as much as the conventions do. Reading all three as no experimental error, therefore nothing to price loses the distinction that matters.

The corollary is about ratios whose two parts have different provenance. A slope, a rate, an efficiency, a per-unit cost: each divides something measured by something derived, and the error analysis conventionally attends to the measured part because that is the part with a published uncertainty. The check that costs nothing is to ask which part of the ratio a plausible change moves further — and here, on two of three predictors, it is the part with no published uncertainty at all.

Who found it, and when #

The propagation of error through a ratio is textbook. Slater’s rules are 1930 and the spin-only formula is older. The measurements are all quoted and all already in this collection. What this essay computes is the classification of the three runs, the model price of the two that admit one, the ratio of the two prices, and the correction to the range that was divided by.

The number worth carrying is not 2.91. It is 5.26 against one: the two readings of one run against two different ranges, in the sentence that argued the run was the load-bearing quantity.

Still open: what family supplies the turning #

The obvious open question is the one the Hückel case leaves in an awkward place. Its 5.26 per cent is a real answer to how much would the run have to change and no answer at all to what would change it, because a graph eigenvalue is not a parameter. What moves it is the form: Hückel theory sets every overlap integral to zero, and restoring a uniform overlap s turns the problem into a generalised one whose eigenvalues are x/(1 + sx), strictly increasing in x and equal to x at s = 0. Every member of that family is a defensible calculation and they disagree about differences. Sweeping it costs nothing and would replace a hypothetical change with a family of real ones.

The nearer question is what that family does to the other instruments, and it is the one worth asking first because the answer may be free. An increasing map of a predictor preserves which pairs are equal and which differences have the same sign, so the exact ties and the discordances ought to be untouched by every member of every such family — while a ratio of differences cannot be. If that holds, the collection’s two instruments have completely different standing: one is a statement about molecules and the other is a statement about molecules on a particular scale, and nothing in three essays of pricing has said so.