A floor on models written in one scale

A denominator was priced and left one case in an awkward place. The Hückel eigenvalue’s run is a graph invariant, so no parameter of Hückel theory moves it, and how much would it have to change had an answer — 5.26 per cent of the predictor’s range — while what would change it had none. What moves a graph eigenvalue is the form of the model, and the form has a family.

Running that family turns out to answer a much larger question than the one it was built for.

Two questions, and only one of them has a scale in it #

Two instruments here are built out of ties, and the difference between them has never been stated in the form that matters.

The exact instrument asks whether two predicted values are equal. Where they are, the prediction becomes a claim with no fitting anywhere in it: whatever f is, two systems with the same predictor get the same answer, so the spread inside a tie is a floor on the error of every model of the form. The continuous instrument extends it two ways. A discordant pair asks whether two differences have the same sign — if the predictor and the measurement order two systems oppositely, no monotone f fits both. A slope floor asks for a ratio of differences — the measured rise over the predicted run, which every model of the form must reach somewhere as a derivative.

Write those three out as questions about the predicted values and the difference is immediate. Are these two equal? and do these two differences have the same sign? are questions about the order of the predicted values. What is the ratio of this measured difference to this predicted one? is a question about their spacing.

A strictly increasing function of the predictor preserves order exactly and preserves no spacing at all. So the first two instruments are invariant under every such change of scale and the third cannot be, and neither of those is a matter of degree.

The family that is a model statement #

An argument from the shape of a question is worth exactly nothing until it is run on something a chemist would defend, so the first family is not a stress test.

Hückel theory sets every overlap integral between atomic orbitals to zero. That is not a parameter choice; it is a structural approximation made to turn a generalised eigenvalue problem into an ordinary one. Restoring a uniform overlap s between bonded atoms puts it back: the problem becomes (αI + βA)c = ε(I + sA)c, and for an eigenvector of the adjacency matrix with eigenvalue x the solution is ε = (α + βx)/(1 + sx). The predictor — the part that is not an affine map on the energy — becomes x/(1 + sx).

That is strictly increasing in x for every s ≥ 0, and s = 0 is the standard theory. Every member of the family is a calculation somebody could publish.

The discordance is one pair at every overlap and the exact ties are two at every overlap. Hexatriene and naphthalene remain discordant; ethene and benzene remain tied, as do butadiene and naphthalene, because x/(1 + sx) is injective and maps equal values to equal values.

And the floor rises from 28.55 to 36.38, twenty-seven per cent across a range of overlap nobody would argue with. The run shrinks from 0.0308 to 0.0242 while the rise stays at 0.88, because the map compresses the large-x end of the predictor harder than the small-x end and the steepest pair sits at the small end.

So a family of models that agree on every order-based verdict disagree by twenty-seven per cent about the number quoted as the largest floor in it.

The family that is a sensitivity test, and is worse #

The overlap family is narrow by construction: it has one parameter with a physical meaning and a range physical meaning restricts. The second family makes no such claim and is labelled accordingly.

(x^λ − 1)/λ is strictly increasing for every λ, equals x − 1 at λ = 1, and tends to log x as λ approaches zero. It is the standard way to ask how much of a conclusion is the scale a quantity was written on. No member is offered as a better model of anything.

The Hückel floor runs from 15.14 at λ = 0.25 to 154.58 at λ = 3. The spin-only floor runs from 4.83 to 0.25. The angle-strain floor runs from 55.86 to 0.005.

Factors of 10.2, 19.1 and 11,746. Four orders of magnitude on one predictor, from a family every member of which makes identical predictions about which pairs are tied and which are discordant.

The angle strain’s four orders of magnitude have an arithmetic cause worth naming, because it is the extreme case of the general one. Its predicted values run from 0.413 to 280.5 — a factor of nearly seven hundred, the widest dynamic range of the three predictors — and a power map’s effect on a difference grows with the ratio of the range’s ends rather than with its span. A predictor whose values span a factor of seven hundred is therefore reparameterised far more violently than one whose values span a factor of two and a half, which is the Hückel eigenvalue’s case.

The angle strain is the largest and it is the control — included precisely because it has no exact ties, so that the exact instrument would have something to be silent about. It has the least robust floor of the three and nothing had asked whether it did.

Which direction a floor moves, and why it is not the same direction #

The three floors do not move together, and reading the directions is where the mechanism becomes visible rather than merely demonstrated.

Under the power family, raising λ compresses the low end of a predictor’s range relative to the high end and expands the high end relative to the low. So a floor rises with λ if its steepest pair sits low in the range and falls if the pair sits high. The Hückel steepest pair — hexatriene and anthracene — sits at the bottom of the eigenvalue range, at 0.414 and 0.445 out of 0.414 to 1.000, so its run compresses as λ rises and its floor climbs from 15 to 155. The spin-only steepest pair sits in the middle and the angle strain’s low pair sits at the top of a range running to 280, so both floors fall.

So the direction of the movement is set by where in its own range a predictor’s near tie happens to be, which is a property of the molecules in the set rather than of the predictor or the reparameterisation. Nothing about a model’s form predicts it, and two collections studying the same predictor on different molecules would find the floor moving in opposite directions under the same family.

The overlap family moves the Hückel floor upward for the same reason and for a reason the power family does not have: x/(1 + sx) compresses everywhere, hardest at large x, so every run shrinks and the runs at small x shrink least. The floor therefore rises on every predictor such a family is applied to, which makes it the tamer of the two families in direction as well as in size — and makes its twenty-seven per cent a one-sided statement. Restoring the overlap can only make the published Hückel floor an understatement.

The pair that is worst is not always the same pair #

There is a second consequence and it reaches further than the floor’s value.

A floor’s price — what was computed earlier, the measurement error that would halve it — is set by the rise of whichever pair is steepest. On the Hückel eigenvalue that pair is hexatriene and anthracene on every one of the thirteen scales, so the price is invariant while the value is not, which is the confusing case. On the spin-only moment it is vanadium(III) and cobalt(II) on some scales and titanium(III) and nickel(II) on others. On the angle strain it takes three different identities: cyclobutane against cyclohexane, cyclopropane against cyclooctane, and cyclopentane against cyclohexane.

So on two of three predictors, the quoted fragility of a slope floor moves as well as its value — and it moves discontinuously, because the identity of the steepest pair jumps. The whole fragility ranking of the previous three essays contains three slope-floor claims, and two of the three have a price that depends on which scale their predictor was written on.

The discordance prices do not. A discordance’s price is the error that reverses one pair, that pair is the cheapest discordant pair, and which pairs are discordant is invariant — so the discordance claims in that ranking are scale-free in both their existence and their price. The ranking mixes two kinds of claim and only now is it clear that they are two kinds.

What was computed, and how #

Thirteen scales per predictor: the unparameterised one, six power exponents, and — for the Hückel eigenvalue alone, since it is the only predictor the family belongs to — seven overlaps. Under each, the whole audit is recomputed: the ties by the predictor’s own tolerance, the unexplained share, every pair’s ratio of differences, the steepest pair, the discordances.

Twelve things are checked. Per predictor: that its exact ties are ties on every scale; that its discordant pairs are discordant on every scale; that the share of the variation its ties leave unexplained is identical to the last bit rather than to a tolerance; and that its slope floor moves by more than half again. Then that every order-based instrument in the collection is invariant, as one statement. Then that the overlap family runs on the predictor it belongs to, that every member of it agrees about the discordance, and that they disagree about the floor.

The refusal is the identity member. The power family must contain a map that is the identity up to an affine shift, and its λ = 1 member must reproduce the unparameterised floor, ties and discordances exactly. A family whose identity member disagreed with the base case would be measuring its own arithmetic, and an invariance demonstrated by such a family would be worth nothing.

Where this stops #

None of this touches the exact instrument’s standing, and that is the cheerful half. A tie bounds every model of a form with no fitting and the bound does not depend on how the predictor is scaled, which is now a checked statement rather than an assumed one. The same is true of a discordance refusing monotonicity. Those are the claims leaned on hardest and they are the ones that survive.

The slope floor is not worthless and the finding is not that it is. It is a correct statement about models written in the scale it was computed in, and a model is written in a scale — somebody choosing to fit ionisation energy against a bare Hückel eigenvalue has made that choice, and the floor tells them what their form costs. What it does not do is bound every model of a class, because the class was never closed under reparameterisation and nobody said so.

And the two families do different work. The overlap family is a claim about chemistry and its 27 per cent is the number to carry; the power family is a sensitivity test and its factors of thousands say how much room there is rather than how much room a chemist would use. Quoting the second as though it were the first would overstate this by two orders of magnitude, which is why they are drawn separately and labelled.

What is genuinely unresolved is whether any scale is privileged. Hückel’s x is the eigenvalue of an adjacency matrix and is natural in a way (x^1.5 − 1)/1.5 is not. Whether that naturalness has any evidential force — whether a bound computed on the natural scale deserves more weight than one computed on an arbitrary one — is a question about what a model is for, and this argument cannot settle it. What it can say is that the naturalness is doing work that was never declared.

The generalisation #

The habit is to ask what group a claim is invariant under, and then to check that the claim was not quietly widened to a larger one.

Every claim in this collection’s tie-based argument is a claim about a set of pairs of numbers. Some of those claims are functions of the order of the numbers and some are functions of their spacing, and the two are invariant under different groups: the order-based ones under every increasing reparameterisation and the spacing-based ones only under affine maps. Nothing in the presentation distinguished them, because both were computed from the same table by the same code and reported in the same list.

The test is cheap and should be automatic. Apply an increasing map with a free parameter, recompute everything, and see what moves. What does not move is invariant under a group larger than anybody claimed; what moves was resting on a choice nobody recorded. Running it with an identity member is the part that makes the answer mean something, and running it with two families — one a modelling claim and one a stress test — is what keeps the answer from being overstated.

The corollary is about the appeal of a number. A verdict — these two are tied, these two are discordant — feels weaker than a bound with a value on it, and the value is what gets quoted. Here the verdicts are exact and scale-free and the value is neither, which is the opposite of the intuition, and the reason is simply that a stronger-sounding claim is a claim about more.

Who found it, and when #

Hückel theory is 1931 and the overlap-retaining generalisation is standard; the Wolfsberg–Helmholz era made it routine. The power family is Box and Cox, 1964, in a different subject for a related purpose. The tie and near-tie instruments were built over the preceding essays. What is computed here is the whole audit of three predictors under thirteen scales, and the separation of its outputs into the ones that are functions of order and the ones that are functions of spacing.

The number worth carrying is not 11,746. It is zero: the number of order-based verdicts that moved, across thirty-nine recomputations of an audit whose headline number moved by four orders of magnitude in the same runs.

Still open: what scale a slope floor should be quoted on #

The obvious open question is whether the choice can be made rather than inherited. A slope floor exists on every scale and the collection quotes the one its predictor happened to be computed in; a defensible alternative is to quote the smallest floor over a declared family, which is the weakest and therefore safest bound the family permits. On the Hückel eigenvalue that is 15.14 at λ = 0.25 against a quoted 28.55, so the safe bound is about half the published one. Whether a bound defined as a minimum over a family is a useful object — it is no longer a statement about any one model — is the question, and it is a question about what the bound is for rather than about arithmetic.

The nearer question is what the invariance buys where it holds. If a discordance is exactly scale-free, then a discordance found on one scale is a discordance on every scale, and the search for discordant pairs need never be repeated when a predictor is recomputed — which is worth knowing because this collection recomputes predictors often. The stronger version is a closure statement: the set of increasing maps is a group, so the order-based verdicts are functions on the quotient of predictors by that group, and the useful object is not a predictor at all but the permutation it induces on the measured values. Whether every order-based instrument in this collection can be rewritten as a statement about that permutation — and whether any of them then becomes cheaper to compute — is a question nobody has asked of instruments that have been running for six essays.