Counting was right except where it mattered

A molecule’s bonding description can be called ambiguous by counting how many distinct answers a localisation search finds from random starts — the measure used since four centres and a pair that will not localise. A nine-vertex cage gives dozens of them — a survey that began when three shapes came out of one search and then found that they were all, to the precision that matters, the same answer — a functional spread of two parts in a hundred thousand across the whole population.

Which raised a question about the measure itself. If a population of forty-four descriptions can be one answer, then a count of descriptions is counting routes rather than destinations, and if that is common across the family the whole use of the count needs restating.

The family is forty-eight cage-and-filling pairs and it had never been looked at this way.

Most of the family has nothing to count #

Thirty-seven of the forty-eight are degenerate by the original criterion — a relative spread under 10⁻⁴. That number sounds like the worry confirmed and it is not, because thirty-five of the thirty-seven find exactly one description. A population of one is degenerate trivially; there is nothing for it to disagree with.

The cases that could tell the measure apart are the thirteen that find more than one, and eleven of those thirteen have descriptions that genuinely differ — spreads from 4.5 × 10⁻³ to 5.3 × 10⁻². So on the family as a whole, a count above one really does mean ambiguity, and the worry does not generalise.

It is worth saying what “one description” means, because thirty-five of forty-eight is a lot of them. A localisation search that finds one basin from two hundred random starts has a functional with a single maximum on the rotation group — every start rolls to the same place. That is the ordinary case, and it is what a chemist means when they say a molecule has a bonding description: not that only one is imaginable, but that the criterion picks one out.

So the family is mostly unambiguous, which is itself worth recording. Five calculations have been spent on cages that are not, and a reader could be forgiven for thinking ambiguity is the normal state of an electron-deficient cage. It is not: it is a quarter of them.

Except at the top #

The two exceptions are the interesting half, and one of them is the case the argument was built on.

The nine-vertex cage at eight electrons finds forty-four descriptions — three times the next case — with a spread of 2.2 × 10⁻⁵. The eleven-vertex cage at twenty electrons finds two, with a spread of 5.3 × 10⁻², two thousand times larger.

By count the first is the most ambiguous system in the family and by a wide margin. By spread it is not ambiguous at all, and the honour goes to a case that finds two answers.

So the measure is not merely imperfect at the extreme; it is inverted there. The one case where counting and spreading disagree most is the one chosen as the headline, and chosen precisely because the count was extreme.

That is the shape of a selection effect. An argument that picks its cases by a measure and then discovers the measure fails on them has not been unlucky — it has selected for the failure, because the cases where a measure is most extreme are where it is most likely to be measuring something else.

One more thing follows and it is about how the cases were chosen at all. The three came from an earlier survey of the whole family and were picked as the extremes of the count — most descriptions, fewest, and one in between. That is a reasonable way to choose three cases from forty-eight and it is exactly the way that puts the measure’s failure in the sample.

The alternative would have been to pick by spread, which was not computed then, or at random, which would have given three cases with one description each and nothing to say. There was no better choice available; what there was, and what is available now, is the check.

Two is enough to be ambiguous #

There is a smaller finding in the thirteen and it cuts against an intuition.

Ten of the thirteen find exactly two descriptions, and those ten include the largest spread in the family. Two descriptions that differ by five per cent of the functional are more ambiguity, in any sense a chemist would recognise, than forty-four that differ by two thousandths of a per cent — because the first is a real choice between two accounts of the bonding and the second is one account reached by many roads.

So the count does not merely mis-rank the extreme; it has the wrong units. A count is a property of the search’s landscape — how many basins there are, which depends on how the functional’s surface is shaped — and the spread is a property of the answers. Those are different objects and the first has been quoted while the second was meant.

What should now be quoted #

The practical output is a change to how a case is described, and it is small.

A case with one description should be reported as having one. A case with more should be reported with both numbers — how many and how far apart — because the pair is what distinguishes real ambiguity from a rugged landscape, and neither number alone does.

For the three headline cases that reads: the twelve-vertex cage at twelve electrons finds fourteen descriptions spread by 3.9 × 10⁻², which is genuinely ambiguous; the twelve-vertex cage at twenty finds three spread by 1.7 × 10⁻⁶, which is one answer; and the nine-vertex cage at eight finds forty-four spread by 2.2 × 10⁻⁵, which is also one answer and is the case quoted most.

None of those three readings is new. What is new is knowing that the pattern across them is not a coincidence of three cases but the family’s, and which of the two numbers to lead with.

What was computed, and how #

Every deltahedron at six, nine, ten, eleven and twelve vertices, at every even filling from two electrons to twice the vertex count — forty-eight pairs, every one of them available. For each, two hundred random starts of the localisation, the distinct basins they reach, the functional at each, and the spread between best and worst as a fraction of the best.

Six things are checked: that the survey covers at least twenty pairs; that every attempted pair is available, so this is a survey of the family rather than of what survived; that all three headline cases are in it; that a large part of the family is degenerate; that there exist cases with many descriptions that are all one answer; and that the two rankings disagree at the top, which is the finding.

Why forty-four descriptions and one answer #

The fact is established and not explained, and the family survey adds a constraint on any explanation.

Forty-four is not a small number of basins and 2.2 × 10⁻⁵ is not a small agreement — these are forty-four genuinely distinct sets of localised orbitals reaching the same functional value to five figures. Whatever relates them has to relate forty-four things, and the only thing on a nine-vertex cage that relates many things is its symmetry group.

What the survey adds is that this is the only case in the family where it happens on any scale: the twelve-vertex cage at twenty electrons has three descriptions agreeing to 1.7 × 10⁻⁶, and nothing else has more than two that agree. So whatever mechanism produces forty-four equivalent answers is not a general feature of these cages at these fillings — it is specific to one of the forty-eight, and a general explanation would have to say why.

That is the shape any explanation has to work in, and it is a narrower target than three cases could show.

Where the model stops #

Two hundred starts is fewer than the four hundred used for the headline cases, and a search that misses a basin under-counts. That biases the count downward and does not bias the spread, since a missed basin is usually a rare one and a rare basin is usually near an already-found one — but the direction of the bias is worth stating and it is the direction that would make the count look better than it is, not worse.

The localisation functional is a Boys-type criterion on a Hückel wavefunction, which is the model used throughout. A different functional would find a different landscape, and nothing here says the number of basins is a property of the molecule rather than of the criterion — the caution one scale from two centres to a cage entered when the measure was first put on a common footing. The spread is closer to being one, since it is a difference of functional values, but it is a difference in the same criterion’s units.

And “degenerate” is a threshold at 10⁻⁴, taken from the headline case, where it was chosen for one molecule rather than for a family. Carrying a one-case convention across forty-eight is the kind of move that deserves checking, and the next section checks it.

What a spread of five per cent means #

The largest spread in the family is 5.3 × 10⁻² on an eleven-vertex cage at twenty electrons, and it is worth asking whether that is large.

The localisation functional is a sum of squared orbital centroids, so a five per cent difference between two maxima means two sets of localised orbitals whose centroids sit measurably differently on the cage — one description putting more weight on some faces and less on others. A chemist looking at the two would draw different bonding pictures: a different set of three-centre bonds, or a different assignment of which vertices share a pair.

Two parts in a hundred thousand, by contrast, is below anything a picture distinguishes. The forty-four descriptions of the nine-vertex cage would all be drawn the same way.

So the threshold at 10⁻⁴ is not arbitrary in kind even if it is in value: it separates differences a reader would see from differences they would not, and that is the right thing for a measure of ambiguity to separate. Whether its exact value matters is a separate question, and it has an answer.

The threshold turns out not to be a choice #

A line drawn at 10⁻⁴ across a continuous quantity invites the obvious objection: move the line and the classification moves with it. If that were true here, every count of degenerate cases in this essay would have to be quoted with its threshold, and the eleven-of-thirteen headline would be a statement about a convention rather than about a family of cages.

It is not true, and the reason is that the quantity is not continuous in the region that matters.

Thirty-five of the forty-eight pairs have a relative spread of exactly zero — one description, or several that the functional cannot separate at all. Of the thirteen with a non-zero spread, two fall below the threshold, at 1.68 × 10⁻⁶ and 2.18 × 10⁻⁵, and eleven fall above it, the lowest at 4.47 × 10⁻³.

Between 2.18 × 10⁻⁵ and 4.47 × 10⁻³ there is nothing. That is a factor of two hundred and five, better than two decades, and the threshold sits inside it. Any value between those two numbers classifies all forty-eight pairs identically, so the line could have been placed a hundred times lower or twenty times higher without moving a single case between the columns.

This is a stronger result than the essay needed and it was not designed for. The threshold was inherited from a single molecule and applied to a family on the assumption that it would roughly work; what the distribution says is that the family sorts itself, and the inherited value happened to land in the gap the family had already made. A different starting molecule would almost certainly have produced a different number and the same classification.

It also disposes of a piece of hedging in the section above. The right thing to say about the eleven-of-thirteen figure is not that it holds at this threshold but that it holds at any threshold in two decades, which is what a robust classification looks like — and which is worth two lines of arithmetic to establish rather than a caveat to carry.

What the gap does not establish is that a different functional would leave one. The spreads are differences in one criterion’s values, and the bimodality is a fact about that criterion on these cages. A criterion that separated descriptions more finely could fill the gap in, and then the threshold would start to matter in exactly the way this section has just ruled out.

The generalisation #

The transferable point is about measures chosen for their extremes.

An argument that ranks cases by a measure and then studies the top of the ranking is doing two things at once: testing a claim about the case, and testing the measure. If the measure has a failure mode, the ranked-first case is where it is most likely to appear — because the ranking is sorting on the measure and a failure inflates it.

So the diagnostic is not “does the measure work” but “does it work at the extreme it was selected on”, and the two questions have different answers here: eleven of thirteen, and no. That is not a reason to abandon the measure. It is a reason to check it separately at the case one has picked, which costs one extra quantity and would have changed how the headline was reported.

The second point is smaller: a count and a spread are different kinds of object, and it is easy to substitute one for the other when the first is cheaper to get. Counting basins comes free from a search that has to be run anyway; computing the spread needs the functional values kept, which is one more line. That asymmetry in cost is why the substitution happens, and it is worth resisting in exactly the cases where it is most tempting. The whole cost here was keeping the functional values a search already computed and discarded, which is about the cheapest correction a measure can have.

Who found it, and when #

Localisation of a Hückel wavefunction, the multiplicity of basins in a Boys criterion, and the ambiguity of bonding descriptions in electron-deficient cages are all older than these essays. The survey, the spreads, the two rankings and everything above are new arithmetic, done to answer a question about a measure.

Still open: symmetry, and a second criterion #

The obvious open question is the symmetry, which nothing here has touched. A degenerate population on a symmetric cage is what a symmetry operation permutes, and the number of members should then be a group order or a divisor of one — forty-four is not an obvious such number for a nine-vertex deltahedron, so either the degeneracy is not symmetry or the search is splitting what a symmetry relates. Applying the cage’s own operations to a description and asking whether the image is in the population would settle it, and the operations are already computed elsewhere.

The nearer question is the one the threshold section opened rather than closed. The gap is a fact about a Boys-type criterion on a Hückel wavefunction, and the whole argument that the classification is robust rests on it. Running the same forty-eight pairs through a second localisation criterion — an Edmiston–Ruedenberg or a Pipek–Mezey functional — and asking whether its spreads are bimodal too would say whether the gap belongs to the cages or to the measure. If it survives, “degenerate” is a property of electron-deficient cages; if the second criterion’s spreads fill the two decades in, then the robustness here is the first criterion’s coarseness, and every count of degenerate descriptions needs its functional named beside it.