The cage is on both sides

A second localisation criterion, run over forty-eight cage-and-filling pairs, leaves a bimodal distribution of spreads just as the first does, and the awkward half of the result is this: the two disagree about six of the forty-eight, in both directions.

That leaves two questions, both about a set of cases once counted the wrong way. What the six are made of — “the descriptions differ in exactly the information one criterion uses and not the other”, which is a reading rather than a measurement. And whether they have a symmetry explanation, since two descriptions with identical orbital centroids and different atomic populations are the sort of thing a permutation of vertices would relate.

The first is exactly right. The second is wrong, and the case that refutes it is the most symmetric cage in the family.

Counting the symmetries #

The relevant group is not a point group. A Hückel localisation knows the adjacency and nothing else — no coordinates enter it — so the operations that could relate two descriptions are the automorphisms of the graph: permutations of the vertices that preserve every edge.

Those are counted here rather than quoted, by backtracking over partial maps with a degree refinement, and the counter is checked on two cages whose answers are known: an octahedron’s graph has 48 automorphisms and an icosahedron’s 120. A counter that was wrong would be wrong on those before anything else.

The family’s five cages give 48, 12, 16, 4 and 120 automorphisms at six, nine, ten, eleven and twelve vertices. The eleven-vertex deltahedron is the least symmetric object in the set by a wide margin — its graph is not regular, and only four permutations preserve it.

The prediction, and what happens to it #

If the disagreements were the cages with room for a permutation to act, they would concentrate at the top of that list. They do not concentrate anywhere.

The icosahedron, with 120 automorphisms, supplies three disagreements out of twelve fillings. The octahedron, with 48, supplies none out of six. The eleven-vertex cage, with four, supplies one out of eleven. The ten-vertex cage, with sixteen, supplies two out of ten.

There is no ordering of the cages by symmetry that puts the disagreeing ones on one side, because the same cage is on both sides. The icosahedron agrees at nine of its twelve fillings and disagrees at three, with the same graph, the same 120 automorphisms and the same edges throughout.

That is the refutation and it needs no threshold. An explanation appealing to a property of the cage predicts that a cage disagrees at every filling or at none. Three of five cages disagree at some fillings and agree at others, so no property of the cage can be what decides it.

What the six actually are #

The other half of that reading survives, and it survives in a stronger form than it was stated.

For the ten-vertex cage at six and at eight electrons, the Pipek–Mezey spread is exactly zero — the search finds one description, every start reaching the same set of participation numbers — and the Boys spread is 7.17 × 10⁻⁴. Identical populations, different centroids.

For the eleven-vertex cage at twenty electrons and the icosahedron at fourteen, sixteen and eighteen, it runs the other way. The Boys spread is 4 × 10⁻¹⁴ or exactly nothing, and the Pipek–Mezey spread is 3.0 × 10⁻² to 5.3 × 10⁻². Identical centroids, different populations.

So in all six, one criterion’s spread is not merely smaller — it is numerically zero, at the last digit the arithmetic carries. There is no case in the family where both criteria see a spread and rank it differently, which is what “half-agree” would look like.

Plotted against each other the family is not a cloud with a diagonal band through it. Thirty-five pairs sit at the origin, seven in the far corner, and the six sit on the axes with one coordinate at the floor.

What the icosahedron’s two descriptions look like #

The clearest of the six is worth writing out, because the anatomy is legible in the numbers.

At fourteen electrons the icosahedron’s Pipek–Mezey search finds two descriptions, and their functionals are 4.52940757 and 4.39165731 — three per cent apart, which is not a marginal difference by any measure used here. Their participation numbers are three orbitals on 1.939 centres and six on 2.012 in one, against three on 1.855 and six on 2.163 in the other.

Boys finds one description, to fourteen decimal places.

So the seven electron pairs sit at the same seven places in space under both descriptions, and how much of each pair each atom is credited with differs by about a tenth of a centre. That is a real chemical distinction — it is the difference between a bond that is slightly more two-centre and one that is slightly more three-centre, which is the scale one measurement of how many centres a pair holds together was built to read — and a criterion that scores by where the pairs sit cannot see it at all.

The reverse case is the same statement read backwards: the ten-vertex cage’s three pairs are shared identically among the atoms in every description, and sit in visibly different places.

What decides it, which is not yet known #

Having refuted the symmetry account it is worth being explicit that nothing replaces it.

The filling decides, not the cage. The icosahedron disagrees at 14, 16 and 18 electrons and agrees at 2, 4, 6, 8, 10, 12, 20, 22 and 24, which is a run of three consecutive even fillings in the middle of the range — the region where a cage’s skeletal electron count is what it is — and the three disagreeing fillings give the same two spreads to three figures, 3.04 × 10⁻² and 4 × 10⁻¹⁴, which says they are one phenomenon rather than three.

That pattern is what a degeneracy in the occupied space would produce: a run of fillings over which the same set of orbitals is being mixed, because the levels entering and leaving are degenerate and the localisation is free in the same subspace throughout, which is the freedom an occupied space’s unitary invariance leaves open. Every level of every cage is already computed and the check is one comparison, but it is a separate question — the point here is that the explanation to look for is one about which orbitals are occupied, and the symmetry guess was about which cage they are on.

The runs are the pattern, on both cages #

The consecutive-filling structure noted above is not the icosahedron’s alone, and that is what turns an observation into a pattern.

The ten-vertex cage disagrees at six and at eight electrons — adjacent even fillings — and its two spreads are 0.00 and 7.17 × 10⁻⁴ at both, identical to three figures. The icosahedron disagrees at fourteen, sixteen and eighteen, and its two spreads are 3.04 × 10⁻² and 4 × 10⁻¹⁴ at all three, again identical. The eleven-vertex cage disagrees at one filling only, and it is the least symmetric cage in the family.

So the two cages with more than one disagreement each produce a contiguous run, and within a run the numbers do not change at all. Two things follow.

First, the six disagreements are not six independent events. They are three — a run of two, a run of three, and a singleton — which makes the sample considerably thinner than a count of six suggests, and any statistic computed over the six as though they were independent is overstating what it has.

Second, a run of fillings over which a quantity is exactly constant is a strong hint about mechanism. Adding two electrons to a cage changes which orbitals are occupied and normally changes everything downstream; a quantity that does not move across two such additions is one that depends on a part of the occupied space the additions are not touching. That is what a partly occupied degenerate set looks like: while the extra pairs are going into levels that are degenerate with each other, the freedom the localisation has is the same freedom, and the description’s ambiguity is unchanged.

That is not verified here, and it would be easy to — the levels are already computed for every cage. What can be said is that the shape of the evidence points at the occupied orbitals rather than at the cage, which is the opposite direction from the one the symmetry guess looked in, and that the right next measurement is a list of degeneracies rather than a list of symmetries.

There is a caution in the same observation. A run of identical numbers is also what a bug produces — a filling parameter not being passed through, say, so that three calculations are one calculation reported three times. That reading is available and is ruled out by the surrounding fillings: the icosahedron’s twelve, twenty and twenty-two electron cases give different spreads under both criteria, so the filling is reaching the calculation and the constancy is a property of the three in the middle.

The surrounding fillings are worth a sentence in their own right, because they show the run is a plateau rather than a step. The icosahedron’s spreads are zero under both criteria at two, four, six, eight and ten electrons; they rise to 6.19 × 10⁻³ and 3.08 × 10⁻⁶ at twelve, which the two criteria still agree about; they sit at the run’s values through fourteen, sixteen and eighteen; they fall to 1.68 × 10⁻⁶ and zero at twenty; and they are zero again at twenty-two and twenty-four. So the ambiguity switches on once, holds flat for three fillings, and switches off — and the disagreement is the middle of that, not its edges. A cage’s descriptions are unambiguous at both ends of its filling range and ambiguous in the middle, which is where the frontier levels are and is exactly where a chemist would look for a bonding description worth arguing about.

What was computed, and how #

The forty-eight pairs, both criteria and both thresholds are those of the two-criterion survey, unchanged. The Pipek–Mezey and Boys lines are each placed in that criterion’s own gap, as before — the arrangement that made the classification robust to its own threshold.

The automorphism counter is new and is the only new computation. It takes the edge list the cage was built from — the 3v − 6 shortest distances between repulsion-minimised points, which is exactly the deltahedron’s edge set — and counts vertex permutations preserving it.

Six results are checked numerically. Two calibrate the automorphism count against an octahedron’s 48 and an icosahedron’s 120. One reproduces the survey’s six disagreements before anything is read off them. One requires that they run in both directions, so neither criterion is the other with more resolution.

And two are the hypothesis, stated so that it could have gone either way. The first asks whether the automorphism order separates the disagreeing pairs from the agreeing ones — every disagreeing cage’s group larger than every agreeing cage’s — and reports the answer whichever it is. The second requires that it does not, which is what makes this a refutation rather than a survey with a null result buried in it — and the second could be stated only once the first had been answered.

Where the model stops #

A spread is a lower bound. It is the difference between the best and the worst maximum a search of finitely many starts happened to find, so more starts can only raise it. The ten-vertex cage’s Boys spread is 7.17 × 10⁻⁴ at twenty starts and at sixty, and 1.07 × 10⁻³ at a hundred and twenty — stable in order and growing slowly, which is what a real multi-basin landscape does. The zeros are the only entries a longer search cannot move, and four of the six disagreements rest on a zero.

Sixty starts, not two hundred. The single-criterion survey used two hundred random starts for Pipek–Mezey alone; running two criteria over forty-eight pairs at that rate is not affordable, and the check that the reduction has not moved anything material is that the gap comes back at 210 against its 205.

Hückel on a cage. One orbital per vertex, one hopping integral, no repulsion. Both criteria are applied to the same wavefunctions, so nothing here compares them as approximations to anything real, and the search’s own stopping rule remains the thing every count of descriptions is a property of.

And the automorphism group is the graph’s. A real borane has hydrogens, substituents and a geometry, and its physical symmetry can be lower than its skeleton’s graph symmetry — which would make the group counted here an upper bound on the symmetry available to relate two descriptions. That direction is the safe one for a refutation: the hypothesis is given the most symmetry it could possibly want and still fails.

The generalisation #

The transferable point is about the grain of an explanation against the grain of the thing it explains.

The survey observed a property of six cage-and-filling pairs, and the natural explanation is in terms of cages. That mismatch is visible before any computation: an explanation one level coarser than the phenomenon predicts that the phenomenon is constant across everything the explanation does not distinguish, and here that is twelve fillings of one cage. The prediction is not subtle and it is not close — nine agreements and three disagreements on one graph.

The habit worth taking is to check the grain first. Before testing a proposed explanation, ask what it holds fixed and whether the effect varies across that. It is one line of arithmetic over data already computed, it costs nothing, and it either kills the explanation or promotes it to something worth testing properly. The same shape appears from the other side where an explanation is checked against a control that varies the wrong thing and the control turns out to rank better than the mechanism.

The second half concerns what the six do have in common, and it is the more useful finding because nothing predicted it. Every one has a numerically zero spread under one criterion. That is a much stronger regularity than the six being scattered near a threshold, and it means the classification is not fragile in the way a set of borderline cases would be: no choice of line moves any of the six, because a line cannot separate a number from zero. That is a firmer footing than the original count of descriptions ever had. The two-criterion classification is therefore more robust than its own gap argument establishes, and the six disagreements are not the places where the two criteria nearly agree — they are the places where one of them is blind.

Who found it, and when #

Boys and Pipek–Mezey localisation are their authors’; graph automorphism counting is elementary and the octahedron’s 48 and the icosahedron’s 120 are standard. The forty-eight-pair survey and the six disagreements come from the two-criterion comparison.

What is new here is the automorphism count, the test, and the observation that the finding’s grain is a filling rather than a cage. The symmetry hypothesis earned its test by being precise enough to be refuted by an automorphism count, which is more useful than a statement that the six were unexplained.

Still open: whether bigger cages favour one criterion #

The obvious continuation is the occupied space. The three icosahedral disagreements are consecutive fillings with identical spreads to three figures, which is the signature of one degeneracy being filled rather than three separate coincidences. The levels are computed already, so listing the degeneracies at each filling and asking whether the disagreeing fillings are exactly the ones where a degenerate set is partly occupied is a read of numbers in memory — and it would replace a refuted explanation with one at the right grain.

The nearer question is the two-way asymmetry. Four of the six are Boys-degenerate and two are Pipek–Mezey-degenerate, and the four are the larger cages while the two are the smaller. That may be a real trend — a bigger cage has more room for populations to differ at fixed centroids than the reverse — or it may be four cases and two. Extending the survey to thirteen and fourteen vertices would say which, and it needs only one more deltahedron per size.