The gap found on purpose

The Bailar twist passes through D₃, a group whose character table was not among the twenty in use until it was written for that path — and how it came to be missing raises a question.

At least one path between two tabulated arrangements goes through an untabulated group. Enumerating the one-parameter paths between all the arrangements in use, and asking which of them pass through groups not on the list, says whether the D3 gap was the last one or the first one found.

This is that enumeration. The direct answer is small — one more gap in two hundred and seventy geometries — and the search turned up a defect in the collection that was not what it was looking for.

Building a path between two arrangements #

Two sets of points on a sphere are the same arrangement seen from different angles unless something says which ligand goes where, so a path between two of them is not simply an interpolation. It is built in two steps.

For each of the ways the second arrangement’s ligands can be assigned to the first’s, the rotation that best superimposes them under that assignment is found, and the assignment with the smallest residual is kept. The path is then the great-circle interpolation between matched pairs.

The assignment is exhaustive rather than heuristic, and that is a decision worth defending. Six ligands give seven hundred and twenty assignments and a wrong one produces a path through arrangements that resemble neither end — which would look exactly like a discovery, since a geometry nobody meant to construct is a good place to find an unexpected group. At these sizes exhaustive costs nothing and a heuristic would be a false economy.

The check on it is at the ends: both endpoints of every path must come back as the arrangements they name. A path built from a bad assignment fails that immediately.

Eighteen paths, and one exclusion #

The collection’s census holds seventeen arrangements from two ligands to six, which give nineteen pairs at equal ligand count. One is excluded and the exclusion is recorded rather than quietly applied.

The linear two-ligand arrangement has a continuous group, and the collection’s standing rule is that such a case is reported as infinite rather than worked in a finite stand-in — a stand-in answers about a different object. Both ends of the linear-to-bent path come back refused, so the path does not join what it names, and it is left out.

That leaves eighteen paths, sampled at fifteen geometries each: two hundred and seventy.

Three kinds of no answer #

243 are named, by one of the collection’s twenty tables — mostly C1, Cs, C2v and the parent groups at the two ends.

25 are refused by the symmetry finder’s own tolerance: it assembles a set of candidate operations, each of which nearly maps the arrangement onto itself, and none holds to a thousandth of an ångström. That is the same failure found in two bands near the ends of the Bailar twist, and it is a statement about how nearly symmetric a geometry is rather than about which tables exist. Eleven of the eighteen paths have at least one such geometry.

2 are a finite group with no table. Both are at t = 1.000, which is the pentagonal pyramidal endpoint of two different paths, and both report an order of ten. No character table in this collection has order ten — the twenty it holds have orders 1, 2, 4, 6, 8, 12, 16, 20, 24 and 48 — and the group of a pentagonal pyramid is C₅ᵥ. It is the arrangement the metal census could not answer for either.

So the answer to the question is: the D3 gap was not the last one, and it was very nearly the last one. Two hundred and seventy geometries hold one further gap, and it is not in a path’s interior.

It is an arrangement, not a stratum #

The two gaps are different in kind and the difference matters more than the count.

The D3 gap was a stratum: the whole open interval of the Bailar twist, every geometry strictly between the prism and the octahedron, eleven of twenty-one sampled points. It was invisible because nobody had reason to ask about the interior of a path — the two ends were tabulated and the count agreed at both.

The C₅ᵥ gap is a single arrangement that two paths happen to end at, and it is in the census in its own right. Nothing about a path was needed to find it: the pentagonal pyramid is one of the seventeen arrangements the collection enumerates and has been all along. The path survey found it by walking into it, and it was sitting in plain sight.

That is a different and slightly embarrassing shape of result. The first gap needed a search because it lived somewhere nobody looked; the second needed a search because of what the collection said about it when anybody did look.

What the collection was saying instead #

This is the part that was not being searched for.

The hypervalency census begins by identifying each arrangement’s group and, if the symmetry search returns no name, reports the arrangement as having an infinite group and stops. That covers two entirely different situations. A linear arrangement genuinely has one. A pentagonal pyramid has C₅ᵥ, of order ten, which the finder assembles perfectly well and cannot name.

So the census reported the pentagonal pyramid as having an infinite group, and every essay that read the census read it that way — including the pricing of arrangements by their repulsion and the formula found failing on flat shapes. Nothing else went wrong. The row was excluded from every count for a reason that was stated and was wrong, and the count it was excluded from is unaffected — an arrangement with no table cannot be reduced whatever the reason.

The distinction now made is the one the Bailar-twist analysis spent a section on, applied one level up:

the decomposition of “no answer” into a missing table and a failed tolerance, which are usually reported as the same thing when they are reported at all.

It had two boxes and there were three. The fix is a condition rather than a flag: the finder assembling nothing — an order of zero — is what a continuous group looks like from here, since no finite set of candidate operations closes; a positive order with no name is a finite group with no table. Every count downstream is unchanged, because both cases were and are excluded, and what changes is what the row says about itself.

Why nothing caught it #

The mechanism is ordinary and is the reason to write it down.

The label infinite was attached by a condition that did not test for infiniteness. It tested for the absence of a name, which is a different property that happens to coincide with infiniteness on the cases it was designed around — the census’s linear arrangements. The pentagonal pyramid was in the census from the start, it was reported as infinite from the start, and there was no second determination of its group to disagree.

Every check on a computed quantity asks whether the quantity is right. None asks whether a reason is right, because a reason is prose attached to a null. The census checks that the formula holds where a count exists and that some arrangements break it; it says nothing about the rows where no count exists, since there is nothing there to compare.

The thing that found it was a survey that classified the same absences by a different route — the finder’s reported order rather than the census’s flag — and got a different answer. Two independent descriptions of the same nulls is what made the disagreement visible, and it is the same discipline the collection applies to its numbers, applied for the first time to its refusals.

Where the refusals fall #

The twenty-five tolerance refusals are worth a paragraph, because they are the majority of the nulls and they carry no information about the collection at all.

Eleven of the eighteen paths have at least one, and twenty-three of the twenty-five sit near an end — nineteen of them in the first three samples of their path. That is the same pattern measured directly on the Bailar twist: near a high-symmetry arrangement the finder builds that arrangement’s operations, and they hold to a residual that grows with the distance from it — so there is always a band, just past the end, where the operations are nearly right and not right enough.

A path between two high-symmetry arrangements therefore has two such bands and a clean interior, and a path between a high-symmetry one and a low-symmetry one has one. The tetrahedron-to-trigonal-pyramidal path has five refusals of fifteen, all in the first five samples, which is the widest band in the survey — and the reason is visible in its ends: a tetrahedron is $T_d$ of order twenty-four and a trigonal pyramid is C₃ᵥ of order six, so the departure from the first is being measured against a great many operations at once.

Two refusals are not near an end and are worth naming rather than averaging away. The trigonal-bipyramid-to-planar-pentagon path refuses at t = 0.36, a third of the way along, and the seesaw-to-trigonal-pyramidal path at t = 0.93 — which is near an end, but the far one. A mid-path refusal is a geometry that is nearly symmetric for a reason neither endpoint supplies, which is the interesting case and is exactly where a further gap would sit if there were one. Neither is untabulated; both are ordinary near-misses.

None of that is a defect and none of it is fixable by writing a table. It is what a tolerance decision costs, spread over eighteen paths, and the reason to count it is that it makes the two genuine gaps legible: two of two hundred and seventy is a small number, and twenty-seven would have been a misleading one.

What was computed, and how #

The arrangements, the symmetry search and its tolerance are the ones used throughout, unchanged. The path construction is new: an exhaustive search over ligand assignments, a best-fit rotation by the iteration $X \leftarrow (X + X^{-\mathsf{T}})/2$ — which converges quadratically to the orthogonal factor of a matrix and needs no decomposition, the same device that cleans a symmetry operation’s matrix before its character is read — and a great-circle interpolation.

Each of the 270 geometries is handed to the finder and sorted by what comes back: a name with a character table, a positive order with no name, or a refusal.

Six results are checked. Every path’s two endpoints are recognised, so each path joins the arrangements it names — the check on the assignment search. The only exclusions are paths with a linear end. All three outcomes occur, which is what makes the classification do work: a survey in which nothing is ever untabulated would make that box decorative, and one in which everything is refused would be a report about the tolerance. Some paths have gaps and others do not, so the survey is a search rather than a census of one answer.

Where the model stops #

Fifteen geometries a path. A gap occupying a single geometry between two samples would be missed, and the D3 gap was found only because it occupies a whole interval. A finer sampling would find more of the first kind and none of the second, since an endpoint is always sampled.

One path per pair. The path built here is the shortest one under the best assignment, and there are others — a pair of arrangements is joined by infinitely many one-parameter families, and a different one could pass through different groups. What the survey establishes is that these paths hold one gap, not that no path between these arrangements does.

Seventeen arrangements. The census is a chosen list of shapes and it is not a survey of coordination geometry. A capped trigonal prism, a bicapped tetrahedron and everything above six ligands are absent, and each would bring its own groups.

And the tolerance is a decision. Twenty-five geometries are refused at a thousandth of an ångström, and a sweep of exactly that decision found a molecule assigned five different groups across a plausible range. A looser tolerance would name more of the 25 and could put some of them in the untabulated box.

The generalisation #

The transferable point is about auditing the reasons a computation declines to answer, and it is a place almost nothing looks.

A body of computation accumulates checks on its outputs. Every number is compared against something; every count has a test that could fail. The cases where a computation returns nothing accumulate no such checks, because there is nothing to compare — and the explanation attached to the null is prose, set by a condition designed around the cases in view at the time.

That makes a null a good hiding place, and the way to search it is the one that worked here: classify the same absences by a second route and require the two to agree. The census said “infinite”; the finder’s own reported order said “ten”. One line of comparison, and a description that had been wrong since the census was written.

There is a sharper version of the same point. The condition that failed was no name standing in for the group is infinite, and the two coincide on every case the census had met. A stand-in that is correct on all current inputs is not an error in any testable sense — it is an error waiting for an input, and the input here had been present all along and had simply never been asked a second question. The same shape appears where a formula is right for exactly the molecules anybody would check it on, and the remedy is the same: run the thing over cases nobody chose.

Who found it, and when #

Point groups, C₅ᵥ and the Kabsch alignment are all standard, and none of them is this collection’s. The survey, the classification and the finding that one flag was covering two situations are its own.

The question came with the shape of its answer — the D3 gap might be the last one or the first one found, and an enumeration rather than a guess would say which — and the enumeration answered it and turned over something else on the way, which is the ordinary reason to run a search whose expected result is boring.

Still open: every other null with a reason attached #

The obvious continuation is C₅ᵥ itself. It is nine lines, exactly as D3 was — four classes, four irreducible representations of dimensions 1, 1, 2 and 2, and the two sums that have to come to ten — and writing it would give the pentagonal pyramid an orphan count for the first time. The census would then have sixteen answerable arrangements instead of fifteen, and the one arrangement that has never been counted is a flat-adjacent five-ligand shape, which is the region the counting formula has been failing in.

The nearer question is the rest of the nulls. This survey examined one label in one census because a path walked into it. Other calculations here return nothing in several places — a minimiser that finds no arrangement, a localisation that finds no basin, a quadrature that declines a pair of ions — and each null carries a reason set by a condition designed around the cases in view at the time. Asking of each condition what else could satisfy it needs no new computation at all, and the pentagonal pyramid is the argument for doing it.