Three chains and ninety orderings ruled out

Two things were found and only one of them put to work. One correlation applied to every measurement multiplies every price by the same number, so it slides the fragility ordering rigidly and cannot scramble it; a correlation assigned per predictor multiplies each predictor’s claims by that predictor’s own factor, so it can reorder claims from different predictors and cannot reorder two claims that share one.

The second half was used to explain why one adjacent pair could not swap. It says considerably more than that, and it says it without any numbers.

Why a shared predictor is a shared factor #

The prices come from a single piece of arithmetic. A claim is a statement about two measurements — that they are ordered a particular way, or that a difference between them is at least so large — and its price is the standard error that would destroy it. For two measurements each with error σ and correlation ρ between them, the difference has error $\sigma\sqrt{2 - 2\rho}$, so a price computed under independence is multiplied by $1/\sqrt{1-\rho}$ when the errors are correlated.

That factor contains nothing about the claim. Not the size of the difference, not the predictor, not the molecules — only ρ. So if every measurement in the collection shares one correlation, every price is multiplied by one number and the ordering is rigid, which is what that essay established.

A correlation belongs to a literature rather than to a collection. Ring strains and ionisation energies are measured by different people with different instruments, and there is no argument that the correlation between the two errors in a difference should be the same in both. So the honest structure is one ρ per predictor — and then two claims priced against the same predictor carry the same factor, whatever it is.

Their ratio is therefore constant. It does not matter what ρ is, whether it is known, or whether it is the same as anybody else’s: the more fragile of the two stays the more fragile. That is a locked pair, and it is locked structurally.

The chains, counted #

The collection prices five claims. The Hückel eigenvalue supplies two — a discordant pair and a slope floor — as does the computed angle strain. The spin-only moment supplies one, its slope floor, because it orders every one of its nine ions correctly and so has no monotonicity claim to destroy.

Chains of two, two and one.

An ordering of five claims that respects each chain’s internal order is an interleaving of the three chains, and interleavings are counted by a multinomial coefficient: 5! divided by 2!·2!·1!, which is thirty. Five claims admit a hundred and twenty orderings.

So ninety of them are forbidden, and the reason is which predictor each claim belongs to. No measured difference is involved, no correlation is assumed, and nothing about the sizes of the prices enters. Three quarters of the possible rankings are ruled out by bookkeeping.

That is the useful form of the result, because it says what a fragility ranking is claiming. A list of five items looks like it makes ten pairwise statements. This one makes two — the two within-chain orders — unconditionally, and the other eight conditionally on somebody’s error correlations.

Which pairs are which #

The two locked pairs are the Hückel eigenvalue’s two claims and the angle strain’s two claims. Everything else is purchasable.

The first of those is the top pair of the ranking, ranks zero and one, which is what was found there and reported as a result about the headline. Stated as a chain it is not a result about a margin at all. Their prices are 3.42 and 10.04 per cent of the measured range, a ratio of 2.93, and the ratio is irrelevant: the two claims share the Hückel literature’s correlation and their order is fixed at every value of it. The headline survives any numbers whatever, which is a much stronger statement than surviving these numbers comfortably.

The other locked pair is the interesting one, because it is not adjacent.

A rigid chain with something loose inside it #

The angle strain’s discordance sits at rank two and its slope floor at rank four. Between them, at rank three, is the spin-only moment’s slope floor — a chain of one, from a third literature.

So the ranking contains a locked pair with a foreign claim inside it. The chain’s own statement holds: the angle strain’s discordance is more fragile than its slope floor at every correlation structure. But the interval of the ranking those two claims span is not rigid, because the thing between them can be moved out of it in either direction.

Lifting it above rank two costs a correlation of 0.736 in the angle-strain literature; dropping it below rank four costs 0.154. Those differ by nearly fivefold, and the cheap one is cheap: a correlation of a sixth between two measurements of ring strain from one source is not a hypothesis anybody would need to defend.

That is the sharpest form of what a chain does and does not say. A chain constrains its own members and nothing else. Reading the rigid part of a ranking as a region of the ranking — the top three are safe, say — is the mistake the decomposition prevents, and nothing in that presentation would have caught it, because there the pairs were examined adjacently and this pair is not adjacent.

Every price, and why reachability is not the question #

All eight purchasable pairs are reachable. Every one of them has a correlation strictly inside the unit interval that transposes it, so can this pair be swapped separates nothing and is not worth asking.

The prices separate them, and they span the interval. The cheapest is 0.154 — the spin-only floor past the angle-strain discordance, two claims 8.7 per cent apart in price. The dearest is 0.990, the Hückel discordance past the angle-strain floor, which are a factor of 9.9 apart.

The arithmetic behind the spread is worth seeing, because it explains why the prices are so unevenly distributed. The factor is $1/\sqrt{1-\rho}$, so the correlation needed to multiply a price by $k$ is $1 - k^{-2}$. For k = 1.1 that is 0.17; for k = 2 it is 0.75; for k = 10 it is 0.99. The function crushes everything above a factor of two into the last quarter of the interval and spreads everything below it across the first three quarters. So a ranking whose adjacent prices differ by tens of per cent is soft, and one whose adjacent prices differ by factors is not — and a list of prices does not make that visible while a list of correlations does.

Two of the three adjacent transpositions cost under 0.75. The middle of this ranking is soft.

What the two rigid statements actually say #

The decomposition leaves exactly two unconditional pairwise statements, and they are worth reading as chemistry rather than as bookkeeping, because they are the only two things this ranking claims without an assumption.

The Hückel chain says a discordance is more fragile than a slope floor. Its discordance is hexatriene against naphthalene: naphthalene has the larger highest-occupied eigenvalue and the smaller ionisation energy, so no monotone model of the form IE = f(x) fits both. Destroying that refutation needs 0.15 electronvolts of error, 3.4 per cent of the set’s range. Its slope floor is hexatriene against anthracene, a rise of 0.88 electronvolts over a run of 0.031, and halving it needs 0.44 electronvolts — three times as much. So the ordering claim is the cheaper one to lose and the magnitude claim the dearer, in one literature, at every correlation.

The angle-strain chain says the same thing about the same kinds of claim, at 16.0 and 33.8 per cent. Two predictors, two literatures, one direction.

That coincidence of direction is not guaranteed by anything and it is the one general remark the two chains permit. A discordance is destroyed by an error large enough to reverse a single pair; a slope floor is halved by an error large enough to halve the largest rise in the set. The second is a bigger difference by construction — a slope floor is chosen as the steepest pair — so it should be the dearer, and on both chains it is. What the decomposition establishes is that the ordering of those two kinds of claim needs no assumption at all, and what the numbers add is that it comes out the same way twice.

The predictor that contributes nothing rigid #

The spin-only moment is a chain of one, and a chain of one makes no unconditional statement.

It is a chain of one because it has no discordance to price. Across nine ions from titanium to copper it orders every measured moment correctly — the four ions whose moments exceed the spin-only value do so without crossing any other pair — so there is no monotonicity claim to destroy and only its slope floor is priced. That is a fact about the predictor’s quality and it has an unwelcome consequence for the ranking: the best-behaved of the three predictors is the one about which the decomposition can say nothing.

So of the three predictors, two supply the ranking’s whole rigid skeleton and the third supplies only a mobile piece — and it is the piece sitting inside the other chain’s interval. The claim whose position is least constrained is the one from the predictor that behaved best, which is the opposite of what a reader would guess from the ranking as a list.

What was computed, and how #

Nothing here is a new measurement. The five prices are the earlier essays’, computed from nine magnetic moments, six ionisation energies and six ring strains, all quoted; each price is the standard error that would destroy its claim, as a fraction of its own set’s measured range, at the one-sigma convention they adopted.

The chain decomposition is a grouping by predictor. The interleaving count is a multinomial coefficient on the chain lengths. Each purchasable pair’s price solves $1/\sqrt{1-\rho} = {}$ ratio of prices for $\rho$, which has a solution in the unit interval exactly when the ratio exceeds one — and since the claims are indexed in price order, it always does for a pair taken in that order.

Six things are checked: that the claims fall into several chains and that more than one chain has an internal order to state, since a decomposition into five singletons would be no decomposition at all; that no pair sharing a predictor returns a correlation, and that the search reports that rather than a number outside the interval; that every pair from different predictors returns one strictly inside it; that the forbidden orderings outnumber the permitted ones; that the top pair is a locked pair, which is that earlier finding restated structurally; and that the purchasable prices span weak correlations to strong ones, so that the softness of the ranking is uneven rather than uniform.

Where this stops #

The decomposition is exact and its premise is a convention. The premise is that a claim’s price is a fraction of its predictor’s own measured range and that all the measurements behind one predictor share a correlation. The first is a normalisation those essays chose so that prices from different units could be compared; a different normalisation would move every price and would not change any chain, since a chain is a statement about which claims share a factor. The second is the substantive assumption, and it is an idealisation: a predictor’s measurements might come from two literatures, in which case its chain breaks into two.

Nothing here says a locked pair is a true pair. It says no correlation structure reorders it. A systematic error that shifted one measurement and not another is not a correlation and is not covered; neither is a wrong measurement. The claim is narrow and it is the claim the calculation can make.

And five claims is a small collection. The chain lengths are two, two and one, which is barely enough for the multinomial to say anything interesting — thirty of a hundred and twenty. A collection with four predictors carrying three claims each would have $12!/(3!)^4 = 369{,}600$ interleavings of 479 million orderings, which is a far more dramatic ratio and a far less useful one, since what matters is not the count but which specific pairs are locked.

The generalisation #

The habit is to ask, of any ranking, which of its pairwise statements are statements about the objects and which are statements about the assumptions.

A ranking presents itself as a total order and invites being read as one, and most of what makes it an order is usually contingent. Here the contingency is the error correlation and the invariant is which predictor a claim came from; the decomposition falls out because the contingent quantity enters every price the same way. That is a general shape rather than a feature of this argument: whenever a nuisance parameter multiplies a family of quantities by one factor, it cannot reorder that family, and a ranking decomposes into the orbits of whatever the nuisance parameters move.

The corollary is the one the straddle makes concrete. Having found the rigid pairs, do not read them as rigid regions. Two claims can be locked relative to each other and both loose relative to something sitting between them, and the ranking then has a rigid skeleton with a mobile filling rather than a rigid top and a soft bottom. Reporting a ranking’s rigid part as the top three are safe is available and is wrong here, and the check that catches it is to look at the non-adjacent locked pairs, which is the half an examination of adjacent swaps never sees.

Who found it, and when #

The propagation of correlated errors through a difference is older than any of this and is in every treatment of measurement. The prices, the discordances and the slope floors come from the earlier essays, computed from measurements already quoted. What this essay computes is the grouping of the claims by predictor, the interleaving count, the price of each of the eight purchasable transpositions, and which locked pair straddles a foreign claim.

The number worth carrying is not thirty. It is 0.154: the correlation that moves a claim out from between the two members of a locked pair, in a ranking whose rigid part had been read as its top.

Still open: the denominator, and a chain that is really two #

The obvious open question is the one deferred three times already, and this essay’s arithmetic makes it sharper rather than answering it. Every price above is a price on a rise — a measured difference. Three of the five claims are slope floors, and a slope floor is a rise over a run, where the run is computed by the predictor rather than measured and so carries no experimental error at all. Nothing above touches it. Pricing it needs a currency the chains say nothing about, and the chains say what that currency cannot be: a nuisance parameter that multiplied all three runs by one factor would leave all three floors’ ratios alone, so whatever prices a run has to be something that moves the three predictors differently.

The nearer question is whether a chain is really a chain. The premise is that all the measurements behind one predictor share a correlation, and the six ionisation energies here are for ethene, butadiene, hexatriene, benzene, naphthalene and anthracene — the acenes and the linear polyenes, which are two families and quite possibly two sources. Splitting that predictor’s measurements into two correlation groups would break its chain into two chains of one, unlock the collection’s top pair, and put a price on the headline for the first time. Which of the six measurements share a source is a question about a literature rather than about arithmetic, and it is the one input this whole decomposition rests on and does not check.