The product a curve measures

How many parameters a curve is worth measured how many parameters a susceptibility curve is worth: with a coupling, a g factor, a monomer fraction and a temperature-independent term, the four directions span a factor of five hundred, and forty points reaching 2 K beat sixteen thousand starting at 20. It ended by naming the question it had set up and not asked:

The singular vector belonging to the smallest singular value is a direction in the four-parameter space, and it is not the χ_TIP axis — it is some mixture, and knowing which mixture would say what a paper’s four quoted numbers are jointly constrained to.

It is a mixture, and there is a reason the answer can be written down rather than drawn. The Jacobian here is logarithmic — every row is a set of exponents, because the parameters are positive and the natural question is by what factor each moves. So a direction is a set of powers, and a combination of parameters is a product.

A covariance ellipse is a matrix. A product of powers is an expression a paper could print.

The four products #

The last row is the answer to the question. The free direction is the temperature-independent term divided by the 0.40 power of the monomer fraction, and its largest component is 0.92 rather than 1 — so it is close to the term’s own axis and not on it.

The first row is the one nobody asks about and it is the more striking. What a susceptibility curve measures best is the g factor divided by the cube root of the coupling, to eight parts in ten thousand, on a curve measured to one part in a hundred. Neither of those numbers is fixed to better than two per cent on its own.

It is worth pausing on why the answer is a product rather than a sum. A direction in an ordinary parameter space is a linear combination — two parts of this plus three of that — and it has units that depend on what is being combined, so it can rarely be printed. In log space the same direction is a set of exponents, and the combination is a product of powers, which is dimensionally sensible whatever it multiplies and is exactly how a physical relation is normally written. That is not a trick of presentation: the logarithmic Jacobian is the right one here because the four parameters are positive quantities spanning six orders of magnitude, and a relative error is the only common currency they have.

The moment a fit invents is the warning about the same arithmetic from the other side, and it is worth reading together with this essay: there, a parameter takes a value that means nothing; here, four parameters take values that are jointly meaningful and separately not.

What the four printed numbers hide #

A paper reports four parameters with four uncertainties. Set those beside the four directions:

The printed numbers are pessimistic and optimistic at once, which is worth separating.

Pessimistic: the coupling is fixed to 3.0 per cent and the g factor to 2.0, and yet a particular ratio of them is fixed to 0.08. A reader told J to three per cent concludes that any statement depending on J carries three per cent, and a statement depending on g/J^⅓ carries a fortieth of that.

Optimistic: the monomer fraction is quoted at 16 per cent and the temperature-independent term at 37, as though each had been measured. Neither has. What was measured is their product to 6.7 per cent, and one combination of them to 40 — and moving along that combination is free.

Free, in the strong sense #

Undetermined can mean poorly determined, so it is worth demonstrating that this one is not.

Take the fitted parameters and move all four together along the free direction by a step of 0.35 in the logarithms — the coupling down by 2.4 per cent, the g factor down by 1.7, the monomer fraction down by 12, the temperature-independent term up by 38. Then recompute the whole curve from the exact spin spectra.

The curve moves by at most 0.87 per cent — under the one per cent it is supposed to be measured to. The same step along the best-determined direction moves it by 126 per cent.

So the two dashed and solid curves in that figure differ in every parameter, by amounts a chemist would argue about, and no measurement of this precision could tell them apart. That is the same demonstration a force field’s flat space gets in a different currency, and it is the only form in which undetermined means anything: a change nobody can see.

What to print instead #

The recommendation is not report fewer parameters. It is that two of the four printed numbers are worse than a product of them, and the product is what the measurement fixed:

ρ · χ_TIP^0.41, to 6.7 per cent.

Against a monomer fraction quoted at 16 and a temperature-independent term at 37. A reader given the product knows something the curve established; a reader given the two numbers knows two things it did not. That is the same recommendation a repeated symmetry species forces in a different corner of this collection — when a description cannot separate two things, the honest report names what it can separate.

There is a check on that recommendation which costs nothing and is worth making explicit. If the product is what the curve fixed, then two fits of the same data that disagree about ρ and about χ_TIP should agree about ρ·χ_TIP^0.41 — and that is a comparison anybody with two published analyses of one compound could run in an afternoon. The same test applied to a force field took four fits from four starting points and found them agreeing to a part in three hundred after projection where they had spanned 0.88 mdyn per ångström before.

That is not a curiosity of this model. Both quantities are corrections that act at the ends of the curve — a monomeric impurity raises χT at the cold end, a temperature-independent term at the warm end — and a curve of finite range trades them against each other. The particular power, 0.41, is what the trade costs.

And the free direction is the window’s #

The other finding was that the temperature window matters more than the number of points. It matters here too, and in a way that changes what the answer is:

The middle rows are where the interesting behaviour is. Between 10 K and 20 K the free direction stops being one parameter and becomes a mixture, and the uncertainty along it doubles — so somewhere in that decade the measurement loses its grip on the monomer fraction and the two corrections start trading. That is the same crossing located by counting parameters, seen as a rotation rather than as a count, and the rotation is the more informative view: a count says a parameter has been lost and a rotation says which one, and into what.

At 2 K the free direction really is the temperature-independent term’s own axis, which is what the parameter count expected and denied. At 20 K it is the mixture above. At 80 K the monomer fraction has become the dominant free direction, at 241 per cent — meaning it is not merely undetermined but unbounded.

So “which combination is free” has no answer that belongs to the model. It belongs to the experiment, and a paper reporting one has to say over what range it measured before the statement means anything.

What the nesting says #

Adding parameters one at a time shows the conditioning worsen. Reading the directions rather than the singular values shows something the counting could not.

With two parameters the directions are g·J^−0.33 and J·g^0.33 — already mixtures, at a condition number of 6.2. Adding the monomer fraction and the temperature-independent term leaves the first two directions almost exactly where they were and adds two much weaker ones.

That is worth stating carefully because it is the opposite of the usual intuition. Adding a parameter is normally described as spreading the information thinner; here the top two directions keep their singular values to three figures — 12.40 and 2.009 with two parameters, 12.41 and 2.026 with four — and the new parameters take almost nothing from them. A new parameter does not take precision away from the old ones evenly; it adds a direction at the bottom. That is a more useful statement than the fit gets worse, and it says why the coupling and the g factor survive a four-parameter fit in good order while the two corrections do not: they are not competing for the same information.

The same instrument, three times #

Three arguments have now met the same object from three directions, and the pattern is worth setting down because the repair is the same each time and the currency is not.

A force field has ten directions no frequency can see, and the repair is a projection: throw away the component along them and the field becomes unique. That works because the flat space there is exactly flat — a null space, not a small singular value.

A substitution structure has three numbers that are not a structure, where the difficulty is that a measured quantity is a difference of two large ones and its error is not the error of either.

And a susceptibility curve has a direction that is nearly flat but not exactly, so nothing can be projected away — the free combination is determined to forty per cent, which is a real number rather than an infinity. The repair here is to report the other three.

The general shape: when a fit’s Jacobian has a small singular value, there are three things one can do — remove the direction, measure more to fill it, or change what is reported. Which is available depends on whether the small value is zero, and that is a question about the model rather than about the data. Two of these three cases have a zero and one does not.

What is quoted, and what is computed #

Nothing is quoted. There is no compound and no measurement: a chain of eight spins, an exchange coupling, a g factor, a monomer fraction and a temperature-independent term, at a stated point in the parameter space, with an assumed relative precision of one per cent on every point of the curve.

The curve itself is exact: the susceptibility comes from the full spin spectrum of the chain, diagonalised once per coupling rather than once per temperature. The Jacobian is a set of logarithmic derivatives taken by finite difference, its singular values and directions come from the eigen-decomposition of its Gram matrix, and the precision along a direction is the assumed measurement error divided by the singular value.

The four directions are checked orthogonal to a part in a thousand million, which is arithmetic rather than a result and is the check that the decomposition was read correctly rather than transposed.

What this cannot say #

The point in parameter space is a choice. All of this is the local behaviour at one set of parameter values — a coupling of 50, a g factor of 2, a monomer fraction of 0.02 and a term of 2 × 10⁻⁴ — and the directions are the directions there. A compound with ten times the impurity would have different ones, and the exponent 0.41 would be a different number.

One per cent is an assumption about the measurement, and it is uniform. A real susceptibility measurement is not equally precise at 20 K and 300 K, and weighting the points differently would change every singular value. What would not change is the existence of a direction the curve cannot see, since that comes from the shapes of the four contributions rather than from the weights.

The free direction is free at this precision. Everything here scales with the assumed one per cent: at a precision of a tenth of a per cent the same direction would be determined to four per cent and would not be free at all. What is fixed independently of the precision is the ordering — the condition number of 500 — and it is the ordering that says which combination to print.

And a product of powers is not a physical quantity. g·J^−⅓ has units nobody would want and no interpretation beyond what this measurement fixed. It is a reporting device, not a discovery — the discovery is that the four numbers usually reported are not four measurements.

What was checked #

The free direction is a mixture, with a largest component under 0.9 in magnitude squared — checked as an upper bound, because the whole finding is that it is not an axis.

The best-determined direction is a mixture too, which is the half that would be missed by anyone looking only for the failure.

The four directions are orthogonal, to a part in a thousand million.

And the refusal is a two-parameter fit. With only the coupling and the g factor the same curve is well conditioned — 6.2 against 500 — and its worst direction is nearly a parameter’s own axis. A routine that reported a badly mixed direction there would be reporting its own arithmetic rather than the curve’s.

Reporting a combination is not an unusual thing to do #

The recommendation this essay ends with — print the determined product instead of four separately undetermined numbers — sounds like a departure from practice, and in this field it is. In neighbouring fields it is the practice.

A crystallographic refinement fits far more parameters than a susceptibility does, and it reports a correlation matrix alongside them, precisely because two parameters that trade off against each other are not two results. A reader who wants to know whether an occupancy and a displacement parameter are separately determined can look, and the answer is in the deposited file rather than in the author’s confidence.

Chemical kinetics does the same thing more bluntly. A rate law that depends on a catalyst concentration and a rate constant only through their product is reported as the product, because separating them requires an experiment that varies one at fixed other — and nobody quotes the two halves of a quantity their data cannot split.

So the proposal here is not a new convention. It is the convention two adjacent fields already use, applied to a curve that has the same problem and does not report it. What is added here is the arithmetic that says which combination to print: the singular directions of the fit, and the powers that make one of them a product.

Still open: a frustrated ring, and the exponents derived #

The obvious open question is the frustrated ring, wanted earlier for a different reason: a ring of an odd number of spins has a low-temperature feature where this analysis says the information is. What the directions add is a sharper version of the question — not does a feature buy back the fourth parameter but does it change which combination is free. If the feature carries information about the monomer fraction specifically, the free direction should rotate towards the temperature-independent term’s axis, and the window table above is the calibration for reading that.

The nearer question is the exponent. 0.41 for the trade between a monomer fraction and a temperature-independent term, and −0.33 for the trade between a g factor and a coupling, are both suspiciously close to simple fractions. The second has a plausible origin — χT at high temperature goes as g² and the coupling enters as a correction — and neither has been derived. Deriving them, on the high-temperature expansion, which can be written down, would turn two fitted powers into two known ones, which is the difference between a reporting device and a result.