When numerical agreement is informative

Two numbers can agree impressively and still provide little independent evidence. A useful comparison identifies what each number represents, how it was obtained and which uncertainties the two calculations share.

Agreement needs a question

Suppose a measured result and a model output both round to 1.37. Before celebrating the match, ask whether they refer to the same observable, preparation, units and scale. A value at one energy or frequency need not be the quantity predicted at another. Rounding can conceal a difference that matters, or display digits that the inputs do not justify.

Independence matters too. If a model parameter was adjusted until its output became 1.37, recovering that number checks the fitting procedure. Evidence for the model's wider reach comes from a further output that was not used to choose the parameter. The earlier guide to circular derivations develops this distinction.

Even genuinely separate calculations can share a calibration. Their uncertainties then cannot always be combined as though every contribution were independent. The common influence may cancel in a difference or make its uncertainty larger, depending on how it enters.

One shared influence, two responses

Consider an explicit dimensionless uncertainty model. Two estimates have zero-mean errors e₁ = c + ε₁ and e₂ = s c + ε₂. The common contribution c has standard uncertainty u_c. The independent contributions ε₁ and ε₂ each have standard uncertainty 0.10. All three contributions are mutually independent before they enter the estimates. Choose s = +1 for equal responses to the common influence, or s = −1 for opposite responses.

The error in their difference is e₁ − e₂ = (1 − s)c + ε₁ − ε₂. Consequently its standard uncertainty is

u_difference = √[0.02 + (1 − s)²u_c²].

If the two total uncertainties were incorrectly treated as independent, the result would instead be √(0.02 + 2u_c²). The missing information is the covariance s u_c². The calculation follows directly by expanding the variance of the difference; it does not require a Gaussian distribution.

The same gap can mean different things

Take an observed absolute difference Δ = 0.30 and u_c = 0.50. With equal sensitivities, the common contribution cancels, leaving uncertainty √0.02, about 0.1414. The difference divided by that uncertainty is about 2.1213. Ignoring covariance would give uncertainty √0.52, about 0.7211, and a ratio of only 0.4160.

With opposite sensitivities, the correct uncertainty is √1.02, about 1.0100, and the ratio is about 0.2970. Here ignoring covariance makes the difference look larger relative to its uncertainty. The same numerical shortcut can therefore distort a comparison in either direction.

The ratio is a standardized discrepancy, not the probability that either theory is true. A probabilistic interpretation would need a distribution, a sampling procedure, a specified hypothesis and an account of how the comparison was selected. The demonstration deliberately supplies no automatic significance verdict.

Small uncertainty is not independent evidence

Correlation is only one part of the audit. Numerical rounding, truncation error, measurement uncertainty and uncertainty about the model's form have different sources. Increasing the number of printed decimals does not reduce any of them. A very accurate computation of a poorly justified equation remains a very accurate answer to that equation.

Repeated comparisons also need a declared scope. Searching many adjustable formulas and reporting only the closest match makes the observed agreement less informative than a prediction specified beforehand. A useful report records the alternatives considered and the data used during selection.

The metrology guide cited below explains how covariance enters uncertainty propagation. Its general rule becomes particularly transparent here because the model is linear and the shared input is shown explicitly. In a real application, the size and sign of the common influence need evidence, not a convenient choice that improves agreement.

From a number to a physical test

For SMM/DTT, a proposed numerical derivation should identify the physical observable, the independently selected inputs, and the response or normalization that connects them. A structural count and a measured coupling are different objects until that connection is derived. The book's discussion of counts and responses makes this a concrete research task.

A stronger test uses a common physical dictionary to predict another quantity and propagates shared uncertainties into the comparison. Agreement then has an inspectable meaning: the same inputs constrain more than the number that motivated their choice.

Sources and further reading

1. JCGM, Guide to the expression of uncertainty in measurement, JCGM 100:2008, especially §5.2 and Annex F.1.2.
2. Mohamed Haj Yousef, Counts, responses, and physical scales and One physical dictionary. The two-estimate model and numerical examples above are teaching constructions.