Version 1 of #4931

It seems to me that "hard-to-vary" is itself the criterion that a theory should be as programmable as possible. As you note, the goal of a theory should be to make it as explicit as possible, and a program is explicitness in its most complete form. Any theory with ambiguous components automatically has a breaking point that is changeable without detection. A programmable theory has strict causal relations all the way from the axioms to the prediction, which makes any change to the components detectable. In other words: a theory is hard to vary to the extent that its components and the couplings between them can be specified as a program. If a theory is vague, you cannot tell when it has been varied.

This gives a concrete operationalization. A breaking point is any place in the formalization where the chain stops being programmable: a primitive with no implementable type, a coupling between components that cannot be specified, or a step that requires implicit theories to fill the explanatory gaps. A mathematical theory with no remaining gaps has zero breaking points and is maximally hard to vary. A theory in natural language is already worse, because words carry ambiguity and vary from mind to mind. This does not rule out better and worse theories in natural language, since we can use more or less ambiguous words and relations. But it does create a hierarchy of hard-to-vary explanations, where the share of the explanation that is programmable, or at least unambiguous, forms the basis for the criterion.

This is probably too crude a formalization. But evaluating the two theories of Demeter's emotions and axial tilt as explanations, you could check how much of each is programmable. Detecting seasons is programmable in both cases through temperature and changes in weather. Demeter's emotions and the causal link from them to the weather, which is the entire explanation, are not programmable. In the axial tilt theory, every component is. So on this measure Demeter scores 25% and axial tilt scores 100%.