C.11.CRC:4.2 - Mathematical and model-routing distinction
| Expression encountered | Exact question | Required boundary |
|---|---|---|
| Finite difference or counterfactual configuration comparison | What changes between realizable S0 and S1 over the declared horizon? | Default here for indivisible, non-smooth, thresholded, path-dependent, or strongly interacting changes. Do not infer additivity. |
| Derivative or gradient | What is the local rate of change with respect to a coordinate under smoothness and small-change assumptions? | It approximates the finite contribution only when the validity region and remainder are adequate for Δ. |
| Sensitivity | How does a result vary with a parameter, assumption, input, model, or scenario? | It supplies robustness or assurance information; it need not describe a realizable configuration change. |
| Shadow price or dual variable | What is the local value of relaxing a formulated active constraint under the primal/dual model? | It depends on formulation, active set, regularity, units, and local region; it is not the contribution of an arbitrary asset or intervention. |
| Functional variation / calculus of variations | How does a functional change when the varied object is a function, path, trajectory, field, control, or shape in a declared admissible variation space? | Name the functional, admissible variations, constraints, boundary conditions, stationarity/extremum claim, sufficiency, and validation. An Euler–Lagrange equation is not a generic contribution claim or proof that a physical System optimizes. |
| Variational inference | Which member of an approximation family best approximates a target probability distribution under a declared divergence or bound? | The result is an approximate distribution and uncertainty account, not an extremal physical trajectory, capability acquisition, or general marginal value. |
| Evolutionary variation | How are retained variants generated and selected in an evolutionary or cultural process? | The shared word variation does not identify the mathematical object or Method above; route to C.18, C.36, or the field practice. |
This is why calculus of variations matters without becoming the default interpretation of marginality. When the candidate is a whole trajectory, field, function, control, or shape, pointwise finite-coordinate reasoning can miss the coupled admissible deformation. C.29 governs the mapping from the world or domain model to that mathematical object, what structure is preserved or lost, and when the lens must stop. Specialist practice governs derivation, discretization, solver choice, optimality and sufficiency checks, and validation.