formulationinvariant

Formulationinvariant is a concept in mathematics, physics, and related fields describing a property that remains unchanged when the underlying problem or model is expressed in a different mathematical formulation. It emphasizes representation independence: the truth of the property is not tied to a specific set of equations, variables, or coordinates but to the underlying structure described by the theory.

In geometry and physics, formulationinvariant quantities include scalar invariants such as geodesic distance and curvature scalars, which do not depend on coordinate choices or parameterization. In physics, observable predictions are required to be invariant under gauge transformations or field redefinitions; for example, physical measurements or S-matrix elements should be formulationinvariant across equivalent formulations of a theory.

In optimization and computer science, a property is formulationinvariant if reformulating the problem—changing variables, rescaling, decoupling constraints, or switching between primal and dual representations—leaves the optimal value and the set of optimal solutions unchanged (up to a natural correspondence). Demonstrating invariance typically relies on transformations that map one formulation to another while preserving the objective and constraints in equivalence.

Formulationinvariant thus represents a general, cross-disciplinary idea tied to the broader notion of invariants: the content that truly matters in a theory or model should not depend on arbitrary representations or formulation details.