An approach whose Hamiltonian cannot distinguish genuine temporal evolution from gauge redundancy fails the minimal criterion of recovering classical dynamics in the appropriate limit.

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Appropriate limit(philosophy and physics): A specific condition or boundary where one theory should reduce to or match another theory; for example, how quantum physics should match classical physics in everyday-sized objects.
Classical dynamics(as used in physics and the history of science): The study of how objects move and interact based on forces, using the physics principles developed before modern quantum mechanics and relativity.
Gauge redundancy(as used in physics): When a physical system can be described in multiple different mathematical ways that all represent the same real situation—like how you can describe a location using different coordinate systems but it's still the same place.
Hamiltonian(Classical mechanics.): A representation of the energetic and other properties of a system that characterises its evolution over time.

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