Energy Bounds Without Codomain Persistence in Time-Dependent Hilbert Structures

This paper establishes a structural non-existence result for terminal states in time-dependent Hilbert frameworks under loss of uniform coercivity. It proves that bounded local energy does not imply extendability when the underlying family of Hilbert structures degenerates, because no compatible Hilbert or reflexive Banach codomain can persist at the terminal time. The obstruction is functional-analytic and independent of blow-up, loss of regularity, or compactness failure.