We consider a non-relativistic space-time $\mathbb{R}^4 \cong \mathbb{R}^3\times\mathbb{R}$ and embed the beam centreline as a two-dimensional world sheet $\mathcal W\subset\mathbb{R}^4$. In analogy to shell theory, we attach a director frame to $\mathcal W$ by lifting the classical beam triad through the canonical embedding of $\mathrm{SO}(3)$ into $\mathrm{SO}(4)$, thereby allowing spatial rotations while keeping the temporal direction fixed. This yields a covariant, shell-like representation of the Lagrangian action for the geometrically exact beam, in which the stationarity condition can be written intrinsically on $\mathcal W$. Based on this formulation, we derive Noether-type identities in both the continuous and discrete settings. In the continuous case, the covariant balance laws written in the strong form directly imply the conservation of linear and angular momentum under vanishing external actions and traction free boundaries. The same applies to the derivation of the local energy balance. In the discrete case, we distinguish algorithmic from asymptotic conservation: invariance of the discrete action under the Euclidean group SE(3) induces mesh independent discrete momentum-map balances (in the sense of a discrete Noether theorem for the space-time Galerkin/IGA discretization),whereas energy conservation is, in general, only recovered asymptotically under temporal refinement due to broken time-translation symmetry on a fixed space-time mesh. We analyze the resulting energy defect and trace it to a symmetry break at the level of admissible discrete test fields: the time-translation generator involves temporal rates (e.g. \(\mathbf{r}^h_{,t}\) and \(\mathbf{\omega}^h\)) that are generally not representable in the chosen $H^1$-conforming space-time ansatz. This mechanism is conceptually distinct from the hyperelastic secant (discrete chain-rule) defect addressed by Gonzalez-type discrete-gradient corrections. Effects on stability and possible remedies are discussed using several benchmark examples.