In conventional quantum mechanics, all unitary evolution takes place within the spacetime Hilbert space ${\mathcal{H}}_{xt}=L^2\left({\mathcal{M}}_{xt}\right)$, with time as the sole evolution parameter. The momentum-energy representation $\varphi (k,E)$ is treated merely as a Fourier reexpression of the same state: kinematically equivalent but dynamically inert. Here we activate a previously frozen dual description between the conjugate pairs $(x,t)$ and $(k,E)$, by extending quantum theory to an enlarged Hilbert-space setting ${\mathcal{H}}_{\mathrm{total}}$ containing two dynamical sectors, ${\mathcal{H}}_{xt}$ and ${\mathcal{H}}_{kE}$, within which the momentum-energy sector ${\mathcal{H}}_{kE}=L^2\left({\mathcal{M}}_{kE}\right)$ carries its own autonomous unitary evolution generated by a self-adjoint operator $\widehat{\mathcal{T}}$. The resulting framework establishes a fundamental duality between two complementary but dynamically incompatible descriptions of a single global quantum state. Physical influence between the two descriptions is mediated through a projection that enforces a consistent choice of evolution parameter. As illustrative examples of this projection mechanism, we show that the dual-manifold framework generically yields a uniform background energy contribution, as well as a characteristic exponential boundary relation that emerges near a singularity. These structures resemble features encountered in certain gravitational contexts, but arise here independently of any gravitational dynamics. They are presented solely as structural consequences of the dual quantum framework, illustrating how nontrivial large-scale features can emerge from the underlying quantum description.

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• Quantum formalism