We present a structural formulation of physical dynamics based on discrete unitary propagation, in which the time evolution is directly described by a unitary operator
W^=f(H^) associated with a self-adjoint structural operator
H^. Starting from the relationship between action and phase,
ϕ=S/ℏ, and the variational construction of structural operators, we show that the propagation can be written exactly as the action of a bilocal kernel
W(x,y) on the state, in continuity with recent developments of the Action-Phase program. We show that the spectral decomposition of
H^ induces a diagonalization of the unitary propagation, in which the state evolution becomes multiplicative in the eigenvector basis of the structural operator. Consequently, the spectral structure organizes the propagation modes of the system, providing a description in which the dynamics is expressed in a decoupled manner between modes. It is evidenced that the distinction between nodal representation and spectral representation emerges naturally from the operator structure, corresponding to different decompositions of the same state in distinct bases of the same Hilbert space. In the case of translational invariance, the decomposition into Fourier modes is recovered as a particular situation. The results presented do not constitute a complete reconstruction of quantum mechanics nor a general gravitational theory. In particular, aspects of measurement, multiparticle structures, or complete covariant formulations are not treated. The objective of this work is more restricted and precise: to establish a formulation in which the dynamics is determined by the propagation operator and organized by its spectral decomposition, providing a structural layer from which more complete physical descriptions can be developed.
We present a structural formulation of physical dynamics in which time evolution is directly described by a unitary propagation operator, rather than being primarily introduced by continuous differential equations. Starting from the relationship between action and phase,
ϕ=S/ℏ, and the variational construction of self-adjoint structural operators, it is shown that propagation can be exactly expressed through a bilocal kernel
W(x,y) associated with an operator
W^=f(H^), with
H^=H^†. We demonstrate that the structural homogeneity of the kernel, characterized by the condition
W(x,y)=W(x−y), implies the spectral diagonalization of the propagation operator. In this regime, the evolution becomes decoupled between modes, and the absence of spectral mixing provides a minimal structural expression of conservation. It is further shown that the off-diagonal structure of the bilocal kernel completely determines the connectivity of propagation between states, allowing the effective locality of the dynamics to be characterized in terms of the support of
W(x,y). The analysis establishes, in a controlled manner, the relationship between the bilocal description and the Lagrangian formulation, indicating that the operatorial structure replaces, at the considered level, the need for a continuous sum over trajectories, without excluding its recovery as an effective limit in appropriate regimes. The obtained results show that dynamics can be entirely organized in terms of the propagation operator and its spectral decomposition, in which the nodal representation and the spectral representation emerge as complementary descriptions of the same structure. The scope of this work is strictly structural: it does not propose a complete reconstruction of quantum mechanics, nor a general gravitational theory, nor a formulation for multiparticle systems or measurement processes. The objective is to establish a minimal layer in which physical propagation is encoded by the unitary operator and its spectral structure.