We propose a unified formulation, at a structural level, in which action occupies the primary role and the phase \(\phi=S/\hbar\) functions as a universal intermediate variable. It is shown that, from this identification, two distinct formal directions emerge naturally. In the first, the phase structure leads to the wave description of matter, to the limits of spectral localization, and to the linear form of quantum dynamics. In the second, the phase determines eikonal propagation, geodesic structure, and, under minimal variational hypotheses, an effective geometry associated with gravity. The objective of this work is not to fully deduce quantum mechanics and general relativity from a single closed microscopic model, but to make explicit that both can be reinterpreted as complementary realizations of the same variational architecture based on action, phase coherence, and effective structure. In this sense, action ceases to be merely an auxiliary functional of dynamics and assumes the role of a primary structure common to the quantum and gravitational regimes.
We propose a unified formulation, at a structural level, in which action occupies the primary role and the phase (\phi=S/\hbar) functions as a universal intermediate variable. It is shown that, from this identification, two distinct formal directions emerge naturally. In the first, the phase structure leads to the wave description of matter, to the limits of spectral localization, and to the linear form of quantum dynamics. In the second, the phase determines eikonal propagation, geodesic structure, and, under minimal variational hypotheses, an effective geometry associated with gravity. The objective of this work is not to fully deduce quantum mechanics and general relativity from a single closed microscopic model, but to make explicit that both can be reinterpreted as complementary realizations of the same variational architecture based on action, phase coherence, and effective structure. In this sense, action ceases to be merely an auxiliary functional of dynamics and assumes the role of a primary structure common to the quantum and gravitational regimes.
In this work, we propose a structural reformulation of the Action--Phase program based on the interpretation of operators as unitary linear filters acting on the physical phase ϕ=S/ℏ. The central thesis is that the various effective sectors of physics can be reinterpreted as regimes of operator action on the phase and its gradient, rather than independent collections of final equations. We start from the chain already established in previous works --- action → phase → episodic compatibility → event network → structural operator --- and show that the emergent local linear dynamics admits a natural reading in terms of filter theory. In particular, the effective network operator can be seen as a transfer function, whose Cayley realization defines a local unitary all-pass filter of strictly unitary modulus. This structure provides a direct interpretation of unitarity as information conservation. We also show that the spatial variation of the local coefficients of the operator induces modulation of the phase response, leading to the definition of a local quasi-energy Ω(x,k). In the slowly varying regime, this structure reproduces an effective dynamics governed by Hamilton--Jacobi-type equations, recovering trajectories previously obtained via the eikonal approach. In this framework, gravity can be reinterpreted as the geometric optics of phase propagation in a structurally inhomogeneous network. As an operational illustration, we present a comparative numerical experiment between discrete unitary evolution (Cayley realization) and explicit integration schemes of the semidiscrete Schrödinger equation. The results show that the unitary formulation preserves the norm at machine-level precision and remains stable in intense potential regimes, while explicit schemes exhibit accumulated drift and more severe time-step restrictions. This evidences that dynamics can be implemented as the iterative application of local unitary operators. The results do not constitute a complete gravitational theory nor do they replace established continuous formulations, but they identify a unifying mathematical structure that organizes the propagation, modulation, and refraction of the phase in a non-ad hoc manner and without circular reasoning. This structure suggests an extension program based on derivations and direct numerical tests, aimed at the explicit recovery of known gravitational results from phase dynamics.
We continue the program in which action is treated as a primary structure and the phase ϕ=ℏS as its associated operational variable. In previous works, the phase structure was connected to quantum dynamics and effective gravitational geometry. In the present article, we focus exclusively on the electromagnetic sector. We show that the electromagnetic coupling can be formulated as a local modulation of the phase by a 1-form Aμ, such that the minimal substitution ∂μϕ⟶∂μϕ−ℏeAμ emerges naturally as a way to preserve phase coherence in the presence of external interaction. In this language, the electromagnetic field acts as a phase connection, gauge invariance expresses the redundancy of the absolute phase reference, and the Aharonov-Bohm effect appears as a direct experimental manifestation of the physical reality of the connection's holonomy. The objective of this work is not to fully derive Maxwell's equations from a new microscopic model, but to make explicit that the formal structure of electromagnetism is compatible with a formulation based on phase, connection, and coherence. In this sense, the electromagnetic interaction can be reinterpreted as an active phase modulation, in contrast to the role of gravity as a passive propagation structure.
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.
This work presents a minimal and unified formulation of physical reality based on a single coupled system of two scalar fields: a phase S(x) and a realization density ρ(x). The dynamics is completely determined by two principles: conservation of a realization current and a geometric constraint linking the norm of the phase gradient to the structure of the density field. Together, these relations form a closed system with no additional dynamical assumptions. From this system, all major physical regimes emerge as limiting cases. When the density is uniform, propagation is null and describes lightlike behavior. When the geometric constraint reduces to a constant, the system reproduces classical Hamilton–Jacobi dynamics for massive particles. When the density varies significantly, the coupled system yields quantum behavior, including the Schrödinger equation in the appropriate limit. In this framework, the wave function is not a fundamental entity but an emergent representation of the underlying geometric structure of realization. Interference arises from phase consistency, and quantum behavior reflects the nontrivial structure of the realization density rather than intrinsic indeterminism. The theory also naturally incorporates gravitation by promoting the spacetime metric to a dynamical field. In the appropriate limit, the stress–energy tensor reduces to that of classical matter, and Einstein’s field equations are recovered. Regions where the realization density approaches zero define a geometric boundary where timelike realization ceases, corresponding to horizon-like structures. The central result is that a single coupled equation governs propagation, mass, quantum behavior, and gravitational structure. Physical reality is determined by global consistency of realization rather than by selection among multiple possibilities. This formulation provides a minimal geometric unification of light, matter, quantum phenomena, and gravity within a single coherent framework.
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.
The possibility of distinct physical systems remaining dynamically compatible over time is often implicitly assumed in various areas of physics, from interferometric descriptions to clock comparisons and the analysis of composite systems. In this work, we show that such continuous compatibility is not a generic property but encounters a structural limitation that stems directly from the variational formulation of dynamics. Starting from only minimal hypotheses—the existence of an accumulative action, monotonic proper parametrizations, and operational dependence on action differences via factors eiS/ℏ—we demonstrate that dynamically independent systems cannot maintain functional coincidence of their actions over open intervals, except under fine-tuning. Consequently, moments of compatibility form a necessarily discrete set, and any coherent coupling is effective only within a time scale determined by the difference between the rates of action accumulation. The result does not depend on a specific microscopic model, nor on dissipative, statistical, or environmental mechanisms, being consistent with classical mechanics, relativity, and quantum field theory. It is shown, in particular, that effects such as the loss of interferometric visibility associated with time dilation can be understood as physical realizations of a more general constraint: the structural incompatibility between evolutions parameterized by distinct actions. Thus, the so-called ``compatibility window'' does not constitute a new dynamic principle, but an inevitable consequence of the variational architecture common to established physical theories.