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 \(\phi = S/\hbar\). 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 \(\to\) phase \(\to\) episodic compatibility \(\to\) event network \(\to\) 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 \(\Omega(x,k)\). 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.
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 (\phi = S/\hbar). 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 (\to) phase (\to) episodic compatibility (\to) event network (\to) 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 (\Omega(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.
This work develops the direct continuation of the Action--Phase program following the reinterpretation of structural operators as unitary filters and the derivation of the scalar delay equation. The objective is to establish, without ad hoc introductions, circular reasoning, or undue duplication of degrees of freedom, the minimal bilocal formulation in which discrete unitary propagation is treated as the action of a spatial convolution kernel. Starting from the chain action⟹phase⟹structural operator⟹unitary filter, it is initially shown that the Cayley realization of a self-adjoint structural operator admits an exact rewriting as propagation by a strictly unitary bilocal kernel W(x,y) on a discrete spatial basis. Next, this kernel is identified as the minimal structural variable of the new formulation, strictly distinguishing it from the propagated state Ψ, the local scalar field Θ(x), and the quasi-energy Ω(x,k), which are now understood at different levels of the theory's physical hierarchy. Under hypotheses of minimal bilocal locality, structural hermiticity, coercivity, minimal differential order, and minimal coupling to the phase state, the variational functional of the kernel is constructed and the corresponding Euler--Lagrange equation is derived. Finally, it is shown that the previous scalar delay theory does not constitute an independent construction but arises as the local/diagonal sector of the bilocal structure obtained here. The result of this stage is, therefore, precise and limited: to establish the minimal bilocal layer of adaptive unitary propagation over a structural network. The work does not intend, at this stage, to provide a complete theory of multipartite entanglement, turbulence, strong gravitational regimes, or covariant extensions. Its role is more fundamental: to provide the explicit variational basis from which such problems can be reformulated and investigated within a unified framework of discrete unitary propagation.
The shape of the classical Lagrangian L=T−V is universally used in analytical mechanics and in the variational formulation of physics. Despite its conceptual centrality, this structure is historically introduced as a working hypothesis, not usually being derived from more fundamental principles. In this work, we present a systematic derivation of this form from a minimal interferential structure based on the action phase. We start from three physical hypotheses widely observed in interferential phenomena: (i) the existence of an accumulative quantity associated with trajectories, identified as action S; (ii) the dependence of physical amplitudes on phase factors of the type eiS/ℏ; and (iii) the possibility of parametrizing evolutions by monotonic proper parameters. We initially show that action differences between systems generate isolated episodic coincidences, implying an operational discretization of interferential compatibility events. These events naturally organize into a discrete network of interactions. Under minimal variational hypotheses—global phase invariance, adjacency locality, and flux conservation—we demonstrate that the structural operator associated with the network coincides with the combinatorial Laplacian K=D−A, where A is the adjacency matrix and D the degree matrix. The linear dynamics of complex amplitudes on this network, imposing linearity, unitarity, and locality, leads to a unitary generator of the form H=αK+diag(V) \cite{vrev}. In the continuous limit, the network Laplacian converges to the differential operator −∇2, recovering the effective Schrödinger equation. The semiclassical decomposition ψ=aeiS/ℏ leads, to leading order in ℏ, to the Hamilton-Jacobi equation ∂tS+2m(∇S)2+V=0, from which we identify the effective Hamiltonian H=2mp2+V. Applying the Legendre transform, we then obtain the Lagrangian L=21mx˙2−V, showing that the T−V decomposition emerges as a direct consequence of the phase's interferential structure and the spectral dynamics of the network operator. As a physical test and validation of the formalism, we extend the result to the relativistic regime, identifying the action phase with proper time. We show that the local term of the network generator acts as a modifier of the effective spacetime metric, naturally leading to the isotropic form g00≈1+2Φ/c2. Under the stationary phase condition for null geodesics, we derive the exact gravitational refractive index. This produces the Einstein light deflection angle (4GM/c2b) purely as a consequence of phase curvature, without postulating the General Relativity field equations. The central result of the work is that classical mechanics (L=T−V) and the geometric structure of gravitation do not need to be assumed as independent postulates, but emerge simultaneously as effective limits of a variational architecture based on phase interference, episodic discretization, and spectral dynamics in networks.
We present a structural derivation of Schrödinger dynamics from a discrete ontology in which physical space is described by an action graph G=(V,E), rather than a primary continuum. The starting point is not an \emph{a posteriori} numerical discretization of an already assumed continuous differential equation, but the direct definition of a local Hamiltonian on the fundamental network. It is first shown that the exact one-step time evolution is given by a discrete unitary propagator U(Δt)=exp(−ℏiΔtH^). In the one-dimensional homogeneous case with nearest-neighbor coupling and constant potential, the exact spatial symbol of this propagator is obtained. Next, we demonstrate that its first-order minimal local expansion produces a three-coefficient bilateral spatial realization, i.e., a minimal FIR. However, it is also proven that this truncated realization is not unitary when iterated, so it cannot represent the exact fundamental dynamics. The structural correction of the problem naturally leads to the Cayley rational unitary realization, UC(Δt)=(I−2ℏiΔtH^)(I+2ℏiΔtH^)−1, which preserves the norm exactly at the discrete level and reveals that fundamental quantum evolution cannot be reduced to a strictly local FIR law of finite support. Thus, the rigorous result of this work is twofold: the FIR form appears as a first-order minimal local realization, while unitarity requires an exact operator or a rational unitary one whose realization is no longer of finite spatial support. Finally, we show that the Δx,Δt→0 limit recovers the continuous Schrödinger equation. The central conclusion is that continuous quantum dynamics must be understood as the macroscopic shadow of a discrete unitary propagator on an action graph, and not as an ontologically primary law.
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.
The objective of this work is to investigate whether the position--momentum duality can be reformulated, without ad hoc hypotheses and without circular reasoning, as a direct consequence of the spectral structure of bilocal unitary propagation. We start from the architecture already established in the Action--Phase program: the physical phase is given by ϕ=S/ℏ; self-adjoint structural operators organize the dynamics on a discrete lattice; such operators admit interpretation as unitary filters via Cayley realization; and the discrete unitary evolution can be written exactly as propagation by a bilocal kernel W(x,y). The central question is to determine under what hypotheses this bilocal kernel can be represented as a function of a self-adjoint structural operator, W^=f(H^),H^=H^†, and how this structure implies the existence of an intrinsic spectral basis in which the propagation becomes diagonal. It is shown, step by step, that the spectral decomposition of H^ induces a corresponding decomposition of the kernel, W(x,y)=∑nf(λn)ϕn(x)ϕn∗(y), so that the unitary evolution begins to act multiplicatively on the spectral modes, Ψn(t+Δt)=f(λn)Ψn(t), without the introduction of additional structures. In the homogeneous case, it is demonstrated that the eigenfunctions of the structural operator are discrete Fourier modes and that the continuous spectral parameter coincides with the usual wavenumber. In this regime, the spectral representation exactly reproduces the momentum sector of standard quantum mechanics. Outside this case, ``momentum'' is reinterpreted more generally as the spectral coordinate of the operator generating the propagation. The result of this stage is, therefore, conceptually precise: the position--momentum duality does not need to be postulated as an ontological duplication, but emerges as a duality between the nodal basis and the spectral basis of the same structural operator that governs the unitary propagation. The work does not intend, at this stage, to completely reconstruct the theory of canonical observables, nor to definitively resolve multiparticle systems or formulate a general covariant theory. Its role is to establish the minimum layer in which the spectral structure of propagation provides the mathematical origin of position--momentum duality, opening a controlled continuation program for these problems.
We present a variational formulation in which the phase is identified as the operational manifestation of the accumulated action, establishing a structural unification between different physical regimes. It is shown that the fundamental phase of any admissible system is given by ϕ=ℏS, from which it follows that the frequency — and, therefore, the operational content of time — emerges directly from the rate of action accumulation, ω=ℏS˙. The analysis of the relative phase evolution between independent systems leads to the demonstration that interferential coherence is, in general, episodic, being characterized by a discrete set of instants of compatibility. This results in a precise formulation of the physical interaction as Eint=Eesp∩Eϕ, in which the occurrence of observable events is filtered by phase compatibility. It is further shown that massive and massless regimes correspond to different parameterizations of the same phase structure, eliminating the need for distinct ontologies for these cases. The introduction of linear classical operators, such as macroscopic electromagnetic systems, allows us to demonstrate that such operators act exclusively on the observed phase, ϕobs=ϕfund+ϕC, shifting the synchronization instants without altering the underlying variational dynamics. As a result, a rigorous distinction is established between the structural level and the observational level: the action governs the fundamental dynamics, while the observed phase corresponds to its operational projection under reading transformations. Finally, the possibility of an extended variational framework in which the experimental apparatus is incorporated into a global functional Stot is discussed, leaving this construction as a program for future work.