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 \[ \text{action} \;\Longrightarrow\; \text{phase} \;\Longrightarrow\; \text{structural operator} \;\Longrightarrow\; \text{unitary filter}, \] 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 \(\Psi\), the local scalar field \(\Theta(x)\), and the quasi-energy \(\Omega(x,k)\), 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.
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 [ \text{action} ;\Longrightarrow; \text{phase} ;\Longrightarrow; \text{structural operator} ;\Longrightarrow; \text{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 (\Psi), the local scalar field (\Theta(x)), and the quasi-energy (\Omega(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.
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.
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 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.
We present a spectral formulation of unitary propagation in which the dynamics is characterized directly by the resolvent structure of a self-adjoint operator. A propagation operator defined by the Cayley transform is considered, whose form allows expressing the evolution in terms of the resolvent of the structural operator, without direct reference to continuous differential equations. It is shown that the bilocal propagation kernel is completely determined by the resolvent, implying that the dynamic connectivity does not reduce, in general, to the local support of the structural operator. In the perturbative regime, a controlled expansion is obtained that characterizes an effective notion of locality, while in the homogeneous regime the evolution is organized by a nonlinear spectral phase associated with the eigenvalues of the operator. From this structure, dynamical quantities such as frequency and group velocity are defined, which emerge as well-defined spectral functions, modulated by nonlinear factors arising from the unitarity of the evolution. Additionally, it is demonstrated that the spectrum responds in a controlled manner to internal perturbations associated with phase parameters, establishing a general mechanism of spectral response. The obtained results are strictly structural and independent of external phenomenological hypotheses. It is concluded that unitary propagation admits a consistent formulation in terms of the resolvent of a self-adjoint operator and its spectral decomposition, providing a minimal level at which connectivity, modal evolution, and response to perturbations are determined internally by the structure of the operator.
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.