Variational Frustration, Gauge Connection, and Magnetic Laplacian: A Structural Extension of the Phase Operator

Jonatan P. camargo

Published Mar 21, 2026• 0 citations

Abstract

Continuing the program in which the action constitutes the primary structure of dynamics and the phase $\phi=S/\hbar$ plays the role of the operational variable, we investigate the relationship between global variational frustration and the emergence of gauge connections in the structural operator of the network. In previous works, it was shown that local extremizing trajectories of the action are only physically realizable when compatible with a global spectral condition, formalized by the Laplacian operator associated with the admissible network, and that electromagnetism can be reinterpreted as a local modulation of the phase by a $U(1)$ connection. The present article unifies these two directions. We show, first, that the Geometric Spectral Stability Principle (GSSP) does not replace the action as a fundamental structure, but acts as a superselection principle: it restricts the set of locally admissible variational configurations to those that remain globally stable. Next, we demonstrate that the incompatibility between local extrema and global coherence --- expressed by the inequality between the global minimum and the naive sum of local minima --- can be encoded as phase holonomy along cycles of the graph. This holonomy naturally leads to the promotion of the real adjacency matrix $A_{ij}$ to a complex Hermitian matrix $\widetilde{A}_{ij}=A_{ij}e^{i\theta_{ij}}$, in which $\theta_{ij}$ represents a discrete $U(1)$ connection. This results in a magnetic Laplacian \[ \widetilde{K}=D-\widetilde{A}, \] whose global quadratic functional \[ \widetilde{S}_{\mathrm{global}}[\Psi] = \frac12\sum_{(i,j)\in E}A_{ij}\left|\psi_i-e^{i\theta_{ij}}\psi_j\right|^2 \] provides the minimal form compatible with gauge, self-adjointness, and spectral stability. We show that the spectral response of the eigenvalues $\widetilde{\lambda}_n$ to the holonomy defines the structural route for the emergence of charged states. Spin is not derived in this work; however, we argue that local frustrations incompatible with a scalar representation naturally point to a multicomponent extension of the formalism. The central result is that topological frustration should not be treated as a defect of the theory, but as the structural source that promotes the real combinatorial operator to a gauge covariant operator. Thus, mass and charge come to be seen as complementary manifestations of the same variational-spectral problem: mass associated with the spectrum of the admissible structural operator, and charge associated with the response of this spectrum to the holonomy connection.

DOI: 10.5281/zenodo.19151461

Related papers

Electromagnetism as Phase Modulation: A Structural Formulation in Terms of Connection, Gauge, and Holonomy

Jonatan P. camargo

We continue the program in which action is treated as a primary structure and the phase \[ \phi=\frac{S}{\hbar} \] 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 $A_\mu$, such that the minimal substitution \[ \partial_\mu \phi \;\longrightarrow\; \partial_\mu \phi - \frac{e}{\hbar}A_\mu \] 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.

Mar 19, 20260 citations

The Emergence of Classical Mechanics and Relativistic Geometry from the Interferential Structure of Phase. From the derivation of L= T - V to the exact deflection of light in dynamic compatibility networks.

Jonatan P. camargo

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 $e^{iS/\hbar}$; 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 = \alpha K + \mathrm{diag}(V) $ \cite{vrev}. In the continuous limit, the network Laplacian converges to the differential operator $-\nabla^2$, recovering the effective Schrödinger equation. The semiclassical decomposition $ \psi = a\,e^{iS/\hbar} $ leads, to leading order in $\hbar$, to the Hamilton-Jacobi equation \[ \partial_t S + \frac{(\nabla S)^2}{2m} + V = 0, \] from which we identify the effective Hamiltonian $ H = \frac{p^2}{2m} + V $. Applying the Legendre transform, we then obtain the Lagrangian \[ L = \frac{1}{2}m\dot{x}^2 - V, \] 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 $g_{00} \approx 1 + 2\Phi/c^2$. Under the stationary phase condition for null geodesics, we derive the exact gravitational refractive index. This produces the Einstein light deflection angle ($4GM/c^2b$) 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.

Mar 5, 20260 citations

Structural Consequences of the Universal Action Principle. Global Phase, Action Constant, and Spectral Stability.

Jonatan P. camargo

Feb 27, 20260 citations

Spectral Mechanics of Phase: Resolvent Structure, Bilocal Connectivity, and the Emergence of Massive States

Jonatan P. camargo

Apr 17, 20260 citations

Observables and Measurement in Spectral Phase Mechanics: A Formulation via Spectral Measure

Jonatan P. camargo

May 2, 20260 citations

Spectral Selection and Admissible Structures: Variational criteria for the physical organization of the spectrum

Jonatan P. camargo

Apr 14, 20260 citations

Variational Frustration, Gauge Connection, and Magnetic Laplacian: A Structural Extension of the Phase Operator

Jonatan P. camargo

Published Mar 21, 2026• 0 citations

Publisher

Abstract

Continuing the program in which the action constitutes the primary structure of dynamics and the phase (\phi=S/\hbar) plays the role of the operational variable, we investigate the relationship between global variational frustration and the emergence of gauge connections in the structural operator of the network. In previous works, it was shown that local extremizing trajectories of the action are only physically realizable when compatible with a global spectral condition, formalized by the Laplacian operator associated with the admissible network, and that electromagnetism can be reinterpreted as a local modulation of the phase by a (U(1)) connection. The present article unifies these two directions. We show, first, that the Geometric Spectral Stability Principle (GSSP) does not replace the action as a fundamental structure, but acts as a superselection principle: it restricts the set of locally admissible variational configurations to those that remain globally stable. Next, we demonstrate that the incompatibility between local extrema and global coherence --- expressed by the inequality between the global minimum and the naive sum of local minima --- can be encoded as phase holonomy along cycles of the graph. This holonomy naturally leads to the promotion of the real adjacency matrix (A_{ij}) to a complex Hermitian matrix (\widetilde{A}{ij}=A{ij}e^{i\theta_{ij}}), in which (\theta_{ij}) represents a discrete (U(1)) connection. This results in a magnetic Laplacian [ \widetilde{K}=D-\widetilde{A}, ] whose global quadratic functional [ \widetilde{S}{\mathrm{global}}[\Psi] = \frac12\sum{(i,j)\in E}A_{ij}\left|\psi_i-e^{i\theta_{ij}}\psi_j\right|^2 ] provides the minimal form compatible with gauge, self-adjointness, and spectral stability. We show that the spectral response of the eigenvalues (\widetilde{\lambda}_n) to the holonomy defines the structural route for the emergence of charged states. Spin is not derived in this work; however, we argue that local frustrations incompatible with a scalar representation naturally point to a multicomponent extension of the formalism. The central result is that topological frustration should not be treated as a defect of the theory, but as the structural source that promotes the real combinatorial operator to a gauge covariant operator. Thus, mass and charge come to be seen as complementary manifestations of the same variational-spectral problem: mass associated with the spectrum of the admissible structural operator, and charge associated with the response of this spectrum to the holonomy connection.

DOI: 10.5281/zenodo.19151461

Related papers

Electromagnetism as Phase Modulation: A Structural Formulation in Terms of Connection, Gauge, and Holonomy

Jonatan P. camargo

We continue the program in which action is treated as a primary structure and the phase $\phi=\frac{S}{\hbar}$ 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 $A_\mu$ , such that the minimal substitution $\partial_\mu \phi \;\longrightarrow\; \partial_\mu \phi - \frac{e}{\hbar}A_\mu$ 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 continue the program in which action is treated as a primary structure and the phase \[ \phi=\frac{S}{\hbar} \] 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 $A_\mu$, such that the minimal substitution \[ \partial_\mu \phi \;\longrightarrow\; \partial_\mu \phi - \frac{e}{\hbar}A_\mu \] 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.

Mar 19, 20260 citations

Jonatan P. camargo

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 $e^{iS/\hbar}$ ; 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 $H = \alpha K + \mathrm{diag}(V)$ \cite{vrev}. In the continuous limit, the network Laplacian converges to the differential operator $-\nabla^2$ , recovering the effective Schrödinger equation. The semiclassical decomposition $\psi = a\,e^{iS/\hbar}$ leads, to leading order in $\hbar$ , to the Hamilton-Jacobi equation $\partial_t S + \frac{(\nabla S)^2}{2m} + V = 0,$ from which we identify the effective Hamiltonian $H = \frac{p^2}{2m} + V$ . Applying the Legendre transform, we then obtain the Lagrangian $L = \frac{1}{2}m\dot{x}^2 - V,$ 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 $g_{00} \approx 1 + 2\Phi/c^2$ . Under the stationary phase condition for null geodesics, we derive the exact gravitational refractive index. This produces the Einstein light deflection angle ( $4GM/c^2b$ ) 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.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 $e^{iS/\hbar}$; 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 = \alpha K + \mathrm{diag}(V) $ \cite{vrev}. In the continuous limit, the network Laplacian converges to the differential operator $-\nabla^2$, recovering the effective Schrödinger equation. The semiclassical decomposition $ \psi = a\,e^{iS/\hbar} $ leads, to leading order in $\hbar$, to the Hamilton-Jacobi equation \[ \partial_t S + \frac{(\nabla S)^2}{2m} + V = 0, \] from which we identify the effective Hamiltonian $ H = \frac{p^2}{2m} + V $. Applying the Legendre transform, we then obtain the Lagrangian \[ L = \frac{1}{2}m\dot{x}^2 - V, \] 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 $g_{00} \approx 1 + 2\Phi/c^2$. Under the stationary phase condition for null geodesics, we derive the exact gravitational refractive index. This produces the Einstein light deflection angle ($4GM/c^2b$) 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.

Mar 5, 20260 citations

Structural Consequences of the Universal Action Principle. Global Phase, Action Constant, and Spectral Stability.

Jonatan P. camargo

Starting from only three ingredients: (i) variational formulation with cumulative action, (ii) interferential dependence on $S$ , and (iii) PFu's episodic compatibility theorem, this paper assumes a single additional hypothesis of the universality of the action constant ( $h$ ) and a minimal hypothesis of macroscopic stability, and derives mathematical consequences in an axiomatic and constructive manner, without introducing new fields, potentials, or \textit{ad hoc} phenomenological terms. Three main results are obtained. First, it is demonstrated that the existence of a universal action constant $h$ is a structural requirement for interferential consistency among multiple systems: the phase $\phi$ must be treated as a global structure, defined modulo $h$ , common to all admissible processes. Second, it is shown that dynamic compatibility between systems can only occur at discrete events of phase coincidence, characterized by $\delta\phi = 0$ ; the observable dynamics are, therefore, reconstructible as sampling over a discrete sequence of compatibility events, whose density is proportional to the rate of variation of $S$ (abstract relativistic sampling theorem). Third, under hypotheses of invariance, extensivity, and coercivity, a global functional $\mathcal{I}$ is constructed that measures the compatibility stability of a configuration of systems, and it is proven that, in the infrared regime, it is equivalent to a quadratic spectral functional $Q = \langle \psi, \mathcal{A} \psi \rangle$ , where $\mathcal{A}$ is a positive self-adjoint operator that encodes the compatibility structure. In the continuous limit, when $\mathcal{A}$ admits a realization as an effective elliptic operator and its spectrum is described by a spectral density $\rho(\lambda)$ , the abstract functional $\mathcal{I}$ is rewritten as a geometric functional of the PESG type, but now interpreted as a consequence of universal action and episodic compatibility, rather than as an independent geometric principle.Starting from only three ingredients: (i) variational formulation with cumulative action, (ii) interferential dependence on $S$, and (iii) PFu's episodic compatibility theorem, this paper assumes a single additional hypothesis of the universality of the action constant ($h$) and a minimal hypothesis of macroscopic stability, and derives mathematical consequences in an axiomatic and constructive manner, without introducing new fields, potentials, or \textit{ad hoc} phenomenological terms. Three main results are obtained. First, it is demonstrated that the existence of a universal action constant $h$ is a structural requirement for interferential consistency among multiple systems: the phase $\phi$ must be treated as a global structure, defined modulo $h$, common to all admissible processes. Second, it is shown that dynamic compatibility between systems can only occur at discrete events of phase coincidence, characterized by $\delta\phi = 0$; the observable dynamics are, therefore, reconstructible as sampling over a discrete sequence of compatibility events, whose density is proportional to the rate of variation of $S$ (abstract relativistic sampling theorem). Third, under hypotheses of invariance, extensivity, and coercivity, a global functional $\mathcal{I}$ is constructed that measures the compatibility stability of a configuration of systems, and it is proven that, in the infrared regime, it is equivalent to a quadratic spectral functional $Q = \langle \psi, \mathcal{A} \psi \rangle$, where $\mathcal{A}$ is a positive self-adjoint operator that encodes the compatibility structure. In the continuous limit, when $\mathcal{A}$ admits a realization as an effective elliptic operator and its spectrum is described by a spectral density $\rho(\lambda)$, the abstract functional $\mathcal{I}$ is rewritten as a geometric functional of the PESG type, but now interpreted as a consequence of universal action and episodic compatibility, rather than as an independent geometric principle.

Feb 27, 20260 citations

Spectral Mechanics of Phase: Resolvent Structure, Bilocal Connectivity, and the Emergence of Massive States

Jonatan P. camargo

We develop a spectral formulation of physical dynamics based on the action phase $\phi = S/\hbar$ , in which the evolution is governed by Cayley-type unitary operators and organized through the resolvent structure of the underlying structural operator. We show that the propagation can be described by a bilocal kernel associated with the resolvent, whose spectral decomposition defines independent modes of evolution. In this formalism, physical connectivity emerges as an intrinsically nonlocal property of the resolvent kernel, and physical states can be interpreted as stabilized spectral modes. We demonstrate that the existence of massive states is conditioned on the presence of a global phase structure, encoded by holonomy, which prevents the variational collapse of the structural coupling. We also show that phase factors associated with gauge connections admit an interpretation as a manifestation of the accumulated phase of the resolvent under discrete reduction. From this structure, we derive finite-dimensional effective operators as infrared spectral projections of the global operator, establishing the emergence of tridiagonal operators as natural approximations of low-energy dynamics. Finally, we demonstrate that the limitations observed in truncated effective models stem from the suppression of the tail of the bilocal kernel, which dominantly affects the description of highly delocalized states. In particular, the discrepancy associated with the ground state is interpreted as a structural consequence of the loss of global connectivity induced by spectral truncation. These results allow formulating the dynamics as a process of spectral filtering of the action phase, in which structural operators act as unitary filters and physical states emerge as stabilized spectral modes of the resolvent. This perspective leads to a framework we call Spectral Mechanics of Phase.We develop a spectral formulation of physical dynamics based on the action phase $\phi = S/\hbar$, in which the evolution is governed by Cayley-type unitary operators and organized through the resolvent structure of the underlying structural operator. We show that the propagation can be described by a bilocal kernel associated with the resolvent, whose spectral decomposition defines independent modes of evolution. In this formalism, physical connectivity emerges as an intrinsically nonlocal property of the resolvent kernel, and physical states can be interpreted as stabilized spectral modes. We demonstrate that the existence of massive states is conditioned on the presence of a global phase structure, encoded by holonomy, which prevents the variational collapse of the structural coupling. We also show that phase factors associated with gauge connections admit an interpretation as a manifestation of the accumulated phase of the resolvent under discrete reduction. From this structure, we derive finite-dimensional effective operators as infrared spectral projections of the global operator, establishing the emergence of tridiagonal operators as natural approximations of low-energy dynamics. Finally, we demonstrate that the limitations observed in truncated effective models stem from the suppression of the tail of the bilocal kernel, which dominantly affects the description of highly delocalized states. In particular, the discrepancy associated with the ground state is interpreted as a structural consequence of the loss of global connectivity induced by spectral truncation. These results allow formulating the dynamics as a process of spectral filtering of the action phase, in which structural operators act as unitary filters and physical states emerge as stabilized spectral modes of the resolvent. This perspective leads to a framework we call Spectral Mechanics of Phase.

Apr 17, 20260 citations

Observables and Measurement in Spectral Phase Mechanics: A Formulation via Spectral Measure

Jonatan P. camargo

In the Action--Phase program, physical dynamics are formulated from the action $S$ as the primary structure, with the phase $\phi = S/\hbar$ playing the role of the universal operational variable. Previous works have established the emergence of quantum dynamics, geometric structure, and gauge interactions as different modes of action upon the phase, as well as the closure of the relationship between spectrum, energy, and momentum. However, it remains to formulate intrinsically the notion of an observable and the measurement process, without the introduction of independent postulates. In this work, we show that this gap can be naturally filled through the spectral theory of self-adjoint operators. In particular, we demonstrate that physical observables are functionals of the spectral measure associated with the state and the relevant structural operator, while measurement corresponds to the effective restriction of this measure to dominant spectral regions. In this framework, the Born rule is recovered as a particular case in the discrete spectrum regime, while in more general situations, physical results are described by spectral regions with finite width. The proposed formulation eliminates the need for additional axioms for observables and measurement, unifying the dynamical and operational structure of quantum mechanics within the same fundamental principle based on action.In the Action--Phase program, physical dynamics are formulated from the action $S$ as the primary structure, with the phase $\phi = S/\hbar$ playing the role of the universal operational variable. Previous works have established the emergence of quantum dynamics, geometric structure, and gauge interactions as different modes of action upon the phase, as well as the closure of the relationship between spectrum, energy, and momentum. However, it remains to formulate intrinsically the notion of an observable and the measurement process, without the introduction of independent postulates. In this work, we show that this gap can be naturally filled through the spectral theory of self-adjoint operators. In particular, we demonstrate that physical observables are functionals of the spectral measure associated with the state and the relevant structural operator, while measurement corresponds to the effective restriction of this measure to dominant spectral regions. In this framework, the Born rule is recovered as a particular case in the discrete spectrum regime, while in more general situations, physical results are described by spectral regions with finite width. The proposed formulation eliminates the need for additional axioms for observables and measurement, unifying the dynamical and operational structure of quantum mechanics within the same fundamental principle based on action.

May 2, 20260 citations

Spectral Selection and Admissible Structures: Variational criteria for the physical organization of the spectrum

Jonatan P. camargo

The spectral formulation of dynamics, based on self-adjoint structural operators, allows organizing the evolution of physical systems in terms of independent eigenmodes. However, the spectral decomposition, although mathematically complete, does not provide, by itself, a criterion of physical admissibility. In this work, a variational principle is introduced in the space of structural operators, in the form of a spectral functional whose stationarity condition imposes restrictions on the spectrum. It is shown that physical admissibility must be formulated as a spectral selection problem, in which only specific subsets of eigenvalues are compatible with conditions of structural stability. It is demonstrated that spectral decomposition does not determine the admissibility of the structure, and that the introduction of an independent variational criterion leads to non-trivial relations between the eigenvalues of the operator. As a consequence, it is obtained that admissibility implies spectral restrictions, establishing a direct connection between the operatorial organization of dynamics and underlying variational principles. The scope of the work is strictly structural. A direct identification between eigenvalues and specific physical quantities is not established, nor is a complete phenomenological theory proposed. The objective is to determine the minimum level at which physical selection can be formulated as a consequence of spectral properties and stability criteria.The spectral formulation of dynamics, based on self-adjoint structural operators, allows organizing the evolution of physical systems in terms of independent eigenmodes. However, the spectral decomposition, although mathematically complete, does not provide, by itself, a criterion of physical admissibility. In this work, a variational principle is introduced in the space of structural operators, in the form of a spectral functional whose stationarity condition imposes restrictions on the spectrum. It is shown that physical admissibility must be formulated as a spectral selection problem, in which only specific subsets of eigenvalues are compatible with conditions of structural stability. It is demonstrated that spectral decomposition does not determine the admissibility of the structure, and that the introduction of an independent variational criterion leads to non-trivial relations between the eigenvalues of the operator. As a consequence, it is obtained that admissibility implies spectral restrictions, establishing a direct connection between the operatorial organization of dynamics and underlying variational principles. The scope of the work is strictly structural. A direct identification between eigenvalues and specific physical quantities is not established, nor is a complete phenomenological theory proposed. The objective is to determine the minimum level at which physical selection can be formulated as a consequence of spectral properties and stability criteria.

Apr 14, 20260 citations

We continue the program in which action is treated as a primary structure and the phase

\phi=\frac{S}{\hbar}

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

A_\mu

, such that the minimal substitution

\partial_\mu \phi \;\longrightarrow\; \partial_\mu \phi - \frac{e}{\hbar}A_\mu

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.

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

e^{iS/\hbar}

; 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 = \alpha K + \mathrm{diag}(V)

\cite{vrev}. In the continuous limit, the network Laplacian converges to the differential operator

-\nabla^2

, recovering the effective Schrödinger equation. The semiclassical decomposition

\psi = a\,e^{iS/\hbar}

leads, to leading order in

\hbar

, to the Hamilton-Jacobi equation

\partial_t S + \frac{(\nabla S)^2}{2m} + V = 0,

from which we identify the effective Hamiltonian

H = \frac{p^2}{2m} + V

. Applying the Legendre transform, we then obtain the Lagrangian

L = \frac{1}{2}m\dot{x}^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

g_{00} \approx 1 + 2\Phi/c^2

. Under the stationary phase condition for null geodesics, we derive the exact gravitational refractive index. This produces the Einstein light deflection angle (

4GM/c^2b

) 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.

Starting from only three ingredients: (i) variational formulation with cumulative action, (ii) interferential dependence on

S

, and (iii) PFu's episodic compatibility theorem, this paper assumes a single additional hypothesis of the universality of the action constant (

h

) and a minimal hypothesis of macroscopic stability, and derives mathematical consequences in an axiomatic and constructive manner, without introducing new fields, potentials, or \textit{ad hoc} phenomenological terms. Three main results are obtained. First, it is demonstrated that the existence of a universal action constant

h

is a structural requirement for interferential consistency among multiple systems: the phase

\phi

must be treated as a global structure, defined modulo

h

, common to all admissible processes. Second, it is shown that dynamic compatibility between systems can only occur at discrete events of phase coincidence, characterized by

\delta\phi = 0

; the observable dynamics are, therefore, reconstructible as sampling over a discrete sequence of compatibility events, whose density is proportional to the rate of variation of

S

(abstract relativistic sampling theorem). Third, under hypotheses of invariance, extensivity, and coercivity, a global functional

\mathcal{I}

is constructed that measures the compatibility stability of a configuration of systems, and it is proven that, in the infrared regime, it is equivalent to a quadratic spectral functional

Q = \langle \psi, \mathcal{A} \psi \rangle

, where

\mathcal{A}

is a positive self-adjoint operator that encodes the compatibility structure. In the continuous limit, when

\mathcal{A}

admits a realization as an effective elliptic operator and its spectrum is described by a spectral density

\rho(\lambda)

, the abstract functional

\mathcal{I}

is rewritten as a geometric functional of the PESG type, but now interpreted as a consequence of universal action and episodic compatibility, rather than as an independent geometric principle.

We develop a spectral formulation of physical dynamics based on the action phase

\phi = S/\hbar

, in which the evolution is governed by Cayley-type unitary operators and organized through the resolvent structure of the underlying structural operator. We show that the propagation can be described by a bilocal kernel associated with the resolvent, whose spectral decomposition defines independent modes of evolution. In this formalism, physical connectivity emerges as an intrinsically nonlocal property of the resolvent kernel, and physical states can be interpreted as stabilized spectral modes. We demonstrate that the existence of massive states is conditioned on the presence of a global phase structure, encoded by holonomy, which prevents the variational collapse of the structural coupling. We also show that phase factors associated with gauge connections admit an interpretation as a manifestation of the accumulated phase of the resolvent under discrete reduction. From this structure, we derive finite-dimensional effective operators as infrared spectral projections of the global operator, establishing the emergence of tridiagonal operators as natural approximations of low-energy dynamics. Finally, we demonstrate that the limitations observed in truncated effective models stem from the suppression of the tail of the bilocal kernel, which dominantly affects the description of highly delocalized states. In particular, the discrepancy associated with the ground state is interpreted as a structural consequence of the loss of global connectivity induced by spectral truncation. These results allow formulating the dynamics as a process of spectral filtering of the action phase, in which structural operators act as unitary filters and physical states emerge as stabilized spectral modes of the resolvent. This perspective leads to a framework we call Spectral Mechanics of Phase.

In the Action--Phase program, physical dynamics are formulated from the action

S

as the primary structure, with the phase

\phi = S/\hbar

playing the role of the universal operational variable. Previous works have established the emergence of quantum dynamics, geometric structure, and gauge interactions as different modes of action upon the phase, as well as the closure of the relationship between spectrum, energy, and momentum. However, it remains to formulate intrinsically the notion of an observable and the measurement process, without the introduction of independent postulates. In this work, we show that this gap can be naturally filled through the spectral theory of self-adjoint operators. In particular, we demonstrate that physical observables are functionals of the spectral measure associated with the state and the relevant structural operator, while measurement corresponds to the effective restriction of this measure to dominant spectral regions. In this framework, the Born rule is recovered as a particular case in the discrete spectrum regime, while in more general situations, physical results are described by spectral regions with finite width. The proposed formulation eliminates the need for additional axioms for observables and measurement, unifying the dynamical and operational structure of quantum mechanics within the same fundamental principle based on action.