We develop a constructive procedure for the explicit extraction of the physical spectrum in systems described by structural operators with non-abelian internal degrees of freedom, within the context of Spectral Phase Mechanics. The formulation is based on representing the dynamics through the resolvent of self-adjoint operators, allowing the evolution to be interpreted as a spectral filtering process. Starting from a structural operator with $SU(3)$ symmetry, we show that the diagonalization of the internal sector induces a decomposition into independent tridiagonal effective operators. This reduction naturally leads to the factorization of the characteristic polynomial into lower-degree factors, whose coefficients are explicitly determined by the structural parameters of the model. The resolvent analysis associated with this decomposition establishes an intrinsic criterion for spectral selection: the physical relevance of the modes is determined by the pole structure of the operator, without the need for externally imposing additional constraints. We demonstrate, within this class of operators and under the established hypotheses, that the reduction of the observable spectrum emerges from the spectral structure of the system itself. The obtained results provide an explicit mechanism for extracting the physical spectrum, in which the interaction between algebraic factorization and spectral filtering determines the effectively observable modes. This construction reinforces the interpretation of dynamics as spectral filtering of the action phase, offering a natural extension of Spectral Phase Mechanics to systems with non-abelian internal structure.
We develop a constructive procedure for the explicit extraction of the physical spectrum in systems described by structural operators with non-abelian internal degrees of freedom, within the context of Spectral Phase Mechanics. The formulation is based on representing the dynamics through the resolvent of self-adjoint operators, allowing the evolution to be interpreted as a spectral filtering process. Starting from a structural operator with SU(3) symmetry, we show that the diagonalization of the internal sector induces a decomposition into independent tridiagonal effective operators. This reduction naturally leads to the factorization of the characteristic polynomial into lower-degree factors, whose coefficients are explicitly determined by the structural parameters of the model. The resolvent analysis associated with this decomposition establishes an intrinsic criterion for spectral selection: the physical relevance of the modes is determined by the pole structure of the operator, without the need for externally imposing additional constraints. We demonstrate, within this class of operators and under the established hypotheses, that the reduction of the observable spectrum emerges from the spectral structure of the system itself. The obtained results provide an explicit mechanism for extracting the physical spectrum, in which the interaction between algebraic factorization and spectral filtering determines the effectively observable modes. This construction reinforces the interpretation of dynamics as spectral filtering of the action phase, offering a natural extension of Spectral Phase Mechanics to systems with non-abelian internal structure.
We develop a spectral formulation of physical dynamics based on the action phase ϕ=S/ℏ, 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 present a structural formulation of physical dynamics in which temporal evolution is described by a unitary propagation operator W^=f(H^), defined from a self-adjoint structural operator H^. In this context, dynamics is not primarily introduced by differential equations or continuous sums over trajectories but is characterized by the spectral structure of the propagation operator. It is shown that the spectral decomposition of H^ induces an organization of the evolution into independent modes, each governed by a spectral factor f(λn). The unitarity of W^ implies that these factors admit a phase representation, leading to a modal description in terms of spectral phases. It is further demonstrated that frequency emerges as a well-defined spectral function associated with the temporal variation of the phase of each mode. Additionally, it is established that dynamical quantities can be formulated as spectral functions of the eigenvalues of H^, providing a minimal spectral dictionary in which phase, frequency, and dynamical scales are defined intrinsically to the formalism. The results obtained are strictly structural and do not involve the introduction of external phenomenological hypotheses. In particular, this work does not establish a direct identification between eigenvalues and specific physical quantities, nor does it construct a complete theory of interactions or measurement. It is concluded that the spectral structure of the propagation operator provides the minimum level at which dynamics can be organized, establishing a formal basis for the development of future extensions of the formalism toward the description of specific physical systems.
In the Action--Phase program, physical dynamics are formulated from the action S as the primary structure, with the phase ϕ=S/ℏ 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.
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 formulation of physical propagation in which dynamics are described by a unitary operator W^=f(H^), associated with a self-adjoint structural operator H^, and by its bilocal representation W(x,y). Time evolution is formulated as the iterative application of this operator, without assuming differential equations as a fundamental starting point. It is shown that the spectral decomposition of H^ induces a complete organization of the evolution into independent eigenmodes, each governed by a spectral factor f(λn). From this structure, it is demonstrated that the bilocal propagation kernel is entirely reconstructed from the spectrum and the eigenvector basis of the structural operator. In the regime of translational homogeneity, it is proven that the propagation operator is diagonalizable in a basis compatible with translations, resulting in the absence of mixing between spectral modes. Outside this regime, it is shown that the support of the kernel W(x,y) completely determines the connectivity of propagation, while its off-diagonal structure encodes relations between distinct nodal states. Additionally, it is established that quantities defined as functions of the structural operator can be evaluated directly in the spectral basis, providing an intrinsic characterization of the dynamics in terms of the spectrum of H^. The results obtained do not constitute a complete physical theory nor do they establish a direct identification between the operator's spectrum and specific empirical observables. The objective of this work is to make explicit the structural level at which propagation is organized by unitary operators and by their spectral decomposition, providing a coherent basis for further developments.
This work addresses the structural problem of defining physical observables in regimes where the spectral decomposition of self-adjoint operators is not purely discrete. In particular, it is demonstrated that, outside the ultraviolet regime, the identification between mass and eigenvalues ceases to be consistent when the associated spectral measure has a continuous component. Starting exclusively from minimal spectral hypotheses -- self - adjoint operator, resolvent, spectral measure, and its density -- it is established, without resorting to ad hoc hypotheses, that the observable physical structure is determined by the spectral measure associated with the state. As a consequence, it is proven that the physical dimension of a system is not given by the dimension of the Hilbert space nor by the cardinality of the spectrum, but by the structure of the dominant spectral regions of the density, defining the notion of effective spectral dimension. In this framework, physical observables emerge as functionals of the spectral measure. In particular, mass is characterized as the first-order spectral moment, while the spectral width quantifies the dispersion associated with the energy distribution. It is further shown that dominant spectral regions act as effective degrees of freedom, providing a distributed description of the dynamics. The developed formalism recovers the discrete limit of quantum mechanics, is compatible with the spectral representation of quantum field theory, and admits a natural interpretation in terms of scale flow. Explicit falsifiability criteria are also established, and the structural limitations of the proposed framework are delineated. The central result is the formulation of a principle: physical dimension and observables emerge from the effective spectral structure, implying the necessary substitution of pointlike descriptions with distributed descriptions in non-discrete regimes.