This work presents the Action--Phase Protocol, or PAF-I, as a methodological framework to formulate interfaces between distinct physical sectors without resorting to the direct stitching of final equations. The objective is not to propose a final theory of unification, but to build a common layer prior to the final formulations of quantum mechanics, general relativity, electromagnetism, and thermodynamics.The starting point is the identification of the operational phase with the dimensionless action,\[ \phi=\frac{S}{\hbar}, \]so that the action provides the common object and the phase provides the variable upon which the physical sectors act. In this language, the free phase is described by\[ D_\mu^{(0)}=\partial_\mu, \]gravity by\[ D_\mu^{(g)}=\nabla_\mu, \]electromagnetism by\[ D_\mu^{(\mathrm{EM})} = \partial_\mu-\frac{q}{\hbar}A_\mu, \]and the minimum quantum mechanics by the decomposition\[ \psi=R e^{i\phi}, \qquad \mathcal Q_R = -\frac{\hbar^2}{2m}\frac{\nabla^2R}{R}. \]The central rule of PAF-I is that interfaces between sectors should not be obtained by the direct sum of final equations, but by the composition of the operators that each sector induces on the common phase. The effective dynamics is then derived from a single effective action,\[ \delta S_{\mathrm{eff}}=0. \]Thus, PAF-I provides a minimum, variational, and non-ad hoc layer to formulate physical interfaces. Its innovation is synthetic and architectural: organizing orthodox tools of theoretical physics into a common language based on action, phase, sectorial operators, and the variational principle.
This work presents the Action--Phase Protocol, or PAF-I, as a methodological framework to formulate interfaces between distinct physical sectors without resorting to the direct stitching of final equations. The objective is not to propose a final theory of unification, but to build a common layer prior to the final formulations of quantum mechanics, general relativity, electromagnetism, and thermodynamics.The starting point is the identification of the operational phase with the dimensionless action,[ \phi=\frac{S}{\hbar}, ]so that the action provides the common object and the phase provides the variable upon which the physical sectors act. In this language, the free phase is described by[ D_\mu^{(0)}=\partial_\mu, ]gravity by[ D_\mu^{(g)}=\nabla_\mu, ]electromagnetism by[ D_\mu^{(\mathrm{EM})} = \partial_\mu-\frac{q}{\hbar}A_\mu, ]and the minimum quantum mechanics by the decomposition[ \psi=R e^{i\phi}, \qquad \mathcal Q_R = -\frac{\hbar^2}{2m}\frac{\nabla^2R}{R}. ]The central rule of PAF-I is that interfaces between sectors should not be obtained by the direct sum of final equations, but by the composition of the operators that each sector induces on the common phase. The effective dynamics is then derived from a single effective action,[ \delta S_{\mathrm{eff}}=0. ]Thus, PAF-I provides a minimum, variational, and non-ad hoc layer to formulate physical interfaces. Its innovation is synthetic and architectural: organizing orthodox tools of theoretical physics into a common language based on action, phase, sectorial operators, and the variational principle.
We propose a unified formulation, at a structural level, in which action occupies the primary role and the phase ϕ=S/ℏ 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.
Modern theoretical physics rests on two highly successful but conceptually distinct frameworks: quantum mechanics and general relativity. While both theories are experimentally confirmed to high precision within their respective domains, they rely on fundamentally different starting points. General relativity takes spacetime as a primitive geometric structure, whereas quantum mechanics is formulated in terms of abstract state spaces without intrinsic spacetime meaning. Most approaches to quantum gravity attempt to reconcile this tension by modifying, quantizing, or discretizing spacetime itself. Even in approaches where spacetime is said to be “emergent,” it typically remains the primary conceptual backdrop on which physical processes are defined. Despite extensive efforts, there is no consensus on whether spacetime should be regarded as fundamental or derivative. In this work, a more radical starting point is explored. Rather than modifying spacetime or its geometry, we ask whether spacetime can be excluded entirely from the fundamental layer of the theory and recovered only at a later, operational stage. The central question is not how spacetime behaves, but whether the notion of “what lies between physical objects” must be spatial or temporal at the most fundamental level. We propose a framework—referred to here as Super Relativity (SR)—based on a single pre-geometric postulate: the existence of a monotonically expanding set of possible quantum configurations, referred to as the flux. Formally, this flux is represented as a monotone, σ-finite family of capacity measures on a common configuration space (introduced in Section 2). This postulate deliberately refrains from assuming spacetime, time ordering, metric structure, causality, or dynamics. Within this pre-geometric core, two derived, dimensionless notions are introduced: utilization, describing the realization of configurations, and saturation, defined as the ratio between utilized and available configurations. A central structural element of SR is the strict separation between a core layer and an interface layer. In the core layer, all quantities are pre-geometric and dimensionless, and no reference is made to space, time, motion, energy, or interaction. Only at the interface layer are operational notions introduced, including clocks, rulers, and signal propagation. Spacetime, in this construction, is not a dynamical entity but an effective bookkeeping structure that encodes relative rates of change once a calibration to physical units is performed. Within this framework, gravitational phenomena do not originate from spacetime curvature taken as fundamental. Instead, they arise from variations in saturation that, at the interface level, manifest as uniform rescalings of physical processes. When expressed in geometric language, these rescalings reproduce the standard predictions of general relativity, including gravitational time dilation and the equivalence between gravitation and acceleration. The invariant local speed of light is not postulated but follows as a derived consequence, since both clock rates and signal propagation are affected identically by the same underlying saturation mechanism. The aim of this paper is not to introduce new empirical predictions beyond those of general relativity, nor to present a complete theory of quantum gravity. Rather, the objective is foundational: to demonstrate that the mathematical structure of general relativity can be reconstructed from a single pre-geometric postulate without assuming spacetime as part of the fundamental ontology. In this sense, SR should be understood as a reconstruction and reinterpretation of known physics, rather than as a competing dynamical theory. An extended and fully explicit formulation of the framework, including all intermediate steps, is available in a companion preprint (version V12) deposited on Zenodo. The present paper restricts itself to the minimal conceptual and mathematical structure required to establish logical consistency and to demonstrate the emergence of general-relativistic behavior at the interface level.
This paper proposes and systematically constructs a novel formulation of quantum mechanics based on complete sets of dual variables and integral operators. Its core postulate elevates the operators of time,spatial position, and phase angle to an equal footing with those of energy, momentum, and action, forming three fundamental dual pairs. Building on this, we define the integral operators ˆT and ˆX for time and space, and derive as a first principle the Quantum Action Balance Equation (QABE)—an integral equation with the action phase eiScl/ℏ as its kernel. We rigorously demonstrate that the Schr¨odinger equation, Dirac equation, and Heisenberg equation naturally emerge as specific limits or corollaries of the QABE under local approximations or expectation value evolution. Furthermore, the Feynman path integral emerges as the direct iterative solution form of this integral equation. We clarify that the quantization of action, phase, and spacetime is not an additional hypothesis but an inevitable mathematical consequence of solving this continuous dynamical framework under inherent algebraic constraints (such as phase compactness) or external physical boundary conditions (such as spacetime periodicity), achieving a conceptual unification for the origin of quantization. Importantly, the inherent non-locality of this integral formulation provides an intrinsic mechanism to circumvent ultraviolet divergences in quantum field theory. This work aims to provide a deeper unified foundation for the different formulations of quantum mechanics and to open new conceptual pathways for bridging quantum theory and gravity
Description This record contains the revised version of Paper 15 in the Fundamental Being Space (FBS) Theory series. In this revision, the updated contents of Papers 12–14 are reflected, and state formation on the Σ side, state projection to the ordinary-spacetime side, observation-output formation after projection, and the classification of multiple identical projected states are reorganized as distinct stages. The basic relation is organized as follows: internal responses in Σ→ψΣ→ψSi→Xi Here, state projection and observation-output formation are not treated as the same process. The classification of exchange, occupation, and effective projected modes for multiple identical projected states is also positioned as a subsequent stage. On this basis, this paper organizes observability, phase and rotational responses, stabilization and retention, consistency with state projection, and constraint conditions as having distinct roles within the minimal effective action of the interface layer Σ. Furthermore, a finite effective sector is extracted from the broad operator-response structure on the Σ side and connected to a finite-dimensional representation as AΣ→PeffAΣPeff→MΣeff As a specific candidate for this finite-dimensional representation, the 4+3 block structure of a 7×7 response matrix is used. By applying the Schur complement, a general form is shown in which the influence of one finite effective sector remains as a correction to the other effective response through off-diagonal responses. However, the dimension 7 is not regarded as having been newly derived from the minimal effective action or from the selection of the finite effective part. Through this revision, the role of Paper 15 within the FBS Theory research program has also been clarified. The observability structure, spinor-type responses, and classification of multiple identical projected states organized in Papers 12–14 are connected in Paper 15 to the framework of the minimal effective action and finite effective sectors. This provides a bridge to the next stage, Paper 16, where the physical specification of the 7×7 response structure and the connection between quantum-type and gravity-type responses will be examined. This paper does not derive the complete internal structure of Σ, standard quantum mechanics, the Dirac equation, or the spin-statistics relation from first principles. Its purpose is to reorganize the response structures developed in the revised Papers 12–14 under the minimal effective action and to present a finite effective structure that connects to Paper 16. Related Papers The present revision of Paper 15 directly reflects the revised contents of Papers 12–14. Paper 12 — From General Σ Responses to Hermitian Pairing: Born-Type Probability and Observability in FBS TheoryConcept DOI: https://doi.org/10.5281/zenodo.20784611 Paper 13 — Spinor Structure and 4π Symmetry as Σ-to-Si Projection in FBS TheoryConcept DOI: https://doi.org/10.5281/zenodo.20793474 Paper 14 — Effective Projected-Mode Classification in the Interface Layer in FBS TheoryConcept DOI: https://doi.org/10.5281/zenodo.20797474 Regarding Paper 16 Paper 16 will be revised to reflect the updated contents of Papers 12–15. After the revision is completed, a link from this Paper 15 record to the revised Paper 16 will be added. Complete FBS Theory Series Paper 1 serves as the main entry point for the overall FBS Theory series, including the series overview, the position of each paper within the research program, and the latest organization of the series. Paper 1 — General Discussion on the Reality of Fundamental Being Space (FBS)Concept DOI: https://doi.org/10.5281/zenodo.18220714 Versioning This preprint may be updated in revised versions as the research progresses. When citing, please specify the DOI and version. Author informationIndependent Researcher, Japanyosihitoookawa@gmail.comORCID: 0009-0006-8294-5738
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 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.