Doron Kwiat
Independent Researcher, Mazkeret Batyia, Israel.
DOI: 10.4236/jhepgc.2026.121033   PDF    HTML   XML   62 Downloads   368 Views   Citations

Abstract

This work explores the connection between the Coupled-Fields (CF) model and gravitation, proposing a unified interpretation in which spacetime curvature arises from gradients in field coupling energy rather than from mass-energy alone. The tension and exchange dynamics of the two real fields produce local distortions equivalent to gravitational curvature, suggesting that gravity is a macroscopic manifestation of the same underlying field interactions that govern quantum behavior. By formulating the CF Lagrangian in curved spacetime, the model naturally incorporates general-relativistic effects while preserving determinism. The analysis shows that Planck’s constant and Newton’s constant G may both originate from the same internal field structure, linking quantum and gravitational scales through the coupling constant $\kappa$ . We conclude, that a quantum theory of gravitation can emerge without quantizing spacetime itself—gravity instead reflects coherent variations in coupled-field energy density. This approach provides a physically grounded route toward unification, bridging quantum mechanics and general relativity within one continuous real-field ontology.

Keywords

Coupled Fields, Quantum Gravity, Planck Density, Spin-Curvature Coupling, Real-Field Fermions

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1. Introduction

In this work I develop a unified framework in which fermions and gravitation arise from two coupled real fields defined on classical spacetime [1]-[32]. The framework is built upon a physically motivated phase-coupling mechanism between the real fields $\varphi \left(x\right),\chi \left(x\right)$ , whose relative phase $\Delta \phi =\text{phase}\left(\varphi \right)-\text{pase}\left(\chi \right)$ , governs internal rotational structure, topological charge, and the observed spin-1/2 behavior. Unlike conventional two-scalar-field models, the present formulation is fundamentally phase-based: the dynamics are dominated not by independent field amplitudes but by a locked internal phase cycle living on $S^1$ . This leads naturally to winding numbers, quantized topological charges, and discrete spin states. The interaction between the two fields is governed by a class of phase-locking potentials (Figure 1 and Figure 2) of the form $U_{lock}=\kappa \left[1-\cos \left(n\Delta \phi \right)\right]$ , with $n=1,3$ , corresponding to the observed quantization patterns of electric charge and fractional charge units. The associated antisymmetric current,$J^{\mu }=\varphi {\partial }^{\mu }\chi -\chi {\partial }^{\mu }\varphi$ ,plays the role of a Noether current whose circulation quantizes naturally through the topology of $S^1\to S^1$ mappings. This provides a local, real-field origin for spin, charge, and internal angular momentum without invoking complex wavefunctions.

The CF formalism also yields a deterministic hidden-variable account of quantum correlations without violating locality. The key variable is the internal phase $\Delta \phi$ , which evolves locally and determines the spin orientation at detection. Measurement settings act as phase filters, selecting sub-ensembles of $\Delta \phi$ without enabling superluminal signaling. This reproduces the quantum correlation curve and the CHSH [33]-[46] value $2\sqrt{2}$ without violating statistical locality, thereby offering a new physical interpretation of Bell-type experiments. Unlike conventional hidden-variable models, CF does not assign arbitrary variables to particles; instead, the hidden variable is the physical internal phase of the coupled fields.

The search for a unified description of gravity and quantum phenomena remains one of the central open problems in fundamental physics. Existing approaches—ranging from string theory to loop quantum gravity [47]-[59], emergent-gravity models, and modified-gravity proposals—provide valuable insights, yet none have delivered a fully satisfactory real-field description of matter that is simultaneously local, deterministic, and compatible with the empirical successes of quantum mechanics. In particular, the microscopic origin of spin, charge, entanglement, and fermionic structure remains conceptually opaque in the standard complex-valued formalism, where these quantities emerge from abstract Hilbert-space axioms rather than from explicit dynamical fields.

A key feature of the coupled-fields (CF) framework is that it remains fully Lorentz-covariant and admits a straightforward coupling to gravitation. The two fields generate an effective stress-energy tensor that supports localized, non-singular fermionic cores—geometric structures closely related to Bardeen-type [56] regular solutions and gravastar-like [57] interiors. This creates a continuous family of objects interpolating smoothly between elementary-particle cores and compact astrophysical objects, thereby linking quantum and gravitational phenomena within the same real-field model. The CF stress-energy contributions modify strong-field curvature while reducing to the Einstein field equations in the weak-field limit, preserving the empirical predictions of general relativity.

To establish the physical viability of the model, I analyze its stability properties, phase dynamics, topological invariants, and energy bounds. Particular attention

Figure 1. Phase-Locking Potential (n = 1) $U\left(\Delta \varphi \right)=\kappa \left[1-\cos \left(\Delta \varphi \right)\right]$ showing the fundamental periodic energy wells that stabilize the internal phase of the CF fields. Minima occur at $\Delta \varphi =0,\pm 2\pi ,\pm 4\pi ,\cdots$ corresponding to topologically equivalent configurations. These wells define the basic locking mechanism responsible for quantized fermionic spin and single-unit electric charge. Minima correspond to stable $\Delta \phi$ values.

Figure 2. Higher-Harmonic phase-locking potential for n = 3 $U\left(\Delta \phi \right)=\kappa \left[1-\cos \left(3\Delta \phi \right)\right]$ . The triple-frequency structure creates three inequivalent minima per 2π cycle. These correspond to fractional topological sectors and provide the CF mechanism for quantized charges of magnitude Q = ±1/3, ±2/3. The deeper confinement produces stronger stability against dephasing. Minima correspond to stable $\Delta \phi$ values.

is given to the emergence of the Planck constant ℏ from internal phase circulation, the normalization condition $c_1^2+c_4^2=1$ of stationary solutions, and the predicted imbalance between spin-up and spin-down populations in partially filtered beams. The resulting framework makes a series of testable predictions across a wide range of scales—from atomic interferometry to strong-field astrophysics—and offers a concrete experimental program designed to differentiate CF from both quantum mechanics and general relativity.

The remainder of the paper is organized as follows. Section 2 introduces the Lagrangian, derives the Euler-Lagrange field equations, and establishes covariance. Section 3 analyzes the internal phase structure, locking potentials, winding numbers, and emergent charge and spin. Section 4 derives the CF stress-energy tensor and presents the resulting gravitational solutions, including connections to regular black-hole cores, Bardeen [56] metrics, and gravastar [57] configurations. Section 5 develops the hidden-variable interpretation and derives the CHSH correlation function. Section 6 presents experimental proposals, including interferometric, spectroscopic, and strong-field tests. Section 7 summarizes the implications and highlights directions for future work.

Overall, the CF framework provides a mathematically consistent and physically motivated real-field foundation for both particle structure and gravitation. By unifying internal phase topology, fermionic behavior, and gravitational self-energy within a single classical field theory, it offers a radically new but testable path toward a unified theory of quantum gravity.

2. Lagrangian Formulation and Field Equations

In the Coupled-Fields (CF) framework, a fermion is represented not by a complex wavefunction or spinor but by two interacting real fields.

$\varphi \left(x\right),\chi \left(x\right)$ , defined on classical spacetime with metric $g_{\mu \nu }$ . The fundamental dynamical quantity is not the individual field amplitudes, but the relative internal phase $\Delta \phi \left(x\right)$ which encodes spin, charge, and topological structure. This section introduces the Lagrangian density, derives the Euler-Lagrange field equations, and establishes Lorentz covariance.

2.1. Lagrangian Density

The CF dynamics follow from the action

$S={\displaystyle \int ℒ-g{\text{d}}^{4}x}$ , where the Lagrangian density takes the form

$ℒ=\frac{1}{2}{g}_{\mu \nu }\left({\partial }^{\mu }\varphi {\partial }^{\nu }\varphi +\partial \mu \chi \partial \nu \chi \right)-U_{lock}\left(\Delta \phi \right)$

The potential $U_{lock}$ enforces local phase coupling between $\varphi$ and $\chi$ . Motivated by classical phase-locking phenomena (Adler equation, Kuramoto synchronization), topological arguments (maps $S^1\to S^1$ ), and phenomenology of charge quantization, two classes of locking interactions are considered:

$U_{lock}\left(1\right)=\kappa \left[1-\cos \left(\Delta \phi \right)\right],U_{lock}\left(3\right)=\kappa \left[1-\cos \left(3\Delta \phi \right)\right]$

• The n = 1 locking yields integer winding numbers → charge ±1.

• The n = 3 locking introduces 3-to-1 symmetry → fractional charge units ±1/3.

Both reduce to the same quadratic form for small $\Delta \phi$ , ensuring stability of the locked configuration.

2.2. Covariant Variational Derivation

The Euler-Lagrange equation for a field X is

${\nabla }_{\mu }\left(\frac{\partial L}{\partial \left(\partial \mu X\right)}\right)-\frac{\partial L}{\partial X}=0$

where ${\nabla }_{\mu }$ is the metric-compatible covariant derivative.

2.2.1. Variation with Respect to $\varphi$

From (2.1):

$\frac{\partial L}{\partial \left({\partial }^{\mu }\varphi \right)}=g_{\mu \nu }{\partial }^{\nu }\varphi$

$\frac{\partial L}{\partial \varphi }=-\frac{\partial U_{lock}}{\partial \varphi }$

Thus,

${\nabla }_{\mu }\left(g^{\mu \nu }{\partial }_{\nu }\varphi \right)+\frac{\partial U_{lock}}{\partial \varphi }=0$

or equivalently,

$\square \varphi =\frac{\partial U_{lock}}{\partial \varphi }$

with $\square ={\nabla }^{\mu }{\nabla }_{\mu }$ .

2.2.2. Variation with Respect to $\chi$

Identical steps yield

$\square \chi =\frac{\partial U_{lock}}{\partial \chi }$

2.3. Explicit Form of the Coupling Terms

Since the potential depends only on $\Delta \phi$ ,

$\frac{\partial U_{lock}}{\partial \varphi }=\frac{\text{d}{U}_{lock}}{\text{d}\left(\Delta \phi \right)}\frac{\partial \left(\Delta \phi \right)}{\partial \varphi }$

The phase is defined implicitly through

$\varphi =R_{\varphi }\cos \varphi ,{\partial }_{\mu }{\phi }_{\varphi }=\frac{\varphi {\partial }_{\mu }\chi -\chi {\partial }_{\mu }\varphi }{R_{\varphi }^2}$ , and analogously for $\chi$ .

The derivative of the potential (Figure 3) is straightforward:

$\frac{\text{d}{U}_{lock}}{\text{d}\left(\Delta \phi \right)}=\{\begin{array}{l}\kappa \sin \left(\Delta \phi \right),n=1\\ 3\kappa \sin \left(3\Delta \phi \right),n=3\end{array}$

Thus, the field equations become

$\varphi =\frac{\text{d}{U}_{lock}}{\text{d}\left(\Delta \phi \right)}\frac{\partial \left(\Delta \phi \right)}{\partial \varphi },\square \chi =\frac{\text{d}{U}_{lock}}{\text{d}\left(\Delta \phi \right)}\frac{\partial \left(\Delta \phi \right)}{\partial \chi }$

These are the coupled-wave equations governing the CF dynamics.

Figure 3. Second Derivative of the Locking Potential.

2.4. Local Lorentz Covariance

Every term in the Lagrangian (2.1) is a scalar under Lorentz transformations:

• $g_{\mu \nu }{\partial }^{\mu }X{\partial }^{\nu }X$ is a scalar quadratic form.

• $U_{lock}\left(\Delta \phi \right)$ depends only on the internal phase, which is a Lorentz scalar.

• The action integral uses the invariant $\sqrt{-g}{\text{d}}^{4}x$ .

Thus, $S\left[\varphi ,\chi \right]$  is manifestly Lorentz invariant. $S\left[\phi ,\chi \right]$ is manifestly Lorentz invariant.

This addresses a common objection in real-field formulations. (S is the magnitude of the particle’s intrinsic angular momentum vector).

2.5. Conserved Current

The antisymmetric bilinear

$J^{\mu }=\varphi {\partial }^{\mu }\chi -\chi {\partial }^{\mu }\varphi$

Obeys ${\nabla }_{\mu }{J}^{\mu }=0$ , a consequence of the field equations and the fact that $U_{lock}$ depends only on the phase difference.

The circulation of this current is quantized:

$Q={\displaystyle \oint J^{\mu }\text{d}{x}_{\mu }}\propto 2\pi n$

giving rise to quantized charge and spin without complex fields.

2.6. Interpretation of the Field Equations

Equations (2.5)-(2.6) describe two real fields coupled through their internal phase.

Key features:

• The coupling is topological, not algebraic.

• The model is not equivalent to two independent scalar fields.

• Phase locking produces stable solitonic cores with quantized internal rotation.

• Internal energy associated with $\Delta \phi$ becomes gravitational mass-energy.

• Stationary solutions satisfy normalization $c_1^2+c_4^2=1$ , fixing the spinor-like structure.

This sets the stage for the gravitational analysis in the next section.

3. Phase-Locking, Topology, and the Origin of Charge and Spin

A distinguishing feature of the Coupled-Fields (CF) framework is that fermionic structure does not arise from complex wavefunctions or spinorial degrees of freedom. Instead, it emerges from the locked internal phase dynamics of two real fields $\varphi \left(x\right)$ and $\chi \left(x\right)$ . The physical degrees of freedom reside not in the individual field amplitudes but in their relative phase, $\Delta \phi$ ,which behaves as a local angular variable living on the compact manifold $S^1$ . This section develops the phase-locking mechanism, identifies the allowed topological sectors, and shows how quantization of internal circulation yields integer and fractional charges, spin-1/2 structure, and 4π rotational symmetry.

3.1. Phase Dynamics and Locking Potential

The relevant part of the Lagrangian (Section 2) is the phase-locking potential

$U_{lock}=\kappa \left[1-\cos \left(n\Delta \phi \right)\right]$

with n = 1 or n = 3.

The equilibrium points are given by

$\frac{\text{d}{U}_{lock}}{\text{d}\left(\Delta \phi \right)}=n\kappa \sin \left(n\Delta \phi \right)=0$ , whose solutions are

$\Delta \phi =2\pi kn,k\in \mathbb{Z}$

This implies:

• For n = 1: stable minima at $\Delta \phi =2\pi k$ → One full cycle per winding → unit electric charge.

• For n = 3: stable minima at $\Delta \phi =2\pi k$ → Three smaller wells → fractional charges $\pm \frac{1}{3}$ .

The curvature of the potential determines stability:

$\frac{{\text{d}}^{2}U}{\text{d}{\left(\Delta \phi \right)}^2}=n^2\kappa \cos \left(n\Delta \phi \right)$

Positive values correspond to stable locked states; negative values to unstable maxima separating topological sectors.

This explicitly produces the quantized charge landscape.

3.2. Topological Sectors: Maps $S^1\to S^1$

The internal phase $\Delta \phi$ is periodic:

$\Delta \phi ~\Delta \phi +2\pi$

so, field configurations are classified by the integer winding number

$\omega =\frac{1}{2\pi }{\displaystyle \oint \text{d}\left(\Delta \phi \right)}$

Thus, the CF model naturally contains:

• Topological solitons

• Quantized internal circulation

• Stable fermionic cores

• Discrete charge states

This construction does not depend on quantization postulates; it arises from the classical topology of the internal phase.

3.3. Emergent Charge from Phase Circulation

The antisymmetric current (Section 2)

$J^{\mu }=\varphi {\partial }^{\mu }\chi -\chi {\partial }^{\mu }\varphi$

is identically conserved:

${\nabla }_{\mu }\text{}{J}^{\mu }=0$

The corresponding charge is

$Q={\displaystyle \int J^0{\text{d}}^{3}x}$

In stationary, locked configurations, $J^0\propto {\partial }_t\left(\Delta \phi \right)$ , and the total charge satisfies

$Q\propto {\displaystyle \oint {\partial }_t\left(\Delta \phi \right)\text{d}t}=2\pi \omega$

Thus:

• $\omega =\pm 1$ → charge ±1 (electron-like).

• $\omega =\pm \frac{1}{3}$ (in the n = 3 locking) → fractional charges, consistent with quark-like values [28]-[32].

This ties electric charge to topological rotation, not to fundamental complex amplitudes.

3.4. Spin-1/2 and the 4π Symmetry

When a configuration undergoes a physical rotation through angle $\theta$ , the internal phase shifts as

$\Delta \phi \to \Delta \phi +\frac{\theta }{2}$

that is only half internal rotation.

This is the key CF result: an external 2π rotation induces only a π shift in the internal phase.

Because the physical state is defined by the pair $\left(\varphi ,\chi \right)$ , not their individual phases, a shift of $\Delta \phi$ by 2π returns the system to itself. Therefore:

• A 2π external rotation does not restore the state.

• A 4π external rotation does.

The model thus yields:

$\text{Spin}=\frac{1}{2}$ Rotational symmetry = 4π,

This matches the defining property of fermions without invoking spinors or complex Hilbert space structures.

3.5. Stability of Fermionic Cores

Expanding the potential around a stable minimum,

$U_{lock}\approx \frac{1}{2}{n}^2\kappa {\left(\Delta \phi -\Delta {\phi }_0\right)}^2$

one finds a restoring “spring constant”

$K_{eff}=n^2\kappa$

The competition between:

1) Gradient energy (spatial variation of the fields)

2) Locking energy (preference for stable $\Delta \phi$ minima)

produces a stable, finite-sized solitonic core.

This core is the CF analogue of a fermion.

Later (Section 4) we show that the stress-energy tensor of this core naturally generates:

• Bardeen-like regular interiors

• Gravastar-type pressure structures

• Smooth transitions to GR weak-field behavior

There is no singularity.

3.6. Summary of Quantization Results

The CF model yields, from classical real-field dynamics alone:

Physical quantity	CF origin	Result
Electric charge	Winding number $\omega$	$Q\propto 2\pi \omega$
Fractional charge	N = 3 phase locking	$Q=\pm \frac{1}{3},\pm \frac{2}{3}$
Spin-1/2	Phase shift under rotations	4π symmetry
Magnetic moment orientation	Sign of $\omega$	Up/down spin
Fermion stability	Balance of gradient + locking energies	Finite solitonic cores
Topological protection	Maps $S^1\to S^1$	Integer invariants

• None of these require:

• Complex wavefunctions

• Hilbert-space axioms

• Spinor structure

• Quantization postulates

They arise from the internal geometry and topology of the coupled real fields.

3.7. Relationship between Electric Charge, Planck’s Constant, and the Locking Coupling κ

A distinctive feature of the CF framework is that electric charge (Q), spin (ħ/2), and the locking-energy scale (κ) all arise from a single internal mechanism: the quantized circulation of the internal phase $\Delta \phi \left(x\right)\in S^1$

3.7.1. Internal Phase Circulation and ħ

From the conserved antisymmetric current

$J^{\mu }=\varphi {\partial }^{\mu }\chi -\chi {\partial }^{\mu }\varphi$

the total internal angular momentum of a stationary CF soliton is

$L_{int}={\displaystyle \int {\left(r\times J\right)}_z{\text{d}}^{3}x}.$

For a fermionic configuration, the internal phase advances by π under a 2π spatial rotation (Section 3.4).

Thus a full 2π return of the physical state requires:

$\Delta \phi \to \Delta \phi +2\pi$ (4π spatial rotation)

This identifies one internal phase cycle with one quantum of intrinsic angular momentum:

$L_{int}=\frac{\hslash }{2}$

In other words:

ħ is the scale of internal phase circulation in the CF soliton.

One full winding of $\Delta \phi$ stores exactly ħ/2 of intrinsic rotation.

This is not imposed; it follows from the locked topology and the normalization of stationary CF solutions (e.g., $c_1^2+c_4^2=1$ ).

3.7.2. Electric Charge as Winding Number

The electric charge derived in Section 3.3 follows from the line integral of the conserved current:

$Q\propto {\displaystyle \oint J^{\mu }\text{d}{x}_{\mu }}=2\pi \omega$

where,

$\omega =\frac{1}{2\pi }{\displaystyle \oint \text{d}\left(\Delta \phi \right)}$

is the integer (or ±1/3) winding number.

Thus, the same internal phase that produces ħ/2 of spin also produces the quantized charge.

They originate from the same topological structure.

3.7.3. Role of the Locking Coupling κ

The locking potential

$U_{lock}=\kappa \left[1-\cos \left(n\Delta \phi \right)\right]$

determines two key quantities:

1) The stiffness of internal phase rotation

$K_{eff}=n^2\kappa$

which controls the soliton’s resistance to internal dephasing.

2) The energy stored per unit winding

$E_{wind}~\kappa {\displaystyle \int \left(1-\cos \Delta \phi \right)\text{d}V}$

To maintain one quantum of intrinsic angular momentum (ħ/2) per topological cycle, the energy cost per full winding must match the rotational energy scale:

$E_{wind}~\frac{\hslash }{2}{\omega }_{int}$

where ${\omega }_{int}$ is the internal rotation frequency.

This links κ to ħ:

$\kappa \propto 4\pi \hslash {\omega }_{int}$

Since electric charge corresponds to discrete windings ($\omega$ ), we obtain the proportionalities:

$Q\propto 2\pi \omega ,S=\frac{\hslash }{2},E_{wind}~\kappa$

Thus, the three constants—charge (Q), spin (ħ/2), and locking strength (κ)—are joined by one physical mechanism: the quantized, energetically stabilized circulation of the internal phase Δφ.

3.7.4. Summary of the Relationship

• Electric charge emerges from the topological integer (or fraction) of phase winding.

• ħ emerges from the angular momentum stored in one internal phase cycle.

• κ controls the energy per cycle, ensuring stability and giving the correct ħ scaling.

In essence:

Q -counts how many times the internal phase winds.

ħ -measures the angular momentum stored in one such winding.

κ -sets the energy scale that stabilizes the winding.

This provides a unified, real-field origin for three constants that are independent in conventional quantum theory (Figure 4).

Schematic illustration of the relationship between electric charge Q, intrinsic spin ħ/2, and the locking-energy scale κ in the Coupled-Fields (CF) model. Electric charge arises from the topological winding number of the internal phase, $Q\propto 2\pi \omega$ , while intrinsic spin corresponds to the angular momentum stored in one full internal phase cycle, S = ħ/2. The locking strength κ determines the energy cost per phase cycle, stabilizing the solitonic configuration and setting the internal rotational energy scale. Together, these relationships show how charge quantization, spin, and the CF coupling constant originate from a single mechanism: quantized, energetically stabilized internal phase rotation.

Figure 4. Unified Relationship Between Electric Charge Q, Spin ħ/2, and the Locking Coupling κ.

4. Gravitational Coupling, Stress-Energy, and Regular Cores

The coupled real fields $\varphi \left(x\right)$ and $\chi \left(x\right)$ contribute to spacetime curvature through their stress-energy tensor. The interaction between internal phase dynamics and gravitation plays a central role in the CF framework: fermions appear as finite-energy regular cores whose stress-energy profile naturally produces non-singular interior geometries. This section derives the stress-energy tensor, describes the resulting gravitational structure, establishes recovery of Einstein gravity in weak fields, and shows how CF interpolates continuously between particle-like cores, gravastar interiors [58], and Bardeen-type regular black-hole solutions.

This interior structure, characterized by positive energy density and negative radial pressure, closely parallels the de Sitter-like cores of gravastars proposed by Mazur and Mottola [60], but here it arises dynamically from the CF phase-locking mechanism rather than from an imposed equation of state. While the resulting interior shares the characteristic de Sitter-like density and negative radial pressure of gravastars, the CF model differs fundamentally in that this structure is not imposed through an ad-hoc equation of state or thin-shell matching conditions but emerges self-consistently from the internal phase-locking dynamics of the coupled fields.

4.1. Stress-Energy Tensor of the Coupled Fields

Varying the action with respect to the metric yields the stress-energy tensor

$T_{\mu \nu }={\partial }_{\mu }\varphi {\partial }_{\nu }\varphi +{\partial }_{\mu }\chi {\partial }_{\nu }\chi -g_{\mu \nu }ℒ$

with $ℒ$ given in Equation (2.1).

Explicitly:

$T_{\mu \nu }={\partial }_{\mu }\varphi {\partial }_{\nu }\varphi +{\partial }_{\mu }\chi {\partial }_{\nu }\chi -g_{\mu \nu }\left[\frac{1}{2}{g}_{\alpha \beta }\left({\partial }_{\alpha }\varphi {\partial }_{\beta }\varphi +{\partial }_{\alpha }\chi {\partial }_{\beta }\chi \right)-U_{lock}\left(\Delta \phi \right)\right]$

Two important observations follow:

1) The locking potential contributes positive internal energy density, but negative pressure in the core.

This is the defining signature of gravastar-like structures.

2) The gradient energy ${\left(\partial \varphi \right)}^2+{\left(\partial \chi \right)}^2$

This mirrors the behavior of regular black-hole cores (Bardeen [53], Bronnikov [55]).

4.2. Coupling to Einstein’s Equations

The full gravitational field equations are

$G_{\mu \nu }=8\pi G{T}_{\mu \nu }$

with $T_{\mu \nu }$ as above and coupled-field dynamics governed by Equations (2.9).

The CF model does not modify the geometric part of GR.

Gravitation remains encoded solely in $G_{\mu \nu }$ .

All new physics enters through the matter sector, specifically through:

• the phase-locked internal energy,

• the quantized solitonic core,

• the stress-energy produced by $\Delta \phi$ gradients.

Thus, GR is recovered automatically in the limit.

It is important to emphasize that the CF framework does not modify the geometric side of Einstein’s equations; all departures from standard GR arise exclusively from the CF stress-energy tensor, while the field equations

$G_{\mu \nu }=8\pi T_{\mu \nu }^{CF}$

retain their usual form.

$\left|\partial \varphi \right|,\left|\partial \chi \right|,U_{lock}\to 0.$

4.3. Static, Spherically Symmetric Solutions

For a stationary fermionic configuration, we take the usual metric ansatz

$\text{d}{s}^2=-{\text{e}}^{2\Phi \left(r\right)}\text{d}{t}^2+\frac{\text{d}{r}^2}{1-2m\left(r\right)/r}+r^2\text{d}{\Omega }^2$

The energy density and pressures follow from (4.2):

$\rho \left(r\right)=T_0^0,p_r\left(r\right)=T_r^r,p_t\left(r\right)=T_{\theta }^{\theta }=T_{\varphi }^{\varphi }$

Because the CF soliton core has finite energy density and negative radial pressure approaching a constant de Sitter value at the center, both the Ricci scalar R and the quadratic invariant $R_{\mu \nu }{R}^{\mu \nu }$ remain finite as $r\to 0$ ; no curvature singularity arises. The Kretschmann invariant $K=R_{\alpha \beta \gamma \delta }{R}^{\alpha \beta \gamma \delta }$ also remains finite because both ρ and p_r approach constants as r→0.

4.3.1. Core Structure

Inside the fermion, the potential term dominates:

$U_{lock}\approx \kappa \left(1-\cos n\Delta {\phi }_0\right)$

while gradients are small.

This, yields:

• positive energy density,

• negative radial pressure,

• finite, constant central density.

Exactly the structure of a gravastar interior (Mazur-Mottola [60]).

4.3.2. Boundary Layer

The gradient terms peak at the soliton edge where $\Delta \phi$ changes between adjacent minima.

This creates a thin shell of positive pressure—analogous to:

• the thin shell in gravastar models [61],

• the “de Sitter core + transition layer” structures in regular black holes [55] [56].

4.3.3. Outer Region

Outside the core:

$\varphi \to {\varphi }_0,\chi \to {\chi }_0,U_{lock}\to 0$ , and spacetime approaches Schwarzschild (or Reissner-Nordström [62] if electromagnetic fields are added).

In the absence of electromagnetic fields, the asymptotic exterior is Schwarzschild; adding Q reproduces the Reissner-Nordström exterior.

4.4. Connection to Bardeen and Regular Black-Hole Solutions

The stress-energy profile produced by the CF soliton (Figure 5) is of the same algebraic type that generates:

• Bardeen regular black holes [53] (non-singular core, de Sitter-like interior),

• Bronnikov-type regular solutions [55] (finite central density, no divergence in curvature invariants),

• Ayón-Beato-García models [61] (stress-energy from nonlinear fields yielding regular cores).

The CF model differs in a key way:

The regular core arises from internal phase topology (Δφ), not from ad-hoc nonlinear electrodynamics.

Thus, CF provides a physical microstructure for regular black-hole cores.

A finite, nearly constant-density core transitions through a stiff gradient layer into an exponentially decaying exterior. This structure corresponds to the regular interior found in Bardeen-like or gravastar-like solutions, but here it arises dynamically from the CF fields without introducing exotic matter. The profile illustrates how the CF soliton avoids singularities while retaining an asymptotically Schwarzschild exterior. Here $\rho \left(r\right)=T_0^0$ with $p_r\left(r\right)=T_r^r$ and $p_t\left(r\right)=T_{\theta }^{\theta }=T_{\varphi }^{\varphi }$ .

Figure 5. Schematic Stress-Energy radial Profile of a CF Soliton energy density.

4.5. Gravastar-Like Behavior

The effective equation of state in the interior satisfies $\rho +p_r<0$ , which is characteristic of gravastar interiors.

But unlike gravastars, where the equation of state is imposed, in CF it is derived dynamically from the phase-locking potential.

This provides a natural, field-theoretic origin for gravastar-like objects.

4.6. Recovery of GR in Weak Fields

In the limit where gradients and locking energy are small:

$\varphi \approx const,\chi \approx const$ , the stress-energy tensor vanishes:

$T^{\mu \nu }\to 0.$

Therefore:

$G^{\mu \nu }=0\Rightarrow GR\text{vacuum}$

This ensures:

• No modification to solar-system tests,

• No deviations in weak lensing,

• No contradiction with classical general relativity.

4.7. Gravitational Predictions of the CF Model

The CF soliton core makes several testable predictions.

Finite Radius of Elementary Fermions

$R_{CF}~{\rho }_0{K}_{eff}$ , giving sub-Compton-scale real structure.

1) Regular center (no singularity)

Curvature invariants remain finite.

2) Deviations in strong-field lensing

CF predicts small corrections to photon-sphere radius.

3) Possible suppression of event horizon formation

For large internal, collapsing cores behave like gravastars.

4) Spin-gravity coupling

The locked phase interacts with curvature scalar RRR, leading to measurable corrections in atomic interferometry.

The ADM mass of the configuration arises from the integrated CF energy density, with the internal phase-locking energy and its associated gradients contributing directly to the total gravitational mass seen by an asymptotic observer.

These predictions distinguish the CF model from both GR and quantum field theory.

4.8. Summary

The CF framework provides the stress-energy of a finite, non-singular, topologically stabilized fermionic core. When coupled to gravity:

• GR is preserved in weak fields.

• The fermion interior behaves like a gravastar core.

• The transition region resembles Bardeen/Bronnikov structures.

• No singularity arises.

• Quantized internal circulation creates mass and charge.

• Strong-field deviations become observable in astrophysical regimes.

Thus the CF model offers a unified real-field explanation for both elementary fermion structure and regular black-hole interiors.

5. Hidden Variable, Local Correlations, and the CHSH Bound in the CF Model

One of the central conceptual achievements of the Coupled-Fields (CF) framework is that it reproduces the experimentally observed Bell-CHSH correlation (Figure 6).

Figure 6. CF Correlation Curve for Bell-CHSH Experiments.

Predicted correlation $E\left(\theta \right)=-\cos \theta$ between spin outcomes measured at relative angle θ. This is the CF model’s local hidden-variable prediction, arising from half-angle internal phase filtering applied to the pre-existing distribution of $\lambda =\Delta \varphi$ . The curve matches the quantum-mechanical cosine law and leads to the Tsirelson value $S=2\sqrt{2}$ in the CHSH test, without invoking nonlocality.

$\left|S\right|=2\sqrt{2}$ (5.1) [34]-[47] while maintaining locality and determinism, without introducing nonlocal influences or superluminal signaling. The key ingredient is that the hidden variable is not an abstract mathematical entity but the physical internal phase $\lambda =\Delta \phi \left(x\right)$ established at the moment of pair creation. This section develops the hidden-variable interpretation of CF, demonstrates locality, explains the filtering mechanism that gives rise to quantum correlations, and derives the CF correlation function.

5.1. The Physical Hidden Variable: Internal Phase

As shown in earlier sections, each CF fermion possesses an internal angular variable $\Delta \phi$ living on the compact manifold $S^1$ . For a pair created in a singlet-like configuration, we have:

• equal and opposite winding numbers,

• equal and opposite internal phases,

• conservation of total internal angular momentum.

Thus, at pair creation:

${\lambda }_1=\lambda ,{\lambda }_2=\lambda +\pi$

This guarantees anti-aligned spins at the source without any indeterminacy.

The distribution of λ is uniform because the creation process does not privilege any phase direction:

$\rho \left(\lambda \right)=\frac{1}{2\pi },\lambda \in \left[0,2\pi \right)$

This satisfies Bell’s requirement that

$\rho \left(\lambda |a,b\right)=\rho \left(\lambda \right)$

i.e., the distribution of hidden variables is independent of the measurement settings.

5.2. Local Measurement Outcomes

For each particle, the measurement setting (e.g., a Stern-Gerlach axis) is represented by an angle a or b.

The CF model assigns deterministic outcomes:

$A\left(a,\lambda \right)=sign\left[\cos \left(\lambda -a\right)\right]$

$B\left(b,\lambda \right)=-sign\left[\cos \left(\lambda -b\right)\right]$

where the minus sign expresses the anti-alignment at creation.

These are local functions:

• A depends only on aaa and λ, never on b.

• B depends only on bbb and λ, never on a.

Thus, the CF model satisfies the locality condition in Bell’s theorem:

$P\left(A,B|a,b,\lambda \right)=P\left(A|a,\lambda \right)P\left(B|b,\lambda \right)$

This is crucial: CF obeys Bell’s factorization assumption.

5.3. Measurement as Phase Filtering

A key CF insight is that Stern-Gerlach measurements do not create spin values.Instead, they select sub-ensembles of the hidden-phase distribution:

• A measurement axis at angle a selects particles whose internal phase aligns or anti-aligns with a.

• Outcomes are determined by whether $\lambda -a$ lies in the “up” or “down” sector of the cycle.

Thus a measurement acts as a filter on the λ -distribution.

This clarifies the meaning of Bell’s assumption:

The settings a and b do not change λ.

They merely select different subsets of the pre-existing λ distribution.

This is how CF preserves locality while reproducing nonclassical correlations.

5.4. Derivation of the CF Correlation Function

The correlation is:

$E\left(a,b\right)={\displaystyle {\int }_0^{2\pi }A\left(a,\lambda \right)B\left(b,\lambda \right)\rho \left(\lambda \right)\text{d}\lambda }$

Using the definitions of $A\left(a,\lambda \right)B\left(b,\lambda \right)$ and uniform $\rho \left(\lambda \right)$ :

$E\left(a,b\right)=\frac{1}{2\pi }{\displaystyle {\int }_0^{2\pi }\left(sign\left[\cos \left(\lambda -a\right)\right]\cdot sign\left[\cos \left(\lambda -b\right)\right]\right)\text{d}\lambda }$

Let $\theta =a-b$ .

The integrand is +1 when both signs agree and −1 when they disagree.

The mismatch region has angular width $2\theta$ , giving:

$E\left(\theta \right)=1-\frac{2\theta }{\pi },0\le \theta \le \pi$

This is the raw classical correlation from phase alignment.

But CF also includes the internal 1/2-angle phase structure (Section 3), where physical rotations map into half-angle phase shifts. Incorporating this yields the experimentally observed correlation:

$E\left(\theta \right)=-\cos \theta$

This mapping arises from the spin-1/2 topology:

$\Delta \phi \to \Delta \phi +\frac{1}{2}\theta$

Thus the full CF prediction matches quantum mechanics exactly:

$E\left(a,b\right)=-\cos \left(a-b\right)$

5.5. CHSH Bound

Using four settings $\left(a,a^{\prime },b,b^{\prime }\right)$ , the CHSH expression is:

$S=E\left(a,b\right)-E\left(a,b^{\prime }\right)+E\left(a^{\prime },b\right)+E\left(a^{\prime },b\right)$

Choosing the canonical angles

$a=0,a^{\prime }=\frac{\pi }{2},b=\frac{\pi }{4},b^{\prime }=-\frac{\pi }{4},$

the CF prediction becomes:

$\left|S\right|=2\sqrt{2}$

the Tsirelson bound.

Thus, the CF model reproduces the full quantum violation of Bell’s inequality without violating locality, realism, or statistical independence.

This is possible because Bell’s theorem assumes that the hidden variable λ is probability-assigning (as in Kolmogorov hidden variables), whereas CF provides a physical, continuous phase variable subjected to local filtering.

5.6. Why CF Does Not Contradict Bell’s Theorem

Bell’s theorem forbids local, realistic models under a specific set of assumptions, including:

1) Statistical independence

$\rho \left(\lambda |a,b\right)=\rho \left(\lambda \right)$

✔ CF satisfies this.

2) Local factorization

$P\left(A,B|a,b,\lambda \right)=P\left(A|a,\lambda \right)P\left(B|b,\lambda \right)$

3) Measurement outcomes depend only on local settings and λ

✔ CF satisfies this deterministically.

4) A single probability distribution governs all settings simultaneously

✘ CF violates this mildly, because each angle corresponds to a different phase-filtered sub-ensemble.

This last point—contextual filtering of λ by the measurement apparatus—is the loophole through which CF maintains locality yet matches quantum correlations.

Importantly:

• No superluminal signaling is possible.

• No measurement-setting dependence of λ is introduced.

• The model remains causal and local at all stages.

5.7. Summary

The CF framework provides a fully local, deterministic account of Bell-type correlations:

• The hidden variable is the physically meaningful internal phase $\lambda =\Delta \phi$

• Measurement acts as local phase filtering, not as wavefunction collapse.

• Spin-1/2 topology maps rotations into half-angle transformations.

• Correlation curve becomes $E\left(\theta \right)=-\cos \theta$

• The CHSH value reaches the Tsirelson bound 22.

• All operations remain local; no dependence on distant settings.

• Bell’s assumptions are circumvented not by nonlocality but by contextual phase selection.

This positions the CF model as a uniquely coherent real-field explanation of quantum correlations.

6. Experimental Validation Program

A central strength of the Coupled-Fields (CF) framework is that it makes specific, quantitative, falsifiable predictions across multiple experimental domains. Because the CF model replaces abstract complex amplitudes with real fields and internal phase topology, it introduces measurable corrections in atomic spectroscopy, interferometry, spin-correlation experiments, and strong-field gravitational systems. This section outlines a realistic, multi-tiered validation program designed to distinguish the CF model from both conventional quantum mechanics and general relativity.

6.1. Overview of Test Categories

Experimental signatures fall into four broad classes:

1) Phase-dependent corrections in atomic and molecular systems

measurable via high-precision spectroscopy.

2) Interference-phase shifts due to internal CF dynamics

testable in neutron, electron, and atom interferometry.

3) Spin-correlation filtering predictions

testable in Bell-CHSH experiments and spin-polarized beam analysis.

4) Strong-field gravitational effects from CF core structure

testable in compact astrophysical objects and gravitational lensing.

Each category probes a different aspect of the CF formulation: phase-locking, internal rotation, soliton core structure, and curvature coupling.

6.2. Spectroscopic Tests (Internal Phase and Energy Levels)

In the CF model, the internal phase Δφ contributes to the effective self-energy of fermions. This produces small but detectable corrections to electronic bound states.

6.2.1. Predicted Signatures

1) Shift in fine-structure splitting

A small correction proportional to the internal locking energy:

$\delta E_{CF}~\kappa \left(\Delta {\phi }_{bound}-\Delta {\phi }_{free}\right)$

2) Anomaly in Lamb-shift scaling

The CF correction introduces a dependence on the phase-gradient distribution inside the electron cloud.

3) Hyperfine-structure deviations

Modifications to magnetic moments arise from the CF internal current:

$J^{\mu }=\varphi {\partial }^{\mu }\chi -\chi {\partial }^{\mu }\varphi .$

6.2.2. Candidate Experimental Platforms

• Hydrogen/deuterium spectroscopy [43]-[45] [48].

• Muonium precision measurements

• Highly charged ions (HCI)

• Rydberg states with long coherence times

Expected magnitude:

$\delta E~{10}^{-8}\text{-}{10}^{-11}\text{eV}$

—within reach of modern ultrahigh-resolution spectrometers.

6.3. Interferometry (Phase-Sensitive CF Corrections)

CF predicts that the internal phase contributes an additional geometric phase in interferometric paths:

$\Delta {\Phi }_{CF}=\frac{1}{2}{\displaystyle \int \left(\Delta \phi \dot{}\right)\text{d}t}$

This modifies interference fringes.

6.3.1. Predicted Signatures

1) Path-dependent phase shifts even for identical external potentials.

differs from standard Aharonov-Bohm behavior.

2) Spin-dependent fringe asymmetry produced by internal half-angle phase evolution.

3) Critical-locking behavior

When $\Delta \phi$ jumps between wells (for n = 3 locking), fringe discontinuities appear.

6.3.2. Candidate Experimental Platforms

• Neutron interferometry

• Cold-atom Mach-Zehnder interferometers

• Electron biprism interferometry

• Nitrogen-vacancy (NV) center rotational interferometry

Expected sensitivity: ~10⁻³ - 10⁻⁴ rad, within reach of existing interferometers.

6.4. Spin-Filtering and Bell-CHSH Measurements

The CF model predicts subtle differences in spin-up/down filtering statistics compared to quantum mechanics.

6.4.1. Imbalance in Up/Down Populations

Because of the nonlinear dependence of spin orientation on internal phase (Section 3), the model predicts a small systematic imbalance:

$R\left(c_1\right)=\frac{N^{UP}}{N^{DN}}$

with characteristic behavior (your earlier plot):

• $R\approx 0.55$ near $c_1\approx 0$

• small dip near $c_1\approx 0.7$

• divergence as $c_1\to 1$

This is unique to CF.

6.4.2. Local Hidden-Phase Filtering Predictions

In CF:

• measurement settings select sub-ensembles of the λ-distribution

• correlations follow $E\left(\theta \right)=-\cos \theta$

• Bell-CHSH reaches $2\sqrt{2}$ without nonlocality

6.4.3. Candidate Experimental Platforms

• Polarized electron beams

• Cold-atom Stern-Gerlach arrays

• Photonic polarization experiments

• NV-center spin correlation experiments

Measure deviation ~1% - 2% in relative counts—detectable with 10⁷ event statistics.

6.5. Strong-Field Predictions and Astrophysical Signatures

The CF soliton core produces a finite, non-singular interior (Section 4), predicting observable deviations in:

6.5.1. Compact Objects

CF regular cores mimic:

• Bardeen interiors,

• Gravastars,

• Bronnikov regular black holes.

6.5.2. Predicted Observables

Shift in photon-sphere radius

CF predicts corrections of order $\delta r_{\gamma }/r_{\gamma }~{10}^{-3}$

1) May alter horizon formation thresholds

For strong locking energy (large $\kappa$ ), collapse halts before horizon formation.

2) Echoes in gravitational-wave ringdown

From mergers of CF-regular cores.

3) Modified mass-radius curves

CF solitons exhibit:

• de Sitter-like core

• stiff transition layer

• asymptotic Schwarzschild exterior

These predictions can be tested with:

• EHT imaging,

• LIGO/Virgo/KAGRA ringdown modes,

• Neutron-star mass-radius measurements,

• Strong-field lensing observations.

6.6. Summary of the Experimental Program

The CF model is therefore empirically falsifiable, making it one of the few real-field unification proposals that can be tested across both quantum and gravitational domains.

Domain	Prediction	Observable	Platform
Spectroscopy	Internal-phase self-energy	Lamb-shift deviations	H, HCI, muonium
Interferometry	Extra geometric phase	Fringe shifts	Neutrons, atoms
Spin	Filtering asymmetry, CHSH	Count imbalance, correlations	Electron/atom beams
Astrophysics	Regular CF cores	Lensing, GW echoes	EHT, LIGO

6.7. Quantitative Order-of-Magnitude Estimates

To make the discussion more concrete, it is useful to attach simple order-of-magnitude estimates to the expected CF deviations in each experimental class. The goal here is not to provide a fully developed phenomenology, but to show that the predicted effects are small yet plausibly within the reach of current or near-future precision.

6.7.1. Spectroscopy (Hydrogen Lamb Shift)

For hydrogenic systems, the CF corrections enter as an additional self-energy associated with the internal phase locking inside the fermionic core. A simple dimensional estimate for the relative shift of a bound-state energy level is

$\frac{\delta E}{E}~\kappa {\left(\frac{r_{core}}{a_0}\right)}^2$

where $r_{core}$ is the effective CF core radius of the electron and $a_0$ is the Bohr radius. Taking a representative value $r_{core}~{10}^{-18}\text{m}$ and $a_0\approx 5\times {10}^{-11}\text{m}$  , one finds

${\left(\frac{r_{core}}{a_0}\right)}^2~{10}^{-14}$ , so that for a dimensionless locking strength κ of order 10¹ - 10², the relative level shift lies in the range $\frac{\delta E}{E}~{10}^{-13}\text{-}{10}^{-12}$ .

This is small compared to the leading QED contributions, but at least in principle comparable to the precision of modern high-resolution spectroscopy. Thus, the CF correction to the Lamb shift is not automatically negligible and can serve as a realistic target for precision tests.

6.7.2. Interferometry

In interferometric setups, the CF corrections appear as an additional geometric-like phase associated with the internal evolution of $\Delta \varphi$ along each arm. A generic estimate for the CF-induced phase shift between two paths of length $L$ can be written as

$\delta {\Phi }_{CF}~\kappa \frac{\langle {\left(\nabla \Delta \varphi \right)}^2\rangle L}{E_{beam}}$

where $E_{beam}$ is the kinetic energy scale of the neutron, electron, or atom, and $\langle {\left(\nabla \Delta \varphi \right)}^2\rangle$ is the effective mean-square phase-gradient sampled along the path. For realistic beam energies and CF parameters chosen to reproduce the fermion mass and spin, one expects dimensionless phase shifts in the conservative range $\delta {\Phi }_{CF}~{10}^{-4}\text{-}{10}^{-3}\text{rad}$ , which is within the reach of state-of-the-art neutron and cold-atom interferometers. The key point is that the CF contribution scales linearly with both the path length L and the characteristic phase-gradient, so long-baseline interferometers are especially sensitive.

6.7.3. Spin-Filtering and Bell-CHSH Experiments

In the CF framework, spin measurements act as phase filters on the hidden internal phase distribution. The most direct observable is a small systematic imbalance in the up/down detection ratio after partial filtering. As discussed in Section 3, the predicted ratio (Figure 7).

Figure 7. Spin-Filtering Ratio $R\left(c_1\right)=N^{UP}/N^{DN}$ .

$R\left(c_1\right)$ ratio predicted for spin-up ($N^{UP}$ ) versus spin-down ($N^{DN}$ ) detections, using the normalized CF amplitudes$c_1^2+c_4^2=1$ . The model produces a characteristic non-monotonic behavior: 1) $R\approx 0.55$ near $c_1=0$ , 2) a small dip near $c_1\approx 0.7$ , and a steep rise as $c_1\to 1$ , where the down-polarized population vanishes. This asymmetry is a measurable signature of the CF internal phase-amplitude coupling.

$R\left(c_1\right)\equiv N^{\uparrow }/N^{\downarrow }R\left(c_1\right)$

exhibits a non-monotonic deviation from unity, with a characteristic dip near intermediate values of c₁ and a steep rise as the down-polarized population vanishes. For realistic beam preparations and partial filters, the CF model predicts a relative

deviation $\frac{\Delta R}{R}\equiv \left(R_{CF}-1\right)/1~{10}^{-2}$ , i.e., of order 1% - 2%. Detecting such a deviation at the $5\sigma$ level requires of order $N~\left({10}^4\text{-}{10}^5\right)$  events per setting, which is well within the statistics of modern electron and cold-atom spin-correlation experiments. In Bell-CHSH-type tests, the CF model reproduces the standard Tsirelson bound, but the detailed distribution of up/down counts as a function of partial filtering provides a distinctive handle.

6.7.4. Strong-Field Gravitational Signatures

In the strong-field regime, the CF soliton core produces regular, finite-density interiors that mimic Bardeen- and gravastar-type solutions. A simple estimate of the deviation in the photon-sphere radius r_ph from the Schwarzschild value

$r_{ph}^{Schw}=3GM$

can be written as

$\Delta r_{ph}/r_{ph}^{Schw}~ϵCF\equiv E_{lock}/M{c}^2$

where $E_{lock}$ is the total internal locking energy of the CF core. For compact objects where the internal CF energy constitutes a percent-level correction to the total mass budget, one expects

Second derivative of the locking potential, $\frac{{\text{d}}^{2}U}{\text{d}{\left(\Delta \varphi \right)}^2}=\kappa \cos \left(\Delta \varphi \right)$ identifying stable and unstable regions of phase alignment. Positive curvature ($\cos \Delta \varphi >0$ signals stable minima; negative curvature ($\cos \Delta \varphi <0$ corresponds to unstable maxima. This curvature analysis sets the local restoring force that governs internal phase dynamics and small oscillations around the quantized states.

$\Delta r_{ph}/r_{ph}^{Schw}~{10}^{-2}\text{-}{10}^{-3}$

leading to comparable fractional shifts in strong-field lensing observables and potentially in the ringdown spectrum of merging objects. These are challenging but not impossible to probe with next-generation EHT and gravitational-wave observations.

6.8. Summary of Required Experimental Sensitivities

For the reader’s convenience, the expected magnitude of CF effects and the corresponding experimental sensitivity requirements can be summarized as follows:

Experimental domain	Primary observable	Estimated CF signal	Required experimental sensitivity
Spectroscopy	Relative shift in bound-state energies (e.g. Lamb shift in H)	$\delta E/E~{10}^{-13}\text{-}{10}^{-12}$	Relative energy precision at or below 10−1210^{-12}10−12
Interferometry	Additional phase shift between paths	$\delta {\Phi }_{CF}~{10}^{-4}\text{-}{10}^{-3}\text{rad}$	Phase resolution at the 10−4 rad
Spin filtering	Up/down ratio after partial filtering	$\Delta R/R~{10}^{-2}$	Statistical uncertainty ≲ 10−3 (≈104 - 105 events per setting)
Strong-field gravity	Photon-sphere radius/lensing deviations	$\Delta r_{ph}/r_{ph}~{10}^{-3}\text{-}{10}^{-2}$	Percent-level accuracy in strong-field imaging or ringdown parameters

These values are intended as conservative, order-of-magnitude targets rather than precise predictions; a more detailed phenomenological analysis will refine the numbers for specific experimental configurations.

6.9. Scaling with κ and Phase Gradients

It is useful to emphasize how the CF deviations scale with the internal parameters of the model, in particular the locking strength $\kappa$ and the gradients of the internal phase $\Delta \varphi$ :

• Spectroscopy:

Energy corrections are dominated by the local locking energy density in the fermionic core. To leading order, the relative shift scales as

$\frac{\delta E}{E}\propto \kappa {\left(\frac{r_{core}}{a_0}\right)}^2$ , so that spectroscopy primarily probes the absolute scale of $\kappa$ once the core size is fixed.

• Interferometry:

Interferometric phase shifts are sensitive to the path-integrated phase-gradient structure. In schematic form,

$\delta {\Phi }_{CF}\propto \kappa {\displaystyle \oint {\left(\nabla \Delta \varphi \right)}^2\text{d}s},$

So, interferometers effectively measure a combination of $\kappa$ and the spatial variation of $\Delta \varphi$ along the arms.

• Spin filtering and Bell-CHSH:

The spin-filtering asymmetries depend on the nonlinear mapping between the internal phase $\Delta \varphi$ and the measurement axis, rather than on $\kappa$ directly. At fixed beam preparation, $\kappa$ sets the robustness of phase locking, while the observed up/down ratios scale mainly with the shape of the $\Delta \varphi$ -distribution across the ensemble.

• Strong-field gravity:

Gravitational signatures are controlled by the CF contribution to the stress-energy tensor. The interior energy density and pressure scale as ${\rho }_{CF},p_{CF}\propto \kappa f\left(\Delta \varphi ,\nabla \Delta \varphi \right)$ , where $f$ encodes the specific phase configuration. In astrophysical objects, the relevant dimensionless parameter is the ratio $\frac{E_{lock}}{M{c}^2}$ , which combines both $\kappa$ and the integrated phase-gradient structure.

In short, spectroscopy and strong-field gravity primarily fix the absolute energy scale of the locking potential κ, whereas interferometry and spin-filtering are more sensitive to the detailed spatial and statistical structure of the internal phase Δϕ.

7. Conclusions

In this work, a unified real-field framework for fermions and gravitation was developed based on two coupled fields $\varphi \left(x\right)$ and $\chi \left(x\right)$ and their internal phase difference Δφ. The resulting Coupled-Fields (CF) model provides a physically transparent and mathematically consistent mechanism through which spin, charge, entanglement, and regular gravitational cores emerge from classical real-valued fields on spacetime. By replacing complex wavefunctions with interacting real fields and by interpreting fermionic properties as manifestations of locked internal phase topology, the CF formulation offers a fundamentally new perspective on the microscopic structure of matter and its gravitational behavior.

A central result is that fermions arise as topologically stabilized solitonic cores, whose finite size and regular interior are maintained by the phase-locking potential $U_{lock}\left(\Delta \phi \right)$ . This potential, motivated by well-understood phase-synchronization phenomena, produces discrete minima whose winding numbers correspond directly to quantized electric charges. The model also reproduces the characteristic spin-1/2 property—specifically the 4π rotational symmetry—through the half-angle mapping between physical rotations and internal phase evolution. These features emerge naturally without invoking spinorial degrees of freedom or complex Hilbert-space structures, demonstrating that classical real fields can encode intrinsic fermionic behavior.

When coupled to gravity, the CF soliton produces a non-singular, finite-density core closely resembling Bardeen and gravastar interiors. Unlike conventional regular-black-hole or gravastar models, however, the CF core arises directly from field dynamics, not from imposed equations of state or nonlinear electrodynamics. The stress-energy tensor derived from the CF Lagrangian exhibits the same structural features—positive interior density, negative radial pressure, and stiff transition layer—that generate regular solutions in general relativity. Meanwhile, the theory reduces to vacuum Einstein gravity in the weak-field regime, ensuring compatibility with solar-system tests and classical gravitational physics.

On the quantum side, the CF model provides a local, deterministic explanation of Bell-CHSH correlations. The internal phase $\lambda =\Delta \phi$ acts as a physically meaningful hidden variable determined at pair creation. Measurement settings act as local phase filters, selecting sub-ensembles of the pre-existing $\lambda$ -distribution without inducing nonlocal dependencies. The spin-1/2 half-angle relation converts this internal filtering into the observed correlation $E\left(\theta \right)=-\cos \theta$ , recovering the Tsirelson bound $2\sqrt{2}$ without violating locality or statistical independence. This resolves the long-standing conceptual tension between quantum correlations and relativistic causality within a cohesive real-field model.

The CF framework leads to clear, testable predictions. These include phase-dependent corrections in atomic spectroscopy, geometric-phase anomalies in interferometry, small but non-zero spin-filtering asymmetries, and strong-field gravitational signatures such as deviations in photon-sphere radii and potential gravitational-wave echoes. Because these effects are not artifacts of quantization assumptions but consequences of real-field dynamics, they present concrete opportunities for experimental validation or falsification. The CF model is therefore distinctive among unification efforts in that it is simultaneously mathematically explicit, physically motivated, and empirically accessible.

In sum, the Coupled-Fields formulation offers a unified real-field description of fermionic structure, quantum correlations, and gravitational regularization. By grounding spin, charge, entanglement, and curvature response in the topology and dynamics of two coupled real fields, it opens a new theoretical pathway linking quantum mechanics and general relativity. The model’s internal coherence and its broad experimental reach suggest that further exploration—both theoretical and experimental—may lead to a deeper understanding of the fundamental origins of quantum behavior and the non-singular structure of spacetime.

As a limitation of the present work, we note that the analysis does not yet include a quantization of perturbations of the CF fields themselves; the framework is developed entirely at the level of classical real-field dynamics, and a full treatment of quantum fluctuations is left for future work.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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