The CF dynamics follow from the action

S = ∫ ℒ − g   d 4 x , where the Lagrangian density takes the form

The potential U l o c k enforces local phase coupling between ϕ and χ . Motivated by classical phase-locking phenomena (Adler equation, Kuramoto synchronization), topological arguments (maps S 1 → S 1 ), and phenomenology of charge quantization, two classes of locking interactions are considered:

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 Δ φ , ensuring stability of the locked configuration.

The Euler-Lagrange equation for a field X is

where ∇ μ is the metric-compatible covariant derivative.

2.2.1. Variation with Respect to ϕ

From (2.1):

Thus,

or equivalently,

2.2.2. Variation with Respect to χ

Identical steps yield

Since the potential depends only on Δ φ ,

The phase is defined implicitly through

ϕ = R ϕ cos ϕ , ∂ μ φ ϕ = ϕ   ∂ μ χ − χ   ∂ μ ϕ R ϕ 2 , and analogously for χ .

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

Thus, the field equations become

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

Figure 3.Second Derivative of the Locking Potential.

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

g μ ν ∂ μ X ∂ ν X is a scalar quadratic form. U l o c k ( Δ φ ) depends only on the internal phase, which is a Lorentz scalar.The action integral uses the invariant − g   d 4 x .

Thus, S [ ϕ , χ ]  is manifestly Lorentz invariant. S [ φ , χ ] 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).

The antisymmetric bilinear

Obeys ∇ μ J μ = 0 , a consequence of the field equations and the fact that U l o c k depends only on the phase difference.

The circulation of this current is quantized:

giving rise to quantized charge and spin without complex fields.

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 Δ φ 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.

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

with n = 1 or n = 3.

The equilibrium points are given by

d U l o c k d ( Δ φ ) = n κ sin ( n   Δ φ ) = 0 , whose solutions are

This implies:

For n = 1: stable minima at Δ φ = 2 π k → One full cycle per winding → unit electric charge.For n = 3: stable minima at Δ φ = 2 π k → Three smaller wells → fractional charges ± 1 3 .

The curvature of the potential determines stability:

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

This explicitly produces the quantized charge landscape.

The internal phase Δ φ is periodic:

so, field configurations are classified by the integer winding number

Thus, the CF model naturally contains:

Topological solitonsQuantized internal circulationStable fermionic coresDiscrete charge states

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

The antisymmetric current (Section 2)

is identically conserved:

The corresponding charge is

In stationary, locked configurations, J 0 ∝ ∂ t ( Δ φ ) , and the total charge satisfies

Thus:

ω = ± 1 → charge ±1 (electron-like). ω = ± 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.

When a configuration undergoes a physical rotation through angle θ , the internal phase shifts as

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 ( ϕ , χ ) , not their individual phases, a shift of Δ φ 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:

Spin = 1 2 Rotational symmetry = 4π,

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

Expanding the potential around a stable minimum,

one finds a restoring “spring constant”

The competition between:

1) Gradient energy (spatial variation of the fields)

2) Locking energy (preference for stable Δ φ 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 interiorsGravastar-type pressure structuresSmooth transitions to GR weak-field behavior

There is no singularity.

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

None of these require:Complex wavefunctionsHilbert-space axiomsSpinor structureQuantization postulates

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

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 Δ φ ( x ) ∈ S 1

3.7.1. Internal Phase Circulation and ħ

From the conserved antisymmetric current

the total internal angular momentum of a stationary CF soliton is

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:

Δ φ → Δ φ + 2 π (4π spatial rotation)

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

In other words:

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

One full winding of Δ φ 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:

where,

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

determines two key quantities:

1) The stiffness of internal phase rotation

which controls the soliton’s resistance to internal dephasing.

2) The energy stored per unit winding

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:

where ω i n t is the internal rotation frequency.

This links κ to ħ:

Since electric charge corresponds to discrete windings ( ω ), we obtain the proportionalities:

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 ∝ 2 π ω , 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.

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

with ℒ given in Equation (2.1).

Explicitly:

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 ( ∂ ϕ ) 2 + ( ∂ χ ) 2

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

The full gravitational field equations are

with T μ ν 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 μ ν .

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

the phase-locked internal energy,the quantized solitonic core,the stress-energy produced by Δ φ 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

retain their usual form.

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

The energy density and pressures follow from (4.2):

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 μ ν R μ ν remain finite as r → 0 ; no curvature singularity arises. The Kretschmann invariant K = R α β γ δ R α β γ δ 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:

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 Δ φ 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:

ϕ → ϕ 0 , χ → χ 0 , U l o c k → 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.

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 ρ ( r ) = T 0 0 with p r ( r ) = T r r and p t ( r ) = T θ θ = T ϕ ϕ .

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

The effective equation of state in the interior satisfies ρ + 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.

In the limit where gradients and locking energy are small:

ϕ ≈ c o n s t , χ ≈ c o n s t , the stress-energy tensor vanishes:

Therefore:

This ensures:

No modification to solar-system tests,No deviations in weak lensing,No contradiction with classical general relativity.

The CF soliton core makes several testable predictions.

Finite Radius of Elementary Fermions

R C F ~ ρ 0 K e f f , 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.

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.

As shown in earlier sections, each CF fermion possesses an internal angular variable Δ φ 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:

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:

This satisfies Bell’s requirement that

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

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:

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:

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

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 λ − 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.

The correlation is:

Using the definitions of A ( a , λ ) B ( b , λ ) and uniform ρ ( λ ) :

Let θ = a − b .

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

The mismatch region has angular width 2 θ , giving:

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:

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

Thus the full CF prediction matches quantum mechanics exactly:

Using four settings ( a , a ′ , b , b ′ ) , the CHSH expression is:

Choosing the canonical angles

the CF prediction becomes:

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.

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

1) Statistical independence

✔ CF satisfies this.

2) Local factorization

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.

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

The hidden variable is the physically meaningful internal phase λ = Δ φ Measurement acts as local phase filtering, not as wavefunction collapse.Spin-1/2 topology maps rotations into half-angle transformations.Correlation curve becomes E ( θ ) = − cos θ 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.

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.

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:

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:

6.2.2. Candidate Experimental Platforms

Hydrogen/deuterium spectroscopy [43]-[45][48].Muonium precision measurementsHighly charged ions (HCI)Rydberg states with long coherence times

Expected magnitude:

—within reach of modern ultrahigh-resolution spectrometers.

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

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 Δ φ jumps between wells (for n = 3 locking), fringe discontinuities appear.

6.3.2. Candidate Experimental Platforms

Neutron interferometryCold-atom Mach-Zehnder interferometersElectron biprism interferometryNitrogen-vacancy (NV) center rotational interferometry

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

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 (Section3), the model predicts a small systematic imbalance:

with characteristic behavior (your earlier plot):

R ≈ 0.55 near c 1 ≈ 0 small dip near c 1 ≈ 0.7 divergence as c 1 → 1

This is unique to CF.

6.4.2. Local Hidden-Phase Filtering Predictions

In CF:

measurement settings select sub-ensembles of the λ-distributioncorrelations follow E ( θ ) = − cos θ Bell-CHSH reaches 2 2 without nonlocality

6.4.3. Candidate Experimental Platforms

Polarized electron beamsCold-atom Stern-Gerlach arraysPhotonic polarization experimentsNV-center spin correlation experiments

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

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

Shiftinphoton-sphereradius

CF predicts corrections of order δ r γ / r γ ~ 10 − 3

1) Mayalterhorizonformationthresholds

For strong locking energy (large κ ), 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 corestiff transition layerasymptotic Schwarzschild exterior

These predictions can be tested with:

EHT imaging,LIGO/Virgo/KAGRA ringdown modes,Neutron-star mass-radius measurements,Strong-field lensing observations.

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.

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

where r c o r e is the effective CF core radius of the electron and a 0 is the Bohr radius. Taking a representative value r c o r e ~ 10 − 18   m and a 0 ≈ 5 × 10 − 11   m  , one finds

( r c o r e a 0 ) 2 ~ 10 − 14 , so that for a dimensionless locking strength κ of order 10¹ - 10², the relative level shift lies in the range δ E E ~ 10 − 13   -   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 Δ ϕ along each arm. A generic estimate for the CF-induced phase shift between two paths of length L can be written as

where E b e a m is the kinetic energy scale of the neutron, electron, or atom, and 〈 ( ∇ Δ ϕ ) 2 〉 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 δ   Φ C F ~ 10 − 4   -   10 − 3   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 ( c 1 ) = N U P / N D N .

R ( c 1 ) ratio predicted for spin-up ( N U P ) versus spin-down ( N D N ) detections, using the normalized CF amplitudes c 1 2 + c 4 2 = 1 . The model produces a characteristic non-monotonic behavior: 1) R ≈ 0.55 near c 1 = 0 , 2) a small dip near c 1 ≈ 0.7 , and a steep rise as c 1 → 1 , where the down-polarized population vanishes. This asymmetry is a measurable signature of the CF internal phase-amplitude coupling.

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 Δ R R ≡ ( R C F − 1 ) / 1 ~ 10 − 2 , i.e., of order 1% - 2%. Detecting such a deviation at the 5 σ level requires of order N ~ ( 10 4   -   10 5 )  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

can be written as

where E l o c k 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, d 2 U d ( Δ ϕ ) 2 = κ cos ( Δ ϕ ) identifying stable and unstable regions of phase alignment. Positive curvature ( cos Δ ϕ > 0 signals stable minima; negative curvature ( cos Δ ϕ < 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.

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.

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

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.

It is useful to emphasize how the CF deviations scale with the internal parameters of the model, in particular the locking strength κ and the gradients of the internal phase Δ ϕ :

Spectroscopy:

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

δ E E ∝ κ ( r c o r e a 0 ) 2 , so that spectroscopy primarily probes the absolute scale of κ once the core size is fixed.

Interferometry:

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

So, interferometers effectively measure a combination of κ and the spatial variation of Δ ϕ along the arms.

SpinfilteringandBell-CHSH:

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

Strong-fieldgravity:

Gravitational signatures are controlled by the CF contribution to the stress-energy tensor. The interior energy density and pressure scale as ρ C F ,   p C F ∝ κ   f ( Δ ϕ , ∇ Δ ϕ ) , where f encodes the specific phase configuration. In astrophysical objects, the relevant dimensionless parameter is the ratio E l o c k M c 2 , which combines both κ 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.