We begin with the standard Hilbert space for a single-particle (or effective single-particle) matter wavefunction,

The inner product is the usual

The matter-sector Hamiltonian is written as

where m is the bare matter mass parameter and U ( x ) a possible external potential. At this stage U plays no dynamical role; its purpose is to allow comparison with the usual Schrödinger picture [7]. All nontrivial interaction with the timeon field will enter through a separate interaction Hamiltonian.

The timeon field is a complex scalar function,

with amplitude t ^ ≥ 0 encoding local time density and phase θ encoding an internal U ( 1 ) degree of freedom. Before introducing dynamics (which will come in Section 3), we specify the kinematic Hilbert space ℋ Φ as the space of square-integrable wavefunctionals over field configurations:

where C Φ is the configuration space of complex fields and D Φ is the formal functional measure. A generic state in this sector is written

a functional of the spatial profile of the timeon field. For classical field configurations, expectation values will later be dominated by stationary-phase contributions corresponding to finite-energy configurations of Φ. The timeon-sector Hamiltonian will be developed fully in Section 3. For now, we denote it symbolically by

emphasizing that it acts on functionals of the field.

The system consisting of matter and timeon field is described on the tensor-product Hilbert space [8]

A general state has the form

but for physical baryons we will be concerned with states that are approximately separable into a localised matter wavefunction and a localised timeon configuration. The inner product on ℋ factorizes:

The dynamics of the joint system are governed by a Hamiltonian of the form

where:

H ψ acts only on the matter wavefunction, H Φ acts only on the timeon sector, and H int couples them locally.

To anticipate later developments, the interaction Hamiltonian will take the form

where g is a coupling constant. This term expresses the intuitive fact that high matter probability density—and, over aeon timescales, chrono-emergent tunnelling—tends to drive the timeon field into high-amplitude, highly stable atomic configurations. Likewise, a region in which the timeon field has adopted the atomic phase produces an effective potential for the matter wavefunction. In classical field realizations, the operator | Φ | 2 becomes simply the squared amplitude t ^ 2 ( x ) ; when quantised, it acts as a local operator multiplying the wavefunctional Ψ Φ .

We now define the key object of the theory: the Baryon Partner State (BPS). A BPS is a joint eigenstate of the composite Hamiltonian (16) of the form

such that

The eigenvalue E BPS defines the effective mass of the composite object,

and the structure of Ψ Φ , BPS determines the internal charge and stability properties. The BPS must satisfy three physical conditions:

1) Finite energy: the timeon configuration must minimise H Φ subject to boundary conditions (derived in Section 4).

2) Self-consistency: the matter wavefunction ψ BPS must be bound to the highly stable timeon configuration that it helps generate.

3) Locality: both ψ BPS and Ψ Φ , BPS must be spatially localised, yielding a finite, well-defined composite excitation.

These conditions distinguish BPS configurations from arbitrary fluctuations of either field sector. Only when the interaction term (17) is included do localised atomic-phase solutions of the timeon field become highly stable, and only then does the matter wavefunction acquire a localised baryonic structure.

While the idealized definition above uses a separable tensor product state ψ ⊗ Ψ Φ , the physical BPS is generally entangled. The presence of the interaction Hamiltonian implies that the true ground-state solution takes the form

with coefficients c i determined by minimizing the total energy functional. The separable form is therefore best understood as the leading-order description in a Born-Oppenheimer-like approximation [9] in which the timeon field provides a slowly varying background for the matter wavefunction. This structure will be made explicit in later sections, where we derive both the static timeon configurations and the coupled evolution equations for ψ and Φ.

We adopt a physical Lagrangian density scaled by a stiffness modulus Z t (units of Force) to ensure the scalar field yields units of Energy Density:

Here, the amplitude field t ^ has units of time-density (s/m) and Z t provides the necessary conversion to Joules/m³.

The two degrees of freedom are thus:

t ^ : a real amplitude field controlling time-density, θ : a Goldstone-like phase variable.

We will see that stable baryonic configurations arise primarily from nonlinear structure in V and localised gradients in t ^ .

The Euler-Lagrange equation for the complex field is:

Working directly with Φ yields the d’Alembertian dynamics [7]:

where □   = ∂ μ ∂ μ is the d’Alembertian. Alternatively, in amplitude/phase variables:

Amplitude equation:

Phase equation:

where V ′ denotes derivative with respect to its argument ( t ^ 2 ) . The phase equation expresses the local conservation of a U ( 1 ) -like current,

which is the Noether current [10] that will play a role in defining effective “charge” of BPS solutions.

To compute energies of static configurations (including BPS cores) we require the Hamiltonian density. For a complex scalar,

where the canonical momentum is

Thus explicitly,

In polar form:

and

For static solutions (all BPS cores considered here), t ^ ˙ = 0 and θ ˙ = 0 , yielding the static Hamiltonian density:

This is the expression we will minimise to obtain baryonic solutions.

To support localised, highly stable atomic-phase reconfigurations, the potential must have at least two distinct minima:

with v atom > v vac ≥ 0 . The minima occur at t ^ = v vac and t ^ = v atom . We refer to these as:

Vacuum phase: t ^ = v vac (low time density)Atomic phase: t ^ = v atom (Condensed Lattice Phase)

The barrier separating them determines tunnelling rates and the nucleation structure of atomic-phase reconfigurations (Figure 1).

Quadratic expansion and mass scales

Expanding around each minimum yields the small-oscillation mass:

Around vacuum phase:

Around atomic phase:

Both masses will appear later in the fluctuation spectrum of BPS cores.

Figure 1.The Timeon Lattice Potential (TLP). The potential energy density V ( t ^ ) features a metastable Vacuum basin and a stable Atomic well. Baryon Partner State formation corresponds to tunnelling through the barrier separating these phases. Scope of this plot: This theoretical landscape is restricted to the formation of discrete Baryon Partner States and intentionally excludes black hole interiors and chrono-shear events (i.e., dense- and shear-time-density phases).

Finite-energy static solutions require that t ^ interpolate between different minima over a finite spatial width. The energy per unit area of such an interface (“domain wall”) is

This quantity plays a central role in critical-radius calculations, atomic-phase nucleation barriers, and BPS core stability. We will use σ in Section 4 and Section 5 to derive baryon mass and tunnelling thresholds.

The timeon field is governed by the Lagrangian (23) and Hamiltonian density (34).The potential must have at least two minima, leading to vacuum and atomic phases.Domain walls have finite surface tension σ given by (40).The small-oscillation masses in each phase determine the reconfiguration spectrum of baryons.

The next section constructs explicit finite-energy static solutions representing Baryon Partner States.

Because baryons are experimentally observed to be approximately spherically symmetric in their lowest-energy state, we impose spherical symmetry on the static timeon configuration:

where r = | x | . Staticity implies ∂ t t ^ = 0 and ∂ t θ = 0 . Thus only spatial gradients contribute to the energy. Under spherical symmetry:

The static energy functional from (34) becomes:

where primes denote derivatives with respect to r .

Finite-energy solutions require:

1) Core condition: The field must approach the atomic phase at r = 0 ,

2) Vacuum exterior: At large r the field returns to vacuum phase equilibrium:

3) Finite energy: The phase gradient must vanish outside the core,

4) Regularity at r = 0 : Terms like r − 2 must not diverge, implying

These conditions uniquely determine the qualitative shape of a BPS profile.

Variation of the energy functional (43) yields the coupled radial Euler-Lagrange equations [7].

Amplitude equation:

Phase equation:

Integrating the phase equation:

where Q is a constant of integration. This constant behaves as a conserved topological charge associated with phase gradients in the timeon field. Thus:

Substituting back into (44) yields the single radial equation:

This nonlinear ODE defines the full static BPS profile.

The quantity Q controls the strength of the phase gradient in the core region. Several key properties follow:

If Q = 0 , the phase is constant and the solution has no internal U ( 1 ) excitation. This describes a neutral baryonic configuration (the analogue of the neutron, if the U ( 1 ) charge is identified with electric charge).If Q ≠ 0 , the core hosts a localised winding of the phase, contributing to a finite U ( 1 ) charge density:

Because the energy density contains

the charge is confined to regions where t ^ is large (atomic phase). This provides a direct algebraic mechanism for baryonic charge confinement.

Clarification of the simultaneous roles ofQ. The quantity Q plays multiple roles simultaneously, and we clarify each:

1) Qas an integration constant. Equation (46) arises by integrating the radial phase equation d d r ( r 2 t ^ 2 θ ′ ) = 0 . In the static, spherically symmetric setting of Section, Q is therefore first and foremost a constant of integration that parameterises the family of solutions. Different values of Q give physically distinct configurations.

2) Qas a Noether charge. The phase equation itself expresses local conservation of the U ( 1 ) Noether current j μ = 2 t ^ 2 ∂ μ θ (derived inAppendixC). The static radial quantity Q measures the total radial phase-current threading each spherical shell, and by the phase equation this is the same on every shell—hence a constant.

3) Qas a topological winding invariant. Unwinding the phase from Q ≠ 0 to Q = 0 requires passing through a configuration in which t ^ = 0 at some radius, which carries infinite energy from the charge-confinement term Q 2 / ( r 4 t ^ 2 ) in the energy functional. Q is therefore also a topological invariant of the BPS.

Summary. Q is simultaneously an integration constant (in the ODE sense), a conserved Noether charge (from the global U ( 1 ) symmetry), and a topological winding invariant (from the energetic barrier against phase unwinding). The integer character of Q for physical BPS states follows from requiring that e i θ be single-valued on each spherical shell, yielding Q ∈ ℤ and thus charge quantisation. The phrase “conserved charge” in the Introduction and the phrase “integration constant” in the present section refer to the same quantity viewed at different levels of description.

Substituting (47) into the static energy functional yields:

Three competing contributions determine the radius and stability of a BPS:

1) Gradient energy: favours slowly varying t ^ .

2) Phase-gradient energy (charge term): grows rapidly as t ^ → 0 , confining charge to the atomic-phase core. Note that Q here denotes the timeon U ( 1 ) Noether charge; its identification with electric charge in units of e is the subject of forthcoming work.

3) Potential energy: favours either vacuum or atomic phase in bulk.

Balancing these terms yields a stable finite-radius atomic-phase core.

When the energy barrier between vacuum and atomic phase is large relative to gradient contributions, the BPS resembles a domain-wall configuration of radius R . In this regime we approximate:

The energy simplifies to:

where Δ V = V ( v atom 2 ) − V ( v vac 2 ) < 0 . Minimizing (50) gives the equilibrium BPS radius:

This cubic equation determines the physical radius as a function of charge Q , potential parameters, and surface tension.

Mass estimate from the thin-wall formula. Setting Q = 0 for simplicity (charge-free limit), minimizing E ( R ) = 4 π R 2 σ + 4 π 3 R 3 Δ V gives the critical radius

and the corresponding BPS mass

Here σ (energy per unit area) and | Δ V | (energy per unit volume difference between phases) are the two free parameters of the thin-wall approximation. For R * ~ 1   fm (proton charge radius) and M BPS ~ 938   MeV (proton mass), dimensional analysis requires σ ~ ( 200   MeV ) 3 and | Δ V | ~ ( 200   MeV ) 4 , coinciding with the QCD confinement scale Λ QCD ≈ 200   MeV . This numerical coincidence is encouraging, though a full parameter fit to the light baryon spectrum—including isospin breaking and coupling-to-matter corrections—is deferred to a forthcoming companion paper.

The BPS core is topologically protected by the structure of the phase variable θ and by the requirement that t ^ remain nonzero inside the atomic region. Key features:

The phase equation mandates that r 2 t ^ 2 θ ′ = Q . If one attempted to split a BPS into smaller charged fragments, this relation would require singularities (where t ^ = 0 ) to maintain total Q . Such configurations have infinite energy.The volume energy density difference Δ V < 0 favours a stable core of atomic phase.The gradient term penalises fragmentation because multiple small atomic-phase regions have larger total surface area than a single region of the same volume.

Thus the BPS is intrinsically confined: it cannot decompose without energy divergence or violation of charge conservation.

We have:

Written the full radial equations for static timeon configurations.Identified the conserved charge Q associated with phase gradients.Derived the energy functional for BPS solutions.Obtained approximate analytic behaviour via the thin-wall limit.Shown that confinement follows directly from the algebraic and topological structure of the timeon field.

These results prepare the ground for computing tunnelling, atomic-phase nucleation, and BPS nucleation in the next section.

Quantum tunnelling in scalar field theory is governed by the Euclidean action, obtained by Wick rotation t → − i τ [11]:

For tunnelling from the vacuum to the atomic phase, the relevant solutions possess O ( 4 ) symmetry in the Euclidean coordinates

leading to the ansatz

The Euclidean action becomes

The Euler-Lagrange equations corresponding to (57) are:

Amplitude:

Phase:

Integrating:

where Q E is a Euclidean analogue of the charge parameter. Most tunnelling calculations assume Q E = 0 , corresponding to a phase-uniform bounce. We will treat this case first, then include nonzero charge.

With θ ′ = 0 , the Euclidean equation reduces to:

Boundary conditions:

The solution interpolates from the atomic minimum near ρ = 0 to the vacuum minimum as ρ → ∞ . The configuration is called the bounce and denoted t ^ B ( ρ ) [12]. The tunnelling exponent is:

The nucleation rate per unit volume is:

where A is a fluctuation determinant prefactor.

When the barrier separating the two minima is large compared to their energy difference, the bounce resembles a thin-walled atomic-phase domain. In this limit, the Euclidean action reduces to:

where:

σ is the surface tension from (40), Δ V = V ( v atom 2 ) − V ( v vac 2 ) < 0 .

Differentiating:

leading to the critical radius:

Only atomic-phase domains with R ≥ R c expand; smaller domains collapse. Substituting back:

This closed-form expression is crucial to estimating BPS nucleation probabilities.

If Q E ≠ 0 , the Euclidean phase gradient contributes a centrifugal-like term:

This term:

suppresses configurations where t ^ is small,drives the interior of the atomic-phase domain toward the atomic phase,increases the effective surface pressure,decreases the tunnelling exponent B .

The corrected thin-wall action becomes:

where C is an order-one geometric constant. Minimizing yields a modified critical radius:

Thus:

charge slightly expands the critical radius,but makes the atomic phase more energetically favoured inside the atomic-phase domain,and typically decreases the tunnelling exponent.

Charged tunnelling therefore increases the likelihood of BPS formation, as does the progression of aeon timescales.

A physically transparent interpretation emerges:

1) The vacuum phase is metastable, adjacent to a more highly stable atomic-phase configuration accessible via probabilistic tunnelling.

2) Quantum fluctuations—and, over aeon timescales, chrono-emergent tunnelling—occasionally nucleate a small region of atomic phase.

3) If the region exceeds the critical radius R c , it expands.

4) The presence of matter (Section 6) reduces R c locally by compressing the timeon field.

5) The expanding atomic region forms the core of a BPS.

Thus baryon formation is not simply imposed—it is a dynamical outcome of timeon-field tunnelling stabilised by matter-wave coupling.

We have:

Formulated vacuum-to-atomic tunnelling using the Euclidean action.Derived the bounce equations and boundary conditions.Obtained the critical radius R c and tunnelling exponent B .Extended the thin-wall approximation to include charge.Established the mechanism by which atomic-phase reconfigurations become stabilised baryonic cores.

The next section introduces the coupling to the matter wavefunction, completing the dynamical picture of Baryon Partner States.

We postulate the simplest local, scalar, bilinear interaction:

where g is a real coupling constant controlling the strength of the compression effect. Writing Φ = t ^   e i θ :

Thus the interaction term becomes:

This is the lowest-order operator capable of:

lowering the effective potential seen by t ^ in regions where | ψ | 2 is large,raising the effective potential of ψ in regions where t ^ is small,and creating a self-localizing composite eigenstate.

The interaction is local, Hermitian, and preserves all global symmetries.

The full Hamiltonian is:

with H ψ and H Φ defined in earlier sections. We now derive the equations governing the coupled dynamics.

6.2.1. Matter Wavefunction Equation

Varying the total action with respect to ψ * yields:

Thus the timeon amplitude t ^ 2 contributes an effective potential:

Regions of high t ^ (atomic phase) attract the matter wavefunction, creating a stable bound state. This is the coupled Schrödinger equation [7].

6.2.2. Timeon Amplitude Equation

Varying the total action with respect to t ^ yields:

The right-hand side shows explicitly:

matter density suppresses the vacuum minimum,and deepens the atomic minimum,effectively tilting the potential energy in favor of the atomic phase.

More precisely, the interaction modifies the potential:

Thus:

if g > 0 : matter pushes the timeon field to larger amplitudes (atomic phase),if g < 0 : matter repels the atomic phase (we focus on g > 0 ).

6.2.3. Timeon Phase Equation

Variation with respect to θ yields the modified current conservation equation:

Because H int depends only on t ^ 2 , the U ( 1 ) phase remains a conserved degree of freedom. The associated charge Q remains valid in the presence of matter.

Since a static BPS is time-independent, we set ∂ t = 0 . The coupled static system is:

These three equations must be solved simultaneously. They express:

Matter binds to the atomic core because t ^ 2 creates an attractive potential.The atomic core forms because matter density lowers the local barrier.Charge remains confined to the core because of the phase equation.

This is the mathematical heart of the Baryon Partner State.

Assuming spherical symmetry:

Equations reduce to:

Matter equation:

Amplitude equation:

Phase equation:

These equations form a nonlinear eigenvalue problem for E ψ , ψ ( r ) , t ^ ( r ) , and θ ( r ) .

The structure is physically transparent:

The matter density | ψ | 2 acts as a source of compression, pushing t ^ toward the atomic minimum.The atomic phase provides a deep potential well that traps ψ , producing a localised matter-state peak.This feedback loop continues until a self-consistent equilibrium is reached, corresponding to a static BPS.The conserved charge parameter Q is forced into the atomic core, because the quantity Q 2 / ( r 4 t ^ 2 ) diverges unless t ^ remains nonzero.The BPS is therefore intrinsically confined: both charge and matter are locked into the core region.

From Section, the critical radius for atomic-phase nucleation was:

Matter modifies the potential:

This reduces the effective energy difference:

Thus the new critical radius becomes:

which is strictly smaller when | ψ | 2 > 0 . Therefore:

Matter catalyses atomic-phase domain expansion.Even small matter densities dramatically increase the likelihood of BPS formation.Baryons form most easily where matter density is highest.

To demonstrate explicitly that the coupled timeon-matter system admits regular, finite-energy, localised solutions, we present a representative numerical solution in the spherically symmetric sector.

Radial system. We work with the static radial equations derived in Section 6.4, cast in dimensionless form via the scaling

where μ is the characteristic inverse length set by the model parameters. In these units the coupled system (83)-(84) takes the form

where Q is the conserved charge parameter of Section 4.4 and { λ i } denotes the set of dimensionless coupling constants. The explicit right-hand sides ℱ and G are obtained directly from Equations (83) and (84) respectively.

Boundary conditions. Regularity at the origin and spatial localization require

with finite central values

and asymptotic decay

Near the origin, both fields admit regular even-power expansions,

with coefficients ψ 2 , t ^ 2 determined by substituting into the field equations. These expansions are used to initialize the numerical integration away from the coordinate singularity at ρ = 0 .

Representative parameter set. We adopt the dimensionless parameter choice

with the double-well potential of Equation (35). The vacuum minimum v vac = 0 simplifies the numerics; the qualitative structure is unchanged for v vac > 0 . These values are chosen to demonstrate the existence of a regular localised configuration and are consistent with the proton-scale heuristic of Section 4.6; they are not intended to represent a unique physical fit.

Numerical method. The shooting parameters ( ψ c , t ^ c ) are adjusted iteratively so that the decaying boundary conditions (92) are satisfied at a finite cutoff ρ max to tolerance ε num . The procedure proceeds in three stages:

1) Initialisation. Set ψ ( 0 ) ( ρ ) = 0 (no matter back-reaction). Solve the decoupled timeon Equation (89) outward from ρ min = 10 − 3 , using the near-origin expansion (94) to start the integration, on a logarithmically spaced grid of N = 2000 points over ρ ∈ [ 10 − 3 , 20 ] . Bisect t ^ c until t ^ ( ρ ) → v vac without divergence, yielding t ^ ( 0 ) ( ρ ) .

2) Matter eigenvalue. Solve the radial Schrödinger Equation (88) with effective potential U eff ( ρ ) = g [ t ^ ( 0 ) ( ρ ) ] 2 as a Sturm-Liouville eigenvalue problem for the lowest bound-state energy E ψ and wavefunction ψ ( 1 ) ( ρ ) , using the Numerov algorithm on the same grid ( 10 4 points for eigenvalue accuracy < 0.1%).

3) Self-consistent iteration. Use | ψ ( 1 ) | 2 as a source term in (89) to obtain a corrected timeon profile t ^ ( 1 ) ( ρ ) . Repeat steps 2 - 3 until ‖ t ^ ( n + 1 ) − t ^ ( n ) ‖ ∞ < 10 − 6 . Convergence is typically reached in 3 - 5 iterations.

Localised solution and characteristic scale. For the parameter set (95), the converged solution is smooth, nodeless, and spatially localised. The timeon profile t ^ ( ρ ) begins at the atomic-phase core value t ^ ( 0 ) ≈ 1.00 , decays through a transition region of half-width δ ρ ≈ 0.8 , and reaches t ^ → 0 exponentially for ρ ≳ 3 . The matter wavefunction ψ ( ρ ) is concentrated within ρ ≲ 2 .

At the integration boundary ρ max = 20 , the numerical solution satisfies

confirming accurate implementation of the asymptotic boundary conditions (92).

A convenient confinement scale is the radius ρ 90 enclosing 90% of the total energy,

where ℰ ( ρ ) is the dimensionless energy density of the coupled system (gradient, phase, potential, and matter contributions). The characteristic BPS radius, defined by the timeon e-folding scale, satisfies

Energy and finiteness. The total dimensionless energy,

evaluates to

which is finite and confirms a localised bound state. The binding energy of the matter sector is

The dominant contributions are gradient and potential energy, consistent with the thin-wall estimate of Section 4.6.

Linearised radial stability. Stability against radial perturbations is assessed by expanding about the static solution ( ψ 0 , t ^ 0 ) :

Substituting into the field equations and linearizing yields the coupled eigenvalue problem (110) for ω 2 . For the representative solution, all computed modes satisfy

confirming the absence of exponentially growing radial instabilities. The lowest non-zero mode,

corresponds to the radial breathing mode of Section 7.5.

Numerical summary. The key outputs of the representative solution are collected in Table 1.

Table 1. Representative numerical solution of the coupled timeon-matter system for the parameter set (95). All quantities are in natural units ( ℏ = c = 1 ).

Physical interpretation. Setting R BPS = 1   fm (proton charge radius, r p = 0.841   fm [13]) to translate to physical units gives v atom ~ Λ QCD / c ~ 200   MeV / c and E BPS ~ 1.8   GeV —within a factor of two of the proton mass, without parameter fitting. The breathing mode frequency ω 1 ≈ 370   MeV is consistent with the lightest baryonic excitations above the ground state. This explicit example demonstrates that the coupled timeon-matter system admits regular, finite-energy, spatially localised solutions with no evidence of radial instability for a concrete representative parameter set.

Code availability. The numerical integration uses Python with scipy.integrate.odeint and scipy.linalg.eig.

We have derived:

the full coupled equations of motion for ψ and Φ,the attractive potential binding ψ to the atomic core,the compression effect driving t ^ into the atomic phase,the self-consistency conditions defining a BPS,and the reduction of the critical radius via matter interaction.

Matter and timeon fields therefore form a tightly bound nonlinear system whose stable joint eigenstates are the Baryon Partner States. The next section analyses the fluctuation spectrum about these solutions, a necessary step in determining stability, internal excitations, and inertial response.

Let the exact static BPS configuration be

We introduce small perturbations:

To linear order, the dynamics separate into amplitude and phase sectors. We write perturbations as normal modes:

Here ω is the mode frequency, determining stability ( ω 2 > 0 ) or instability ( ω 2 < 0 ).

First consider small oscillations around the vacuum minimum t ^ = v vac . Expanding the potential:

where

Thus the amplitude perturbation satisfies:

Similarly, around the atomic minimum:

We interpret m atom as the intrinsic mass scale governing small internal vibrations of the BPS core. Phase fluctuations:

Are massless in the absence of charge,But become massive in the presence of a nonzero phase gradient, And are confined to the atomic core due to the charge term t ^ 2 ( ∇ θ ) 2 .

These homogeneous spectra form the asymptotic basis for fluctuations around the full BPS profile.

We linearize the amplitude Equation (78) around t ^ 0 ( r ) , holding the background phase θ 0 fixed. Define the radial fluctuation operator:

with effective potential:

The amplitude perturbation satisfies:

This is a Schrödinger-type eigenvalue problem with potential U t ( r ) . Key features:

Near the core, U t ( r ) is dominated by large positive terms coming from the charge confinement term Q 2 / ( r 4 t ^ 0 4 ) .In the exterior, U t ( r ) → m vac 2 .There is a finite number of discrete bound states.

Thus the spectrum consists of: 1) A zero mode (translation), 2) One or more internal breathing modes, 3) A continuum starting at ω 2 = m vac 2 .

Because the total energy is invariant under spatial translations, the BPS supports a zero-frequency mode:

To verify:

This mode corresponds physically to moving the baryon slightly in space. It is crucial for understanding inertial mass, constructing collective coordinates, and quantizing the motion of the BPS as a whole.

The lowest nonzero mode corresponds to a radial compression or expansion:

where R is the effective core radius. This “breathing mode” has:

a frequency ω 1 of order m atom ,spatial support concentrated in the transition region,a key role in BPS excitations.

This is the internal vibrational degree of freedom analogous to energy levels in bound quark systems, but arising purely from timeon dynamics. These resonances give rise to Breit-Wigner peaks [14].

Perturbations of the phase satisfy the linearised form of (79):

Letting

the equation becomes a 1D Schrödinger-like equation:

with