Gyromagnetic Ratio of Electrically Neutral Particles: Case of the Neutron

1. Introduction

This is the third instalment of the series — Toward a Unified and Universal Dirac Equation. Papers (I) and (II) [1,2] developed a framework in which the anomalous $\mathrm{g}$-factor of a charged spin-$\frac{1}{2}$ particle arises from the coupling of the Dirac spinor to a hypothetical cosmic vector field $\mathbf{H}$, yielding:

$${\mathrm{g}}_{\mathrm{P}}=2\left(1+{\displaystyle \frac{{\zeta }_{\mathrm{P}}{m}_{★}{r}_{\mathrm{P}}{c}_0}{\hslash }}\right).$$

(1)

This formula applies directly to charged particles such as the electron and proton. The neutron, however, poses an immediate difficulty: it is electrically neutral, so the minimal coupling through which the $\mathbf{H}$-field enters Equation (1) vanishes identically when $q=0$. Yet the neutron’s measured $\mathrm{g}$-factor, ${\mathrm{g}}_{\mathrm{N}}=-3.82608545\left(9\right)$ [3], departs dramatically from the Dirac prediction of zero for a neutral particle. The bare Dirac equation therefore fails completely for the neutron, regardless of its quark substructure.

Two complementary approaches to this problem are explored in the present paper:

1.

EDM Coupling (§3): The neutron is treated as an effectively charged particle by assigning it an effective charge proportional to its electric dipole moment (EDM). Although the neutron’s net charge is zero, it may possess an internal charge distribution whose dipole component couples to the ambient field.

2.

Leptonic Composite Model (§5): Revisiting Rutherford’s historical proton–electron composite model [4,5] and extending it to a three-body system that resolves the original spin objection. In this picture the neutron is a quantum superposition of a tauon $\left({\tau }^{-}\right)$, a de-excited proton $\left(p_1^{+}\right)$, and a neutrino $\left({\overline{\nu }}_{p_1}\right)$, and its $\mathrm{g}$-factor is a weighted average of the constituents’ anomalies.

Scope and Limitations

This paper is exploratory. The EDM coupling [§(3)] is physically motivated but relies on an effective charge whose value is not known, since no permanent neutron EDM has yet been observed [6]. The leptonic composite model [§(5)] is speculative in several respects itemised in §(8): the identification of the tauon as an excited electron, the assignment of a specific de-excited proton mass, and — most critically — the required neutrino magnetic anomaly of $\Delta {\mathrm{g}}_3\simeq -12.37$, which lies ten orders of magnitude beyond Standard Model expectations. The model makes testable predictions [listed in §(7)], but these await experimental scrutiny before any of its claims can be considered established or viable.

Relation to the Standard Model

We do not claim to replace the Standard Model’s successful quark model of the neutron, which — amongst others — successfully accounts for many neutron properties including the ratio ${\mu }_p/{\mu }_{\mathrm{N}}\simeq -3/2$ [7] and is supported by lattice QCD calculations [8]. The leptonic composite model is presented as an alternative theoretical perspective that, if correct, would require a fundamental revision of our understanding of nuclear structure and possibly nuclear physics itself. The two pictures — quarks versus leptonic composites — make overlapping but distinct predictions, some of which are discussed in §(7).

2. Review of Papers (I) and (II)

Before we proceed with matters of the paper article, we shall make a recap of Paper (I) and (II).

2.1. Modified G-Factor Formula

In Paper (I) [1], the Dirac spinor $\psi$ was coupled to a hypothetical all-pervading, all-permeating and non-ponderable neutral scalar field ${\varphi }_{\mathrm{H}}$, promoting the scalar mass term to a $4\times 4$ matrix. The resulting modified Dirac equation yields:

$${\mathrm{g}}_{\mathrm{P}}=2\left(1+\Delta {\mathrm{g}}_{\mathrm{P}}\right),\Delta {\mathrm{g}}_{\mathrm{P}}={\displaystyle \frac{{\zeta }_{\mathrm{P}}{m}_{★}{r}_{\mathrm{P}}{c}_0}{\hslash }},$$

(2)

where $r_{\mathrm{P}}$ is the particle radius, $m_{★}=q{\mathsf{\phi }}_{\mathrm{P}}{\kappa }_{\mathrm{H}}\hslash /c_0$, and ${\zeta }_{\mathrm{P}}$ is the dimensionless alignment parameter.

2.2. Dyson Series from Classical Self-Energy

In Paper (II) [2], ${\zeta }_{\mathrm{P}}$ was identified with the dimensionless stored electromagnetic self-energy of the particle (the ASTE-model), giving the Dyson expansion:

$$\Delta {\mathrm{g}}_{\mathrm{P}}={\displaystyle \frac{{\alpha }_0}{2\pi }}\sum _{\ell =0}^{\infty }{\left({\displaystyle \frac{{\alpha }_0}{2\pi }}\right)}^{\ell }{a}_{\ell },$$

(3)

with $a_0=1$ fixed by matching to Schwinger’s result [9]. This identification required fixing two proportionality constants — a step that is explicitly acknowledged as a normalisation to known QED results rather than a true first-principles derivation.

2.3. Current Status of Neutron as a udd-Quark System

For completeness, we summarise the current status of the neutron in the Standard Model. In the Standard Model of Particle Physics, the neutron is a composite of one up quark (charge $+2e/3$) and two down quarks (charge $-e/3$), denoted $udd$. The neutron’s magnetic moment arises from the spin and orbital motion of these quarks, and the SU(6) symmetric quark model predicts ${\mu }_p/{\mu }_{\mathrm{N}}=-3/2$, in agreement with experiment to within 3% [7]. Lattice QCD (LQCD) provides the only known method for exact first-principles calculations from QCD, with recent progress in nucleon axial coupling and form factors [8].

Significant challenges remain. Measurements of the neutron’s spin polarizabilities at low momentum transfer show strong discrepancies with chiral effective field theory predictions [10]. The neutron’s charge radius relative to the proton has been measured via parity-violating electron scattering [11]:

$$r_{\mathrm{N}}-r_p=0.{33}_{-0.18}^{+0.16}\mathrm{fm},$$

(4)

but precision measurements of the free neutron radius remain challenging.

The search for a permanent neutron EDM is one of the most sensitive probes of physics beyond the Standard Model [12]. The current best limit from PSI [6] is:

$$|d_{\mathrm{N}}|\lesssim 1.8\times {10}^{-26}e·\mathrm{cm} (90\% \mathrm{C}.\mathrm{L}.),$$

(5)

many orders of magnitude above Standard Model predictions, leaving room for new physics.

3. Extension to Electrically Neutral Particles: EDM Coupling

The framework of Papers (I) and (II) couples to particles through their electric charge q. For the neutron, $q=0$ and the coupling vanishes. We introduce the following extension.

  Hypothesis 1

(Effective EDM charge). For an electrically neutral particle possessing an internal electric dipole moment $d_{\mathrm{EDM}}^{\mathrm{N}}$, an effective charge is defined:

$$q_{\mathrm{N}}^{\mathrm{eff}}={\kappa }_{\mathrm{H}}{d}_{\mathrm{EDM}}^{\mathrm{N}},$$

(6)

where ${\kappa }_{\mathrm{H}}$ is the inverse length scale of the CFV-field introduced in Paper (I). This effective charge allows the neutron to couple to the ambient magnetic vector potential $\mathit{A}$ in analogy with a charged particle.

With $q\to q_{\mathrm{N}}^{\mathrm{eff}}$ in Equation (2), the framework of Papers (I) and (II) applies to the neutron. The modified $\mathrm{g}$-factor becomes:

$${\mathrm{g}}_{\mathrm{N}}=2\left(1+{\displaystyle \frac{{\zeta }_{\mathrm{N}}{m}_{★}^{\left(\mathrm{N}\right)}{r}_{\mathrm{N}}{c}_0}{\hslash }}\right),m_{★}^{\left(\mathrm{N}\right)}=q_{\mathrm{N}}^{\mathrm{eff}}{\mathsf{\phi }}_{\mathrm{N}}{\kappa }_{\mathrm{H}}\hslash /c_0.$$

(7)

 Remark 1.

Hypothesis (1) faces an immediate empirical difficulty: no permanent neutron EDM has been observed. The limit $|d_{\mathrm{N}}|\lesssim 1.8\times {10}^{-26}e·\mathrm{cm}$ [Equation (5)] is consistent with $d_{\mathrm{N}}=0$. If $d_{\mathrm{N}}=0$ exactly, then $q_{\mathrm{N}}^{\mathrm{eff}}=0$ and the coupling vanishes, rendering this approach inapplicable.

Two resolutions are possible. First, a non-zero $d_{\mathrm{N}}$ at or below the current experimental limit could provide sufficient coupling: the n2EDM experiment at PSI aims for an order-of-magnitude improvement in sensitivity [13], and a detection would directly test this hypothesis. Second, the EDM referred to in Equation (6) may be an internal or dynamical EDM associated with the charge distribution within the neutron, distinct from the permanent EDM probed by spin-precession experiments. The relationship between these two quantities within the present framework requires further theoretical development.

4. General Spin Dirac Equation

The composite model of §(5) draws on a generalisation of the Dirac equation to particles of arbitrary spin, developed in [14,15,16]. We summarise the relevant results here.

4.1. Generalised Dirac Hamiltonian

The generalised Dirac Hamiltonian for a particle with spin parameter s is:

$$H_{\mathrm{D}}\left(s\right)=-ıs\hslash c_0{\mathsf{\gamma }}^0{\mathsf{\gamma }}^j{\partial }_j+{\mathsf{\gamma }}^0{m}_{0s}{c}_0^2,$$

(8)

leading to the generalised Dirac equation:

$$ı\hslash {\mathsf{\gamma }}_s^{\mu }{\partial }_{\mu }\psi =m_{0s}{c}_0\psi ,$$

(9)

where ${\mathsf{\gamma }}_s^0={\mathsf{\gamma }}^0$ and ${\mathsf{\gamma }}_s^k=s{\mathsf{\gamma }}^k$. The total angular momentum $J_i\left(s\right)=L_i\left(s\right)+S_i\left(s\right)$ satisfies $[H_D\left(s\right),J_i\left(s\right)]=0$, confirming that this describes a particle of spin $s/2$.

4.2. Mass Scaling

A key result of [16] is that the rest mass scales with s:

$$m_{0s}=s{m}_{01},$$

(10)

where $m_{01}$ is the mass in the lowest spin state $\left|s\right|=1$. Higher-spin states therefore have proportionally higher masses, providing a possible explanation for the observed mass hierarchy of elementary particles.

 Remark 2

(On the spin–statistics connection). The spin parameter s determines the statistics: odd s gives fermions, even s gives bosons. While this is consistent with the spin–statistics theorem [17], the identification $s=2$ for the photon and the implied mass $m_{02}=2m_{01}$ for the corresponding fermionic partner requires further justification that goes beyond the present paper.

4.3. Modified G-Factor for General Spin Particle

Applying the $4\times 4$ matrix mass term of Paper (I) to the generalised Dirac equation gives the $\mathrm{g}$-factor for a particle with spin parameter $s_{\mathrm{P}}$:

$$g_{\mathrm{P}}^s=2s_{\mathrm{P}}\left(1+\Delta {\mathrm{g}}_{\mathrm{P}}\right),$$

(11)

where $\Delta {\mathrm{g}}_{\mathrm{P}}$ is the same anomalous excess as in Equation (2) and $s_{\mathrm{P}}=\pm 1,\pm 2\dots etc$. The anomalous departure from $2s_{\mathrm{P}}$ is:

$$\Delta g_{\mathrm{P}}^s=(s_{\mathrm{P}}-1)+s_{\mathrm{P}}\Delta {\mathrm{g}}_{\mathrm{P}}.$$

(12)

For $s_{\mathrm{P}}=1$ (ordinary spin-$\frac{1}{2}$ particles), Equation (11) reduces to Equation (2).

 Remark 3

(On gravitational light deflection). Reference [16] shows that for a photon with spin parameter $s=2$, the deflection angle in a gravitational field is $\delta =4GM/c_0^{2}R$, matching Einstein’s General Relativity prediction. This is an independent validation of the general spin framework, though it does not bear directly on the neutron model and is noted here for completeness.

5. A Leptonic Composite Model of the Neutron

5.1. Historical Background: Rutherford’s Conjecture

Long before the discovery of quarks, Rutherford speculated in his 1920 Bakerian Lecture [4] that a neutral particle might form from the “intimate union” of a proton and an electron. As Chadwick later recounted [5], Rutherford thought that a proton and electron “might unite in a much more intimate way than they do in the hydrogen atom, and so form a particle of no nett charge and with a mass nearly the same as that of the hydrogen atom.”

This hypothesis was abandoned for a compelling reason: the electron and proton are both spin-$\frac{1}{2}$ fermions. Their combination can only yield total spin 0 or 1 (integer), whereas the neutron has spin $\frac{1}{2}$ [3]. This violates the spin-statistics theorem [17], and was precisely the objection raised by Pauli at the time [18]. The neutrino hypothesis of Fermi [19] subsequently provided a more successful framework for nuclear $\beta$-decay, and the quark model [20] completed the displacement of the Rutherford model.

5.2. Resolving the Spin Problem: A Three-Body Extension

The fatal flaw of the two-body model is that two spin-$\frac{1}{2}$ fermions cannot combine to give spin $\frac{1}{2}$. A three-body system of fermions, however, can: three spin-$\frac{1}{2}$ particles can combine to give total spin $\frac{1}{2}$ (e.g., two spins anti-aligned and one aligned). Orbital angular momentum $\ell =1$ between the constituents provides an additional degree of freedom to achieve the required half-integer total spin.

We therefore propose, within the generalised spin framework of §(4), that the neutron is a quantum superposition of three component states ${\psi }_k$ ($k=1,2,3$):

1.

State $k=1$: The excited electron (tauon ${\tau }^{-}$), with spin parameter $s_1=+1$, mass $m_1=1776.86$ MeV/$c_0^2$ [3], and magnetic anomaly $\Delta {\mathrm{g}}_1^{+1}=+0.002319304362$.

2.

State $k=2$: A de-excited proton $p_1^{+}$, with spin parameter $s_2=-1$, mass $m_2=8.69$ MeV/$c_0^2$ (predicted in Paper (VIII) [21]), and magnetic anomaly $\Delta {\mathrm{g}}_2^{-1}=-5.5856946893$.

3.

State $k=3$: An associated neutrino ${\overline{\nu }}_{p_1}$, with spin parameter $s_3=-1$, mass $m_3\simeq 8.63\times {10}^{-9}$ MeV/$c_0^2$ [3], and magnetic anomaly $\Delta {\mathrm{g}}_3^{+1}$ (to be determined).

 Remark 4

(On the identification of State $k=1$ with the tauon). The identification of State $k=1$ with the tauon ($m=1776.86$ MeV/$c_0^2$) as an “excited electron” relies on the mass scaling relation Equation (10) from the general spin framework. This is a hypothesis of the present series, not an established result. The mass ratio $m_{\tau }/m_e\simeq 3477$ is not reproduced by a simple integer ratio of spin parameters, and a detailed derivation of the lepton mass spectrum within this framework is deferred to Papers (VI)–(X). The identification is used here motivationally and should be treated as tentative.

 Remark 5

(On the de-excited proton mass). The mass $m_2=8.69$ MeV/$c_0^2$ for the de-excited proton is a prediction of Paper (VIII) [21] and is not independently verified at this stage. The composite model results in §(6) depend sensitively on this value.

5.3. Dirac Equation for the Composite Neutron

We postulate that the total neutron wavefunction is a quantum linear superposition:

$${\psi }_N=\sum _{k=1}^3{c}_k{\tau }_k{\psi }_k,$$

(13)

where ${\tau }_k$ are $4\times 4$ matrices ensuring correct spinor combination, and the coefficients $c_k\in \mathbb{R}$ satisfy:

$$\sum _{k=1}^3{c}_k^2=1.$$

(14)

Each component ${\psi }_k$ satisfies the modified Dirac equation with mass matrix ${\mathcal{M}}_k$:

$$ı\hslash {\mathsf{\gamma }}^{\mu }{\partial }_{\mu }{\psi }_k={\mathcal{M}}_k{c}_0{\psi }_k.$$

(15)

5.4. Mass Relation

Substituting Equation (13) into the neutron Dirac equation and using the orthogonality of component states (assumed, as they represent distinct particle species), the expectation value of the mass operator is:

$${\mathcal{M}}_{\mathrm{N}}=\sum _{k=1}^3{c}_k^2{\mathcal{M}}_k.$$

(16)

Squaring and separating diagonal from cross terms, the observed scalar mass satisfies:

$$m_{\mathrm{N}}^2-m_B^2=\sum _{k=1}^3{P}_k^2{m}_k^2,$$

(17)

where $P_k=c_k^2$ and the binding mass $m_B$ is defined by the cross terms:

$$m_B^2=\sum _{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^3{P}_i{P}_j{m}_i{m}_j.$$

(18)

In the present calculation we take $m_B=m_{\mathrm{N}}-(m_e+m_p)$, the mass deficit when assembling the neutron from a free electron and proton (the remaining constituent, the neutrino, has negligible mass).

5.5. Magnetic Anomaly Relation

By an analogous averaging principle, the total neutron magnetic anomaly is:

$$\Delta {\mathrm{g}}_{\mathrm{N}}^{s_N}=(s_N-1)+s_N\sum _{k=1}^3{P}_k\Delta g_k^{s_k},$$

(19)

where $s_N=+1$ for the neutron (spin $\frac{1}{2}$).

5.6. System of Constraint Equations

Writing $P_k=c_k^2$ and using the data in the table below:

Particle	$s_k$	Mass (MeV/$c_0^2$)	$\Delta g_k^{s_k}$	Source
Neutron	$+1$	$939.565420$	$-5.82608552\left(90\right)$	[3]
Tauon	$+1$	$1776.860$	$+0.002319304362$	[3]
$p_1^{+}$	$-1$	$8.690$	$-5.5856946893$	[21]
${\overline{\nu }}_{p_1}$	$-1$	$\approx 0$	unknown	—

the constraint equations are:

$$\begin{array}{cc}\hfill P_1+P_2+P_3& =1,\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill m_{\mathrm{N}}^2-m_B^2& =m_1^2{P}_1^2+m_2^2{P}_2^2\left(\mathrm{neglecting}{m}_3^2\right),\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill \Delta {\mathrm{g}}_{\mathrm{N}}^{+1}& =P_1\Delta {\mathrm{g}}_1^{+1}+P_2\Delta {\mathrm{g}}_2^{-1}+P_3\Delta {\mathrm{g}}_3^{+1}.\hfill \end{array}$$

(22)

 Remark 6

(On the determinacy of the system). Equations (20)–(22) constitute three equations in four unknowns: $P_1$, $P_2$, $P_3$, and $\Delta {\mathrm{g}}_3$. The system is therefore underdetermined. The approach taken here is to use Equation (21) to constrain $P_1$ and $P_2$, use Equation (20) to obtain $P_3=1-P_1-P_2$, and then solve Equation (22) for $\Delta {\mathrm{g}}_3$. The neutrino anomaly $\Delta {\mathrm{g}}_3$ is thus a fitted quantity determined by requiring the model to reproduce the observed neutron $\mathrm{g}$-factor, not an independent prediction. This is an important limitation acknowledged explicitly in §(8).

6. Solution and Results

6.1. Probability Coefficients

With $m_{\mathrm{N}}^2-m_B^2\simeq 8.8278\times {10}^5$ ${\mathrm{MeV}}^2/c_0^4$, $m_1^2=3.157\times {10}^6$ ${\mathrm{MeV}}^2/c_0^4$, and $m_2^2=75.52$ MeV${\mathrm{MeV}}^2/c_0^4$, Equation (21) becomes:

$$8.8278\times {10}^5=3.157\times {10}^6{P}_1^2+75.52P_2^2.$$

(23)

Since the coefficient of $P_2^2$ is negligibly small compared to that of $P_1^2$, any non-zero $P_2$ has a negligible effect on Equation (23). Numerical exploration shows that $P_2$ must be effectively zero to avoid exceeding the left-hand side. Thus:

$$P_1\simeq \sqrt{{\displaystyle \frac{8.8278\times {10}^5}{3.157\times {10}^6}}}\simeq 0.5287,P_2\simeq 0,P_3\simeq 0.4713.$$

(24)

6.2. Neutrino Magnetic Anomaly

Substituting Equation (24) into Equation (22):

$$\begin{array}{cc}\hfill -5.82608552& =0.5287\times (+0.002319304362)+0\times (-5.5856946893)+0.4713\times \Delta {\mathrm{g}}_3,\hfill \\ \hfill -5.82608552& \simeq 0.001226+0.4713\Delta {\mathrm{g}}_3,\hfill \\ \hfill \Delta {\mathrm{g}}_3& \simeq -{\displaystyle \frac{5.82731}{0.4713}}\simeq -12.37.\hfill \end{array}$$

(25)

6.3. Summary of Results

The neutron’s internal composition in the leptonic composite model is approximately:

• Tauon (${\tau }^{-}$, $s=+1$): $P_1\simeq 52.87\%$
• De-excited proton ($p_1^{+}$, $s=-1$): $P_2\simeq 0\%$
• Neutrino (${\overline{\nu }}_{p_1}$, $s=-1$): $P_3\simeq 47.13\%$

with the effective neutrino magnetic anomaly $\Delta {\mathrm{g}}_3\simeq -12.37$.

7. General Discussion

The results presented in §6 offer a radically different picture of the neutron from that of the conventional quark model. Before examining the physical implications, it is worth restating the key findings and emphasising their tentative character.

• Summary of key results.

Solving the system of constraint equations (20)–(22) with the input data from Table 1 yields:

• Probability coefficients: $P_1\simeq 0.5287$, $P_2\simeq 0$, $P_3\simeq 0.4713$;
• Neutrino effective magnetic anomaly: $\Delta g_3\simeq -12.37$ (fitted, not predicted);
• Implied internal composition: tauon (${\tau }^{-}$) $\sim 53\%$, neutrino $\sim 47\%$, de-excited proton $\sim 0\%$.

• The proton as spectator.

The most striking result is $P_2\simeq 0$: the de-excited proton contributes negligibly to the neutron’s bulk properties. This does not mean the proton is irrelevant to the neutron’s existence. Rather, its role may be indirect — acting as a catalyst or binding agent for the tauon–neutrino bound state, encoded in the structure of the binding potential rather than appearing explicitly in the probability distribution. This refines Rutherford’s original intuition of an “intimate union” between proton and electron, while resolving the spin objection that doomed the two-body version.

• The neutrino anomaly as the central puzzle.

The fitted value $\Delta {\mathrm{g}}_3\simeq -12.37$ is the most problematic aspect of the model. Standard Model predictions for free-neutrino magnetic moments are of order ${10}^{-10}{\mu }_B$ or smaller [3], corresponding to $|\Delta {\mathrm{g}}_{\nu }|\lesssim {10}^{-9}$. The required value exceeds this by approximately ten orders of magnitude. Within the present framework, this is attributed to enhancement by the $\mathcal{H}$-field when the neutrino is confined inside the neutron — but this argument is qualitative. A quantitative derivation of $\Delta {\mathrm{g}}_3$ from first principles within the $\mathcal{H}$-field dynamics is the most urgent open problem of this work. Until it is resolved, the model cannot be regarded as quantitatively predictive.

• Relation to the quark model.

The leptonic composite model is not presented as a replacement for the Standard Model’s successful quark description, which accurately accounts for many neutron properties, including the ratio ${\mu }_p/{\mu }_{\mathrm{N}}\simeq -3/2$ [7] and is supported by lattice QCD calculations [8]. Rather, it is offered as an alternative theoretical perspective that, if correct, would require a fundamental revision of our understanding of nuclear structure. The two pictures make overlapping but distinct predictions, some of which are discussed below.

• Outline of the discussion.

In what follows, we examine the physical interpretation of these results [§(7.1)], compare with the quark model in more detail [§(7.2)], explore the reinterpretation of beta decay [§(7.3)], discuss astrophysical implications [§(7.4)].

7.1. Physical Interpretation

The most striking result is $P_2\simeq 0$: the de-excited proton contributes negligibly to the neutron’s bulk properties. Within the model, this does not mean the proton is irrelevant to the neutron’s existence. Rather, its role is indirect: it may act as a catalyst or binding agent for the tauon–neutrino bound state, encoded in the structure of the binding potential rather than appearing in the probability distribution. This is reminiscent of Rutherford’s intuition that the proton and electron are united “in a much more intimate way” than in hydrogen [4,5], though the present model refines this picture substantially. The near-equal sharing between tauon ($\sim 53\%$) and neutrino ($\sim 47\%$) suggests a deep symmetry. Both are leptons, both are spin-$\frac{1}{2}$ fermions, and neither carries colour charge. Their approximately equal weighting implies that if the model is correct, the neutron is fundamentally a leptonic composite, with baryon number emerging from the combined quantum state rather than from individual constituent quarks.

7.2. Neutrino’s Effective Anomaly

The fitted value $\Delta {\mathrm{g}}_3\simeq -12.37$ is the most problematic aspect of the model. The Standard Model predicts neutrino magnetic moments of order ${10}^{-10}{\mu }_B$ or smaller [3], corresponding to $|\Delta g_{\nu }|\lesssim {10}^{-9}$. The required value exceeds this by approximately ten orders of magnitude. Within the present framework, this is attributed to the enhancement of the neutrino’s effective magnetic moment by the CFV-field when the neutrino is confined within the neutron. At femtometer scales, the CFV-field strength is estimated [in Paper (II)] to be $H_0\sim 150$, providing significant amplification. However, this argument is qualitative and does not constitute a quantitative prediction. A detailed calculation of the confined neutrino’s effective magnetic moment from the CFV-field dynamics is an essential open problem: until this is resolved, the model cannot be regarded as quantitatively predictive.

7.3. Beta Decay Re-Interpretation

If the neutron contains a tauon and a neutrino as structural components, beta decay can be reinterpreted as a rearrangement:

$$n\equiv \left[{\tau }^{-}+p_1^{+}+{\overline{\nu }}_{p_1}\right]⟶p^{+}+e^{-}+{\overline{\nu }}_e.$$

(26)

This involves two simultaneous transitions, each individually violating lepton number conservation (LNC), that together conserve LNC:

$$\begin{array}{cc}\hfill {\tau }^{-}& ⟶e^{-}+{\overline{\nu }}_e(\mathrm{de}-\mathrm{excitation}\mathrm{of}\mathrm{tauon}),\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill p_1^{+}+{\overline{\nu }}_{p_1}& ⟶p^{+}(\mathrm{de}-\mathrm{excitation}\mathrm{of}\mathrm{proton}\mathrm{absorbing}\mathrm{neutrino}).\hfill \end{array}$$

(28)

The combination of Equations (27) and (28) gives the observed beta decay mode. The long neutron lifetime ($\sim 880$ s) would then be explained by the small overlap between initial and final states, requiring tunnelling through the CFV-field potential.

This reinterpretation has potentially testable consequences:

1.

Electron–antineutrino angular correlations in beta decay may differ from Standard Model predictions.

2.

The electron energy spectrum near the endpoint may show subtle deviations.

3.

The decay rate may depend weakly on environmental factors (magnetic field strength, density) that affect the CFV-field configuration.

These predictions are at or beyond the edge of current experimental sensitivity and await dedicated experimental searches.

7.4. Comparison with the Quark Model

The leptonic composite model differs from the Standard Model quark description in several key respects, summarised in Table 1.

The quark model is the established, experimentally validated description of the neutron. The leptonic composite model is a speculative alternative that would require a fundamental revision of nuclear physics if correct.

7.5. Astrophysical Implications

If neutrons are predominantly tauon–neutrino composites, neutron stars would consist largely of tauon–neutrino bound states rather than dense quark matter. This would significantly affect:

• The equation of state and the mass-radius relationship, potentially detectable by NICER, LIGO/Virgo/KAGRA, and future gravitational wave observatories.
• Cooling rates, since neutrinos as structural components may be more readily emitted, leading to enhanced cooling observable in X-ray telescopes.
• Magnetar magnetic field generation, given the large effective neutrino anomaly $\Delta {\mathrm{g}}_3\simeq -12.37$.

These implications are speculative at the present stage and cannot be quantified without resolving the neutrino anomaly problem.

8. Limitations and Open Questions

We summarise the primary limitations of the present paper.

• 1. Underdetermined system.

As noted in Remark 6, the system of equations is underdetermined. The neutrino anomaly $\Delta {\mathrm{g}}_3\simeq -12.37$ is a fitted quantity, not a prediction. The model will become genuinely predictive only when $\Delta {\mathrm{g}}_3$ can be derived independently — for example, from a full CFV-field treatment of the confined neutrino.

• 2. Neutrino anomaly magnitude.

The required $\Delta {\mathrm{g}}_3\simeq -12.37$ exceeds Standard Model predictions for free neutrinos by $\sim 10$ orders of magnitude. The CFV-field enhancement argument is qualitative only. A quantitative calculation of the confined neutrino’s effective magnetic moment is the most urgent open problem of this paper.

• 3. Tauon as excited electron.

The identification of the tauon as an excited electron state (Remark 4) relies on the general spin framework of §(4), whose derivation of the lepton mass spectrum is deferred to Papers (VI)–(X). Until these papers are completed, this identification remains an unverified hypothesis.

• 4. De-excited proton mass.

The mass $m_2=8.69$ MeV/$c_0^2$ for the de-excited proton (Remark 5) is a prediction of Paper (VIII) [21], not yet published. The composite model depends sensitively on this value, and the results of §(6) should be revisited when this prediction is established.

• 5. EDM coupling.

Hypothesis 1 requires a non-zero neutron EDM, while no such EDM has been observed. The relationship between the permanent EDM searched for experimentally and the internal dynamical EDM invoked here is not yet specified.

• 6. CFV-field dynamics.

Throughout this paper, the CFV-field is treated as a static background. Its equation of motion, propagation, and coupling to matter beyond the leading-order alignment condition remain unspecified.

• 7. Quark model consistency.

The model does not address how the baryon number of the neutron is encoded in a leptonic composite, nor how the Standard Model’s successful SU(3) flavour symmetry predictions emerge from this picture. Compatibility with established nuclear physics must be demonstrated before the model can be taken seriously as a candidate for physical reality.

9. Conclusions

We have extended the framework of Papers (I) and (II) to electrically neutral particles via an effective EDM charge coupling, and proposed a speculative leptonic composite model of the neutron. The main results are:

1.

EDM coupling: An effective charge $q_{\mathrm{N}}^{\mathrm{eff}}={\kappa }_{\mathrm{H}}{d}_{\mathrm{EDM}}^{\mathrm{N}}$ allows the neutron to couple to the CFV-field in analogy with charged particles. This approach is testable in principle by precision EDM measurements [13].

2.

Leptonic composite model: The neutron is proposed to be a quantum superposition of a tauon, a de-excited proton, and a neutrino, with probability coefficients $P_1\simeq 0.53$, $P_2\simeq 0$, $P_3\simeq 0.47$.

3.

Proton as spectator: The de-excited proton contributes negligibly to the neutron’s bulk properties, with the tauon and neutrino dominating nearly equally.

4.

Fitted neutrino anomaly: The neutrino’s effective magnetic anomaly $\Delta {\mathrm{g}}_3\simeq -12.37$ is required to reproduce the observed neutron moment. This value is not predicted but fitted, and its magnitude presents the most serious challenge to the model’s physical plausibility.

5.

Beta decay reinterpretation: Neutron decay is reinterpreted as a rearrangement of constituent states, with potentially testable deviations from Standard Model predictions for electron–antineutrino correlations.

The model is explicitly speculative and rests on several unverified assumptions. Its primary value at this stage is as a motivated proposal that identifies specific experimental and theoretical targets: determining $\Delta {\mathrm{g}}_3$ from first principles within the CFV-field framework, deriving the lepton mass spectrum from the general spin equation, and testing the beta decay predictions. Until these are achieved, the leptonic composite model should be regarded as an interesting exploratory hypothesis rather than an established alternative to the quark model.