A Lorentz-Covariant Quaternionic Double Coupled Field Theory from Real Coupled Fields to SU(2), Chirality, and the Dirac Limit

Abstract

This paper develops a Lorentz-covariant Double Coupled Field (DCF) framework in which a relativistic matter carrier is represented by eight real degrees of freedom organized as two quaternionic coordinates. Starting from coupled real fields, the construction establishes a real Cl(1,3) representation, an internal unit-quaternion symmetry Sp(1) ≅ SU(2), where Sp(1) denotes the group of unit quaternions, together with chirality and controlled Dirac and Schrödinger–Pauli limits. The work investigates the hypothesis that complex structures appearing in standard quantum formulations may represent compact descriptions of underlying coupled real degrees of freedom. The present results establish a consistent classical framework and its limiting correspondence with established relativistic quantum structures; they do not constitute a complete Poincaré-covariant quantum field theory, do not derive the spin–statistics connection, and do not claim experimentally distinguished predictions from the linear Dirac limit. Nonlinear DCF sectors, internal dynamics, and possible two-particle extensions are identified as directions for future investigation. The manuscript distinguishes three theoretical levels: the classical DCF ontology, a conditional full-Sp(1) extension, and a quantized two-particle extension; these levels are kept explicitly separate throughout the analysis.

1. Introduction

The purpose of this work is to examine whether the familiar structure of a relativistic spin-½ particle can arise from a system of coupled real fields rather than being introduced from the outset as a complex quantum spinor. The starting point is physical: two real fields can exchange excitation and execute an internal rotation. Such a pair has the kinematics of a single phase degree of freedom. The central question is what happens when this elementary coupled-field mechanism is doubled and then required to propagate relativistically.

Motivation from complex quantum formulations. The present work is motivated by the observation that the complex structures appearing in the Schrödinger and Dirac formulations [1,2] may be interpreted as compact representations of underlying coupled real degrees of freedom. While the equivalence between complex and doubled real representations is mathematically established, the present approach investigates whether the coupled-real structure can be promoted from a representational choice to a physical organizing principle for the dynamics. The aim is therefore not merely to rewrite complex variables in real form, but to examine whether the coupled exchange of real degrees of freedom can serve as a primitive structural hypothesis from which the familiar quantum and relativistic structures emerge.

The construction below follows that question step by step. Two coupled real-field pairs provide four real internal amplitudes. Their simplest associative organization with a norm-preserving three-generator internal rotation is quaternionic [3,4], giving the SU(2) geometry required for spin. A single quaternion, however, is not yet a relativistic fermion. Relativistic completion requires an eight-real-component carrier, represented here by two quaternionic coordinates. A real Clifford action on this carrier produces the Lorentz spin transformation and, after diagonalizing the resulting chirality operator, the familiar left- and right-chiral variables.

Organization of the article. The sections are retained as one article because they form one dependency chain rather than two independent claims: the coupled-real construction fixes the quaternionic carrier and chiral coordinates; those structures define the covariant action; the action supplies the Dirac and low-energy limits; and the later symmetry, branch, detector, and two-particle analyses use the same carrier, bilinears, and conventions. To keep this unified presentation readable, technical classification proofs and extended constructions are placed in the appendices, while the main text explicitly states which results belong to the classical DCF ontology and which belong only to conditional or quantized extensions.

Notation used in the overview. J_eff denotes an effective inter-DCF exchange coupling, J_n denotes the selected unit internal quaternion-generator direction used by the fixed-direction theory, and J denotes the pair-exchange coefficient in the two-DCF spin interaction. The companion coupled-field parameters κ_s and τ_s denote, respectively, the exchange coupling and characteristic field tension.

Only after this kinematic structure has been established is a dynamical action introduced. Its linearized vacuum equation is shown to reduce to the Dirac equation, and the low-energy limit then gives the Pauli and Schrödinger equations. Particle/antiparticle branches, a residual U(1) Noether charge, gravitational coupling, nonlinear locking mechanisms, and the two-particle sector are developed as subsequent consequences or extensions. For clarity, the manuscript distinguishes three theoretical levels. (i) The classical DCF ontology comprises the real carrier, the covariant action and its Dirac/Pauli limits, and the classical joint-mode/energetic structure of a coupled two-DCF system. (ii) A conditional full-Sp(1) extension supplies one possible positive effective exchange coupling J_eff > 0 and is not part of the minimal action with fixed internal direction J_n. (iii) The covariant source vertex used to calculate two-particle production amplitudes belongs to a quantized two-particle extension of the DCF carrier. This last level is introduced explicitly because an antisymmetric pointwise bilinear of a single classical c-number field vanishes. The three levels are therefore complementary but must not be conflated.

Reference target equations. To make the destination of the construction explicit from the outset, the two familiar equations recovered in the controlled limits of DCF are displayed here in their conventional textbook form. They are reference targets, not starting postulates of the DCF construction.

Relativistic target—Dirac equation

iℏγ^μ∂_μψ − mcψ = 0

Nonrelativistic target—Schrödinger equation

iℏ ∂ψ/∂t = [−ℏ²/(2m) ∇² + V]ψ

Remark 1.

The relativistic target is written here for a free Dirac field in flat spacetime, so the derivative is ∂μ. Gauge or gravitational interactions are introduced by replacing ∂μ with the appropriate covariant derivative Dμ; this generalization is treated later and is not assumed in the reference target displayed here.

Here i denotes the conventional scalar complex unit used in the standard textbook notation of the target equations. In the DCF construction below, the corresponding complex structure is represented instead by the real matrix J_n, satisfying J_n² = −I, rather than introduced as a primitive scalar imaginary unit. The logical chain developed in the manuscript is therefore coupled real fields → quaternionic internal structure → Lorentz-covariant real carrier → Dirac limit → Pauli and Schrödinger limits. The displayed equations identify the endpoint of that construction and are not assumed to derive its primitive real-field structure.

The aim is, therefore, not to present a catalog of mathematical structures. It is to develop and assess the internal consistency of one physical hypothesis: that the standard fermionic structures normally written with complex spinors can be represented as the effective relativistic description of a deeper system of real coupled fields. Mathematics is introduced only where it is needed to advance that construction; detailed algebraic proofs and classification arguments are retained in the appendices.

Status and scope of the real-field interpretation. The real coupled fields are proposed as a candidate for classical physical ontology. At this level, the established results are the real Lorentz/quaternionic carrier, chiral coordinates, covariant action, controlled Dirac and Schrödinger–Pauli limits, and the classical singlet/triplet joint-mode structure. The full-Sp(1) gauge construction and the quantized pair-production vertex are separate extensions, as specified above. This explicit hierarchy defines the present boundary of the classical ontology. Agreement with established Dirac, Pauli, and singlet algebra remains a consistency requirement rather than, by itself, an experimental distinction.

Relation to established real and quaternionic Dirac formulations. The use of eight real components to represent a four-component complex Dirac spinor, and the existence of real (including Majorana-type) and Clifford-algebra formulations of relativistic spinors are established features of Dirac theory [5,6,7,8,9]. Likewise, quaternionic formulations of the Dirac equation and Dirac Lagrangian have been studied previously [10,11]. These representation-theoretic ingredients are established and are not claimed individually as new. The novelty claimed for DCF is, instead, the physical ontology and constructive hierarchy from which those structures are reached. Its starting observation is not the realification of a pre-existing Dirac spinor. Rather, the Schrödinger equation itself, when its complex amplitude is resolved into real and imaginary parts, is equivalent to two mutually coupled real evolution equations. DCF promotes this coupled-pair structure from a mathematical decomposition to a physical hypothesis: the primitive entities are real fields undergoing reciprocal exchange, characterized in the underlying coupled-field model by quantities such as the exchange coupling κ_s and characteristic field tension τ_s. A single real coupled pair supplies the planar phase structure; doubling the primitive pair supplies four real amplitudes on which the quaternionic Sp(1) ≅ SU(2) carrier is realized; relativistic completion then produces the eight-real-component Lorentz carrier, chirality operator, and derived left- and right-chiral coordinates. Only after this sequence has been constructed, a vacuum complex direction selected, and the dynamics linearized does the conventional complex Dirac spinor appear. The explicit real-to-complex map of Section 5 therefore proves representational equivalence of the linearized endpoint to ordinary Dirac theory; it does not prove the equivalence of the starting ontology or of the full nonlinear dynamics. Put differently, the Dirac map establishes the equivalence of the limiting representation, not the identity of the underlying coupled-field hypothesis. The full DCF proposal embeds the same primitive real variables in nonlinear vacuum-locking, branch-selection, residual U(1)J_n, detector, and composite dynamics that are absent from the linear Dirac correspondence. Accordingly, the central novelty asserted here is ontological and constructive: a candidate real coupled-field substrate from which the standard fermionic spinor structure is recovered as an effective relativistic limit. A second constructive distinction appears in the two-particle sector, but with an explicit boundary between classical and quantized claims. At the classical DCF level, the coupled pair has collective singlet/triplet modes and, for J > 0, a definite energetic ordering with the singlet as the unique ground mode. A conditional full-Sp(1) extension supplies one possible J_eff > 0. The stronger statement that a local spin-zero decay vertex creates the antisymmetric two-particle state is formulated only in the quantized two-particle extension of Appendix C, Covariant Pair-Production Mechanism for the Antisymmetric State; it is not attributed to the single-field classical action. A third, potentially deeper constructive distinction concerns the quantum action scale itself. In conventional quantum mechanics and Dirac theory, the reduced Planck constant ℏ is inserted as a fundamental dynamical constant. In the coupled-field ontology motivating DCF, the companion two-field model of the subsection “Possible Coupled-Field Interpretation of the Action Scale” instead yields ℏ = γcτ_s/κ_s, where c is the speed of light and γ is the dimensionless closure factor defined in the subsection “Possible Coupled-Field Interpretation of the Action Scale” from the geometric and evaluation-interval factors of the companion model. This relation follows from the characteristic exchange dynamics and scale of the primitive real fields. Thus, the CF ontology proposes a dynamical origin of the action scale rather than taking its fundamental status as part of the starting ontology. The present relativistic formulation still uses ℏ as an input because the microscopic map from κ_s, τ_s, and γ to the parameters of the covariant DCF action (Equation (18)) has not yet been completed; establishing that map is therefore a central unfinished step rather than an assumption hidden in the relativistic reduction. Whether these deeper nonlinear, production, composite, and action-scale structures ultimately yield experimentally distinguishable predictions remains a separate question. At the linearized level, no observable distinction from ordinary Dirac theory is claimed; physical inequivalence, if established, must arise from additional DCF nonlinear or internal dynamics beyond that linear correspondence.

Relation of the action to standard Dirac-spinor notation. Here and throughout, the Dirac adjoint is defined by $\overline{\mathrm{\Psi }}$ ≡ ΨᵀΓ_C⁰. Equations (18) and (20)–(23) can, at the level of the linearized one-particle sector and of ordinary Lorentz-scalar interactions, be rewritten in conventional complex Dirac-spinor notation. No claim of physical inequivalence is made at that level. In particular, Π corresponds to the parity-even Dirac scalar $\overline{\mathrm{\Psi }\mathrm{\Psi }}$, while Σ is the second real Lorentz-scalar bilinear selected by the DCF construction, and an independent real scalar s with a Yukawa-type coupling can also be introduced in an ordinary Dirac theory. The purpose of the quaternionic-real formulation is, therefore, not to obtain an interaction that is algebraically forbidden in standard Dirac notation. Its claimed content is constructive: the Lorentz carrier, its complex structure, chirality coordinates, residual U(1), and the nonlinear DCF sectors are organized from primitive coupled real degrees of freedom. Consequently, the Dirac map establishes the equivalence of the linear endpoint, whereas any test of equivalence must come from additional nonlinear or internal DCF dynamics rather than from a mere change of representation.

Beyond the linear correspondence. The Dirac correspondence is the consistency limit of DCF rather than the end of the program. Possible extensions of the DCF construction to closed C₂ and C₃ networks are reserved for future work. Any attempt to associate such composite structures with physical particle multiplets must be formulated consistently with the constraints on combining spacetime and internal symmetries, including the Coleman–Mandula theorem [12]. No nontrivial unification of the Poincaré algebra with internal symmetries is constructed or claimed in the present work.

2. From Coupled Real Fields to Quaternionic Spin

A single coupled pair supplies only a planar internal rotation. To represent spin, the internal motion must be enlarged without abandoning the real-field description. The next step is, therefore, to combine two such pairs into four real amplitudes and ask for the simplest associative structure that supports the required three-generator internal rotations.

Real-matrix quaternion algebra. Throughout this work, the quaternionic structure is represented entirely over the real numbers. No primitive scalar imaginary units are introduced. Let J₁, J₂, J₃ be real matrices in Mat_d(ℝ), with d ≥ 4 (the minimal faithful real realization is d = 4), satisfying J$\mathrm{a}{\mathrm{J}}_{\mathrm{b}}$ = −δ_ab I_d + ε_ab^c J_c. Here δ denotes the Kronecker delta, ε the Levi-Civita symbol, and I_d the d × d identity matrix. These matrices realize the quaternion algebra as real linear operators. All references below to quaternionic multiplication mean multiplication in this real matrix representation.

$${\mathrm{J}}_{\mathrm{a}}{\mathrm{J}}_{\mathrm{b}}=\text{-}{\mathsf{\delta }}_{\mathrm{ab}}{\mathrm{I}}_{\mathrm{d}}+{\mathsf{\epsilon }}_{\mathrm{ab}}^{\mathrm{c}}{\mathrm{J}}_{\mathrm{c}},{\mathrm{J}}_{\mathrm{a}}\in {\mathrm{Mat}}_{\mathrm{d}}\left(\mathrm{R}\right),\mathrm{d}\ge 4$$

(1)

$${\mathrm{J}}_1^2={\mathrm{J}}_2^2={\mathrm{J}}_3^2=\text{-}{\mathrm{I}}_{\mathrm{d}},{ \mathrm{J}}_1{\mathrm{J}}_2={\mathrm{J}}_3,{\mathrm{J}}_2{\mathrm{J}}_3={\mathrm{J}}_1,{\mathrm{J}}_3{\mathrm{J}}_1={\mathrm{J}}_2$$

(2)

For real coefficients q₀, q₁, q₂, q₃, a quaternionic element is represented in the real matrix realization as

$$\mathrm{Q}\equiv {\mathrm{q}}_0{\mathrm{I}}_{\mathrm{d}}+{\mathrm{q}}_1{\mathrm{J}}_1+{\mathrm{q}}_2{\mathrm{J}}_2+{\mathrm{q}}_3{\mathrm{J}}_3,{\mathrm{q}}_{\mathrm{a}}\in \mathrm{R}$$

(3)

Conjugation and norms. In the real-matrix realization define Q† = q₀I_d − q₁J₁ − q₂J₂ − q₃J₃. For the standard orthogonal realization J_a^T = −J_a, hence Q† = Q^T. The real matrix norm is the real scalar N(Q) = q₀² + q₁² + q₂² + q₃², equivalently Q†Q = N(Q)I_d.

$$\bar{\mathrm{Q}}\mathrm{Q}=\mathrm{Q}\bar{\mathrm{Q}}={\mathrm{q}}_0^2+{\mathrm{q}}_1^2+{\mathrm{q}}_2^2+{\mathrm{q}}_3^2{\equiv \Vert \mathrm{Q}\Vert }^2$$

(4)

For a second quaternionic element P = p₀I_d + p₁J₁ + p₂J₂ + p₃J₃, let p = (p₀,p₁,p₂,p₃)ᵀ denote its real coefficient vector.

Real inner product. I_dentifying Q with its real coefficient vector q = (q₀,q₁,q₂,q₃)^T, the Euclidean inner product is ⟨P,Q⟩ = p^Tq. Equivalently, in the minimal d = 4 matrix realization, it is proportional to Tr(P^TQ).

$$⟨P,Q⟩=p^{T}q=p_0{q}_0+p_1{q}_1+p_2{q}_2+p_3{q}_3$$

(5)

Adjoint identity. Each real generator Jₐ is skew-symmetric, Jₐᵀ = −Jₐ. Therefore, its action is skew-adjoint with respect to the real Euclidean inner product.

2.1. Unit Quaternions and SU(2)

Unit-quaternion group in real form. Define Sp(1) as the set of real matrices U = q₀I_d + q₁J₁ + q₂J₂ + q₃J₃ with q₀² + q₁² + q₂² + q₃² = 1. Multiplication is ordinary real-matrix multiplication. This group is isomorphic to SU(2) [3,4,13], while the DCF ontology itself remains entirely real.

Real matrix representation. No complex 2 × 2 representation is required or used in the DCF ontology. A quaternionic element is represented by the real matrix Q = q₀I_d + q₁J₁ + q₂J₂ + q₃J₃. In the minimal faithful realization d = 4, the three generators may be chosen as real 4 × 4 skew-symmetric matrices obeying Equation (1).

$$\mathrm{Q}={\mathrm{q}}_0{\mathrm{I}}_4+{\mathrm{q}}_1{\mathrm{J}}_1+{\mathrm{q}}_2{\mathrm{J}}_2+{\mathrm{q}}_3{\mathrm{J}}_3,{ \mathrm{J}}_{\mathrm{a}}^{\mathrm{T}}=\text{-}{\mathrm{J}}_{\mathrm{a}}, {\mathrm{J}}_{\mathrm{a}}{\mathrm{J}}_{\mathrm{b}}=\text{-}{\mathsf{\delta }}_{\mathrm{ab}}{\mathrm{I}}_4+{\mathsf{\epsilon }}_{\mathrm{ab}}^{\mathrm{c}}{\mathrm{J}}_{\mathrm{c}}$$

(6)

Here, all entries and operators are real. The notation J₁, J₂, J₃ denotes matrix generators, not scalar imaginary numbers. Higher-dimensional faithful real realizations may be used, but the minimal realization is 4 × 4.

The unit quaternions Sp(1) form a group isomorphic to SU(2); the short proof is given in Appendix A.1.

Lie algebra. The real matrix generators span a three-dimensional Lie algebra isomorphic to SU(2), with commutators [J_a, J_b] = 2ε_ab^cJ_c. The isomorphism concerns the real matrix algebra; no scalar imaginary unit is introduced.

$$\left[{\mathrm{J}}_{\mathrm{a}}, {\mathrm{J}}_{\mathrm{b}}\right]={\mathrm{J}}_{\mathrm{a}}{\mathrm{J}}_{\mathrm{b}}\text{-}{\mathrm{J}}_{\mathrm{b}}{\mathrm{J}}_{\mathrm{a}}=2{\mathsf{\epsilon }}_{\mathrm{ab}}^{\mathrm{c}}{\mathrm{J}}_{\mathrm{c}}$$

(7)

This result supplies an exact internal SU(2) structure, consistent with the standard unit-quaternion parametrization used elsewhere to coordinate the Bloch sphere [13]. It does not yet identify that internal group with the spatial spin group. That distinction is addressed in Section 2.3.

2.2. Why Quaternions? Minimality and the Coupled-Field Correspondence

A single coupled pair carries only a U(1) phase. Doubling the pair gives four real amplitudes: the minimum real dimension of an SU(2) fundamental doublet. Organizing those amplitudes as one quaternion preserves associativity and supplies the unit-quaternion Sp(1) ≅ SU(2) action used for spin. This choice is motivated rather than unique: ℝ⁴ admits other structures, but the quaternion is the simplest associative normed organization that extends the single-pair phase rotation to the required three-generator internal rotation.

If q₂ = q₃ = 0, the internal algebra restricts the real two-dimensional subspace generated by {I₄,J₁}; because J₁² = −I₄, this subspace carries the same SO(2) rotation law as a conventional U(1) phase without introducing a scalar imaginary unit. The second coupled pair enlarges this planar rotation to the full three-generator SU(2) geometry.

2.3. Real DCF Field Content and Its Chiral Coordinates

A single quaternion supplies the internal SU(2) geometry but not the Lorentz transformation law of a relativistic fermion. A four-dimensional Dirac field has eight real components, so the relativistic DCF carrier is taken to contain two quaternionic coordinates. These are primitive coordinates of the real field space; the physical particle and antiparticle branches will arise later as positive- and negative-frequency solutions of the dynamics.

Real DCF field content. The fundamental relativistic DCF carrier space is an eight-real-dimensional space coordinated by two real quaternions, X_F and X_{$\overline{\mathrm{F}}$}, which will be referred to here neutrally as the primitive F-type and $\overline{\mathrm{F}}$-type coordinates:

$${\mathrm{X}}_{\mathrm{DCF}}\left(\mathrm{x}\right)=\left(\begin{array}{c}{\mathrm{X}}_{\mathrm{F}}\left(\mathrm{x}\right)\\ {\mathrm{X}}_{\acute{\mathrm{F}}}\left(\mathrm{x}\right)\end{array}\right),{\mathrm{X}}_{\mathrm{F}},{ \mathrm{X}}_{\acute{\mathrm{F}}}\in {\mathrm{R}}^4\simeq \mathrm{H},{\mathrm{X}}_{\mathrm{DCF}}\in {\mathrm{R}}^4\oplus {\mathrm{R}}^4\simeq {\mathrm{R}}^8$$

(8)

X_F and X_{$\overline{\mathrm{F}}$} are independent quaternionic coordinates of the eight-real-dimensional carrier space. The labels F and $\overline{\mathrm{F}}$ are mnemonic primitive sector labels only: they provide the real variables on which the Lorentz spin representation acts, but at this stage they are not identified with physical fermion and antifermion states, are not two simultaneously occupied constituents of one localized particle, and are not yet the dynamically defined positive- and negative-frequency particle/antiparticle branches. Their proposed relation to those physical branches is addressed only after the dynamics have been introduced in Section 7.

To connect this real carrier to relativistic fermion physics, we construct a real chirality operator and use its eigenspaces as the left- and right-chiral coordinates. The same orthogonal transformation must then be applied to every Clifford operator, so the primitive X basis and the derived chiral C basis remain unambiguous.

Chirality operator and the derived chiral basis. On the primitive DCF space, define the selected internal real matrix complex structure J_n ≡ n₁J₁ + n₂J₂ + n₃J₃, with n₁² + n₂² + n₃² = 1. Because the J_a satisfy the quaternion algebra, J_n² = −I₄ and J_n^T = −J_n. On the eight-real-dimensional carrier, define J_X ≡ diag(J_n, J_n). Thus, J_n is a real 4 × 4 matrix (or the corresponding block in any higher faithful real representation), not an imaginary scalar. Using the explicit real Clifford representation Γ_x^a of Appendix A.2, define the chirality involution as below. Let R_Jn denote the real 4 × 4 operator of right multiplication by J_n. Let Γ_X^a denote the real 8 × 8 Clifford generators acting on the primitive X basis, and define Ω_X ≡ Γ_X⁰Γ_X¹Γ_X²Γ_X³; the chirality operator is χ_X ≡ J_xΩ_x.

$${\mathsf{\Omega }}_{\mathrm{X}}\equiv {\mathsf{\Gamma }}_{\mathrm{X}}^0{\mathsf{\Gamma }}_{\mathrm{X}}^1{\mathsf{\Gamma }}_{\mathrm{X}}^2{\mathsf{\Gamma }}_{\mathrm{X}}^3,{\mathsf{\Omega }}_{\mathrm{X}}^2=\text{-}{\mathrm{I}}_8,{\mathrm{J}}_{\mathrm{X}}^2=\text{-}{\mathrm{I}}_8,{\mathsf{\chi }}_{\mathrm{X}}\equiv {\mathrm{J}}_{\mathrm{X}}{\mathsf{\Omega }}_{\mathrm{X}},{\mathsf{\chi }}_{\mathrm{X}}^2={\mathrm{I}}_8$$

(9)

Because right multiplication commutes with every left quaternionic multiplication, J_X commutes with each Γ_X^a and hence with Ω_X. Appendix A gives the explicit calculation Ω_X² = −I₈. Therefore χ_X = J_XΩ_X is a real symmetric orthogonal involution with eigenvalues ±1, each of multiplicity four. In the primitive basis, it has the block form ${\mathsf{\chi }}_{\mathrm{x}}=\left[\begin{array}{cc}0& {\mathrm{J}}_{\mathrm{n}}\\ \text{-}{\mathrm{J}}_{\mathrm{n}}& 0\end{array}\right]$. Its normalized +1 and −1 eigencoordinates are

$${\mathrm{Q}}_{\mathrm{L}}\equiv (1/\surd 2\left)\right({\mathrm{X}}_{\mathrm{F}}+{\mathrm{R}}_{\mathrm{Jn}}{\mathrm{X}}_{\overline{\mathrm{F}}}),{\mathrm{Q}}_{\mathrm{R}}\equiv (1/\surd 2\left)\right({\mathrm{X}}_{\mathrm{F}}\text{-}{\mathrm{R}}_{\mathrm{Jn}}{\mathrm{X}}_{\overline{\mathrm{F}}})$$

(10)

We write X_DCF = (X_F,X_{$\overline{\mathrm{F}}$})ᵀ for the primitive carrier column and Ψ_C = (Q_L,Q_R)ᵀ for the corresponding chiral-basis column.

Q_L and Q_R are therefore the normalized +1 and −1 eigencoordinates of ${\chi }_X$. Equivalently, with Ψ_C = $\left(\begin{array}{c}{\mathrm{Q}}_{\mathrm{L}}\\ {\mathrm{Q}}_{\mathrm{R}}\end{array}\right)$ and X_DCF = $\left(\begin{array}{c}{\mathrm{X}}_{\mathrm{F}}\\ {\mathrm{X}}_{\overline{\mathrm{F}}}\end{array}\right)$,

$$\mathrm{U}=(1/\surd 2) \left[\begin{array}{cc}{\mathrm{I}}_4& {\mathrm{J}}_{\mathrm{n}}\\ {\mathrm{I}}_4& \text{-}{\mathrm{J}}_{\mathrm{n}}\end{array}\right]$$

(11)

One confirms directly that UU^T = I₈ and U⁻¹ = U^T. Thus, U is a genuine orthogonal basis rotation, not an involution in general. It diagonalizes the primitive-basis chirality operator,

$$\mathrm{U}{\mathsf{\chi }}_{\mathrm{X}}{\mathrm{U}}^{\mathrm{T}}={\chi }_C\equiv \mathrm{diag}({\mathrm{I}}_4, -{\mathrm{I}}_4).$$

The Clifford generators in the chiral basis are consequently not the untransformed matrices (Equation (14)), but Γ_C^a≡UΓ_X^aU^T, given explicitly in Equation (15) below. This resolves the basis ambiguity: Γ_X^a, Ω_X, ${\chi }_X$ act on (X_F,X_{$\overline{\mathrm{F}}$}), while Γ_C^a and χ_C act on (Q_L, Q_R).

No complex Dirac spinor is assumed here: X_F and X_{$\overline{\mathrm{F}}$} are real, the Clifford generators and the internal generators J₁,J₂,J₃ are real matrices, and the gauge potential is real. The explicit Clifford construction is independently proved in Appendix A.2. Physically, Q_L and Q_R are the two chirality eigensectors of one relativistic DCF field. The selected operator J_n supplies the real internal quarter-turn structure; conventional complex notation is only a compact external representation of the same real two-plane rotations and is not part of the DCF ontology.

The mass interaction introduced below couples these sectors, just like the mass term couples left- and right-handed components in the ordinary Dirac equation.

Thus, the construction proceeds from the primitive real coordinates (X_F,X_{$\overline{\mathrm{F}}$}), through the real Clifford and complex structures, to the chirality operator χ_X and the orthogonal basis rotation U. The resulting (Q_L,Q_R) are derived chiral coordinates, not a relabeling of the primitive pair.

In the following conceptual paragraph, J refers to the eight-dimensional real complex-structure action J_X, Ω refers to Ω_X, and χ refers to χ_X.

Conceptual significance of the complex structure. The operator J is the internally defined complex structure of the real DCF carrier: within the eight-real-dimensional space it supplies the operation represented by multiplication by i in the conventional complex formulation. Consequently, with Ω corresponding to the real-representation analog of iγ⁵, the relation χ = JΩ recovers the usual γ⁵ chirality grading without assuming a complex Dirac spinor at the starting point. This emergent complex/chiral structure is one of the central structural results of the DCF construction.

Let κ_F and κ_{$\overline{\mathrm{F}}$} denote the effective branch-label coupling constants assigned, respectively, to the fermion and antifermion branches.

The sign relation κ_F = −κ_F relates the effective branch-label couplings of the fermion (F) and antifermion ($\overline{\mathrm{F}}$) branches and is used as a convention; its derivation from the nonlinear frequency branches and from the nonrelativistic exchange coupling remains open.

2.4. Coupled-Field Origin of the Quaternionic Components

The four real components of one quaternion have a direct coupled-field reading: they form two real pairs: CF₁ = (q₀, q₁) and CF₂ = (q₂, q₃). Each pair alone supports a U(1) internal rotation; together they form the SU(2) doublet described above.

$$\mathrm{CF}1=({\mathrm{q}}_0,{\mathrm{q}}_1),\mathrm{CF}2=({\mathrm{q}}_2,{\mathrm{q}}_3)$$

(12)

are each ordinary coupled-field sectors. A first-order kinetic coupling within one pair generates an SO(2) rotation of that real pair. Taken separately, either pair supplies only this single planar rotation. When the two real pairs are combined into q = (q₀, q₁, q₂, q₃)^T and acted on by the three real generators J₁,J₂,J₃, the full three-generator Sp(1) ≅ SU(2) action appears. Thus, four real components, organized by the real matrix quaternion algebra, are the minimal structure carrying the required SU(2) doublet geometry.

3. Relativistic Completion and DCF Dynamics

The quaternionic construction supplies the internal spin geometry, but it does not yet determine relativistic propagation. We now require the eight real DCF amplitudes to transform as a Lorentz spinor and then introduce the dynamics that govern their propagation and coupling.

Scope note. The present DCF construction is formulated at the classical c-number field level. The field variables are not operator-valued quantum fields; this statement does not imply commutativity of the internal quaternion algebra, which is noncommutative. The present construction establishes the Lorentz-covariant spinor structure of the DCF carrier; it does not attempt a complete Poincaré-group classification or a quantized Poincaré-covariant field theory. Accordingly, the spin–statistics connection is not derived here and remains part of a future quantized extension, as discussed further in Section 14.

Scope and limitations. The DCF framework developed here should be distinguished from a complete relativistic quantum field theory. The present manuscript constructs a classical c-number field carrier with Lorentz covariance and establishes its correspondence with the Dirac equation in a controlled linearized limit. A complete treatment based on the full Poincaré group, including quantized field operators, creation and annihilation operators, and the spin–statistics connection, is not derived from this work. Likewise, the identification of nonlinear DCF sectors with physical particle properties, charge quantization, or experimentally distinguishable deviations from standard Dirac/QED dynamics remains an open problem. The purpose of the present framework is, therefore, to investigate a possible deeper coupled-real organization underlying familiar complex relativistic structures, rather than to replace the complete standard quantum field-theoretic formulation at this stage.

3.1. Lorentz and Quaternionic Transformation Laws

It is useful to distinguish the internal action in the two bases. On the primitive field X_DCF, the global internal Sp(1) acts by simultaneous right multiplication on the two quaternionic components, while Lorentz transformations act through the real spin representation built from Γ_X^a. In the chiral coordinates Ψ_C = UX_DCF, the same transformations are conjugated by U. Here Λ denotes a Lorentz transformation and V∈Sp(1) an internal unit quaternion. S_X(Λ) and ρ_X(V) denote, respectively, the Lorentz-spin and right-Sp(1) actions in the primitive basis; S_C and ρ_C denote their conjugated actions in the chiral basis. For V = exp(θJ_n), θ is a real group parameter, and R_V denotes right multiplication by V.

$$\begin{gathered} X_{DC\mathrm{F}}\to S_X\left(\mathrm{\Lambda }\right)X_{DC\mathrm{F}},X_{DC\mathrm{F}}\to {\rho }_X\left(V\right)X_{DC\mathrm{F}},{\rho }_X\left(V\right)=diag\left(R_V,R_V\right),V\in Sp\left(1\right) \\ {\mathrm{\Psi }}_C\to S_C\left(\mathrm{\Lambda }\right){\mathrm{\Psi }}_C,{\mathrm{\Psi }}_C\to {\rho }_C\left(V\right){\mathrm{\Psi }}_C \end{gathered}$$

(13)

The symbol R_n used below denotes the same right-multiplication operator R_Jn defined above.

Since S_X and ρ_X commute in the primitive basis, their conjugates S_C and ρ_C commute in the chiral basis. For V in the stabilizer U(1)J_n = {exp(θJ_n)}, ρ_X(V) commutes with U and ρ_C(V) reduces again to ordinary componentwise right multiplication. For a generic Sp(1) rotation, this simplification does not hold because U itself contains R_n; the full internal action in chiral coordinates is the conjugated operator ρ_C(V), not a bare right multiplication on each chiral component.

We use the Minkowski metric η_ab = diag(1, −1, −1, −1) and define Γ_X^ab ≡ ½[Γ_X^a, Γ_X^b]. The antisymmetric parameters ω_ab are the Lorentz-generator parameters.

Lorentz representation. There is an isomorphism of real associative algebras Cl(1, 3) ≅ Mat₂(ℍ), realized on the primitive DCF coordinate space by the real 8 × 8 matrices

$${\mathrm{\Gamma }}_{{\mathrm{X}}^0}=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right], {{\mathsf{\Gamma }}_{\mathrm{X}}}^{\mathrm{r}}=\left[\begin{array}{cc}{\mathrm{J}}_{\mathrm{r}}& 0\\ 0& \text{-}{\mathrm{J}}_{\mathrm{r}}\end{array}\right] \mathrm{r}=1,2,3$$

(14)

acting on X_DCF = $\left(\begin{array}{c}{\mathrm{X}}_{\mathrm{F}}\\ {\mathrm{X}}_{\overline{\mathrm{F}}}\end{array}\right)$∈ℍ², with every quaternionic matrix entry acting by left multiplication. These satisfy Γ_X^aΓ_X^b + Γ_X^bΓ_X^a = 2η^abI₈, η = diag(1, −1, −1, −1). Here Λ denotes a Lorentz transformation and ${\omega }_{ab}$ its antisymmetric Lorentz parameters. The associated real spin representation S_X(Λ) = exp(¼ω_abΓ_X^ab) commutes with the primitive global right-Sp(1) action ρ_X(V), because left and right quaternionic multiplication commute by associativity.

The proof of the algebra identification, the explicit Clifford relations, and the commutation with the primitive right-Sp(1) action is given in Proposition A1 of Appendix A.2. The chirality construction then supplies the orthogonal matrix U that diagonalizes the real chirality involution. The same abstract Clifford representation in the chiral coordinates is obtained by similarity transformation.

For each quaternion generator J_r, L_Jr denotes its real matrix action by left multiplication.

Clifford generators in the chiral basis. With ΨC = UXDCF and U of Equation (11), define ΓCᵃ = UΓXᵃUT. Using Rn² = −I4 and the commutation of Rn with every left multiplication LJᵣ gives:

$${\mathrm{\Gamma }}_C^0=\left(\begin{array}{cc}0& R_n\\ -R_n& 0\end{array}\right),{\mathrm{\Gamma }}_C^r=\left(\begin{array}{cc}0& L_{J_r}\\ L_{J_r}& 0\end{array}\right),r=\mathrm{1,2},3$$

(15)

These again satisfy $\{\mathrm{\Gamma }{C}^{\mathrm{a}},\mathrm{\Gamma }{C}^{\mathrm{b}}\}=2{\eta }^{\mathrm{ab}}{I}_8$. Moreover ${\chi }_C=U\chi X{U}^T=diag(I_4,-I_4)$, so $Q_L$ and $Q_R$ are genuinely the +1 and −1 chirality eigencoordinates, respectively. All occurrences of Γᵃ in Section 3, Section 4, Section 5, Section 6, Section 7, Section 8, Section 9 and Section 10 henceforth mean ${\mathrm{\Gamma }}_C^{\mathrm{a}}$ unless an X subscript is shown explicitly.

A covariant derivative in the chiral coordinates is therefore written as: Here, A_μ denotes the residual internal gauge-potential element, $R_{A_{\mu }}$ denotes right multiplication by $A_{\mu },A_{\mu ,C}$ denotes the corresponding connection transported to the chiral basis, and $g_1$ denotes the residual Abelian gauge coupling.

$${\mathrm{D}}_{\mathsf{\mu }}{\mathsf{\Psi }}_{\mathrm{C}}={\partial }_{\mathsf{\mu }}{\mathsf{\Psi }}_{\mathrm{C}}+{\displaystyle \frac{1}{4}}{\mathsf{\omega }}_{\mathsf{\mu }\mathrm{ab}}{\mathsf{\Gamma }}_{\mathrm{C}}^{\mathrm{ab}}{\mathsf{\Psi }}_{\mathrm{C}}+{\mathrm{A}}_{\mathsf{\mu },\hspace{0.17em}\mathrm{C}}{\mathsf{\Psi }}_{\mathrm{C}}$$

(16)

where Γ_C^ab = ½[Γ_C^a, Γ_C^b] and $A_{\mu ,C}$ is the internal connection transported from the primitive right-action representation,

$$A_{\mu ,C}\equiv \mathrm{U} \mathrm{diag}(R_{A_{\mu }},R_{A_{\mu }}){\mathrm{U}}^{\mathrm{T}}.$$

For the residual U(1)J_n selected by fixed J_n this reduces to the familiar componentwise right action ΨCAμ. Whenever the compact notation ${\mathrm{\Psi }}_{A_{\mu }}$ is used below, it denotes the action of the corresponding transported internal connection in the chiral basis. It represents literal componentwise right multiplication only in the primitive basis or, after restriction, in the residual $U{\left(1\right)}_{J_{\mathrm{n}}}$ sector.

$${\mathrm{A}}_{\mathsf{\mu }}^{\left(\min \right)}={\mathrm{g}}_1{\mathrm{A}}_{\mathsf{\mu }}{\mathrm{J}}_{\mathrm{n}}$$

(17)

3.2. Unified Lorentz-Covariant DCF Action

The relativistic DCF dynamics must provide four ingredients already motivated above: propagation of the real spinor carrier, coupling between its chiral sectors, interaction with the residual electromagnetic U(1), and a stable nonlinear vacuum; gravity is included through the standard generally covariant term. The action should, therefore, be read as a minimal dynamical completion of the kinematic structure established above, not as a uniqueness theorem. We adopt the following fixed-J_n local form:

Action notation. Let M denote spacetime, g_μν its metric, g ≡ det(g_μν), e^μ_a a vierbein, and $ℒ$_DCF the DCF Lagrangian density. The curved Clifford matrices are Γ^μ ≡ e^μ_aΓ^a, and D_μ is the covariant derivative containing the spin and residual internal connections. The real Dirac-conjugate row spinor is denoted by $\overline{\mathrm{\Psi }}$. For gravity, κ_G ≡ 8πG/c⁴ is the gravitational coupling, R is the Ricci scalar, and Λ is the cosmological constant. For the residual electromagnetic sector, A_μ is the gauge potential and F_μν its field strength. The symbol s denotes an independent real scalar amplitude field and y its real Yukawa-type coupling to the DCF field. Finally, Σ and Π denote the two independent Lorentz-scalar quadratic DCF bilinears, V(Σ,Π) their vacuum potential, and V_s(s) the scalar-field potential; their explicit forms are given immediately below.

The general covariant action is S_DCF = ∫M d⁴ × √(−g) $ℒ$_DCF, where g = det(${\mathrm{g}}_{\mathsf{\mu }\mathsf{\nu }}$), Γ^μ = eμ_aΓ^a is formed with the vierbein, and D_μ contains the spin connection and the residual internal connection defined in Equation (16). The integration measure is therefore the invariant curved-spacetime measure d⁴ × √(−g).

$$\begin{gathered} L_{DC\mathrm{F}}={\displaystyle \frac{1}{2{\kappa }_G}}\left(R-2\mathrm{\Lambda }\right)-{\displaystyle \frac{1}{4}}{\mathrm{F}}_{\mu \nu }{\mathrm{F}}^{\mu \nu } \\ {\displaystyle \frac{+1}{2}}\left[\overline{\mathrm{\Psi }}{\mathrm{\Gamma }}^{\mu }\left(D_{\mu }\mathrm{\Psi }\right)J_{\mathrm{n}}-\left(D_{\mu }\overline{\mathrm{\Psi }}\right){\mathrm{\Gamma }}^{\mu }\mathrm{\Psi }{J}_{\mathrm{n}}\right] \\ {\displaystyle \frac{+1}{2}}\left({\nabla }_{\mu }s\right)\left({\nabla }^{\mu }s\right)-ys\overline{\mathrm{\Psi }}\mathrm{\Psi }-V\left(\Sigma ,\mathrm{\Pi }\right)-V_s\left(s\right) \end{gathered}$$

(18)

For clarity, the terms in Equation (18) have three different logical statuses. (i) The Lorentz-covariant kinetic structure, the general covariant measure and spin connection, and the residual U(1)J_n gauge form are fixed by the stated carrier, covariance, and symmetry requirements once the fixed real matrix direction J_n is adopted. (ii) The scalar Yukawa completion and Maxwell/Einstein sectors are minimal dynamical choices used to recover the regular Dirac, electromagnetic, and classical-gravity limits. (iii) The detailed scalar and DCF vacuum potentials, branch-locking terms, and their numerical coefficients are phenomenological or optional nonlinear additions. This classification is intended to make explicit that the action is a motivated minimal completion, not a uniqueness theorem or a derivation independent of the required limiting physics.

Dirac adjoint. In the real chiral representation used here, the Dirac-adjoint row spinor is defined explicitly below. The overbar notation is used consistently throughout the manuscript.

Conventions and gauge action. The real Dirac conjugate is the row spinor defined by the Lorentz-invariant bilinear form of the real Clifford representation and transported to the chiral basis by U; equivalently, it is the conjugation for which $\overline{\mathrm{\Psi }\mathrm{\Psi }}$ = Π in Equation (19). Under the local residual transformation $U_{\theta }\left(x\right)=exp\left[\theta \left(x\right)Jₙ\right]$, the matter field is transformed by right action, $\mathrm{\Psi }\to \mathrm{\Psi }{U}_{\theta }$, while the Abelian connection transforms in the usual compensating way so that $D\mu \mathrm{\Psi }\to \left(D\mu \mathrm{\Psi }\right)U_{\theta }$. The fixed J_n is not varied as an independent dynamical field in the minimal action: it specifies a chosen internal complex direction and is imposed by definition to satisfy J_nᵀJ_n = I₄ and J_n² = −I₄, while n₁² + n₂² + n₃² = 1, and reduces the locally realized symmetry to its stabilizer U(1)J_n. A fully local Sp(1) theory would instead require J_n to transform (or be promoted to a dynamical adjoint field) together with the additional gauge directions described in Appendix C. The present minimal theory does not derive a dynamical physical mechanism selecting the internal matrix direction J_n; its selection is part of the assumed vacuum structure, and deriving such a mechanism remains open. Explicitly, in the chiral basis, $\overline{\mathrm{\Psi }}$ ≡ Ψ^TΓ_C⁰, so $\overline{\mathrm{\Psi }\mathrm{\Psi }}$ = Ψ^TΓ_C⁰Ψ = Π.

Ontological role of gravity. Gravity is included here at the level of the same general covariant action that governs the DCF matter field, rather than appended only as an external Newtonian potential to an otherwise quantum-mechanical model. Variation with respect to the metric therefore yields the Einstein equation sourced by the DCF stress–energy tensor, while the matter equation reduces to Dirac and then Schrödinger–Pauli dynamics in the appropriate limits. The intended hierarchy is consequently DCF matter plus classical spacetime geometry → Dirac/Einstein regime → Schrödinger–Pauli weak-field regime. This unifies the proposed matter ontology with classical gravitation at the action level; it does not constitute a quantum theory of gravity.

The nonlinear potential must be built from Lorentz-invariant quadratic combinations of the real DCF field. Within the Clifford-generated quadratic ansatz used here, two independent combinations are required. They have different parity properties: Π is the parity-even Dirac scalar, while Σ is invariant under the proper Lorentz group but parity-odd. The next equations give their explicitly transported forms in the chiral basis.

$$\Sigma =2\mathrm{Sc}\left({\mathrm{Q}}_{\mathrm{L}}^{†}{\mathrm{Q}}_{\mathrm{R}}\right),\mathsf{\Pi }=2\mathrm{Sc}\left({\mathrm{Q}}_{\mathrm{L}}^{†}{\mathrm{Q}}_{\mathrm{R}}{\mathrm{J}}_{\mathrm{n}}\right)=\overline{\mathsf{\Psi }}\mathsf{\Psi }$$

(19)

These expressions are the chiral-basis images of the two invariant bilinears classified in Appendix D. In the primitive X basis, they are respectively |X_F|²−|X_{$\overline{\mathrm{F}}$}|² and 2Sc(X_F†X_{$\overline{\mathrm{F}}$}). After the nontrivial rotation U, they become the two cross-bilinears in Equation (19). This point is important: the genuine chirality grading is χ_C = diag(I₄,−I₄), so the Euclidean norm difference |Q_L|²−|Q_R|² is a chirality asymmetry but is not the Lorentz-scalar Σ used in the action. The symbol Σ below therefore denotes the invariant scalar of Equation (19), not the raw difference of chiral norms.

$$\mathrm{V}\left(\Sigma ,\hspace{0.17em}\mathsf{\Pi }\right)={\displaystyle \frac{{\mathsf{\lambda }}_{\Sigma }}{4{\mathrm{v}}^4}}{\left({\Sigma }^2\text{-}{\mathrm{v}}^4\right)}^2+{\displaystyle \frac{{\mathsf{\lambda }}_{\mathsf{\Pi }}}{2}}{\left(\mathsf{\Pi }\text{-}{\mathsf{\Pi }}_0\right)}^2$$

(20)

In the potential below, v > 0 is the DCF vacuum scale, λ_Σ and λ_Π are real potential coefficients (taken positive for local stability), and Π₀ is the preferred locking value of Π.

The first term is even under Σ → −Σ and has two degenerate minima at Σ = ±v²; the second locks the independent scalar Π about Π₀. A physical vacuum may therefore choose either sign spontaneously. In the branch used for the linearization below, we take Σ = +v² without inserting that sign preference explicitly into the action. To keep the mass sector real and regular on the full DCF field space, the mass scale is carried by an independent real scalar amplitude field s rather than by a square root of Σ.

$$\mathrm{M}=\mathrm{ys},{\mathrm{V}}_{\mathrm{s}}\left(\mathrm{s}\right)={\displaystyle \frac{{\mathsf{\lambda }}_{\mathrm{s}}}{4}}{\left({\mathrm{s}}^2\text{-}{\mathrm{v}}_{\mathrm{s}}^2\right)}^2$$

(21)

$${\nabla }_{\mathsf{\mu }}{\nabla }^{\mathsf{\mu }}\mathrm{s}+{\mathsf{\lambda }}_{\mathrm{s}}\mathrm{s}\left({\mathrm{s}}^2\text{-}{\mathrm{v}}_{\mathrm{s}}^2\right)+\mathrm{y}\mathsf{\Pi }=0$$

(22)

For the scalar completion, define the field-dependent mass scale M(s) ≡ ys; λ_s is the scalar self-coupling, and v_s is the preferred nonzero scalar amplitude.

Relation to the Standard Model scalar sector. The field s is not identified here with the Standard Model Higgs doublet. It is a single real Lorentz scalar introduced as a regular phenomenological mass-amplitude completion, and y is therefore a Yukawa-type parameter rather than a derivation of a Standard Model Yukawa coupling. In a one-generation Standard Model description, the Higgs field is an SU(2)L electroweak doublet with hypercharge, and its Yukawa couplings connect the appropriate left-handed doublet and right-handed singlet representations. Those electroweak gauge representations, the Higgs mechanism, and the measured fermion Yukawa hierarchy are not derived by Equation (18). Accordingly, the s sector should be read as a minimal phenomenological scalar completion used to obtain a regular mass term and the Dirac limit, not as a physical approximation to the full Standard Model Higgs sector. Establishing such an embedding is an open extension of the framework.

The scalar completion is introduced for regularity, not as a uniqueness claim. Its potential, coupling y, and preferred scale v_s remain phenomenological parameters of the proposed action, just as the quartic completion V(Σ,Π) and its coefficients remain motivated choices subject to the symmetry constraints summarized in Section 12 and Appendix D.

The quadratic DCF invariant sector is fixed within the Clifford-generated ansatz by the classification in Appendix D.2; the scalar completion, quartic potentials, and numerical coefficients remain motivated rather than unique.

4. Field Equation and Internal Symmetry

I now vary the action with respect to the DCF field. The purpose of this step is to obtain the full nonlinear relativistic DCF equation; the ordinary Dirac equation is not assumed here and will emerge only after linearization about a stationary vacuum in Section 5. Let B_Σ denote the linear operator obtained by varying the bilinear Σ with respect to $\overline{\mathrm{\Psi }}$. The variation gives:

$${\mathsf{\Gamma }}^{\mathsf{\mu }}\left({\mathrm{D}}_{\mathsf{\mu }}\mathsf{\Psi }\right){\mathrm{J}}_{\mathrm{n}}\text{-}\mathrm{ys}\mathsf{\Psi }\text{-}{\displaystyle \frac{\partial \mathrm{V}}{\partial \Sigma }}{\mathrm{B}}_{\Sigma }\mathsf{\Psi }\text{-}{\displaystyle \frac{\partial \mathrm{V}}{\partial \mathsf{\Pi }}}\mathsf{\Psi }=0$$

(23)

Here BΣ denotes the linear operator associated with the invariant bilinear Σ. Equation (23) is the DCF matter equation in the regular scalar completion; it is accompanied by the scalar Equation (22).

Internal Sp(1) Symmetry and the Residual Electromagnetic U(1)

Why gauge the internal symmetry, and how far? The quaternionic carrier space has an exact global right-Sp(1) action. Promoting that entire action to a local gauge symmetry would be a natural Yang–Mills completion [14], but the present minimally motivated action also contains a fixed, non-transforming real matrix complex-structure operator J_n. As shown below, that choice reduces the symmetry actually respected by the fermion kinetic sector to the stabilizer U(1)J_n. Accordingly, the minimal theory gauges only U(1)J_n. A fully local Sp(1) completion, with J_n transforming in the adjoint and with the additional Wμ^r gauge directions, is a separate extension rather than part of Equation (18).

Before a preferred matrix direction J_n is selected, the matter sector of $ℒ$_DCF is invariant under global right multiplication

$$\mathsf{\Psi }\to \mathsf{\Psi }\mathrm{U},\mathrm{U}\in \mathrm{Sp}\left(1\right)$$

(24)

Because Sp(1) ≅ SU(2) exactly (the unit-quaternion SU(2) result of Appendix A.1), the quaternionic carrier space possesses an exact internal Lie-group action. Before the fixed matrix direction J_n is inserted, this is the natural global symmetry of the J_n-independent structures. For the fixed-J_n theory, only the stabilizer U(1)J_n is gauged in the minimal motivated action. If one instead promotes the full Sp(1) action to a local redundancy in an extended theory, the corresponding connection obeys the standard inhomogeneous transformation law

$${\mathrm{A}}_{\mathsf{\mu }}\to {\mathrm{U}}^{\text{-}1}{\mathrm{A}}_{\mathsf{\mu }}\mathrm{U}\text{-}{\mathrm{U}}^{\text{-}1}{\partial }_{\mu } \mathrm{U}$$

(25)

The selected internal operator is J_n = n₁J₁ + n₂J₂ + n₃J₃ with n₁² + n₂² + n₃² = 1. It acts through the real matrix representation and satisfies J_n² = −I₄. Lorentz transformations act through the corresponding real left-action matrices; associativity of the represented algebra ensures commutation of the left and right actions. Therefore, the kinetic term remains Lorentz covariant. Internal gauge covariance restricts local transformations to the one-parameter subgroup generated by J_n, U(1)J_n = {exp(θJ_n)}.

Internal gauge covariance imposes a further restriction. Under Ψ → ΨU(x), the factor (D_μΨ)J_n transforms covariantly only when U(x)J_n = J_nU(x). The allowed local transformations are therefore the stabilizer of J_n, U(1)J_n = {exp(θJ_n)}, identified below with electromagnetism.

Thus, the minimal motivated fixed-J_n action retains the global carrier-space Sp(1) structure but has only a local U(1)J_n gauge symmetry. A fully local Sp(1) completion would require J_n itself to transform in the adjoint and would introduce additional gauge fields; that extension is treated separately in Appendix C.

5. Emergence of the Dirac Equation and Mass

The preceding construction has not assumed the Dirac equation. We can therefore ask whether ordinary relativistic fermion dynamics emerge from it. Linearizing the DCF equation about its vacuum provides the decisive test.

The reduction proceeds in three steps. First, choose a stationary vacuum in which Σ, Π, s, and J_n take constant vacuum values. Second, suppress nonlinear fluctuations about that vacuum. Third, identify the constant Yukawa scale ys₀ with the fermion mass m. The resulting small-amplitude equation must then have the real-matrix form corresponding to the ordinary Dirac equation. This is a consistency test: if the reduction failed, DCF would already disagree with established relativistic fermion dynamics. Any experimentally distinguishable departure must therefore arise from the nonlinear or extended sectors, not from this linear correspondence.

Consider a stationary vacuum with Σ = v², Π at its locked value, J_n frozen at its vacuum direction, and the scalar field at s = s₀. At that stationary point, the derivatives of the locking potential vanish. Neglecting fluctuations δΣ, δΠ, δs, and δJ_n in Equation (23) therefore removes the nonlinear potential terms, while the Yukawa factor becomes the constant ys₀. Defining the linearized rest mass by m ≡ ys₀ gives

$${\mathsf{\Gamma }}^{\mathsf{\mu }}\left({\mathrm{D}}_{\mathsf{\mu }}\mathsf{\Psi }\right){\mathrm{J}}_{\mathrm{n}}\text{-}\mathrm{m}\mathsf{\Psi }=0,\mathrm{m}=\mathrm{y}{\mathrm{s}}_0$$

(26)

Equation (26) is already a real eight-component Dirac equation. The equivalence to the conventional complex four-component form can be demonstrated explicitly, rather than stated only by analogy.

Explicit real-to-complex representation map. Fix J_n. Because J_n² = −I₄ and J_nᵀ = −J_n, each four-real-dimensional chiral sector decomposes into two invariant real two planes on which J_n acts as the standard 90-degree rotation. Define the identification 𝓘_n from the real DCF carrier to the conventional complex Dirac carrier by pairing the two real coordinates in each such plane into one complex coordinate. By construction, right multiplication by J_n becomes multiplication by the conventional scalar i. The same identification intertwines the real Clifford action with the conventional complex gamma matrices:

𝓘_n(ΨJ_n) = i𝓘_n(Ψ), 𝓘_n(Γ^μΦ) = γ^μ𝓘_n(Φ), ψ ≡ 𝓘_n(Ψ)

(27)

The covariant derivative is built from the same real representation and therefore commutes with this change of representation in the sense 𝓘_n(DμΨ) = Dμψ. Applying 𝓘_n term by term to Equation (26) gives

𝓘_n[Γ^μD_μΨJ_n − mΨ] = iγ^μD_μψ − mψ = 0

(28)

Thus the scalar imaginary unit of the conventional Dirac equation is not introduced as a primitive object: it is the complex notation for the real matrix action J_n on the invariant two planes. Restoring ℏ and c in the convention used for the reference target equation in the Introduction gives

iℏγ^μD_μψ − mcψ = 0

(29)

For a free field in flat spacetime, Dμ→∂μ, so Equation (29) becomes exactly the relativistic target displayed in the Introduction. Equations (27)–(29) therefore exhibit the DCF → Dirac reduction explicitly: vacuum linearization produces Equation (26), and the real-to-complex identification converts that equation into the standard Dirac equation without adding a new dynamical postulate.

5.1. Left–Right Coupling and Mass

In two-component chiral notation, the coupled field equations following from Equation (23) are. Here, σ^μ denotes the standard two-component chiral matrix four-vector in the selected real representation.

$${\mathrm{J}}_{\mathrm{n}}{\mathsf{\sigma }}^{\mathsf{\mu }}{\mathrm{D}}_{\mathsf{\mu }}{\mathrm{Q}}_{\mathrm{R}}=\mathrm{m}{\mathrm{Q}}_{\mathrm{L}},{\mathrm{J}}_{\mathrm{n}}{\acute{\mathsf{\sigma }}}^{\mathsf{\mu }}{\mathrm{D}}_{\mathsf{\mu }}{\mathrm{Q}}_{\mathrm{L}}=\mathrm{m}{\mathrm{Q}}_{\mathrm{R}}$$

(30)

so that the mass term represents a continuous exchange between the left and right quaternionic sectors, in direct analogy with the two-component chiral form of the ordinary Dirac equation. In the massless limit m → 0 the two chiral sectors decouple entirely. In this completion the mass scale is set by the DCF Yukawa coupling and the scalar vacuum value, m = ys₀, rather than by a square-root function of the DCF bilinear Σ.

5.2. Parity and the Spontaneous Selection of the Vacuum Σ = v²

The revised potential Equation (20) is invariant under Σ → −Σ and has two degenerate minima at Σ = ±v². The transformation Σ → −Σ is induced by ordinary parity, also in the chiral basis, but parity is no longer a bare component swap because Γ_C⁰ contains R_n. Thus, the action need not select the sign of Σ explicitly; choosing the +v² branch for the vacuum expansion amounts instead to spontaneous selection of one of two parity-related vacua.

With Γ_C⁰ = $\left[\begin{array}{cc}0& {\mathrm{J}}_{\mathrm{n}}\\ \text{-}{\mathrm{J}}_{\mathrm{n}}& 0\end{array}\right]$ and Γ_C^r = $\left[\begin{array}{cc}0& {\mathrm{J}}_{\mathrm{r}}\\ {\mathrm{J}}_{\mathrm{r}}& 0\end{array}\right]$, direct multiplication gives

$${\mathsf{\Gamma }}_{\mathrm{C}}^0{\mathsf{\Gamma }}_{\mathrm{C}}^0({\mathsf{\Gamma }}_{\mathrm{C}}^0{)}^{\text{-}1}={\mathsf{\Gamma }}_{\mathrm{C}}^0,{\mathsf{\Gamma }}_{\mathrm{C}}^0{\mathsf{\Gamma }}_{\mathrm{C}}^{\mathrm{r}}({\mathsf{\Gamma }}_{\mathrm{C}}^0{)}^{\text{-}1}=\text{-}{\mathsf{\Gamma }}_{\mathrm{C}}^{\mathrm{r}}(\mathrm{r}=1,2,3).$$

(31)

Therefore, the transformation generated by Γ_C⁰ reverses the spatial Clifford generators and becomes a symmetry of the kinetic operator when combined with x → −x. In the chiral coordinates, the parity operation is

$${\mathsf{\Psi }}_{\mathrm{C}}‘(\mathrm{t},\mathrm{x})={\mathsf{\Gamma }}_{\mathrm{C}}^0{\mathsf{\Psi }}_{\mathrm{C}}(\mathrm{t},\text{-}\mathrm{x}),\mathrm{i}.\mathrm{e}.,({\mathrm{Q}}_{\mathrm{L}},{\mathrm{Q}}_{\mathrm{R}})\to ({\mathrm{R}}_{\mathrm{Jn}}{\mathrm{Q}}_{\mathrm{R}},\text{-}{\mathrm{R}}_{\mathrm{Jn}}{\mathrm{Q}}_{\mathrm{L}}\left)\mathrm{at}\right(\mathrm{t},\text{-}\mathrm{x}).$$

(32)

Using the transported invariant bilinears in Equation (19), one finds under Equation (32)

$$\Sigma \to \text{-}\Sigma ,\mathsf{\Pi }\to \mathsf{\Pi }$$

(33)

Π is parity even, as required for the ordinary Dirac scalar, while Σ is parity-odd. With the symmetric choice Equation (20), the potential respects this transformation:

$$\mathrm{V}\left(\text{-}\Sigma ,\hspace{0.17em}\mathsf{\Pi }\right)={\displaystyle \frac{{\mathsf{\lambda }}_{\Sigma }}{4{\mathrm{v}}^4}}{\left({\Sigma }^2\text{-}{\mathrm{v}}^4\right)}^2+{\displaystyle \frac{{\mathsf{\lambda }}_{\mathsf{\Pi }}}{2}}{\left(\mathsf{\Pi }\text{-}{\mathsf{\Pi }}_0\right)}^2=\mathrm{V}\left(\Sigma ,\hspace{0.17em}\mathsf{\Pi }\right)$$

(34)

Thus, parity is a symmetry of the covariant kinetic sector, the regular scalar mass sector, and the revised vacuum potential. The physical interpretation is that Σ is the parity-odd Lorentz scalar of Equation (19), not the Euclidean norm difference |Q_L|²−|Q_R|². Selecting the Σ = +v² vacuum for the expansion therefore chooses one sign of a parity-odd order parameter spontaneously; it should not be described as a direct preference for a larger left- than right-chiral norm.

This parity statement concerns the full covariant kinetic structure together with the scalar mass and vacuum sectors. The internal gauge field and spin connection transform in the conventional way, A₀(t,x) → A₀(t,−x) and A_i(t,x) → −A_i(t,−x), with the analogous transformation of ωμ^ab, while s(t,x) → s(t,−x). Under these assignments the kinetic, Yukawa, and revised V(Σ,Π) terms are parity invariant. Choosing one of the degenerate vacua breaks parity spontaneously at the level of the realized vacuum, not explicitly in the action. No identification of this vacuum-sector spontaneous parity breaking with the observed parity violation of the weak interaction is made in the present work.

6. Low-Energy Limit and the Action Scale

The next reduction introduces no new DCF postulate; it checks the low-energy behavior of the Dirac limit obtained above. I apply the standard large/small-component expansion to the real-matrix Dirac-form equation. Eliminating the small component should produce the Pauli equation, including the magnetic spin term, and then the Schrödinger equation when spin and gravity are suppressed. In a weak gravitational field, factor out the positive rest-frequency oscillation and split Ψ into large and small components. Denote the large two-component field by ψ and the small component by ε.

$$\mathsf{\Psi }\left(\mathrm{x},\hspace{0.17em}\mathrm{t}\right)={\mathrm{e}}^{\text{-}{\mathrm{J}}_{\mathrm{n}}\mathrm{m}{\mathrm{c}}^2\mathrm{t}/\hslash }\left(\begin{array}{c}\mathsf{\psi }\left(\mathrm{x},\hspace{0.17em}\mathrm{t}\right)\\ \mathsf{\epsilon }\left(\mathrm{x},\hspace{0.17em}\mathrm{t}\right)\end{array}\right)$$

(35)

with the lower component slaved to the upper one at leading order,

$$\mathsf{\epsilon }\approx \text{-}{\displaystyle \frac{{\mathrm{J}}_{\mathrm{n}}\hslash }{2\mathrm{mc}}}\mathsf{\sigma } \cdot \mathrm{D}\mathsf{\psi }$$

(36)

For the nonrelativistic reduction, σ = (σ₁,σ₂,σ₃) denotes the Pauli-matrix vector, D the spatial gauge-covariant derivative, q the residual U(1) charge coupling, and B the magnetic field.

At the same order, the upper-component equation contains the combination (σ·D)(σ·D)ψ. Using the Pauli identity (σ·D)² = D² + (q/ℏ)σ·B with the sign fixed by the covariant-derivative convention of Equation (54), substituting Equation (36) eliminates ε and leaves the kinetic term −ℏ²D²/(2m) together with the magnetic term −(qℏ/2m)σ·B. Retaining the electrostatic and weak-gravity contributions then gives Equation (37). This is the standard controlled 1/m reduction of the Dirac-form equation, expressed in the selected real complex structure J_n.

Substituting into the covariant field equation and expanding to lowest nonrelativistic order gives Equation (37). In Equation (37), Φ denotes the electrostatic potential, Φ_G the weak gravitational potential, and V_eff any retained effective scalar potential term.

$${\mathrm{J}}_{\mathrm{n}}\hslash {\mathrm{D}}_{\mathrm{t}}\mathsf{\psi }=\left[{\displaystyle \frac{\text{-}{\hslash }^2}{2\mathrm{m}}}{\mathrm{D}}^2+\mathrm{q}\mathsf{\Phi }+\mathrm{m}{\mathsf{\Phi }}_{\mathrm{G}}\text{-}{\displaystyle \frac{\mathrm{q}\hslash }{2\mathrm{m}}}\mathsf{\sigma } \cdot \mathrm{B}+{\mathrm{V}}_{\mathrm{eff}}\right]\mathsf{\psi }$$

(37)

With the selected real matrix J_n kept explicitly, Equation (37) is the real-matrix form corresponding to the conventional Pauli equation. In the spin-independent, gravity-free, free-particle limit it reduces to the real two-component form corresponding to the ordinary Schrödinger equation.

$${\mathrm{J}}_{\mathrm{n}}\hslash {\partial }_{\mathrm{t}}\mathsf{\psi }={\displaystyle \frac{\text{-}{\hslash }^2}{2\mathrm{m}}}{\nabla }^2\mathsf{\psi }$$

(38)

The Schrödinger equation therefore appears as the low-energy, positive-frequency, direction-selected limit of the same relativistic DCF field equation, completing the reduction DCF → Dirac → Pauli → Schrödinger.

Section 5 and Section 6 together establish a single reduction hierarchy, DCF → Dirac → Pauli → Schrödinger: the full covariant DCF field equation reduces, in the linearized vacuum, to the Dirac equation (Section 5); the Dirac equation reduces, in the nonrelativistic limit, to the Pauli equation with an explicit gravitational term (Equation (37)); and the Pauli equation reduces further, on dropping spin and gravity, to the ordinary Schrödinger equation (Equation (38)). Each arrow in this chain is a controlled limit of the one before it, so that DCF’s agreement with textbook quantum mechanics at low energy is inherited rather than separately re-derived at each stage.

Possible Coupled-Field Interpretation of the Action Scale

The relativistic theory above uses ℏ in the same way as ordinary Dirac theory. A companion nonrelativistic coupled-field model [15] suggests a possible dynamical interpretation: the exchange coupling κ_s fixes an internal frequency ω_s, and that frequency fixes a characteristic mode length L_s through the propagation speed c. In the companion model, η denotes the dimensionless geometry factor relating mode length to exchange frequency.

$${\mathsf{\omega }}_{\mathrm{s}}^2\equiv {\mathsf{\kappa }}_{\mathrm{s}},{\mathrm{L}}_{\mathrm{s}}\equiv {\displaystyle \frac{\mathsf{\eta }\mathrm{c}}{{\mathsf{\omega }}_{\mathrm{s}}}}=\mathsf{\eta }\mathrm{c}{\mathsf{\kappa }}_{\mathrm{s}}^{\text{-}1/2}$$

(39)

Here η is a dimensionless geometric factor. To distinguish one exchange time from a complete oscillation cycle, define an evaluation interval Δt_s = ζ/ω_s, with ζ a fixed convention factor.

$$\Delta {\mathrm{t}}_{\mathrm{s}}\equiv \mathsf{\zeta }/{\mathsf{\omega }}_{\mathrm{s}}$$

(40)

$${\int }_{\Delta {\mathrm{t}}_{\mathrm{s}}}{\mathsf{\kappa }}_{\mathrm{s}}\mathrm{dt}={\mathsf{\kappa }}_{\mathrm{s}}\Delta {\mathrm{t}}_{\mathrm{s}}=\mathsf{\zeta }{\mathsf{\omega }}_{\mathrm{s}}$$

(41)

The mode energy is T_s = τ_sL_s = ηcτ_s/ω_s. Equating it to ℏ times the accumulated exchange over the same interval gives

$$\hslash ={\displaystyle \frac{\mathsf{\eta }\mathrm{c}{\mathsf{\tau }}_{\mathrm{s}}}{\mathsf{\zeta }{\mathsf{\omega }}_{\mathrm{s}}^2}}={\displaystyle \frac{\mathsf{\eta }\mathrm{c}{\mathsf{\tau }}_{\mathrm{s}}}{\mathsf{\zeta }{\mathsf{\kappa }}_{\mathrm{s}}}}$$

(42)

Define the dimensionless closure factor γ ≡ η/ζ, where η is the geometric factor relating the characteristic mode length to the exchange frequency and ζ specifies the evaluation interval in units of 1/ω_s. Then

$$\hslash ={\displaystyle \frac{\mathsf{\gamma }\mathrm{c}{\mathsf{\tau }}_{\mathrm{s}}}{{\mathsf{\kappa }}_{\mathrm{s}}}}$$

(43)

Thus, in the companion two-field model, once the exchange dynamics, characteristic scale ω_s² = κ_s, and the closure/evaluation interval specified above are adopted, the action associated with one characteristic internal exchange process is obtained internally as ℏ = γcτ_s/κ_s rather than inserted as a primitive constant. In this precise sense, the underlying CF ontology supplies a candidate dynamical closure and interpretation of the quantum action scale within that model. What has not yet been derived is the microscopic map from the companion-model parameters κ_s, τ_s, and γ to the parameters of the relativistic covariant DCF action Equation (18). Consequently the present relativistic formulation still uses ℏ as an input, while completion of that parameter map remains open.

7. Particle and Antiparticle Branches

Two questions must be kept separate. The linearized relativistic equation already contains positive- and negative-frequency solutions; these define the usual particle and antiparticle branches. What the linear equation does not provide is a mechanism explaining why a localized nonlinear DCF excitation should settle into one branch. This section first establishes the branch decomposition and then examines the distinct, proposed nonlinear selection mechanism. Accordingly, the general free solution contains both frequency branches. In Equation (44), x is the spacetime point, p is the four-momentum, u(p) and v(p) are the positive- and negative-frequency branch amplitudes, and J_n⁽⁸⁾ ≡ diag(J_n,J_n) is the eight-dimensional action of the selected real complex structure.

$$\mathsf{\Psi }\left(\mathrm{x}\right)=\mathrm{u}\left(\mathrm{p}\right)\exp \left[\text{-}{\mathrm{J}}_{\mathrm{n}}^{\left(8\right)}\mathrm{p}·\mathrm{x}\right]+\mathrm{v}\left(\mathrm{p}\right)\exp \left[+{\mathrm{J}}_{\mathrm{n}}^{\left(8\right)}\mathrm{p}·\mathrm{x}\right],({\mathrm{J}}_{\mathrm{n}}^{\left(8\right)}{)}^2=\text{-}{\mathrm{I}}_8$$

(44)

Identified, respectively, with particle and antiparticle solutions. The real matrix J_n supplies the internal quarter-turn generator that conventional complex notation represents by a scalar phase symbol; the DCF formulation itself contains only real matrices and real field components. These positive- and negative-frequency sectors define the physical particle/antiparticle branch decomposition used in this paper. The free DCF Lagrangian alone does not, however, dynamically select one branch: it is linear in Ψ and admits arbitrary superpositions of the two frequency sectors. A nonlinear branch-selection mechanism is needed to explain why a physically located configuration occupies one sector rather than a generic superposition; one candidate mechanism is proposed in Section 7.2.

Physical interpretation. The two frequency sectors are alternative branches of the same relativistic DCF solution space. Their simultaneous appearance in the general linear solution means that the equations admit both possibilities; it does not mean that a single realized localized excitation physically contains a fermion and an antifermion at the same time. In the proposed DCF ontology, a stable localized excitation is expected to realize one branch rather than a persistent superposition of both, but this exclusivity has not yet been derived from the nonlinear equations. The primitive X_F/X_{$\overline{\mathrm{F}}$} coordinate decomposition of Section 2.3 is conceptually distinct from this frequency decomposition; establishing their precise dynamical correspondence is part of the branch-selection problem rather than an assumption. Pair creation is a different situation: it may produce two distinct localized excitations on opposite branches. The present theory does not explain the observed cosmological excess of matter over antimatter; that population asymmetry is left as a separate open problem, plausibly determined by early-universe dynamics.

7.1. Relation of the Primitive Sector Labels to Charge Conjugation

Once the positive- and negative-frequency branches have been defined, the primitive coordinates X_F and X_{$\overline{\mathrm{F}}$} can be compared with a discrete exchange of the two primitive sectors. The simplest candidate is

$$\mathrm{C}:({\mathrm{X}}_{\mathrm{F}},{\mathrm{X}}_{\overline{\mathrm{F}}})\mapsto ({\mathrm{X}}_{\overline{\mathrm{F}}},{\mathrm{X}}_{\mathrm{F}})$$

(45)

This map is an involution and exchanges the primitive sector labels. In the chiral basis, it becomes

$$\mathrm{C}:({\mathrm{Q}}_{\mathrm{L}},{\mathrm{Q}}_{\mathrm{R}})\mapsto ({\mathrm{R}}_{\mathrm{Jn}}{\mathrm{Q}}_{\mathrm{R}},\text{-}{\mathrm{R}}_{\mathrm{Jn}}{\mathrm{Q}}_{\mathrm{L}})$$

(46)

The transformed map is again an involution. Whether this primitive sector exchange is dynamically identical to standard Dirac charge conjugation Ψ → C$\overline{\mathrm{\Psi }}$^T, and whether the full nonlinear DCF action is invariant under the same operation remains open. The relevant Dirac intertwiner is given in Appendix C.

7.2. Proposed Nonlinear Sector-Selection Mechanism

To turn the two frequency alternatives into dynamically stable sectors, introduce the scalar frequency-orientation density. Denote this real scalar density by C. For the pure-branch expression below, ω > 0 denotes the magnitude of the branch angular frequency.

$$\mathrm{C}\equiv \overline{\mathsf{\Psi }}{\mathsf{\Gamma }}^0{\mathrm{J}}_{\mathrm{n}}^{\left(8\right)}{\partial }_0\mathsf{\Psi },{\mathrm{J}}_{\mathrm{n}}^{\left(8\right)}\equiv \mathrm{diag}({\mathrm{J}}_{\mathrm{n}},{\mathrm{J}}_{\mathrm{n}})\in {\mathrm{Mat}}_8\left(\mathrm{R}\right)$$

(47)

Here J_n = n₁J₁ + n₂J₂ + n₃J₃ is the selected real 4 × 4 generator combination and J_n⁽⁸⁾ = diag(J_n,J_n) is its action on the eight-real-component DCF carrier. Since all factors are real, C is a real scalar and no scalar-part operator is needed. For a pure branch Ψ±(t) = exp(∓ωJ_n⁽⁸⁾t)Ψ±(0), ∂₀Ψ ±= ∓ωJ_n⁽⁸⁾Ψ±; because (J_n⁽⁸⁾)² = −I₈, the sign of C reverses between the two frequency orientations. Thus Equation (47) can serve as the proposed branch-orientation order parameter, subject to the dynamical caveats stated below.

And augment the Lagrangian with a branch-locking potential. Denote the potential by V_branch(C); λ_C > 0 is its locking-strength coefficient and C₀ > 0 is the target magnitude of the branch order parameter.

$${\mathrm{V}}_{\mathrm{branch}}={\displaystyle \frac{{\mathsf{\lambda }}_{\mathrm{C}}}{4}}{\left({\mathrm{C}}^2\text{-}{\mathrm{C}}_0^2\right)}^2$$

(48)

whose two degenerate minima, C = +C₀ and C = −C₀, are proposed to correspond respectively to the positive-frequency particle branch (C > 0) and negative-frequency antiparticle branch (C < 0). A branch-conserving pair-production ansatz may then be represented by $C_{particle}+C_{a\mathrm{n}tiparticle}=0$, consistent with the usual opposite-branch bookkeeping of pair production. This relation is an ansatz at the present stage, not a consequence derived from the nonlinear field equations. Demonstrating that these nonlinear minima dynamically enforce the same branch decomposition as the frequency sectors of Equation (44), and determining how that selection is represented in the primitive X_F/X_{$\overline{\mathrm{F}}$} coordinates, remain part of the full dynamical fluctuation and branch-selection analysis.

7.3. Local Potential Stability of the Branch-Locking Vacua

The local stability implied by the proposed branch potential can be assessed without assuming the stronger, still-open identification of its two minima with the positive- and negative-frequency Dirac branches. Differentiating Equation (48) gives

$${\displaystyle \frac{\mathrm{d}{\mathrm{V}}_{\mathrm{branch}}}{\mathrm{dC}}}={\mathsf{\lambda }}_{\mathrm{C}}\mathrm{C}\left({\mathrm{C}}^2\text{-}{\mathrm{C}}_0^2\right),{\displaystyle \frac{{\mathrm{d}}^2{\mathrm{V}}_{\mathrm{branch}}}{\mathrm{d}{\mathrm{C}}^2}}={\mathsf{\lambda }}_{\mathrm{C}}\left(3{\mathrm{C}}^2\text{-}{\mathrm{C}}_0^2\right)$$

(49)

Hence, for λ_C > 0 and C₀ ≠ 0, the stationary points C = ±C₀ have positive curvature, while C = 0 has negative curvature:

$${{\mathrm{V}}_{\mathrm{branch}}}^{″}\left(\pm {\mathrm{C}}_0\right)=2{\mathsf{\lambda }}_{\mathrm{C}}{\mathrm{C}}_0^2>0,{{\mathrm{V}}_{\mathrm{branch}}}^{″}\left(0\right)=\text{-}{\mathsf{\lambda }}_{\mathrm{C}}{\mathrm{C}}_0^2<0$$

(50)

Thus C = ±C₀ are strict local minima in the collective C direction. Writing C = ±C₀ + δC gives V_branch = λ_C C₀²(δC)² + O(δC³), so the local restoring curvature of this order-parameter direction is κC = 2λ_C C₀².

The same test can be applied to the other local collective invariants. For the potential-energy density U = V(Σ,Π) + Vs(s) + y_sΠ + V_branch(C), a stationary vacuum satisfies Σ* = ±v², C* = ±C₀ together with

$${\mathsf{\lambda }}_{\mathsf{\Pi }}(\mathsf{\Pi }*\text{-}{\mathsf{\Pi }}_0)+{\mathrm{y}}_{\mathrm{s}}*=0,{\mathsf{\lambda }}_{\mathrm{s}}\mathrm{s}*({\mathrm{s}}^2\text{-}{\mathrm{v}}_{\mathrm{s}}^2)+\mathrm{y}\mathsf{\Pi }*0.$$

(51)

In the collective coordinates (Σ,Π,s,C), the local Hessian is block diagonal between Σ, C, and the coupled (Π,s) sector: Let H_coll denote the full collective Hessian and H_Πs its coupled (Π,s) block.

$${\mathrm{H}}_{\mathrm{coll}}=\mathrm{diag}(2{\lambda }_{\Sigma },{\mathrm{H}}_{\Pi \mathrm{s}},2{\lambda }_{\mathrm{C}}{\mathrm{C}}_{0^2}),\mathrm{H}{\mathsf{\Pi }}_{\mathrm{s}}=\left[\begin{array}{cc}\mathsf{\lambda }\mathsf{\Pi }& \mathrm{y}\\ \mathrm{y}& \mathsf{\lambda }\mathrm{s}(3{\mathrm{s}}^2-{\mathrm{v}}_{\mathrm{s}}^2)\end{array}\right].$$

(52)

A sufficient set of local energetic-stability conditions is therefore λ_Σ > 0, λ_C > 0, λ_Π > 0, and λ_Πλ_s(3s*² − v_s²) − y² > 0. These conditions establish positive curvature of the reduced collective potential about the chosen vacuum.

This is not yet the complete propagating fluctuation spectrum of the underlying eight real DCF components. The quantities Σ, Π, and C are composite bilinears of Ψ rather than independent canonically normalized scalar fields; obtaining physical eigenfrequencies requires linearizing the full Ψ–s field equations about an explicit solution and mapping δΨ into the collective invariant variations. The calculation above therefore establishes local potential stability, while the full dynamical spectrum, localized nonlinear solutions, and the identification of C = ±C₀ with the Dirac frequency branches remain open.

8. Electric Charge as a Noether Charge

The residual U(1) rotation about J_n acts on the DCF field as Let α(x) denotes the local residual phase parameter.

$$\mathsf{\Psi }\to \mathsf{\Psi }{\mathrm{e}}^{\mathrm{q}{\mathrm{J}}_{\mathrm{n}}\mathsf{\alpha }\left(\mathrm{x}\right)/\hslash }$$

(53)

with covariant derivative

$${\mathrm{D}}_{\mathsf{\mu }}\mathsf{\Psi }={\nabla }_{\mathsf{\mu }}\mathsf{\Psi }+{\displaystyle \frac{\mathrm{q}}{\hslash }}\mathsf{\Psi }{\mathrm{J}}_{\mathrm{n}}{\mathrm{A}}_{\mathsf{\mu }}$$

(54)

By Noether’s theorem [16], the associated Noether current is denote this current by J^μ.

$${\mathrm{J}}^{\mathsf{\mu }}={\displaystyle \frac{\mathrm{q}}{2}}\mathrm{Sc}\left[\overline{\mathsf{\Psi }}{\mathsf{\Gamma }}^{\mathsf{\mu }}\mathsf{\Psi }{\mathrm{J}}_{\mathrm{n}}\text{-}{\mathrm{J}}_{\mathrm{n}}\overline{\mathsf{\Psi }}{\mathsf{\Gamma }}^{\mathsf{\mu }}\mathsf{\Psi }\right]$$

(55)

which is conserved on shell, ∇μJμ = 0, giving the conserved electric charge Define Q_em as the conserved charge obtained by integrating J^μ over a spacelike hypersurface, with dΣ_μ the oriented hypersurface element.

$${\mathrm{Q}}_{\mathrm{em}}={\int }_{\Sigma }{\mathrm{J}}^{\mathsf{\mu }}\mathrm{d}{\Sigma }_{\mathsf{\mu }}$$

(56)

This identifies electric-charge conservation with the Noether generator of rotations about the selected internal matrix direction J_n. It therefore provides a geometric interpretation of the conserved electromagnetic charge within the DCF framework. It does not derive the numerical elementary charge e, prove that all physical charges occur in integer units, or determine the coupling q. Such quantization requires additional topology or dynamics, neither of which is established here. For the winding relation below, N∈ℤ denotes the winding number.

$${\displaystyle \frac{1}{2\mathsf{\pi }}}\oint \left(\mathrm{d}\mathsf{\theta }\text{-}{\displaystyle \frac{\mathrm{q}}{\hslash }}{\mathrm{A}}_{\mathsf{\mu }}\mathrm{d}{\mathrm{x}}^{\mathsf{\mu }}\right)=\mathrm{N}\in \mathrm{Z}$$

(57)

under which distinct winding sectors N carry discrete charge labels. No mechanism fixing the physical value of q or forcing N to be quantized dynamically, rather than assumed, is derived here.

9. Two-Particle States and Bell Correlations: A Consistency Test

Bell notation. S denotes the standard CHSH combination of four two-setting correlations; the Bell-local bound is |S| ≤ 2 and the Tsirelson value for the singlet is |S| = 2√2.

Nothing in this section is presented as evidence that the classical DCF dynamics itself produce quantum entanglement. This section is not used to establish the primary DCF construction. Its purpose is to test whether the two-particle sector is consistent with the Bell constraints once the single-particle Lorentz-covariant framework has been established. Two distinct constructions must be kept separate. The Bell-local shared internal phase model obeys |S| ≤ 2. A non-factorizable antisymmetric joint state gives the standard singlet correlation and |S| = 2√2 conditional on production of that state. Nothing in this section claims a local hidden-variable derivation of the Tsirelson value.

9.1. DCF Description of a Classically Correlated Pair

Consider two DCF excitations produced by a common local source and subsequently separated to spacelike distance, each described by its own doublet field of the form of the DCF field content of Equation (8). We denote their fields by Ψ⁽¹⁾ and Ψ⁽²⁾.

$${\mathsf{\Psi }}^{\left(1\right)}\left(\mathrm{x}\right)=\left(\begin{array}{c}{\mathrm{Q}}_{\mathrm{L}}^{\left(1\right)}\\ {\mathrm{Q}}_{\mathrm{R}}^{\left(1\right)}\end{array}\right),{\mathsf{\Psi }}^{\left(2\right)}\left(\mathrm{x}\right)=\left(\begin{array}{c}{\mathrm{Q}}_{\mathrm{L}}^{\left(2\right)}\\ {\mathrm{Q}}_{\mathrm{R}}^{\left(2\right)}\end{array}\right)$$

(58)

DCF-correlated pair. At the source, the internal quaternionic orientations of Ψ⁽¹⁾ and Ψ⁽²⁾ are locked antipodal on a common great circle of the internal Sp(1) fiber, parametrized by a single shared phase λ common to both particles. λ is not fixed by the production process; it varies from pair to pair with distribution ρ(λ), taken to be uniform, ρ(λ) = 1/2π on [0,2π), by the same isotropy reasoning used elsewhere in this paper to fix free distributions in the absence of a preferred internal direction (cf. Section 4. This is a postulated initial condition on the pair’s internal degrees of freedom, offered in the same spirit as the DCF field content of Equation (8): it is not derived from an explicit local pair-production interaction term added to $ℒ$_DCF, and no source interaction is supplied for this Bell-local shared-phase model. This is intentional: Section 9.2, Section 9.3, Section 9.4, Section 9.5 and Section 9.6 test that model as stated. The distinct antisymmetric joint-state construction of Section 9.7 is different; its classical SU(2)-invariant pair interaction and singlet/triplet energetic ordering are established in Appendix B, while a distinct Lorentz-covariant singlet-production vertex is formulated in the quantized two-particle extension of Appendix C. What remains open there is normalization into a complete production probability or an explicit classical nonlinear source solution.

This source assumption plays the role that singlet preparation plays in ordinary quantum mechanics, but it is stated here as a classical correlation of a hidden internal variable λ shared by two spacelike-separated local field configurations—not as a single inseparable joint quantum state. Section 9.2, Section 9.3, Section 9.4 and Section 9.5 develop only the local consequences of this shared classical variable at each wing separately, so that the detector model constructed below is, by design, a local hidden-variable model in Bell’s sense; Section 9.6 makes that status precise.

9.2. Quaternionic Synchronization Between the Two DCFs

For the spin degree of freedom relevant to a Bell experiment, it suffices to work in the nonrelativistic, positive-frequency sector of Section 6, where each particle is described by the two-component doublet ψ of Equations (35)–(38) rather than the full eight-component Ψ. Fix a measurement plane and represent the internal orientation associated with the shared phase λ by the unit vector Denote the unit Bloch-sphere orientation associated with the shared phase by $\hat{\mathrm{n}}$(λ).

$$\hat{\mathrm{n}}\left(\mathsf{\lambda }\right)=\left(\begin{array}{c}\cos \mathsf{\lambda }\\ \sin \mathsf{\lambda }\\ 0\end{array}\right)$$

(59)

a physical 3-vector on the Bloch sphere carried by the SU(2) doublet representation of the unit-quaternion SU(2) result of Appendix A.1; $\hat{\mathrm{n}}$(λ) is not to be confused with the fixed real matrix direction J_n of Section 3.1, which selects the electromagnetic U(1) and play_s no role in this construction. Let ψ⁽¹⁾(λ) denote the eigenstate of $\hat{\mathrm{n}}$(λ)·σ with eigenvalue +1, where σ are the same Pauli matrices already appearing in the spin–magnetic coupling term of Equation (37). The shared-phase source assumption of Section 9.1 is realized by taking the second particle’s internal orientation to be $\hat{\mathrm{n}}$(λ + π) = −$\hat{\mathrm{n}}$(λ), so that ψ⁽²⁾(λ) is the eigenstate of −$\hat{\mathrm{n}}$(λ)·σ with eigenvalue +1. Because Sp(1) ≅ SU(2) is exact (the unit-quaternion SU(2) result of Appendix A.1), this Bloch-sphere parametrization of the internal orientation is an exact statement about the doublet representation, not a doubled-angle approximation of the kind used in the scalar-field construction.

9.3. Local Detector Interaction

Before introducing the macroscopic pointer, the spin orientation and analyzer response can be written directly in the four real components of one quaternion.

Orientation vector in purely real coordinates. Let q = (q₀,q₁,q₂,q₃)^T be a normalized real quaternion-coordinate vector, q^Tq = 1. Define the physical orientation directly by the real Hopf map, with no complex amplitudes: $\hat{\mathrm{n}}$₁ = 2(q₀q₂ + q₁q₃), $\hat{\mathrm{n}}$₂ = 2(q₀q₃−q₁q₂), and $\hat{\mathrm{n}}$₃ = q₀² + q₁² − q₂² − q₃². Then |$\hat{\mathrm{n}}$| = 1. This is the real-coordinate form of the same SU(2) orientation map used by the analyzer algebra below.

$${\hat{\mathrm{n}}}_1=2({\mathrm{q}}_0{\mathrm{q}}_2+{\mathrm{q}}_1{\mathrm{q}}_3),{\hat{\mathrm{n}}}_2=2({\mathrm{q}}_0{\mathrm{q}}_3\text{-}{\mathrm{q}}_1{\mathrm{q}}_2),{\hat{\mathrm{n}}}_3={\mathrm{q}}_0^2+{\mathrm{q}}_1^2\text{-}{\mathrm{q}}_2^2\text{-}{\mathrm{q}}_3^2$$

(60)

A direct computation gives $\hat{\mathrm{n}}$₁² + $\hat{\mathrm{n}}$₂² + $\hat{\mathrm{n}}$₃² = (q₀² + q₁² + q₂² + q₃²)² = ‖Q‖⁴, so a unit quaternion (‖Q‖ = 1) carries $\hat{\mathrm{n}}$(Q)∈S², the physical spin-orientation sphere. Equation (60) is the general form of the Bloch vector $\hat{\mathrm{n}}$(λ) of Equation (59): since $\hat{\mathrm{n}}$(λ)·σ has ψ(λ) as its +1 eigenstate by construction, $\hat{\mathrm{n}}$(λ) coincides with Equation (60) evaluated on ψ(λ); Equation (60) simply extends this to an arbitrary doublet ζ, not restricted to an eigenstate of a fixed analyzer direction. The measured spin direction is therefore not one of the four real fields individually but a bilinear combination of all four.

Real form of the analyzer operator. For a unit vector a = $\left(\begin{array}{c}{\mathrm{a}}_1\\ {\mathrm{a}}_2\\ {\mathrm{a}}_3\end{array}\right)$, the complex analyzer operator a·σ of Equations (62) and (63) has the following real 4 × 4 matrix realization acting on the coordinate vector Qᵥ ≡ $\left(\begin{array}{c}{\mathrm{q}}_0\\ {\mathrm{q}}_1\\ {\mathrm{q}}_2\\ {\mathrm{q}}_3\end{array}\right)$:

$$\mathsf{\Gamma }(\mathrm{a}) = \left[\begin{array}{cccc}{\mathrm{a}}_3& 0& {\mathrm{a}}_1& {\mathrm{a}}_2\\ 0& {\mathrm{a}}_3& \text{-}{\mathrm{a}}_2& {\mathrm{a}}_1\\ {\mathrm{a}}_1& \text{-}{\mathrm{a}}_2& \text{-}{\mathrm{a}}_3& 0\\ {\mathrm{a}}_2& {\mathrm{a}}_1& 0& \text{-}{\mathrm{a}}_3\end{array}\right]$$

(61)

Involution. Γ(a)² = I₄ for every unit vector a; consequently, Γ(a) has real eigenvalues ±1 only, each with multiplicity two.

Proposition 1 (Real pre-measurement bias).

QᵥT Γ(a) Qᵥ = a·$\hat{\mathrm{n}}$(Q), with $\hat{\mathrm{n}}$(Q) the orientation vector of (60).

By Proposition 1, the analyzer expectation value can, therefore, be written entirely as the real quadratic formQᵥTΓ(a)Qᵥ = a·$\hat{\mathrm{n}}$(Q). The operator Γ(a), however, already fixes the ideal spin observable: for every analyzer direction, a, its only eigenvalues are ±1. These two eigenchannels are properties of the DCF/Pauli spin algebra and do not depend on the microscopic construction of a particular detector.

The two-particle construction of Section 9.7, DCF⁽¹⁾ = (CF₁⁽¹⁾,CF₂⁽¹⁾) and DCF⁽²⁾ = (CF₁⁽²⁾,CF₂⁽²⁾), is the tensor product of two independent copies of the single-DCF real field space ℝ⁴ introduced here.

A Stern–Gerlach-type analyzer must couple the internal spin orientation to a controllable laboratory direction. For an analyzer angle α, we therefore write Let $\hat{\mathrm{n}}$_α be the unit analyzer direction at angle α and let f_D(t) be the time-dependent magnetic-field envelope, so B_α(t) = f_D(t)$\hat{\mathrm{n}}$_α.

$$\mathrm{B}\mathsf{\alpha }\left(\mathrm{t}\right)=\mathrm{fD}\left(\mathrm{t}\right)\hat{\mathrm{n}}\mathsf{\alpha }$$

(62)

so that the analyzer–DCF interaction is not a new postulate but the restriction of the spin–magnetic coupling already present in Equation (37), −(qℏ/2m)f_D(t) $\hat{\mathrm{n}}$α·σ, to this configuration. The pre-registration bias this coupling exerts on a particle prepared with internal orientation $\hat{\mathrm{n}}$(λ) is the expectation value of this operator in ψ(λ),

$$\mathrm{a}(\mathsf{\alpha },\mathsf{\lambda })=⟨\mathsf{\psi }\left(\mathsf{\lambda }\right)|\hat{\mathrm{n}}\mathsf{\alpha }·\mathsf{\sigma }|\mathsf{\psi }\left(\mathsf{\lambda }\right)⟩=\hat{\mathrm{n}}\left(\mathsf{\lambda }\right)·\hat{\mathrm{n}}\mathsf{\alpha }=\cos (\mathsf{\lambda }\text{-}\mathsf{\alpha })$$

(63)

The quantity a(α,λ) = cos(λ − α) is a continuous pre-registration bias only in the deliberately Bell-local shared-phase model of Section 9.1, Section 9.2, Section 9.3, Section 9.4, Section 9.5 and Section 9.6. It must not be identified with a universal law of spin detection. To test that specific classical shared-λ model, we introduce one illustrative bistable pointer whose macroscopic registration follows the sign of this bias. This construction is therefore a model of amplification for the Bell-local toy sector, not the definition of the spin observable and not an ingredient of the successful antisymmetric singlet calculation of Section 9.7. Couple a bistable pointer coordinate ξ_D(t) to the bias through the quartic double-well form let ξ_D(t) denote the collective pointer coordinate, V_D(ξ_D) its double-well potential, γ_D its overdamped coefficient, κ_D its instability/curvature scale, and g_D the detector–bias coupling.

$${\mathsf{\gamma }}_{\mathrm{D}}{\dot{\mathsf{\xi }}}_{\mathrm{D}}=\text{-}{\displaystyle \frac{\partial {\mathrm{V}}_{\mathrm{D}}}{\partial {\mathsf{\xi }}_{\mathrm{D}}}}+{\mathrm{g}}_{\mathrm{D}}{\mathrm{f}}_{\mathrm{D}}\left(\mathrm{t}\right)\mathrm{a}\left(\mathsf{\alpha },\hspace{0.17em}\mathsf{\lambda }\right),{\mathrm{V}}_{\mathrm{D}}\left({\mathsf{\xi }}_{\mathrm{D}}\right)={\displaystyle \frac{{\mathsf{\kappa }}_{\mathrm{D}}}{4}}{\left({\mathsf{\xi }}_{\mathrm{D}}^2\text{-}1\right)}^2$$

(64)

Microscopic status of the pointer dynamics. Equation (64) is a phenomenological effective overdamped equation for one possible collective detector coordinate ξ_D, introduced only to realize the Bell-local shared-phase threshold model. It is not a direct Euler–Lagrange equation obtained from the minimal DCF action Equation (18), and it is not asserted to be the universal microscopic mechanism of spin measurement. The ideal analyzer observable Γ(a) has the same ±1 eigenchannels for any physically adequate detector implementing the same analyzer direction. Detector-specific parameters κ_D, γ_D, and g_D may affect latency, amplification, efficiency, noise, or inconclusive events, but they must not change the ideal spin eigenvalues or the singlet correlation associated with a fixed analyzer setting.

Accordingly, no detector-dependent modification of the ideal Bell correlation is claimed from Equation (64). A microscopic completion may determine device-dependent registration properties, but the ideal spin statistics of the antisymmetric state are fixed by the state and analyzer algebra, not by κ_D, γ_D, or g_D.

A microscopic completion would enlarge the theory to denote the total, detector, and interaction Lagrangian densities by $ℒ$_total, D_at, and L_int, respectively.

$${\mathrm{L}}_{\mathrm{total}}={\mathrm{L}}_{\mathrm{DCF}}+{\mathrm{L}}_{\det }+{\mathrm{L}}_{\mathrm{int}}$$

(65)

where L_aet contains the detector degrees of freedom and L_int couples a collective pointer mode to a local DCF observable, such as the analyzer bias or an appropriate DCF bilinear. After projecting the detector dynamics onto the unstable collective coordinate and eliminating the remaining detector and environmental modes, the reduced equation has the general form In the reduced equation, M_D denotes the effective pointer inertia and Γ_D(t − t′) the memory/damping kernel.

$${\mathrm{M}}_{\mathrm{D}}{\ddot{\mathsf{\xi }}}_{\mathrm{D}}+{\int }_0^{\mathrm{t}}{\mathsf{\Gamma }}_{\mathrm{D}}\left(\mathrm{t}\text{-}{\mathrm{t}}^{\prime }\right){\dot{\mathsf{\xi }}}_{\mathrm{D}}\left({\mathrm{t}}^{\prime }\right){\mathrm{dt}}^{\prime }\text{-}{\mathsf{\kappa }}_{\mathrm{D}}{\mathsf{\xi }}_{\mathrm{D}}={\mathrm{g}}_{\mathrm{D}}{\mathrm{f}}_{\mathrm{D}}\left(\mathrm{t}\right)\mathrm{a}\left(\mathsf{\alpha },\hspace{0.17em}\mathsf{\lambda }\right)+\mathrm{O}\left({\mathsf{\xi }}_{\mathrm{D}}^3\right)$$

(66)

Here M_D is the effective inertia of the collective pointer mode and Γ_D is its memory/damping kernel. In the low-frequency Markovian and overdamped limits,

$${\mathsf{\Gamma }}_{\mathrm{D}}\left(\mathrm{t}\text{-}{\mathrm{t}}^{\prime }\right)\to 2{\mathsf{\gamma }}_{\mathrm{D}}\mathsf{\delta }\left(\mathrm{t}\text{-}{\mathrm{t}}^{\prime }\right),{\mathrm{M}}_{\mathrm{D}}{\ddot{\mathsf{\xi }}}_{\mathrm{D}}\ll {\mathsf{\gamma }}_{\mathrm{D}}{\dot{\mathsf{\xi }}}_{\mathrm{D}}$$

(67)

this reduces Equation (64).

In such a completion, the instability coefficient is the negative curvature of the effective pointer potential at its unstable stationary point. Denote the effective pointer potential by U_eff(ξ_D).

$${\mathsf{\kappa }}_{\mathrm{D}}=\text{-}{\left.{\displaystyle \frac{{\mathrm{d}}^2{\mathrm{U}}_{\mathrm{eff}}}{\mathrm{d}{\mathsf{\xi }}_{\mathrm{D}}^2}}\right|}_{{\mathsf{\xi }}_{\mathrm{D}}=0}$$

(68)

while the damping coefficient is determined by the low-frequency dissipative part of the retarded self-energy of the collective detector mode, denote the retarded detector self-energy by Σ_D^R(ω).

$${\mathsf{\gamma }}_{\mathrm{D}}=\text{-}\underset{\mathsf{\omega }\to 0}{\lim }{\displaystyle \frac{\mathrm{Im}{\Sigma }_{\mathrm{D}}^{\mathrm{R}}\left(\mathsf{\omega }\right)}{\mathsf{\omega }}}$$

(69)

Therefore the characteristic detector timescale may be written microscopically as denote this timescale by τ_D.

$${\mathsf{\tau }}_{\mathrm{D}}={\displaystyle \frac{{\mathsf{\gamma }}_{\mathrm{D}}}{{\mathsf{\kappa }}_{\mathrm{D}}}}={\displaystyle \frac{\text{-}\underset{\mathsf{\omega }\to 0}{\lim }\mathrm{Im}{\Sigma }_{\mathrm{D}}^{\mathrm{R}}\left(\mathsf{\omega }\right)/\mathsf{\omega }}{\text{-}{{\mathrm{U}}_{\mathrm{eff}}}^{″}\left(0\right)}}$$

(70)

Equations (68)–(70) specify the required bridge to microscopic dynamics, but they do not yet give a parameter-free prediction: evaluating U_eff and Σ_D^R requires an explicit detector–DCF interaction and a model of the detector environment.

Near the unstable ridge ξ_D = 0 separating the two wells, −∂V_D/∂ξ_D≈κ_Dξ_D, and with ξ_D(0) = 0 the equation integrates to ξ_D(t) ≈ K_D(t)a(α,λ) with K_D(t) = (g_D/γ_D)∫₀^t e(κ_D/γ_D)(t−t′)f_D(t′)dt′, which grows without bound over the measurement window. The linear instability amplifies whatever sign a(α,λ) carries until nonlinear saturation of V_D locks ξ_D into one of its two minima, yielding the dichotomy registration Define K_D(t) ≡ (g_D/γ_D)∫₀ᵗexp[(κ_D/γ_D)(t−t′)]f_D(t′)dt′.

$$\mathrm{A}(\mathsf{\alpha },\mathsf{\lambda })=\mathrm{sign}\left(\mathrm{a}\right(\mathsf{\alpha },\mathsf{\lambda }\left)\right)=\mathrm{sign}\left(\cos \right(\mathsf{\lambda }\text{-}\mathsf{\alpha }\left)\right)$$

(71)

Within this illustrative Bell-local shared-phase detector model, and only within it, the threshold rule (Equation (71)) follows from Equation (64) rather than being separately postulated. Thus A(α,λ) = sign[cos(λ−α)] is not proposed as a universal DCF measurement law. The same illustrative construction applied to particle 2, with analyzer angle β and internal orientation −$\hat{\mathrm{n}}$(λ), gives the registered outcome for particle 1 by A(α,λ), and for particle 2 by B(β,λ), where β is the second analyzer angle.

$$\mathrm{B}(\mathsf{\beta },\mathsf{\lambda })=\mathrm{sign}(-\cos (\mathsf{\lambda }-\mathsf{\beta }\left)\right)=-\mathrm{sign}\left(\cos \right(\mathsf{\lambda }-\mathsf{\beta }\left)\right)$$

(72)

9.4. Dichotomic Observable in the Bell-Local Shared-Phase Test Model

For the shared-λ test model of Equations (71) and (72), A(α,λ),B(β,λ)∈{−1,+1} identically. This establishes a clean Bell-local benchmark without any Pearson rescaling. It should not be confused with the general rule for spin measurement. In the full DCF singlet construction of Section 9.7, the dichotomy outcomes are instead the ±1 eigenchannels of the analyzer operators, exactly as required by the SU(2) spin algebra. For the present shared-phase benchmark, the marginal probability follows from (Equation (71)): for λ uniform on [0,2π), cos(λ−α) > 0 on an interval of length π, so

$${\mathrm{P}}_{\hspace{0.17em}+\left(\mathsf{\alpha }\right)=\mathrm{P}\left(\mathrm{A}\left(\mathsf{\alpha },\hspace{0.17em}\mathsf{\lambda }\right)=+1\right)={\displaystyle \frac{1}{2}}}$$

(73)

independent of α. This 1/2 marginal is a property of the uniform shared-phase benchmark. It is not used to derive the singlet result of Section 9.7.

Detector universality. For an ideal spin analyzer, the physical content is the operator Γ(a) (equivalently a·σ) and its two eigenchannels ±1. Different detector technologies may realize the amplification and registration of those channels by different microscopic routes, but after calibration they are presumed to measure the same spin observable at the same analyzer angle. The phenomenological bistable pointer of Section 9.3 is therefore only an illustrative realization for the Bell-local shared-λ model; it does not imply detector-dependent spin values or detector-dependent ideal Bell correlations.

9.5. Correlation Function

The joint correlation of the two dichotomy outcomes, averaged over the shared hidden phase λ with the uniform distribution of the shared-phase source assumption of Section 9.1, is denoted the correlation in E(α,β).

$$\mathrm{E}\left(\mathsf{\alpha },\hspace{0.17em}\mathsf{\beta }\right)={\int }_0^{2\mathsf{\pi }}{\displaystyle \frac{\mathrm{d}\mathsf{\lambda }}{2\mathsf{\pi }}}\mathrm{A}\left(\mathsf{\alpha },\hspace{0.17em}\mathsf{\lambda }\right)\mathrm{B}\left(\mathsf{\beta },\hspace{0.17em}\mathsf{\lambda }\right)=\text{-}{\int }_0^{2\mathsf{\pi }}{\displaystyle \frac{\mathrm{d}\mathsf{\lambda }}{2\mathsf{\pi }}}\mathrm{sign}\left(\cos \left(\mathsf{\lambda }\text{-}\mathsf{\alpha }\right)\right)\mathrm{sign}\left(\cos \left(\mathsf{\lambda }\text{-}\mathsf{\beta }\right)\right)$$

(74)

The integral of two ±1 square waves phase-shifted by θ = α − β is the standard piecewise-linear (triangular) correlation of two threshold detectors on a circle, giving

$$\mathrm{E}(\mathsf{\alpha },\mathsf{\beta })=-1+\frac{2|\alpha -\beta |}{\pi },|\mathsf{\alpha }-\mathsf{\beta }|\le \mathsf{\pi }$$

(75)

extended periodically outside this range. At α = β, E = −1, exact anticorrelation, matching the physical expectation for the antipodal locked pair of the shared-phase source assumption of Section 9.1 measured along the same axis. At |α−β| = π/2, E = 0, uncorrelated. E(α,β) is piecewise linear in the setting difference rather than sinusoidal; Section 9.8 compares it directly with the quantum-mechanical prediction, while Section 9.7 shows that the nonfactorizable antisymmetric joint state reproduces the standard quantum correlation.

9.6. CHSH Inequality and Tsirelson Bound

Proposition 2 (CHSH bound for the DCF detector model).

For any four analyzer settings α,α′,β,β′ and the detector model of Equations (71) and (72), the CHSH combination S = E(α,β) + E(α,β′) + E(α′,β)−E(α′,β′) satisfies |S| ≤ 2.

Fix λ. Since A(α,λ),A(α′,λ),B(β,λ),B(β′,λ) ∈ {−1, +1}, A(α,λ)B(β,λ) + A(α,λ)B(β′,λ) + A(α′,λ)B(β,λ) − A(α′,λ)B(β′,λ) = A(α,λ)[B(β,λ)+B(β′,λ)] + A(α′,λ)[B(β,λ)−B(β′,λ)]. Because B(β,λ),B(β′,λ)∈{−1,+1}, exactly one of [B(β,λ) + B(β′,λ)] and [B(β,λ) − B(β′,λ)] vanishes and the other equals ±2; hence the whole expression equals ±2 for every λ, independent of α,α′. Averaging this expression over the normalized distribution ρ(λ), and applying the triangle inequality, gives.

This is the elementary form of Bell’s original argument (as sharpened by Clauser, Horne, Shimony and Holt [17] and placed in its general local-hidden-variable form by Fine [18]): any model in which the two outcomes are deterministic functions of their own local setting and a shared variable λ is bound by |S| ≤ 2, for every choice of settings, not merely the ones realized in (Equation (75)). The construction of Section 9.3 and Section 9.4 is manifestly of this form—A depends only on α and λ, B only on β and λ—so Proposition 2 applies directly to it, without needing to evaluate S at any particular set of angles. The DCF detector model derived here therefore does not, and by Proposition 2 cannot, exceed the classical CHSH bound, regardless of how the analyzer directions are chosen.

Explicit standard-angle comparison. For the standard CHSH settings α = 0, α′ = π/2, β = π/4, β′ = −π/4, Equation (75) gives E(α,β) = E(α,β′) = E(α′,β) = −1/2 and E(α′,β′) = +1/2, hence S = −2. By contrast, the joint-state correlation of Equation (79) gives the corresponding values $-1/\sqrt{2}$, $-1/\sqrt{2}$, $-1/\sqrt{2}$, $+1/\sqrt{2}$, and therefore $\mathrm{S}=-2\sqrt{2}$. Thus the same analyzer settings display the distinction directly: the Bell-local shared-λ construction saturates the classical bound, whereas the antisymmetric joint state saturates the Tsirelson bound—a normalized quaternionic analyzer algebra does not by itself evade Bell locality; every factorized shared-variable model stays bounded by |S| ≤ 2, and the value $2\sqrt{2}$ comes only from the non-factorizable antisymmetric state.

9.7. Joint Pair Modes and the Boundary Between Classical and Quantized DCF

This subsection separates two statements that use the same SU(2) carrier but have different theoretical statuses. At the strictly classical DCF level, two coupled DCF carriers form a joint internal system whose exchange operator decomposes the pair into singlet and triplet sectors. This classical joint-mode analysis establishes the available collective modes and their energetic ordering; it does not by itself provide a Fock-space production amplitude. The local covariant source vertex introduced later in Appendix C belongs instead to a quantized two-particle extension, where separately labeled outgoing mode operators make the antisymmetric production amplitude nonzero. The Bell/Tsirelson calculation below is therefore conditional on preparation of the corresponding nonfactorizable two-particle state and should not be read as a derivation of that state from the single-field classical action.

Status of the three-level two-particle construction. Level 1—classical DCF ontology: the coupled pair possesses singlet and triplet joint modes, and Appendix B proves that for H₁₂ = J σ₁·σ₂ with J > 0 the singlet has E_singlet = −3J while the triplet sector has E_triplet = +J, so ΔE = 4J > 0. The singlet is therefore the unique minimum-energy joint mode. Consequently, if pair preparation or subsequent relaxation is constrained to the lowest available internal-energy state, energy conservation together with J > 0 forces the pair into the singlet sector; a triplet can occur only when the source or environment supplies and retains the additional internal energy 4J. Energy conservation alone does not forbid such excited triplets, which remain legitimate prepared modes. Level 2—conditional full-Sp(1) extension: Appendix C gives a possible gauge-exchange realization with J_eff > 0, but this is not a consequence of the minimal fixed-J_n action. Level 3—quantized two-particle DCF extension: Appendix C introduces a covariant spin-zero source vertex in a mode/operator description and shows that its production amplitude is purely antisymmetric. Because this third result uses second-quantized machinery, it is not claimed as a theorem of the strictly classical single-field theory. Once the singlet state is prepared for that extension, the standard joint-state SU(2) algebra gives E(a,b) = −a·b and |S| = 2√2, independently of detector microphysics. A classical nonlinear two-packet production solution that would reproduce the same selection without quantization has not yet been constructed.

Let the analyzer act on the full doublet, rather than only producing the scalar bias (Equation (63)), as the matrix Denote the resulting analyzer matrix by Γ($\hat{\mathrm{n}}$_α).

$$Â\left(\mathsf{\alpha }\right)=\hat{\mathrm{n}}\mathsf{\alpha }·\mathsf{\sigma }$$

(76)

the same SU(2) generator already used to build (Equation (63)), with eigenstates |$\hat{\mathrm{n}}$α; ± ⟩ of eigenvalue ±1. For a pair of DCFs there are then four CF channels in play, two per particle, DCF⁽¹⁾ = (CF₁⁽¹⁾,CF₂⁽¹⁾) and DCF⁽²⁾ = (CF₁⁽²⁾,CF₂⁽²⁾). The joint two-doublet space contains both singlet and triplet sectors. For the spin-zero source construction of Appendix C, the invariant antisymmetric contraction selects the normalized singlet composite Denote the normalized antisymmetric two-doublet state by |Ω₁₂⟩.

$${\mathsf{\Omega }}_{12}⟩={\displaystyle \frac{1}{\sqrt{2}}}\left(\left|\widehat{\mathrm{z}};+{⟩}^{\left(1\right)}\otimes \right|\widehat{\mathrm{z}};\text{-}{⟩}^{\left(2\right)}\text{-}\left|\widehat{\mathrm{z}};\text{-}{⟩}^{\left(1\right)}\otimes \right|\widehat{\mathrm{z}};+{⟩}^{\left(2\right)}\right)$$

(77)

built from a fixed reference axis ẑ, rather than a shared phase λ. Unlike the classical pair ($\hat{\mathrm{n}}$(λ),−$\hat{\mathrm{n}}$(λ)), which singles out a definite axis for every realization, |Ω₁₂⟩ is invariant under any joint rotation R⊗R applied to both doublets together; it is the DCF analog of the singlet, and no hidden phase λ or its distribution ρ(λ) appears in its definition. The joint correlation is then the matrix element Here R denotes an arbitrary common SU(2) doublet rotation in the statement R⊗R.

$$\mathrm{E}(\mathsf{\alpha },\mathsf{\beta })=⟨{\mathsf{\Omega }}_{12}|Â(\mathsf{\alpha })\otimes \hat{\mathrm{B}}(\mathsf{\beta }\left)\right|{\mathsf{\Omega }}_{12}⟩$$

(78)

rather than the integral ∫dλ ρ(λ)A(α,λ)B(β,λ) of Equation (74). Evaluating (Equation (78)) on the antisymmetric composite (Equation (77)) by the standard SU(2) singlet identity gives

$$\mathrm{E}(\mathsf{\alpha },\mathsf{\beta })=\text{-}\hat{\mathrm{n}}\mathsf{\alpha }·\hat{\mathrm{n}}\mathsf{\beta }=\text{-}\cos (\mathsf{\alpha }\text{-}\mathsf{\beta })$$

(79)

identically, for every pair of settings, with no averaging step and no piecewise-linear approximation. For the standard CHSH settings α = 0, α′ = π/2, β = π/4, β′ = −π/4, Equation (79) gives S = E(α,β) + E(α,β′) + E(α′,β)−E(α′,β′) = $-2\sqrt{2}$, saturating the Tsirelson bound exactly.

The CHSH bound of Section 9.6 is not contradicted, because Equation (78) is a joint-state expectation value rather than an average of factorized local outcomes A(α,λ)B(β,λ). The value 2√2 appears only after the source assumption changes from a shared classical variable to a non-factorizable two-particle state.

Relation to the standard quantum-mechanical picture. Ordinary quantum mechanics allows both singlet and triplet two-spin states. DCF presently reaches this sector in stages. The classical DCF theory supplies the carrier-space joint modes and, for the stated positive coupling, their singlet/triplet energetic ordering. The optional full-Sp(1) extension supplies a conditional microscopic mechanism for positive exchange. The actual covariant creation amplitude of a singlet pair is then written in a quantized two-particle extension, using the same DCF one-particle carrier but ordinary mode/operator machinery. The manuscript therefore does not claim that classical DCF alone has derived quantum entanglement or the full production probability. Its narrower classical contribution is the joint-mode structure and energetic preference; the quantized extension demonstrates how the same carrier can enter a covariant source process that selects the antisymmetric state.

Appendix B gives a local SU(2)-invariant interaction for which, when J > 0, the antisymmetric mode is the unique ground mode, while Appendix C analyzes a possible positive coupling and a covariant production vertex. Detailed second-quantized bookkeeping and its limitations are confined to Appendix C.

9.8. Comparison with Quantum Mechanics

Ordinary quantum mechanics predicts for a spin-1/2 singlet E_QM(alpha,beta) = −cos(alpha-beta), with CHSH reaching |S| = 2√2 [19,20]. DCF distinguishes three logically separate ingredients. The Bell-local shared-phase benchmark of Section 9.2, Section 9.3, Section 9.4, Section 9.5 and Section 9.6 belongs to a classical hidden-variable construction and necessarily gives |S| <= 2. The classical coupled-pair analy_sis of Section 9.7 and Appendix B supplies singlet/triplet joint modes and their energetic ordering but does not itself create a quantum two-particle state. Appendix C gives a conditional full-Sp(1) stabilization mechanism with J_eff > 0. Finally, Appendix C uses a quantized two-particle DCF extension to construct a covariant spin-zero production vertex selecting the antisymmetric state. Conditional on that preparation, Equation (79) gives E = −cos(alpha-beta) and |S| = 2√2. Thus the Tsirelson result is neither generated by sign[cos(lambda-alpha)] nor claimed as a consequence of the strictly classical single-field action.

The central open question is therefore sharper than in a conventional Bell consistency check: can the strictly classical DCF field equations themselves generate separating, nonfactorizable two-packet solutions with a dynamically selected joint internal mode, or is quantization an indispensable additional layer for production and statistics? The present manuscript does not decide that question. It establishes the classical joint-mode energetics, exhibits a conditional positive-exchange extension, and then shows in a clearly labeled quantized two-particle extension how a covariant spin-zero source selects the singlet. Proposition A4 is independent of Proposition A3; the full-Sp(1) exchange mechanism is therefore optional for the production selection itself and serves only as a possible energetic ordering/stabilization mechanism.

10. Coupling to Gravitation

Gravitation is incorporated at the level of the action, following the standard coupling of matter to the Einstein–Hilbert term [21,22], rather than appended to a nonrelativistic equation afterward. To be explicit about scope: this is not a new theory of gravity. The gravitational sector is unmodified general relativity; what is new is the matter content sourcing it, namely the DCF stress-energy tensor of (Equation (82)) in place of, or alongside, conventional matter fields.

$${\mathrm{L}}_{\mathrm{G}}={\displaystyle \frac{1}{2{\mathsf{\kappa }}_{\mathrm{G}}}}\left(\mathrm{R}\text{-}2\Lambda \right),{\mathsf{\kappa }}_{\mathrm{G}}={\displaystyle \frac{8\mathsf{\pi }\mathrm{G}}{{\mathrm{c}}^4}}$$

(80)

is already present in the unified action, so that gravitation is not appended phenomenologically after the matter sector is fixed. Variation of the total action with respect to g_μν gives

$$\mathrm{G}\mathsf{\mu }\mathsf{\nu }+\Lambda \mathrm{g}\mathsf{\mu }\mathsf{\nu }={\mathsf{\kappa }}_{\mathrm{G}}(\mathrm{T}\mathsf{\mu }{\mathsf{\nu }}_{\mathrm{DCF}}+\mathrm{T}\mathsf{\mu }\mathsf{\nu }\mathrm{gauge})$$

(81)

with the DCF matter stress-energy tensor obtained in the standard way from

$${\mathrm{T}}_{\mathsf{\mu }\mathsf{\nu }}^{\mathrm{DCF}}={\displaystyle \frac{\text{-}2}{\sqrt{\text{-}\mathrm{g}}}}{\displaystyle \frac{\mathsf{\delta }\left(\sqrt{\text{-}\mathrm{g}}{\mathrm{L}}_{\mathrm{matter}}\right)}{\mathsf{\delta }{\mathrm{g}}^{\mathsf{\mu }\mathsf{\nu }}}}$$

(82)

with Tμνgauge the standard Yang–Mills contribution. Because J_n is a fixed non-dynamical background in the present action rather than an independent field, no separate J_n-field stress tensor is introduced. Spacetime curvature and the DCF spinor and gauge fields are thereby solutions of one coupled system rather than independently specified structures.

11. Experimental Consequences

The linearized, direction-selected classical DCF theory was constructed to reproduce ordinary Dirac and Schrödinger–Pauli dynamics. Agreement in that sector is therefore not an experimental success unique to DCF. The two-particle program has a similarly explicit hierarchy: classical DCF predicts a joint-mode structure and conditional singlet/triplet energetic ordering; the full-Sp(1) extension offers a possible J_eff > 0; the present covariant source-amplitude calculation is quantized. A decisive future result would be either (a) a classical nonlinear DCF production solution that generates and separates the relevant joint mode without importing Fock-space machinery, or (b) a fully specified quantized DCF theory whose production rates and statistics follow from the DCF action rather than being appended phenomenologically. Detector-specific registration is secondary to this question.

Predictive status and extensions. The Dirac and Schrödinger–Pauli limits and the SU(2) singlet correlation are consistency requirements. The specifically classical two-particle result presently established is the collective singlet/triplet mode structure and its energetic ordering for the assumed positive interaction. The positive J_eff mechanism is conditional on the separate full-Sp(1) extension. The covariant singlet-production vertex and normalized two-particle amplitudes belong to a quantized extension and therefore do not yet constitute a prediction of the minimal classical DCF action. This boundary defines a concrete research target: derive the production process either directly from nonlinear classical DCF dynamics or from a principled quantization of DCF. Other potentially discriminatory consequences arise from nonlinear branch locking, thresholds, metastability, hysteresis, and the companion composite program. The candidate experimental discriminators and their present status are summarized in Table 1.

Table 1. Candidate experimental discriminators and present status.

Discriminator	Dirac/QM Baseline	DCF Departure	Needed to Establish a Prediction	Status
Ordered noncommuting phase operations	U(1) phases commute; order reversal gives no intrinsic commutator signal.	Distinct quaternion-axis interactions can be order-sensitive.	Identify physical couplings to J1 and J2 and derive the observable phase shift from the minimal action.	Consistency check only; ordinary SU(2) spin already has noncommuting generators.
Detector registration dynamics	Finite detector bandwidth, efficiency, noise, and latency.	The Section 9.3 pointer can represent finite registration time or inconclusive events without changing the ideal ±1 channels.	Specify a microscopic detector–DCF coupling and isolate any DCF-specific contribution from ordinary detector response.	Phenomenological; not used to derive ideal spin outcomes or singlet Bell correlations.
Nonlinear sector locking	Linear Dirac evolution has no classical branch-locking threshold or hysteresis.	The proposed branch potential may yield thresholds, metastability, hysteresis, or rare branch transitions.	Complete the stability analysis, connect the mechanism to observable, and calculate rates.	Proposed mechanism; no quantitative prediction yet.
Pair-state preparation/Bell correlations	Classical DCF gives singlet/triplet carrier modes with energetic ordering for H12 = J σ1·σ2.	Full-Sp(1) extension: possible Jeff > 0 stabilization. Quantized extension: a spin-zero source selects the antisymmetric channel.	Derive nonlinear classical pair production, or provide a principled quantization with normalized production rates.	Classical energetic ordering established; Jeff conditional; source vertex quantized; normalized production open.
Curvature-dependent internal dynamics	Minimal curved-spacetime Dirac theory provides the standard baseline.	Additional nonminimal curvature–internal couplings could modify interferometric phases.	Specify a covariant operator and derive its weak-field observable limit.	Extension only; not predicted by the minimal action.

Noncommuting real-matrix quaternionic rotations. For U = exp(J₁α) andV = exp(J₂β), with J₁,J₂ real matrices obeying Equation (1), UV − VU = 2J₃ sinα sinβ; hence UV ≠ VU whenever sinα sinβ ≠ 0. This is a directly real-matrix statement.

Using J₁² = J₂² = −I₄, exp(J₁α) = I₄cosα + J₁sinα and exp(J₂β) = I₄cosβ + J₂sinβ. Together with J₁J₂ = J₃ and J₂J₁ = −J₃, direct multiplication gives UV − VU = 2J₃sinαsinβ = 2J₃αβ + O(α³β,αβ³). All operators in this calculation are real matrices.

11.1. Validation Test: Ordered Quaternionic Interactions

The chirality construction supplies the algebraic core of this test. For U = exp(J₁α) and V = exp(J₂β), reversing the sequence changes the combined transformation because UV − VU = 2J₃αβ + O(α³β,αβ³). This noncommutativity is already implied by the Sp(1) ≅ SU(2) structure established in Section 2.1 together with the ordinary Pauli/Dirac limits of Section 5 and Section 6: U and V are mathematically identical to ordinary spin rotations Rx(α), Ry(β), and their failure to commute is standard SU(2) behavior, already familiar with NMR and ESR. An interferometer in which two independently controlled interactions implement rotations about distinct internal axes would compare the sequences UV and VU while holding path length, timing, and ordinary dynamical phase fixed. Reproducing the noncommutator therefore does not by itself distinguish DCF from ordinary quantum mechanics; a null result confirms that the DCF quaternionic construction faithfully reproduces known spin algebra, and is reported here as a consistency check on the formalism, not as a proposed discriminator. Section 11.2 identifies the dynamical sector, absent from ordinary Dirac theory, in which a genuine departure could appear.

Were this test nonetheless pursued for its bounding value, the minimally motivated DCF action introduced in Section 3.2 would need to be supplemented or specialized so that identifiable laboratory controls couple separately to J₁ and J₂. Until those operators are given, no numerical phase shift or energy scale ΛQ is derived. Dimensional estimates such as δφ∼(E/ΛQ)² are therefore only parameterizations of sensitivity, not predictions. This wording avoids assigning an anomalously low ΛQ merely to place the effect within current instrumental reach.

11.2. Detector Registration Dynamics as a Secondary, Device-Dependent Test

The bistable pointer of Section 9.3 should not be interpreted as a fundamental DCF law determining spin up or spin down. The ideal spin is observable and its ±1 eigenchannels are detector-independent. What may depend on detector microphysics is only the macroscopic registration process: latency, amplification, efficiency, noise, and the probability of an inconclusive event in a finite observation window. The phenomenological pointer equation can therefore motivate secondary tests of registration dynamics, but such effects are device-specific and are not part of the derivation of the singlet Bell correlation.

Section 9.3 solved the linearized detector equation introduced there for a constant envelope f_D(t) = f0 over a measurement window of duration T, giving ξ_D(t) ≈ K_D(t)a(α,λ) with

K_D(T) = (g_D f₀/κ_D)(eκD T/γ_D − 1)

A definite registration within the window requires the pointer to reach the saturation region of VD, |ξ_D(T)| > 1, i.e., |a(α,λ)| > ac(T)≡1/K_D(T). Because a(α,λ) = cos(λ−α) as defined in Section 9.3, this threshold fails for λ within a band of half-width Δλ(T) ≈ ac(T) around λ = α±π/2, where cos(λ−α) is itself small. Since K_D(T) grows exponentially in T, this excluded band shrinks.

Averaging over the uniform hidden-variable distribution ρ(λ) introduced in the shared-phase model of Section 9.1, the probability that a pair drawn at random fails to register a dichotomy outcome within the window T is therefore

CHSH bound restored: |S| ≤ 2.

Within the illustrative shared-phase threshold model, this calculation predicts a finite integration time for resolving very small biases. Any such click-time or inconclusive-event distribution would be detector-dependent through γ_D and κ_D and, therefore, cannot, by itself, be regarded as a universal DCF prediction. A meaningful test would require a microscopic detector–DCF model and comparison across different calibrated detector technologies while holding the measured spin observable fixed.

11.3. Nonlinear Branch-Locking Tests

The proposed potential V_branch introduces dynamics absent from the linear Dirac equation. Once a stable localized solution is established, the most characteristic signatures would be threshold behavior, hysteresis under a slowly cycled control parameter, metastable residence times, or rare transitions between opposite branch orientations. These would be stronger discriminators than small corrections to the Dirac spectrum because they test the nonlinear mechanism directly. The local potential analysis of Section 7.3 now establishes positive curvature of the collective branch coordinate at C = ±C₀ and gives sufficient Hessian conditions for local energetic stability in the reduced invariant sector. The complete dynamical fluctuation spectrum of the underlying eight real fields, localized nonlinear solutions, and observable transition rates remain open, and coupling those modes to a concrete spectroscopic or interferometric observable is still required.

11.4. Pair-State Preparation: Classical, Full-Sp(1), and Quantized Levels

Bell experiments expose the boundary between the three levels particularly clearly. The shared classical phase model is Bell-local and gives |S| ≤ 2. The classical coupled-pair DCF analysis instead establishes singlet/triplet joint modes and, for J > 0, a singlet ground mode separated by 4J, but it does not by itself supply a nonzero same-species antisymmetric production vertex or normalized pair statistics. The conditional full-Sp(1) extension can provide J_eff > 0 and energetic protection. The covariant spin-zero production vertex of Appendix C is formulated in a quantized two-particle extension and selects the antisymmetric channel independently of Proposition A3. Once that state is prepared, E(a,b) = −a·b and |S| = 2√2 follow from the joint-state algebra, not detector microphysics. The experimental/theoretical priority is, therefore, to determine whether DCF can derive the production and statistics from its classical dynamics or from a principled quantization of the DCF carrier.

11.5. Gravitational and Curvature-Dependent Tests

The minimal gravitational coupling contained in the DCF action introduced in Section 3.2 reproduces the standard curved-spacetime spinor baseline and is not by itself a distinctive prediction. Distinguishable gravitational effects require a specified additional operator, for example a curvature-dependent internal-locking term, followed by a controlled weak-field reduction. Neutron or atom interferometry would then be natural platforms because both phase and ordering can be measured precisely. Such terms are possible extensions of the framework, not consequences of the minimal action, and should be presented and tested separately rather than counted as present predictions.

11.6. Experimental Priority

The highest theoretical priority is to close the boundary identified above. In the classical program, this means constructing an explicit nonlinear source-and-two-packet solution and testing whether its joint internal mode can be selected and preserved without operator quantization. In the quantized program, it means specifying a principled quantization of DCF, deriving the covariant production vertex and normalizing singlet/triplet rates from that theory, and establishing its relation to spin-statistics. The conditional full-Sp(1) exchange mechanism should be treated separately as a possible stabilization/energetic mechanism rather than as a prerequisite for Proposition A4. Parallel priorities are the stability analysis of V_branch and explicit laboratory operators for noncommuting internal rotations.

12. Scope, Constraints, and Open Problems

A reader examining Equation (18) may reasonably ask why this action, rather than another invariant action, was chosen. The answer is deliberately limited. The terms were selected as low-order local, covariant, real structures compatible with the stated field content and with the required Dirac and gauge limits. Thus Equation (18) is a minimally motivated candidate, not a uniqueness theorem. This section makes the selection criteria explicit and separates what is fixed by symmetry and limiting requirements from what remains phenomenological or open.

The Constraints

Five requirements were imposed in building the fixed-J_n minimal motivated action (Equation (18)): (a) locality; (b) general covariance; (c) covariance under the local Spin(1,3) spin bundle together with the residual local U(1)J_n selected by fixed J_n, while retaining the exact global carrier-space Sp(1) structure before that selection; (d) reality of the classical field content; and (e) lowest derivative order. A fully local Sp(1) gauge theory is not among the constraints of the fixed-J_n action, because it would require J_n to transform and therefore defines a separate extension.

The detailed, term-by-term justification of each piece of Equation (18) against these constraints, together with a catalog of what remains a free choice rather than a derived necessity, is given in Appendix D.

13. Relation to Existing Frameworks

DCF differs in specific, checkable ways from the constructions it most resembles. Against the ordinary Dirac theory, the difference is one of construction rather than content: a quaternionic Dirac equation has been written down before as a postulated construction on a quaternionic module [10,11], whereas here it is obtained as the linearized vacuum limit of a proposed covariant action, whose form is motivated independently at the level of field content and symmetry, but constrained in part by the required Dirac limit, with mass and a conserved internal U(1) Noether charge—subsequently identified with electromagnetic charge in Section 8, though not its numerical value—following from the same Lagrangian rather than being added by hand. Against the Standard Model, DCF shares the gauge-field and Yang–Mills structure but replaces the postulated complex Dirac spinor with a real quaternionic doublet whose internal Sp(1) is geometric rather than an externally imposed gauge group; DCF does not reproduce the full Standard Model gauge group SU(3) × SU(2) × U(1) and makes no claim to. Against Einstein–Cartan theory, DCF also augments general relativity with additional geometric structure, but the addition is an internal quaternionic symmetry acting on matter rather than spacetime torsion; unlike Einstein–Cartan, DCF leaves the gravitational field equations themselves in standard Einstein form (Section 10) and modifies only the matter sector. Against Kaluza–Klein theory, both approaches derive internal gauge structure from extra geometric degrees of freedom, but Kaluza–Klein compactifies additional spacetime dimensions, while DCF’s extra structure is an internal quaternionic space attached at each spacetime point, with no additional dimensions of spacetime itself. Against prior quaternionic quantum mechanics [23,24], where ℍ replaces ℂ as the amplitude field of the wave function itself, DCF instead uses ℍ as a classical internal target space, multiplying an ordinary real Lorentz-covariant field; this distinction is established in Section 2.1, Section 2.2 and Section 2.3, and it is why DCF inherits none of the measurement-theory and experimental constraints associated with quaternionic amplitudes [25,26,27,28].

14. Discussion

The construction above should be read in three layers. The first is the classical DCF ontology: an eight-real-dimensional quaternionic carrier, real Lorentz representation, proposed general covariant action, Dirac and Schrödinger–Pauli limits, residual U(1) charge, and classical two-carrier singlet/triplet joint modes with an energetic ordering for H₁₂ = J σ₁·σ₂. The second is the conditional full-Sp(1) extension, which can generate J_eff > 0 but is not contained in the minimal fixed-J_n action. The third is the quantized two-particle extension used for the covariant source vertex and production amplitudes. The last layer is deliberately not presented as a consequence of the classical action: for one classical c-number same-species field, the relevant antisymmetric pointwise bilinear vanishes, so independently labeled mode operators are required in the present construction. This separation makes the present boundary of the theory explicit. Spin-statistics, charge quantization, and a classical derivation or a principled quantization yielding normalized pair-production probabilities remain open.

Bell-sector status. The classical DCF theory does not presently derive the observed Bell correlations from a Bell-local hidden variable, nor does it presently derive a nonzero same-species antisymmetric production amplitude from the single-field classical c-number action. Its classical result is narrower: a coupled pair has collective singlet/triplet modes and, for positive J, the singlet is the unique ground mode. A conditional full-Sp(1) extension can supply positive exchange. The covariant spin-zero vertex selecting the singlet is, instead, a result of the explicitly labeled quantized two-particle extension, and Proposition A4 does not depend on Proposition A3. Conditional on that preparation, the Tsirelson correlation follows from standard SU(2) joint-state algebra. Whether an equivalent preparation mechanism can be obtained from purely classical nonlinear DCF dynamics is an open and experimentally consequential question.

On the status of spin. A key distinction should be made here. The real Dirac generators Γ^a and the spin representation S(Λ) of Section 3.1 are now given by an explicit construction (the Clifford construction of Appendix A.2), not merely introduced as fixed structure on Ψ; what is not derived from a more primitive requirement is the choice to represent Ψ in this particular Clifford module at all, as opposed to some other real representation of Spin(1,3). The present DCF construction therefore provides an explicit real representation of spin-½, rather than deriving the Lorentz spin representation from more primitive DCF dynamics. Once that representation is assumed, the Dirac equation, the Schrödinger–Pauli limit, and the particle/antiparticle structure follow from the subsequent construction. This is a legitimate and nontrivial result, but a different claim from spin being emergent. The quaternionic Sp(1) ≅ SU(2) internal symmetry of Section 2.1 and Section 4 is, by contrast, genuinely obtained from the quaternion product rather than assumed; the caveat above concerns the Lorentz spin structure specifically.

DCF is therefore not yet a completed fermion theory: the action remains a classical c-number field theory unless a quantization procedure or another mechanism supplies Fermi–Dirac statistics. The quaternionic carrier accounts for the SU(2) geometry and 4π spinorial property, while the full relativistic fermion requires the eight-real-component doublet. The nonlinear branch-selection potential of Section 7.2 is a proposed dynamical mechanism for which local potential stability is established in Section 7.3; its full dynamical stability and its correspondence with the positive- and negative-frequency branches remain to be established.

The residual U(1) Noether current is derived from the action and is identified here with electromagnetic charge, but the numerical value and quantization of the elementary charge are not derived. Likewise, promoting the full carrier-space Sp(1) action to a local gauge symmetry would require a transforming internal matrix direction J_n and an enlarged gauge sector; that extension is kept separate from the minimal fixed-J_n theory.

Two extensions suggested by this separation are natural topics for future work. A Deser-type current–field bootstrap could be investigated as a possible dynamical route from a global Sp(1) ≅ SU(2) carrier symmetry to a fully local non-Abelian Yang–Mills sector, while keeping that construction distinct from the present fixed-J_n U(1)J_n theory. Likewise, a first-order tetrad/Palatini–Einstein–Cartan formulation could allow the DCF spin current to source torsion and could distinguish the Lorentz spin connection from an independent internal quaternionic connection. Neither extension is derived or claimed as a result of the present manuscript.

Table 2 collects, in one place, the logical status of every principal result claimed above.

Table 2. Summary of established results and open issues.

Objective	Established/Included Here	Open Issue
Lorentz covariance	Built in through the spin bundle and Γμ.	—
Quaternionic Sp(1) ≅ SU(2)	Exact global right action on the carrier before fixing Jn; the fixed-Jn kinetic sector retains residual local U(1)Jn.	A consistent local extension beyond U(1)Jn requires transforming Jn and defines the separate full-Sp(1) extension.
Dirac fermion equation	Obtained by linearizing the vacuum equation for Ψ (Equations (26)–(29)).	Full nonlinear solution theory.
Particle/antiparticle branches	Positive- and negative-frequency solutions (Equation (44)).	Dynamical branch selection in Section 7.2 remains proposed, not proved.
Gravitation	Obtained by metric variation (Equation (81)).	Explicit Tμν for the complete field content.
Schrödinger–Pauli limit	Nonrelativistic positive-frequency limit (Equations (37) and (38)).	—
Origin of ℏ	Candidate exchange closure: ℏ = γcτs/κs, with γ = η/ζ, extending the companion two-string relation [26] (Equations (39)–(43)).	Map κs, τs, η, ζ to the relativistic action. The closure remains conditional on ωs2 = κs and the chosen evaluation interval.
Electric charge	Residual U(1)Jn Noether current and charge (Equations (55) and (56)).	Numerical value of e and dynamical charge quantization.
Fermi–Dirac statistics	Not addressed.	Requires quantization or a new classical spin–statistic mechanism.
Bell/entanglement correlation	Classical joint-mode energetics; conditional full-Sp(1) Jeff > 0; singlet source vertex only in the quantized two-particle extension.	Strictly classical same-species pair production and normalized probabilities remain open; alternatively, derive principled quantization and spin–statistics.
Vacuum selection and parity	Kinetic, scalar-mass, and revised V(Σ,Π) sectors are parity invariant; Σ = ±v2 are degenerate vacua, and choosing +v2 is spontaneous selection.	Determine vacuum-selection dynamics and observable consequences; no identification with weak-interaction parity violation is claimed.
Primitive XF/X$\overline{\mathrm{F}}$ ↔ chiral coordinates	XF and $\overline{\mathrm{XF}}$ are independent primitive quaternionic coordinates. Chirality is derived from χ = JΩ, with QL and QR its normalized ±1 eigencoordinates (Section 2.3; Equations (8)–(11)).	—
Discrete charge conjugation	Proposed primitive swap ℭ:(XF,$\overline{\mathrm{XF}}$)↦($\overline{\mathrm{XF}}$,XF) (Equation (45); Section 7.1); its relation to physical frequency branches remains open.	Check invariance of the full action and reconcile ℭ with the standard C-conjugate construction of Appendix C.
Gauge status of n	For fixed Jn, the kinetic sector is Lorentz covariant and locally U(1)Jn invariant; full carrier-space Sp(1) remains global.	A dynamical position-dependent n with gauge-covariant transformation and kinetic term belongs to a future full-Sp(1) extension.

Important limitation. A classical quaternionic doublet formulated with c-number field variables can transform correctly as a Lorentz spinor without possessing Fermi–Dirac statistics or anticommuting field components; this paper’s classical Lagrangian does not derive the spin–statistics connection. The c-number qualification concerns the field-theoretic status of the variables, not the noncommutative multiplication law of the internal quaternion algebra.

The established result of the present paper is structural and limiting: the real/quaternionic DCF carrier admits the stated Lorentz-covariant construction and reproduces conventional Dirac and Schrödinger–Pauli dynamics under the specified assumptions and limits. The DCF framework should, therefore, be viewed as a candidate coupled-real foundation and organizing principle, rather than as a completed alternative formulation of quantum field theory. The linearized correspondence does not by itself imply experimentally observable deviations from ordinary Dirac theory; such distinctions, if they exist, must arise from nonlinear internal dynamics beyond the demonstrated limit.

A Lorentz-covariant Double Coupled Field framework based on an eight-real-dimensional quaternionic carrier provides a coherent mathematical structure connecting coupled real degrees of freedom, internal Sp(1) ≅ SU(2) organization, chirality, and the Dirac limit. The manuscript explicitly separates established structural results from open physical questions. In particular, a complete Poincaré-covariant quantized theory, the spin–statistics connection, charge quantization, and experimentally verified departures from standard quantum dynamics are not claimed here and remain subjects for future work.

A Lorentz-covariant Double Coupled Field built on an eight-real-dimensional quaternionic carrier provides a coherent classical framework with Dirac, Schrödinger–Pauli, electromagnetic Noether-charge, and Einstein limits. The present manuscript now draws an explicit boundary around its two-particle claims. At the classical DCF level, two coupled carriers possess collective singlet and triplet modes; for H₁₂ = J σ₁·σ₂ with J > 0 the singlet is the unique ground mode, E_singlet = −3J versus E_triplet = +J. Hence, a pair that is produced or relaxes into the lowest available internal-energy state must occupy the singlet sector; producing or maintaining a triplet requires the additional energy 4J, so energy conservation plus lowest-energy preparation selects the singlet, while energy conservation alone does not exclude excited triplets. A separate full-Sp(1) extension provides a conditional possible origin of J_eff > 0 and can energetically stabilize the mode ordering. The stronger production statement belongs to a third level: Appendix C formulates a quantized two-particle DCF extension in which a covariant spin-zero source vertex selects the antisymmetric channel. This quantized vertex is not claimed to follow from the strictly classical single-field action, because the corresponding same-field antisymmetric c-number bilinear vanishes pointwise. Once the singlet is prepared in the quantized extension, E(a,b) = −a·b and |S| = 2√2 follow from the joint-state SU(2) algebra independently of detector microphysics. The unresolved task is to determine whether pair creation and statistics can be derived from nonlinear classical DCF itself or require a principled quantization of the theory. The independent scalar s used in the regular mass completion is phenomenological and is not identified with the Standard Model Higgs doublet; embedding the DCF carrier and scalar sector in the full electroweak representation content remains open.

Possible extensions of the DCF construction to closed C₂ and C₃ networks are reserved for future work and are intended to test whether the primitive DCF ontology has consequences beyond the linear Dirac correspondence. Any attempt to associate such composite structures with physical particle multiplets must be formulated consistently with the constraints on combining spacetime and internal symmetries, including the Coleman–Mandula theorem [12]. No nontrivial unification of the Poincaré algebra with internal symmetries is constructed or claimed in the present work. The electromagnetic normalization and quantitative confrontation with hadronic data remain open.

The present results should, therefore, be read as a structured, test-oriented field proposal with a deliberately explicit hierarchy of claims. The classical DCF ontology establishes the relativistic real-field carrier, its controlled Dirac and Schrödinger–Pauli limits, and the classical two-carrier joint-mode/energetic structure. In addition, the companion coupled-field construction supplies a candidate dynamical origin of the quantum action scale, ℏ = γcτ_s/κ_s, rather than postulating ℏ at the primitive ontological level; the microscopic parameter map to the relativistic DCF action remains to be completed. The conditional full-Sp(1) extension supplies a possible positive exchange mechanism. The covariant source vertex and two-particle production amplitudes are part of a quantized extension and are not counted among the results of the minimal classical action. Neither the known real/quaternionic representation of the Dirac carrier nor the standard singlet algebra is presented as a novel prediction. The immediate theoretical challenge is to close the gap between the classical ontology and the quantized production sector, either by constructing a purely classical nonlinear pair-production solution or by deriving a consistent quantization, including spin-statistics and normalized production probabilities, from DCF itself.

Appendix A.1. Elementary Quaternion I_dentities

For completeness, the elementary identities used in Section 2 are collected here so that the main text can remain focused on the physical construction.

Multiplicativity of the norm. For all P,Q∈ ℍ, ‖PQ‖² = ‖P‖²‖Q‖².

Quaternion conjugation reverses order: $\overline{\mathrm{PQ}}$ = $\overline{\mathrm{Q}} \overline{\mathrm{P}}$. Therefore ‖PQ‖² = $\overline{\mathrm{PQ}}$PQ) = $\overline{\mathrm{Q}}$($\overline{\mathrm{P}}$P)Q = ‖P‖²$\overline{\mathrm{Q}}$Q = ‖P‖²‖Q‖², because ‖P‖² is real and commutes with Q. □

Using Re(AB) = Re(BA) for quaternions and ē_a = −J_a, one has ⟨PJ_a,Q⟩ = Re($-{\overline{\mathrm{PJ}}}_{\mathrm{a}}$Q) = Re((−J_a)$\overline{\mathrm{P}}$Q) = −Re($\overline{\mathrm{P}}$Qe_a) = −⟨P,Qe_a⟩. □

In the real representation, closure follows directly from Equation (1): products of q₀I₄ + q_aJ_a remain in the same four-dimensional real algebra. Unit-norm elements act orthogonally on R⁴, and their three-generator Lie algebra obeys [J_a,Jb] = 2ε_ab^cJ_c, establishing the standard Sp(1) ≅ SU(2) group structure without a complex 2 × 2 representation.

Appendix A.2. Real Lorentz Representation

The real Lorentz representation used in Section 3.1 is Cl(1,3) ≅ Mat₂(ℍ), realized by Γ_X^a on X_DCF = (X_F,X_{$\overline{\mathrm{F}}$})^T. The associated real spin representation commutes with the primitive global right-Sp(1) action. The proof follows.

Proposition A1 (Real Lorentz–quaternionic carrier and chirality).

The eight-real-dimensional DCF carrier admits a real Cl(1,3) representation Cl(1,3) ≅ Mat2(ℍ) whose Lorentz-spin action commutes with the primitive global right-Sp(1) action. With the selected real complex structure Jn, the chirality operator χX = JXΩX is an involution with four-dimensional ±1 eigenspaces, and the orthogonal transformation U of Equation (11) carries these eigenspaces to the chiral coordinates (QL,QR), while transporting the Clifford generators to ΓCa = UΓXaUT.

Proof. 

(i) Algebra identification. dimℝCl(1,3) = 16. By the standard mod-8 classification, p − q = 1−3 ≡ 6 (mod 8), hence Cl(1,3) ≅ Mat₂(ℍ). (ii) Explicit primitive-basis generators. Write Mat₂(ℍ) ≅ ℍ⊗ℝMat₂(ℝ) and let τ₁ and τ₃ be the two real 2 × 2 matrices defined in Equation (14). Setting Γ_X⁰ = I₄⊗τ₁ and Γ_X^r = J_r⊗τ₃ reproduces (Equation (14)). Since J_r² = −I₄, J_rJ_s + J_sJ_r = 0 for r≠s, τ₁² = τ₃² = I₂, and τ₁τ₃ + τ₃τ₁ = 0, these matrices obey the Cl(1,3) relations. Their products span the full sixteen-real-dimensional algebra. (iii) Commutation with primitive right Sp(1). Every Γ_X^a entry acts by left quaternionic multiplication. For any p,q,V∈ℍ, associativity gives p(qV) = (pq)V; hence Γ_X^a(X_DCF V) = (Γ_X^aX_DCF)V. The same holds for Γ_X^ab and S_X(Λ), proving commutation with ρ_X(V). (iv) Transport to the chiral basis. Because U is real orthogonal, Γ_C^a = UΓ_X^aU^T obeys the same Clifford relations and S_C = US_XU^T. The conjugated internal action ρ_C(V) = Uρ_X(V)U^T then commutes with S_C. For V∈U(1)J_n, ρ_C(V) reduces to ordinary componentwise right multiplication because R_V commutes with J_n and therefore with U. □

Remark (Explicit chirality calculation, referenced in Proposition A1).

In the primitive basis, Ω_X ≡ Γ_X⁰Γ_X¹Γ_X²Γ_X³ = (J₁J₂J₃) ⊗(τ₁τ₃) = −I₄ ⊗(τ₁τ₃), so Ω_X² = −I₈. J_X ≡ diag(R_n,R_n) commutes with Ω_X and satisfies J_X² = −I₈; therefore χ_X ≡ J_XΩ_X obeys χ_X² = I₈. Explicitly χ_X = $\left[\begin{array}{cc}0& {\mathrm{J}}_{\mathrm{n}}\\ \text{-}{\mathrm{J}}_{\mathrm{n}}& 0\end{array}\right]$. The matrix U of (Equation (11)) is orthogonal and gives Uχ_XU^T = diag(I₄,−I₄). Applying the same U to the Clifford generators gives: Γ_C⁰ = $\left[\begin{array}{cc}0& {\mathrm{J}}_{\mathrm{n}}\\ \text{-}{\mathrm{J}}_{\mathrm{n}}& 0\end{array}\right]$, Γ_C^r = $\left[\begin{array}{cc}0& {\mathrm{J}}_{\mathrm{r}}\\ {\mathrm{J}}_{\mathrm{r}}& 0\end{array}\right]$, which verifies directly that the operator represented off-diagonally in the primitive basis is diagonal in the chiral basis. The previous ambiguity arose from writing the primitive matrices Γ_X^a as though they already acted on (Q_L,Q_R); the X/C subscripts now remove that ambiguity.

Appendix D.1. Complex Structure, Chirality Coupling, and Branch-Label Convention

Remark (Complex structure J and chirality coupling K).

The chiral coordinates of Proposition A1 also support further separation.

$$\mathrm{K}\equiv \mathsf{\kappa }{{\mathsf{\Gamma }}_{\mathrm{C}}}^0{\mathrm{J}}_{\mathrm{C}}$$

(A17)

built from the mass-like coupling κ, the time-like Clifford generator Γ_C⁰ in the chiral basis, and J_C ≡ UJ_XU^T. The three objects have distinct roles: the Clifford generators supply Lorentz covariance, J_C supplies the selected internal complex structure, and the mass sector couples the two chirality eigenspaces. Keeping the basis labels X and C explicit prevents these roles from being conflated.

This separation also sharpens where the present construction stops. Section 7.2’s branch potential Equations (47) and (48) introduces two candidate nonlinear sectors C > 0 and C < 0, proposed to correspond respectively to the particle and antiparticle frequency branches of Section 7. Consistent with that proposal, the present remark permits an effective coupling constant on each branch, denoted κ_F for the fermion branch and κ_{$\overline{\mathrm{F}}$} for the antifermion branch, with

$${\mathsf{\kappa }}_{\overline{\mathrm{F}}}=\text{-}{\mathsf{\kappa }}_{\mathrm{F}}$$

(A18)

The sign relation κ_{$\overline{\mathrm{F}}$} = −κ_F is used as a branch-label convention; deriving it from the nonrelativistic exchange coupling remains open.

Appendix D.2. Term-by-Term Accounting

Gravity, (1/2κ_G)(R − 2Λ): the unique two-derivative diffeomorphism-invariant scalar built from g_μν alone, up to the cosmological constant—the same statement underlying the uniqueness of the Einstein–Hilbert action itself.

DCF kinetic term ½[$\overline{\mathrm{\Psi }}$Γ^u(DμΨ)J_n − (Dμ$\overline{\mathrm{\Psi }}$)Γ^uΨJ_n]: the direct DCF analog of the minimal Dirac kinetic bilinear, invariant under Spin(1,3) and, for fixed J_n, under the residual U(1)J_n ⊂ Sp(1). A classification of all possible first-derivative vector bilinears is not established here. In particular, this fixed-J_n kinetic term should not be described as locally Sp(1)-invariant.

Yukawa and scalar mass sector: the quadratic DCF invariant space is basis-independent, but its coordinate expressions change under U. In the primitive Mat₂(ℍ) representation, the two invariant symmetric bilinear generators are represented by Γ_X⁰ and − Γ_X¹Γ_X²Γ_X³. In the chiral basis, they become BΠ,C = UΓ_X⁰U^T = Γ_C⁰ and BΣ,C = U(−Γ_X¹Γ_X²Γ_X³)U^T = −Γ_C¹Γ_C²Γ_C³ = $\left[\begin{array}{cc}0& {\mathrm{I}}_4\\ {\mathrm{I}}_4& 0\end{array}\right]$. Consequently the two invariant DCF scalars are Π = 2Sc(Q_L†Q_R n) = $\overline{\mathrm{\Psi }\mathrm{\Psi }}$ and Σ = 2Sc(Q_L†Q_R), as stated in Equation (19). The regular mass completion couples the Lorentz scalar Π to an independent real scalar s through −y_sΠ and gives s its own standard kinetic term and quartic potential Vs(s). This avoids imposing any square-root function on Σ; the quadratic-invariant classification itself is unchanged by adding the Lorentz scalar s.

Concretely, start in the primitive basis with the Euclidean pairing ⟨X,X′⟩ = Re(X_F†X_F′ + X_{$\overline{\mathrm{F}}$}†X_{$\overline{\mathrm{F}}$}′). There, ⟨X,Γ_X⁰X⟩ = 2Sc(X_F†X_{$\overline{\mathrm{F}}$}) and ⟨X,−Γ_X¹Γ_X²Γ_X³X⟩ = |X_F|²−|X_{$\overline{\mathrm{F}}$}|². Transporting both the field and the bilinear generators by U preserves their numerical values. In the chiral coordinates, this yields

⟨Ψ_C,Γ_C⁰Ψ_C⟩ = 2Sc(Q_L†Q_R n) = Π,

and

⟨Ψ_C,−Γ_C¹Γ_C²Γ_C³Ψ_C⟩ = 2Sc(Q_L†Q_R) = Σ.

Thus the invariant classification is consistent with the derived chirality basis. The separate Euclidean quantity |Q_L|²−|Q_R|² = ⟨Ψ_C,χ_CΨ_C⟩ is the chirality grading itself; it is not one of the two Lorentz-scalar quadratic bilinears used in the action.

Potential V(Σ,Π): the lowest-order (quartic in Ψ) Sp(1)-invariant polynomial in the two independent scalar bilinears Σ and Π identified in Section 3.2. No cubic Sp(1)-invariant exists for this field content. Whether every further quartic invariant reduces, under the quaternion algebra relations, to a combination of Σ- and Π-type terms already present is not established here: Appendix D.2 shows that the quadratic classification does not by itself guarantee this reduction, and identifies the completed quartic classification as an open problem.

The fixed real matrix J_n is part of the background structure of the minimal theory and is imposed, by definition, to satisfy J_nᵀJ_n = I₄ and J_n² = −I₄, while n₁² + n₂² + n₃² = 1. Because J_n is not varied in the fixed-J_n action, no separate constraint potential or kinetic term for J_n is included. If J_n is promoted to a genuinely dynamical matrix field J_n(x) in a future full-Sp(1) extension, its gauge-covariant transformation law, kinetic term, and potential would have to be specified consistently (Section 4, Appendix D.2).

Appendix D.3. What Remains a Choice, Not a Derivation

Quadratic-invariant classification (proved). Consider real quadratic bilinears of the form ⟨X,BX⟩, with B ∈ Cl(1,3) in the primitive quaternionic representation Γ_X^a. Requiring Spin(1,3) invariance gives a two-dimensional simultaneous solution space represented by Γ_X⁰ and Γ_X¹Γ_X²Γ_X³. Orthogonal transport by U preserves the dimension and invariance of this space, giving in chiral coordinates the generators BΠ,C = Γ_C⁰ and BΣ,C = −Γ_C¹Γ_C²Γ_C³. Hence every Lorentz-invariant real quadratic scalar bilinear in the Clifford-generated ansatz is a linear combination of Π and Σ of Equation (19). The full primitive right-Sp(1) action commutes with the Clifford representation; in the chiral basis, the corresponding statement uses the conjugated action ρ_C(V) = Uρ_X(V)U^T. With fixed J_n, the kinetic sector retains only the residual local U(1)J_n symmetry discussed in Section 4.

What this theorem does and does not establish. It proves that Σ and Π exhaust the invariant quadratic scalar bilinears within the Clifford-generated ansatz. It does not prove that every quartic invariant is generated by products of these two bilinears; primitive quartic invariants may still exist, and a separate invariant-theoretic classification remains open. The basis bookkeeping does not change that status. It does, however, remove the earlier identification of Σ with |Q_L|²−|Q_R|²: that norm difference is the chirality grading ⟨Ψ_C,χ_CΨ_C⟩, whereas the invariant Σ entering V and M is 2Sc(Q_L†Q_R).

The remaining freedom in the proposed action should be kept distinct from the parts fixed by symmetry. Within the quadratic Clifford-generated DCF ansatz, the invariant scalar content is fixed as described above. The specific potentials V(Σ,Π) and Vs(s), together with their numerical coefficients, Yukawa coupling y, and vacuum parameters, are motivated choices of the minimally motivated completion introduced in Section 3.2 rather than consequences of a unique selection theorem. The constraints in Section 12 constrain which operator structures are admissible, but it does not fix all coefficients or exclude every additional quartic invariant. Reducing this residual freedom remains an important next theoretical task, potentially through additional symmetry, a dynamical fixed-point condition, compositeness relations, or field-redefinition redundancies.