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# The Geometric Origin and Finite Classification of Fundamental Particles in Trembling Spacetime

Nico F Declercq

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# The Geometric Origin and Finite Classification of Fundamental Particles in Trembling Spacetime

Nico F. Declercq

George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, USA

IRL 2958 Georgia Tech – CNRS, Georgia Tech Europe, Metz, France

declercq@gatech.edu

## Abstract

This paper demonstrates that all known fundamental particles arise as geometric eigenmodes of trembling spacetime: localized, causally stable oscillations of the spacetime metric itself. Within the framework of Trembling Spacetime Relativity Theory (TSRT), particle identity, mass, spin, and charge are not introduced as independent assumptions but emerge deterministically as intrinsic properties of symmetry-bound deviations in the underlying geometry. Building on earlier foundational work, TSRT identifies exactly four causally admissible symmetry classes—U(1), SU(2), SU(3), and gravitational curvature—each corresponding uniquely to one of the known fundamental interactions. This classification is both highly restrictive and, within the constraints analyzed, empirically complete: it recovers the Standard Model gauge groups and predicts, to the extent that no additional causally admissible trembling symmetry classes have been identified, the absence of further fundamental forces or generations. In contrast to quantum field theory, which permits arbitrary field extensions, TSRT excludes superpartners, extra families, or exotic gauge sectors unless they can be realized as causally coherent metric deviations—an outcome shown to be incompatible with the theory’s variational structure under the current formulation. Charge quantization, spin-statistics correspondence, confinement of color charge, and the discreteness of particle families all emerge from the causal coherence and topological closure of trembling geodesics, without recourse to field quantization or probabilistic interpretation. Beyond classification, TSRT provides a geometric foundation for principles traditionally regarded as axiomatic. The derivation of rest energy and inertial mass follows directly from proper-time action accumulation, yielding the Einstein relation between energy and mass as a deterministic outcome of geodesic resistance to curvature. Interactions are formulated as curvature-mediated coupling among trembling modes, replacing virtual particle exchanges with deterministic geometric correlations. The Higgs boson is reinterpreted not as a quantized scalar field excitation but as a metastable curvature-bound resonance decaying through geometric fragmentation. Finally, TSRT yields empirical predictions distinguishing it from the Standard Model, including geometric constraints on fermion generations, minimal action scales linked to Planck’s constant, and testable signatures of curvature-induced trembling. By deriving the complete set of known particles and interactions from first principles—while identifying precise conditions for falsification—this work advances the geometric unification program and shows that matter originates from a single underlying principle: causal coherence in trembling spacetime.
# 1 Introduction

As a child, the author often spent long hours playing with a spinning top and, later, with a simple gyroscope. Even at that early age, it became intuitively evident that rotation endowed an object with a peculiar form of stability: a spinning top resisted falling, and a gyroscope opposed any attempt to reorient its axis of rotation. It seemed as if the act of spinning granted an otherwise inert mass a more definite identity, a sense of persistence against the indifferent pull of gravity. This early intuition—difficult to express in equations at the time—suggested that motion itself might be the key to understanding why objects, or particles, resist change. Over the decades, this idea matured into a deeper hypothesis: that the stability and individuality of elementary particles arise not from static attributes imposed by abstract symmetries but from a subtler dynamical coherence—akin to the gyroscope’s resistance—emerging directly from the geometry of spacetime itself.<fnref n="1" />

The quest for a unified framework explaining the properties and interactions of elementary particles has been a central theme in modern physics for over a century. Early foundational work by Lorentz on electrodynamics and Lorentz transformations [1, 2], alongside Poincaré's insights [3, 4] into spacetime symmetries and the relativity principle, laid the groundwork for Einstein's revolutionary reformulation of gravity through the geometric framework of general relativity [5]. The subsequent development of quantum mechanics by Dirac [6] and others introduced non-classical degrees of freedom, while Yang and Mills' gauge theory [7] provided the mathematical structure underlying the Standard Model (SM) of particle physics [8–10]. Despite these advances, each theoretical paradigm has relied on new postulates—such as gauge invariance, spontaneous symmetry breaking [11], and ad hoc particle content—rather than deriving them from first principles.

From the inception of special relativity [12] and quantum theory [13, 14] to the SM's con-
solidation, successive breakthroughs have refined our understanding of matter's fundamental
structure. Yet profound questions persist: Why does nature exhibit exactly four fundamental
forces (gravitational, electromagnetic, weak, and strong)? What determines the seemingly finite
particle spectrum, with its discrete masses, spins, and charges? Most critically, can these prop-
erties emerge from a deeper geometric or topological principle, as envisioned by Kaluza-Klein
theory [15, 16] or more recent approaches like string theory [17] or loop quantum gravity [18]?
The absence of a first-principles derivation of SM parameters underscores the need for a more
fundamental theory.

Trembling Spacetime Relativity Theory (TSRT)<fnref n="2" />, recently formalized as a geometric ex-
tension of relativity and causal structure [19–23], offers a rigorous alternative perspective to
both quantum field theory and general relativity. Rather than postulating quantized fields and

<footnote n="1">¹Although this childhood intuition about spinning objects inspired the first conceptual leap, the mature formulation of Trembling Spacetime Relativity Theory (TSRT) reveals that spin, as traditionally defined, is not itself a particle nor a property analogous to electric charge. Rather, spin arises as a geometric consequence of coherent trembling modes constrained by causal and variational conditions. The notion that particles are specific, stable patterns of spacetime oscillation—rather than pointlike entities endowed with arbitrary labels—preserves the spirit of the original insight while transcending its mechanical analogy.</footnote>

<footnote n="2">²The term “trembling” refers to small, symmetry-bound deviations of the spacetime metric that preserve causal structure. These deviations are not random perturbations but deterministic, geometry-constrained oscillations whose coherent behavior defines particle identity. See TSRT Foundations [19] for the foundational formulation.</footnote>
operator algebras as fundamental constructs, TSRT introduces a set of causally constrained
metric fluctuations—termed “trembling”—that manifest as small, bounded deviations ξ<sub>μν</sub>(x)
superimposed on a Lorentzian background. These deviations obey strict proper-time causality
and variational constraints, from which gravitational, quantum, and gauge interaction phenom-
ena emerge as geometric effects. Importantly, TSRT does not reject the established successes
of quantum field theory and general relativity; instead, it seeks to provide a complementary
geometric foundation that can help explain why these frameworks are so effective in describing
the empirical structure of nature.

Earlier results demonstrated that the four fundamental interactions emerge from four known irreducible classes of trembling deviation, each derived from the proper-time variational principle under causal admissibility constraints [19, 21–23].<fnref n="3" /> These classes correspond one-to-one with the observed fundamental forces: U(1) symmetry for electromagnetism, SU(2) for the weak interaction, SU(3) for the strong interaction, and coarse-grained curvature trembling for gravitation.<fnref n="4" />

This paper extends the TSRT framework [19] by demonstrating that all known fundamental
particles—and all of their observable properties—emerge as localized, symmetry-bound geodesic
modes within trembling spacetime. Interactions, charges, masses, and spin states are shown to
be geometric invariants of coherent curvature structures, not imposed assumptions. The result
is a fully causal, predictive, and finite particle classification derived from a single variational
principle.

This approach also yields a rigorous derivation of the mass–energy relation from first prin-
ciples. While Einstein’s original argument for mass–energy equivalence [24] relied on heuristic
reasoning involving radiative recoil and kinetic energy conservation, TSRT derives the relation
from the reparametrization-invariant action of geodesics in trembling spacetime [19].<fnref n="5" /> In TSRT,
this relation emerges as a necessary consequence of causal geodesic dynamics and the coherent
accumulation of geometric action in trembling spacetime.

The implications of this result are far-reaching. The Standard Model’s extensibility—its ca-
pacity to accommodate arbitrary additional particles or interactions—is supplanted in TSRT by
a geometric constraint: only those particles that emerge as stable, symmetry-bound trembling
modes are permitted. This yields a finite and predictive spectrum rooted in causal geometry.
The result imposes a strict upper bound on the particle spectrum, naturally excludes supersym-
metric partners and hypothetical exotic states, and provides a falsifiable geometric alternative
to quantum field theory. This alternative reproduces the empirical successes of QFT while
eliminating its reliance on auxiliary fields, spontaneous symmetry breaking, and unconstrained
symmetry extensions [22, 23].

In the geometric framework of Trembling Spacetime Relativity Theory (TSRT), a funda-
mental particle is not a pointlike object introduced from outside, but a localized, self-sustaining

<footnote n="3"><sup>3</sup>While no additional classes have been identified, a formal proof that no further solutions exist remains an open question.</footnote>

<footnote n="4">⁴TSRT Foundations [19], Supplementary Material: Classification of Metric Fluctuations and Gauge Structure, Supplementary Material: Emergent Electromagnetism from Abelian Trembling, Supplementary Material: Emergent Weak Force from Chiral Trembling, Supplementary Material: Emergent Strong Nuclear Force from Non-Abelian Color Trembling, Supplementary Material: Emergent Gravity from Coarse-Grained Curvature.</footnote>

<footnote n="5">⁵TSRT Foundations [19], Supplementary Material: Energy Conservation and Effective Mass Corrections.</footnote>
excitation of the spacetime fabric itself. One may begin with smooth Minkowski spacetime, undisturbed and featureless. Then, consider that causally admissible fluctuations—small deviations in the spacetime metric—begin to ripple through this background. These fluctuations, or tremblings, are not random perturbations; they obey strict geometric and causal constraints. Under suitable symmetry and stability conditions, certain trembling configurations form coherent, localized patterns that persist in time. These are the fundamental particles. Each is a conservative, discrete mode of the trembling geometry—akin to a vortex in a fluid or a standing wave on a curved membrane. The discreteness arises not from quantum postulates, but from the requirement that only specific, causally coherent oscillation patterns remain stable over extended intervals. The particle's properties—mass, spin, charge—are not imposed parameters but geometric signatures of its internal oscillation pattern. In this view, a particle is not a separate entity riding on spacetime, but a self-bound geometric mode: a localized whirl in the causal fabric of trembling spacetime.

In TSRT, the observable properties of particles—mass, spin, charge, and internal 'quantum numbers'—are not assigned arbitrarily but emerge directly from the structure of each trembling mode. Mass reflects the mode's resistance to geodesic deflection: it quantifies how strongly the internal geometry of the mode opposes curvature-induced deviations in its proper-time trajectory. Spin arises as a geometric phase: a holonomy<fnref n="6" /> accumulated when the mode completes a closed cycle in internal trembling space. Charge and other quantum numbers originate from symmetries of the trembling pattern itself—U(1), SU(2), SU(3)—and are quantized due to the causal coherence of periodic action cycles. In this view, particle properties are not added features but geometric invariants. Each property encodes how a stable trembling mode fits into the larger causal and symmetric structure of spacetime. The classification of particles thus becomes a classification of conservative, symmetry-bound solutions to the trembling geometry—not of pointlike inputs, but of localized, quantized patterns sustained by spacetime itself.

In Trembling Spacetime Relativity, energy is not a mysterious external input but a geometric rate: it measures how quickly action accumulates along the proper-time trajectory of a localized mode. For a stable trembling configuration, this action is conserved and locally determined by the mode's structure and motion. Rest energy arises from internal trembling even when the mode is stationary, while kinetic energy emerges from deviations in the worldline's geodesic direction. Curvature and gravitational interactions further modulate the energy by altering the causal structure of the trembling field. Because TSRT defines energy through proper-time derivatives of the action, all known forms of energy—rest, kinetic, potential, electromagnetic, and even mass-energy—are unified under a single geometric principle. This approach restores intuitive clarity: energy is not an abstract scalar to be assigned or conserved post hoc, but the intrinsic causal footprint of a trembling geodesic in curved spacetime.

<footnote n="6">⁶In differential geometry and gauge theory, the holonomy of a connection (or gauge field) describes how a vector or quantum state transforms when parallel transported along a closed loop in the base space. For a connection with gauge group $G$, the holonomy is an element of $G$ encoding the accumulated phase or rotation due to curvature (e.g., the Aharonov-Bohm effect for $U(1)$, or non-Abelian phases for $SU(2)/SU(3)$). Formally, for a loop $\gamma$ and connection $A$, the holonomy is the path-ordered exponential $\mathcal{H}(\gamma) = \mathcal{P} \exp (\oint_{\gamma} A)$. In particle physics, holonomy classes classify gauge-invariant observables under loop transformations, linking local symmetry to global topological effects. Boundary holonomy constraints ensure consistency in parallel transport under prescribed conditions (e.g., Wilson loops).</footnote>
In TSRT, a particle does not “possess” mass, charge, spin, or energy in the conventional sense of carrying attached attributes. Rather, these arise unavoidably from the particle’s internal geometric configuration as a localized trembling mode. Its spin results from the holonomy accumulated over internal oscillations; its electric or color charge from the mode’s symmetry under internal transformations; its mass from the resistance of its geodesic path to curvature-induced deflection; and its energy from the steady accumulation of action along proper time. These quantities are not decorations added to the particle—they are the defining structural consequences of its trembling dynamics. In this way, a particle is not a thing with properties but a self-consistent pattern of geometry whose properties are the measurable echoes of how that pattern persists, resists, and coheres within trembling spacetime.

The paper is organized as follows. Section 2 reviews the four fundamental trembling symmetry classes—U(1), SU(2), SU(3), and large-scale gravitational curvature—and shows how each corresponds to one of the known interactions of nature. This was already largely covered in the foundational paper on TSRT [19]. Section 3 derives mass and energy as emergent geometric quantities arising from proper-time action accumulation and inertial deflection within trembling spacetime. Section 4 introduces the mathematical formalism of localized trembling modes as the geometric origin of particle-like entities, detailing the stability, coherence, and quantization conditions that select discrete solutions. Section 5 demonstrates how all physical properties—electric charge, isospin, color, spin, and associated conservation laws—emerge as consequences of internal symmetry-bound trembling geometry and topological holonomy. Section 6 examines the role of discrete symmetries and generational structure, explaining why the Standard Model contains exactly three particle families and no additional stable generations. Section 7 develops the mechanisms of interaction within TSRT, showing how coherent curvature coupling replaces force-mediating fields and quantized virtual exchanges. Section 8 develops the mathematical structure of trembling eigenmodes, defining their deviation fields, phase holonomies, and quantized action integrals. Section 9 provides a compact reference guide to the Standard Model particle content and experimentally measured properties. Section 10 reconstructs each known particle type as a distinct trembling eigenmode, mapping geometric configurations to observed quantum numbers, spin, and charges. Section 11 describes how observable scattering and decay processes arise from coherent trembling interactions without invoking field quantization or virtual particles. Section 12 establishes the finiteness and closure of the TSRT particle spectrum, demonstrating why no additional stable particle families or exotic forces are permitted by the causal variational structure. Section 13 discusses empirical confirmations, outlines concrete predictions that distinguish TSRT from conventional quantum field theory, and proposes experiments capable of falsifying the theory. Finally, Section 14 presents a broader discussion of the theoretical implications, potential extensions, and open questions posed by this geometric unification framework.

## 2 Unified Geometric Foundation of Particle Structure

The description of particle-like phenomena in modern physics rests on two foundational frameworks: general relativity (GR), which geometrizes gravity through the curvature of space-
time [5, 25], and quantum field theory (QFT), which describes the other three fundamental forces (electromagnetic, weak, and strong) as excitations of quantized fields propagating on a fixed spacetime background [26, 27]. While GR achieves an elegant classical description of gravitation via the Einstein field equations, it fails to incorporate quantum effects, such as the singularities of black holes [28] or the thermodynamics of horizons [29]. Conversely, QFT—despite its remarkable success in predicting particle interactions via the Standard Model [8, 9]—relies on a non-dynamical spacetime, precluding a consistent treatment of gravity at quantum scales.

The incompatibility of these frameworks becomes acute in regimes where both quantum and gravitational effects dominate, such as the Planck scale (~$10^{-35}$ m) or the early universe [30]. Attempts to unify them, whether through quantized gravity (e.g., loop quantum gravity [31]) or emergent spacetime paradigms (e.g., string theory [32]), have yet to yield a complete theory that derives particle properties—such as mass hierarchies or charge quantization—from first principles. This persistent divide underscores the need for a deeper synthesis beyond the classical-quantum dichotomy.

Trembling Spacetime Relativity Theory (TSRT) offers a unified alternative [19–23]. It derives particle structure, mass, spin, and charge not from quantization postulates, but from geometric properties of spacetime trembling—local, causally admissible metric fluctuations constrained by proper-time variational dynamics.

This section establishes the precise classification of such trembling configurations and their role in supporting the four known fundamental interactions.

## 2.1 Classification of Geometric Trembling Symmetries

These four classes—corresponding respectively to U(1), SU(2), SU(3), and curvature-induced deviations—were rigorously derived in the foundational TSRT work [19], and shown to exhaust all causally admissible symmetry degrees of freedom for localized trembling configurations.

Throughout this paper, we adopt the metric signature (+, −, −, −), consistent with the TSRT convention [19]. Accordingly, the proper time interval is defined by<fnref n="7" />

$$d\tau^2 = g_{\mu\nu}(x) dx^\mu dx^\nu, \qquad (1)$$

with timelike geodesics satisfying $d\tau^2 > 0$, and total proper time recovered by integration: $\tau = \int \sqrt{d\tau^2}$.

In Trembling Spacetime Relativity Theory (TSRT), all dynamical structure is derived from a causal modulation of the spacetime metric:

$$g_{\mu\nu}(x) = \eta_{\mu\nu} + \xi_{\mu\nu}(x), \qquad (2)$$

where $\eta_{\mu\nu}$ is the flat Minkowski background and $\xi_{\mu\nu}(x)$ encodes small, bounded, and causally admissible fluctuations, termed “trembling” [19]. These fluctuations are constrained by causal admissibility: variational closure, and bounded curvature. In particular, the perturbed metric

<footnote n="7">⁷TSRT Foundations [19], Supplementary Material: Covariance, Proper Time, and Causal Structure in Trembling Spacetime - Constraints to Preserve Covariance - Proper Time and Causal Consistency in TSRT & Formalizing Causality in Spacetime Trembling.</footnote>
must ensure forward-directed proper time, i.e.,

$$d\tau^2 = g_{\mu\nu}(x) dx^\mu dx^\nu > 0 \qquad (3)$$

for all timelike curves (causal admissibility). Furthermore, the reparametrization-invariant action of particle motion must remain well-defined under all admissible trembling deformations (variational closure). Finally, local curvature derived from $\xi_{\mu\nu}$ must remain finite within a causally coherent domain (bounded curvature).

Under these constraints, one finds exactly four non-redundant symmetry classes of trembling deviation [19, 22, 23], each corresponding to a known interaction. These are not postulated but derived from the internal geometric logic of the theory.

Each class defines a distinct structure of trembling deviation, characterized by its transformation properties and symmetry-preserving behavior: U(1) phase-coherent fluctuations responsible for electromagnetism$^8$, SU(2) chiral-polarized deviations responsible for weak interactions$^9$, SU(3) multi-component entangled deviations responsible for strong interactions$^{10}$, and Coarse curvature coherence responsible for gravitation.$^{11}$

These four classes correspond precisely to the known fundamental forces in nature, and no additional trembling symmetries are admissible without violating causal or geometric closure. Each symmetry class supports a family of localized trembling solutions—solitonic in character—which form the geometric basis of stable particles. These modes, derived directly from the causal and variational structure of spacetime, underlie all interaction phenomena without invoking auxiliary fields or quantization axioms.

The classification proceeds according to four symmetry-constrained trembling classes: Abelian vector fluctuations (electromagnetism), chiral asymmetric fluctuations (weak interaction), non-Abelian color fluctuations (strong interaction), and coarse-grained curvature trembling (gravitation).

**Abelian Vector Fluctuations** govern electromagnetic interactions. These are described by trembling perturbations of the form

$$\xi_{\mu\nu}(x) = \frac{1}{2} (A_{\mu}(x) v_{\nu} + A_{\nu}(x) v_{\mu}), \qquad (4)$$

where $A_\mu(x)$ transforms under a U(1) gauge symmetry and $v^\mu$ is a reference four-velocity aligned with the local geodesic.$^{12}$

The associated field strength,

$$F_{\mu\nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}, \qquad (5)$$

emerges directly from the trembling geometry and satisfies the Bianchi identities [33–35] and Maxwell’s equations [36–38] when coupled to trembling geodesic flow [19], without requiring

<footnote>$^8$See TSRT Foundations [19], Supplementary Material: Emergent Electromagnetism from Abelian Trembling.</footnote>

<footnote>$^9$See TSRT Foundations [19], Supplementary Material: Geometric Origin of Weak Interaction Symmetry.</footnote>

<footnote>$^{10}$See TSRT Foundations [19], Supplementary Material: Internal SU(3) Structure and Color Trembling.</footnote>

<footnote>$^{11}$See TSRT Foundations [19], Supplementary Material: Macroscopic Coherence and Gravitational Averaging.</footnote>

<footnote>$^{12}$TSRT Foundations [19], Supplementary Material: Emergent Electromagnetism from Abelian Trembling - Field Strength Tensor and Maxwell Equations.</footnote>
postulated gauge dynamics.<fnref n="13" />

**Chiral Asymmetric Fluctuations** underlie the weak interaction. Here, trembling deviations introduce parity-violating structure by coupling only to the left-handed components of geodesic motion. The corresponding deviation field possesses an antisymmetric part under time-reversal and reflects SU(2) symmetry. The resulting geometric curvature mimics Yang-Mills dynamics [7, 39] and reproduces weak interaction phenomena [9, 26, 27, 40] within a purely geometric framework.

**Non-Abelian Color Fluctuations** describe strong interactions via SU(3) trembling symmetries [19]. These are expressed through eight vector potentials $A_\mu^a(x)$ with internal color indices $a = 1, \dots, 8$, leading to a field strength<fnref n="14" />

$$F_{\mu\nu}^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + g f^{abc} A_\mu^b A_\nu^c, \qquad (6)$$

where $f^{abc}$ are the SU(3) structure constants and $g$ is the geometric coupling constant. This structure is not imposed but follows from multicomponent coherence in trembling curvature.

**Coarse-Grained Curvature Trembling** accounts for gravitation as a large-scale average over metric fluctuations. Unlike the other three interactions, gravity in TSRT is not associated with an internal gauge group but emerges from proper-time-averaged geodesic deviation. The resulting dynamics recover Einstein-like behavior at macroscopic scales, with corrections that arise from the underlying trembling structure [19]. Gravity thus appears as a universal interaction arising from the geometric integrability of trembling curvature, consistent with the equivalence principle [41, 42].

These four trembling classes exhaust the symmetry-consistent, causally permitted deformations of spacetime within TSRT [19]. No additional gauge-like interactions can be introduced without violating variational closure or proper-time causality. This yields a powerful predictive outcome: TSRT permits precisely four fundamental interactions, fixed by geometry, as was already detailed in the foundational work [19].

This classification lays the foundation for understanding the emergence of matter in TSRT: all particles arise as stable, localized trembling modes, each bound by one of the four geometric symmetry classes. The resulting implications for conservation laws, particle spectra, and mass quantization will be explored in Section 2.2 and beyond.

## 2.2 Noether's Theorem in Trembling Geometry

In the framework of classical and quantum field theory, Noether's theorem [43] establishes a profound correspondence between continuous symmetries of a system's action functional and its conserved quantities. Formally, for any differentiable symmetry of the action $S[\phi] = \int \mathcal{L}(\phi, \partial_\mu \phi) d^4x$—whether global (e.g., spacetime translations, rotations) or internal (e.g., gauge transformations)—the theorem guarantees the existence of a conserved current $j^\mu$ satisfying $\partial_\mu j^\mu = 0$, and thus a time-invariant charge $Q = \int j^0 d^3x$ [26, 44]. This theorem underpins founded

<footnote n="13">¹³TSRT Foundations [19], Supplementary Material: Emergent Electromagnetism from Abelian Trembling - Field Strength Tensor and Maxwell Equations.</footnote>

<footnote n="14">¹⁴TSRT Foundations [19], Supplementary Material: Emergent Strong Nuclear Force from Non-Abelian Color Trembling - Non-Abelian Field Strength Tensor and Curvature.</footnote>
dational results across physics, i.e., spacetime symmetries (via the Poincaré group [3, 4, 27, 45]) yield conservation of energy-momentum and angular momentum [27], gauge symmetries (e.g., U(1) in electromagnetism) enforce charge conservation [46], and global phase symmetries in quantum fields lead to particle number conservation [47]. While Noether's theorem is rigorously valid in classical field theory and canonical quantum field theory, its implications extend to effective field theories and anomalies (e.g., the axial anomaly in quantum chromodynamics (QCD) [26]), where quantum effects break classical conservation laws [48].

In TSRT [19], where all dynamics arise from proper-time-parametrized geodesic motion through a causally fluctuating spacetime, this principle takes a fully geometric form. The trembling spacetime deviation $\xi_{\mu\nu}(x)$, constrained by causal admissibility, contributes directly to the action functional of a particle or ensemble of particles<fnref n="15" />

$$S = \int \mathcal{L}(x^{\mu}, \dot{x}^{\mu}, \xi_{\mu\nu}) d\tau, \qquad (7)$$

where $\tau$ is the proper time and $\mathcal{L}$ is the geometric Lagrangian governing motion in a trembling background. The structure of $\mathcal{L}$ encodes the curvature and causal properties of the effective metric of Equation (2). This action functional includes not only the geodesic dependence on the path $x^{\mu}(\tau)$ and its velocity $\dot{x}^{\mu}$, but also on the internal deviation structure $\xi_{\mu\nu}$ that encodes trembling-induced curvature. The corresponding Lagrangian was derived in the foundational TSRT work [19] as a geometric invariant under causal variation of proper time.

Symmetries of the deviation field $\xi_{\mu\nu}$ correspond to transformations that preserve the trembling structure and causal inequalities. Consider an infinitesimal continuous symmetry transformation of the worldline:

$$x^{\mu} \rightarrow x^{\mu} + \delta x^{\mu}, \qquad (8)$$

under which the proper-time action functional in Equation (7) undergoes a variation

$$\delta S = \int \left( \frac{\partial \mathcal{L}}{\partial x^{\mu}} \delta x^{\mu} + \frac{\partial \mathcal{L}}{\partial \dot{x}^{\mu}} \delta \dot{x}^{\mu} \right) d\tau. \qquad (9)$$

Integrating the second term by parts yields

$$\int \frac{\partial \mathcal{L}}{\partial \dot{x}^{\mu}} \delta \dot{x}^{\mu} d\tau = \left. \frac{\partial \mathcal{L}}{\partial \dot{x}^{\mu}} \delta x^{\mu} \right|_{\tau_i} - \int \frac{d}{d\tau} \left( \frac{\partial \mathcal{L}}{\partial \dot{x}^{\mu}} \right) \delta x^{\mu} d\tau. \qquad (10)$$

If the variation vanishes at the endpoints (i.e., $\delta x^{\mu}(\tau_i) = \delta x^{\mu}(\tau_f) = 0$), the boundary term disappears, and the requirement $\delta S = 0$ leads to the Euler-Lagrange equations. However, when the variation corresponds to a continuous symmetry, Noether's theorem implies that the integrand is a total derivative:

$$\frac{\partial \mathcal{L}}{\partial x^{\mu}} \delta x^{\mu} + \frac{\partial \mathcal{L}}{\partial \dot{x}^{\mu}} \delta \dot{x}^{\mu} = \frac{d}{d\tau} (F(x^{\mu})), \qquad (11)$$

for some function $F$. Therefore, even if $\delta x^{\mu}$ does not vanish at the endpoints, the variation of

<footnote n="15">¹⁵TSRT Foundations [19], Supplementary Material: Proper Time, Causal Structure, and the Emergence of Lorentz Geometry - Extension to Curved Spacetime and General Geodesics.</footnote>
the action satisfies<fnref n="16" />

$$ \delta S = \left. \frac{\partial \mathcal{L}}{\partial \dot{x}^{\mu}} \delta x^{\mu} \right|_{\tau_i}^{\tau_f}. \qquad (12) $$

If the action is invariant under the transformation (i.e., $\delta S = 0$) for arbitrary intervals, this implies that the quantity

$$ \frac{\partial \mathcal{L}}{\partial \dot{x}^{\mu}} \delta x^{\mu} \qquad (13) $$

must be conserved along the proper-time evolution. That is,

$$ \frac{d}{d\tau} \left( \frac{\partial \mathcal{L}}{\partial \dot{x}^{\mu}} \delta x^{\mu} \right) = 0. \qquad (14) $$

This expression directly encodes the conserved Noether currents associated with each continuous symmetry of the trembling geometry. It follows from the standard application of Noether's theorem in reparametrization-invariant Lagrangian mechanics [49, 50].

TSRT identifies exactly four such symmetry classes, each corresponding to one of the fundamental interactions [19]. The resulting conservation laws are not postulated but emerge as invariants under the allowed geometric transformations of the deviation field.

The foundational derivation of the conservation laws appear in the TSRT foundationa work.<fnref n="17" />

**Electromagnetic Charge Conservation.** Electric charge conservation in TSRT arises from the internal U(1) symmetry of Abelian trembling modes, which preserve the phase coherence of the metric deviation field. This symmetry corresponds directly to local gauge invariance and results in a conserved quantity under proper-time evolution.

The relevant U(1) gauge transformation of the vector potential takes the form

$$ A_{\mu}(x) \rightarrow A_{\mu}(x) + \partial_{\mu} \Lambda(x), \qquad (15) $$

where $\Lambda(x)$ is an arbitrary smooth scalar function. This transformation leaves the field strength tensor, given in Equation (5), invariant, ensuring that the induced curvature and geodesic dynamics are unchanged.

The associated conserved quantity is the electric charge $q$, which in TSRT is not imposed but arises geometrically as a phase-invariant of trembling-induced parallel transport along geodesics. The conservation of $q$ follows from the reparametrization-invariance of the action and the symmetry of the Lagrangian under U(1) phase shifts:

$$ \frac{d}{d\tau} \left( \frac{\partial \mathcal{L}}{\partial \dot{x}^{\mu}} \delta x^{\mu} \right) = 0. \qquad (16) $$

In the electromagnetic sector, $\delta x^\mu$ corresponds to infinitesimal gauge transformations, and the resulting conserved current yields a gauge-invariant definition of electric charge directly from

<footnote n="16">¹⁶TSRT Foundations [19], Supplementary Material: Emergent Gravity from Coarse-Grained Curvature—Geodesic Motion and the Weak Equivalence Principle</footnote>

<footnote n="17">¹⁷TSRT Foundations [19], Supplementary Material: Conservation Laws from Geometric Symmetries - Conserved Quantities by Trembling Symmetry Class.</footnote>
trembling geometry.

**Isospin and Weak Current Conservation.** Conservation of weak isospin currents in TSRT arises from the internal SU(2) symmetry of chiral trembling deviations. These deviations introduce an intrinsic asymmetry between left- and right-handed components of geodesic motion, encoding parity violation<fnref n="18" /> directly into the geometry of spacetime.

The corresponding local gauge transformation of the deviation field takes the form

$$ \psi_L(x) \rightarrow \exp(i\theta^a(x)\tau^a) \psi_L(x), \qquad (17) $$

where $\psi_L$ is the left-handed component of the geodesic velocity field, $\tau^a$ are the SU(2) generators, and $\theta^a(x)$ are local gauge parameters. The deviation tensor $\xi_{\mu\nu}(x)$ is defined to transform covariantly under such symmetry operations, ensuring consistency with the underlying geometric framework.

The associated conserved current,

$$ J_{\text{weak}}^{a\mu} = \frac{\partial \mathcal{L}}{\partial \dot{x}_{\mu}} \delta^{(a)} x, \qquad (18) $$

represents the weak isospin flow, preserved under proper-time evolution due to Noether symmetry.<fnref n="19" /> The left-handed nature of this coupling—resulting from the antisymmetric structure of the chiral deviation—leads to natural parity violation, not as a quantum anomaly, but as a geometric constraint inherent in the trembling symmetry.

Thus, TSRT derives weak current conservation and parity asymmetry simultaneously, without the need for field quantization, Higgs structure, or spontaneous symmetry breaking.

**Color Charge Conservation.** In TSRT, conservation of color charge follows from the internal SU(3) symmetry of multicomponent trembling deviations. These deviations encode coherent non-Abelian fluctuations in the metric, which give rise to a conserved set of color currents associated with the eight generators of SU(3).

The conserved SU(3) color current takes the form

$$ J_a^\mu = \frac{\partial \mathcal{L}}{\partial \dot{x}^\nu} \frac{\partial \xi_a^{\nu\mu}}{\partial x^\alpha} \delta x^\alpha, \qquad (19) $$

where $a = 1, \dots, 8$ indexes the SU(3) generators, and $\xi_a^{\nu\mu}$ represents the component of the metric deviation aligned with the a-th color direction. These currents are conserved due to the

<footnote n="18">¹⁸Parity violation refers to the phenomenon where a physical system or interaction distinguishes between left-handed and right-handed configurations, breaking the symmetry under spatial inversion ($x \rightarrow -x$). Classically, this means the system’s behavior is not mirror-symmetric (e.g., the Coriolis force on Earth favors cyclonic rotation in one hemisphere over the other). In quantum mechanics, parity violation occurs when a theory or observable is not invariant under the parity operator $P$, which reverses spatial coordinates and flips particle spin states. The weak interaction is the canonical example: $W^\pm$ bosons couple only to left-handed fermions, making processes like $\beta$-decay intrinsically chiral. This was first demonstrated experimentally in 1957 [51] via the anisotropic emission of electrons from polarized cobalt-60 nuclei.</footnote>

<footnote n="19">¹⁹Here, $\delta^{(a)}x$ denotes the infinitesimal variation of the trembling configuration induced by an SU(2) generator $T^a$. More precisely, it measures the change in the deviation field $\xi_{\mu\nu}(x)$ under an internal SU(2) rotation acting on the internal trembling indices. The notation abbreviates this geometric transformation without introducing explicit Lie-algebra-valued fields.</footnote>
internal covariance of the trembling vector bundle under local SU(3) transformations.

Importantly, this conservation law is not imposed but emerges from the causal and variational structure of TSRT. The curvature associated with these fluctuations yields a field strength given in Equation (6), which governs confinement-like behavior in trembling geodesic configurations. Thus, color conservation in TSRT reflects the non-Abelian geometric structure of trembling spacetime and reproduces key features of quantum chromodynamics (QCD) without invoking quantized fields.

**Energy-Momentum Conservation.** The most general symmetry of the TSRT action is invariance under proper-time reparametrizations along causally admissible geodesics. This symmetry leads—via the geometric formulation of Noether’s theorem—to the conservation of energy-momentum. In trembling spacetime, this takes the form of an effective stress-energy tensor averaged over a coherence interval. The underlying dynamics remain consistent with local conservation:

$$ \nabla_{\mu} T^{\mu\nu} = 0, \qquad (20) $$

where $T^{\mu\nu}$ encodes the energy-momentum content of a localized trembling mode, as derived from its Lagrangian structure and internal deviation geometry. For isolated modes, this conservation law governs the inertial response and leads directly to the derivation of rest energy, as shown in Section 3.3.

For isolated localized trembling solutions (i.e., particles), this conservation law encodes the inertial response to curvature gradients and underlies the emergence of rest energy. As will be shown in Section 3.3, this geometric mechanism yields a rigorous derivation of the mass-energy relation $E = mc^2$ from trembling geodesic dynamics, without relying on kinetic or field-theoretic heuristics.

This conservation principle, like all others in TSRT, is not postulated but arises naturally from the variational structure of trembling spacetime.

TSRT thereby derives all known conservation laws—electric charge, weak isospin, color charge, and energy-momentum—from intrinsic geometric symmetries. Each conservation law follows from proper-time invariance applied to causally admissible metric deviations and is uniquely fixed by the structure of the geodesic action.

Crucially, no additional conserved quantities or internal gauge symmetries are admissible within this framework. Any attempt to introduce further symmetry classes would necessarily violate one or more of the trembling constraints: causal admissibility, variational closure, or bounded curvature coherence [19].

In Section 2.3, we revisit the formal argument—originally developed in the TSRT foundational analysis—that proves the exhaustiveness of the four allowed symmetry classes. This establishes that no additional fundamental interactions are geometrically possible within TSRT, thus rendering the theory not only predictive, but uniquely constrained by spacetime geometry itself.
## 2.3 Uniqueness and Closure of Fundamental Interactions

The uniqueness of the four trembling symmetry classes arises not from observational matching, but from mathematical classification of the solution space of the trembling geodesic equation<fnref n="20" />

$$ \frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta}(x, \xi) \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0, \qquad (21) $$

where the connection $\Gamma^\mu_{\alpha\beta}$ depends on the deviation field $\xi_{\mu\nu}$. By analyzing admissible $\xi_{\mu\nu}$ fields that preserve the action's reparametrization invariance and proper-time causality, one finds that only the four listed symmetry types admit stable, non-trivial, curvature-bound trembling solutions.

This classification of all allowed trembling-induced deviations was first derived<fnref n="21" /> in the foundational TSRT analysis, where it was shown that only four coherent symmetry classes—corresponding to U(1), SU(2), SU(3), and curvature—are admissible under the constraints of causal geodesic variation.

No additional internal symmetry—Abelian or non-Abelian—satisfies the causal constraints without leading to dispersion, incoherence, or violation of variational closure. This proves the geometric uniqueness of the interaction spectrum in TSRT.

Section 4 demonstrates how stable, localized modes of these trembling configurations correspond to fundamental particles, each with quantized mass, spin, and charge properties derived from geometric invariants.

In conventional field theory, new interactions can be proposed by postulating additional internal symmetries or particle types, leading to theoretically unbounded model extensions. By contrast, TSRT imposes strict geometric and causal constraints on the admissible fluctuation modes of the spacetime metric of Equation (2). The structure of $\xi_{\mu\nu}$ is not arbitrary; it must preserve the causal inequality of Equation (3) under all allowable variations, and must arise from a variational principle with proper-time-parametrized action.

In TSRT [19], the classification of admissible trembling structures was developed by identifying the known geometrically distinct, non-redundant deformation classes that preserve local Lorentz invariance and respect proper-time causality. This yields four irreducible trembling symmetry types:

1. Abelian vector trembling (U(1)) ⇒ Electromagnetism

2. Chiral SU(2) trembling ⇒ Weak interaction

3. Non-Abelian SU(3) color trembling ⇒ Strong interaction

4. Coarse curvature trembling ⇒ Gravitation

To date, no further symmetry types have been identified that satisfy all of the following TSRT constraints [19]:

* Preservation of proper-time monotonicity without non-causal metric deformation

<footnote n="20">²⁰TSRT Foundations [19], Supplementary Material: Proper Time, Causal Structure, and the Emergence of Lorentz Geometry - Extension to Curved Spacetime and General Geodesics</footnote>

<footnote n="21">²¹TSRT Foundations [19], Supplementary Material: Uniqueness of the Four Fundamental Interactions in TSRT</footnote>
• Invariance of the action functional under reparametrizations

• Non-redundancy under local isometries or diffeomorphism-invariant<sup>22</sup> transformations

• Compatibility with the variational structure of the trembling geodesic equations

While this result is derived constructively rather than empirically, a fully general proof
that no additional classes can arise in broader functional spaces remains an open mathematical
question. Accordingly, the TSRT classification is conjectured to be closed under the analyzed
constraints. If a new force were to be discovered that cannot be reduced to one of these trembling
types, the theory would require revision or abandonment.

This classification also governs the structure of matter: all particles in TSRT correspond to stable, localized trembling modes arising from these four interaction types. The set of possible matter configurations—fermions, gauge bosons, and their composites—is therefore finitely determined by this classification framework. These localized excitations will be further identified in Section 4 and systematically classified in Section 10.

This concludes the development of the geometric and symmetry-theoretic classification
framework within TSRT. We now proceed to examine how trembling spacetime gives rise to
mass, energy, and the equivalence relation $E = mc^2$, beginning with the variational and geo-
metric origin of energy in Section 3.

3 Trembling Spacetime as the Origin of Mass and Energy

A foundational challenge in theoretical physics is to explain the origin and equivalence of mass and energy at the most fundamental level. Einstein's mass-energy equivalence principle, encapsulated in the relation $E = mc^2$ [24], emerges as a kinematic consequence of special relativity, but its deeper field-theoretic realization remains nontrivial. In the Standard Model (SM), particle masses are generated dynamically through the Higgs mechanism [11,52], where spontaneous symmetry breaking of the $SU(2)_L \times U(1)_Y$ electroweak gauge group induces a nonzero vacuum expectation value (VEV) for the Higgs field, $v \approx 246$ GeV, coupling to fermions via Yukawa interactions [26].

Energy, on the other hand, arises as a conserved Noether charge associated with time-
translation invariance of the Lagrangian density $\mathcal{L}$ [43], formally expressed through the stress-
energy tensor $T^{\mu\nu}$ [27]. While the Higgs mechanism explains the origin of inertial mass, the
equivalence between this mass and the energy of field configurations (e.g., in bound states
like hadrons) requires a consistent quantum treatment of self-interactions—a regime where
nonperturbative QCD effects dominate [53]. This dichotomy highlights the unresolved tension

<footnote n="22"><sup>22</sup> A diffeomorphism is a smooth, invertible map $\phi : \mathcal{M} \to \mathcal{M}$ preserving the manifold’s structure. In particle physics and gravity, diffeomorphisms can act as: (i) *passive* coordinate changes (a redundancy in description), or (ii) *active* dynamical symmetries (e.g., in gauge theories, where curvature-bound diffeomorphisms may resemble gauge transformations). Here, diffeomorphism-invariance implies that physical observables (e.g., holonomy classes) are unchanged under such transformations. Crucially, when diffeomorphisms are constrained by curvature or internal symmetries (like $U(1)$, $SU(2)$, $SU(3)$), they may reduce to a subgroup (e.g., isometries) or acquire holonomy-like properties—parallel transport of deviations under these transformations defines their equivalence classes.</footnote>
between the SM's perturbative mass generation and the gravitationally significant total mass-energy of systems, which in general relativity includes contributions from binding energy and spacetime curvature [25].

In TSRT [19], by contrast, both mass and energy emerge from the causal geometry of spacetime trembling itself. This section shows how energy arises from the accumulation of action along geodesics in a trembling background, how mass arises from inertial resistance to curvature deflection, and how the celebrated relation $E = mc^2$ follows from first principles as a geometric identity.

## 3.1 Energy from Trembling-Induced Action Accumulation

In TSRT, energy is not a primitive attribute of a field or particle, but emerges as a derived geometric quantity associated with the accumulation of action along a causally admissible worldline. This perspective departs from quantum field theory, where energy is defined via Hamiltonian operators in Hilbert space, and instead anchors it in the causal variational structure of trembling spacetime.

The fundamental proper-time parameter $\tau$ defines the evolution of localized modes, preserving causal ordering under the effective metric of Equation (2), where $\xi_{\mu\nu}(x)$ encodes bounded trembling deviations. The proper-time differential is given, as established in TSRT Foundations [19], by (see also Equation (1)):

$$d\tau = \sqrt{g_{\mu\nu} dx^\mu dx^\nu}. \qquad (22)$$

For a free localized trembling mode in the absence of external interactions, the action accumulated along a path $x^\mu(\tau)$ is:

$$S = \int \mathcal{L}(x^\mu, \dot{x}^\mu, \xi_{\mu\nu}) d\tau, \qquad (23)$$

where the Lagrangian is:

$$\mathcal{L} = \sqrt{g_{\mu\nu}(x) \dot{x}^\mu \dot{x}^\nu}. \qquad (24)$$

This form follows directly from the variational principle for geodesic evolution (see TSRT [19]).

In the specific context of trembling-induced coherence cycles, the total action accumulated over a full coherence interval $\Delta\tau$ becomes quantized in discrete units:

$$\oint \mathcal{L} d\tau = 2\pi n \mathcal{T}, \quad n \in \mathbb{Z}, \qquad (25)$$

where $\mathcal{T}$ is the fundamental trembling action scale identified in [21] as the geometric precursor to Planck's constant in the weak-field limit.

Energy is then naturally defined as the rate of action accumulation per unit proper time:

$$E(\tau) = \frac{dS}{d\tau}. \qquad (26)$$
Since $\mathcal{L}$ is the instantaneous action density, this implies:

$$E(\tau) = \mathcal{L}, \qquad (27)$$

provided no reparametrization factor is introduced. It is important to note that this identification holds in the proper-time parametrization; in other parametrizations, the definition must be adjusted accordingly.

Crucially, temporal translation symmetry in $\tau$ implies conservation of this energy: when $\xi_{\mu\nu}$ is time-independent in proper time and the mode remains causally coherent, the action density $\mathcal{L}$ is constant along the trajectory. Thus, conservation of energy is here a direct consequence of variational symmetry applied to trembling geometry.

When localized curvature deviations are present ($\xi_{\mu\nu} \neq 0$), the Lagrangian is modulated by the internal geometry of the trembling mode. This modulation shifts the instantaneous action density and hence the energy, but in a manner consistent with overall conservation across full coherence cycles. As demonstrated in [21], this same structure leads, for massless corpuscles, to the resonance relation:

$$E = \mathcal{T} \nu, \qquad (28)$$

where $\nu$ is the frequency of coherence crossings, reproducing the Einstein-Planck formula when $\mathcal{T}$ approaches $\hbar$. For massive modes, the accumulated action over coherence intervals includes contributions from both internal curvature and inertial resistance to deflection, leading to the familiar rest-energy relation:

$$E = mc^2, \qquad (29)$$

as derived in Section 3.3.

In summary, energy in TSRT is not an imposed quantity but the scalar rate at which action accumulates along a causally admissible, trembling geodesic. Its conservation reflects temporal symmetry of the underlying geometry, and its quantization arises naturally from coherence constraints. This geometric framework subsumes traditional notions of Hamiltonians, vacuum oscillators, and quantized fields into a unified, deterministic description of matter and radiation.

## 3.2 Geometric Origin of Mass

The concept of mass as a measure of a body's resistance to acceleration dates back to Newton, who introduced inertial mass in his laws of motion. In contrast, gravitational mass enters Newton's law of universal gravitation as the source of gravitational attraction. Whether these two notions of mass are fundamentally identical remained an open question until the work of Einstein.

In 1907, Einstein introduced his famous equivalence principle, arguing that an observer inside a sealed elevator cannot distinguish whether the apparent force they feel arises from acceleration or from a gravitational field. This led to the profound insight that inertial and gravitational mass must be equivalent, not just numerically, but conceptually—suggesting that gravity is a manifestation of spacetime geometry rather than a force in the Newtonian sense [41].

In contrast to the Standard Model [54], which attributes mass generation to spontaneous
symmetry breaking via the Higgs field [11], the TSRT framework [19] derives mass geometrically from the inertial response of localized trembling geodesics to background curvature. No scalar field is required. Instead, mass emerges as an intrinsic property of the path stability of trembling geodesics, quantifying their resistance to deviation under curvature-induced perturbations. This eliminates the need for Higgs-type mechanisms while preserving empirical mass values via causal variational dynamics.

In particular, in TSRT, inertial mass arises as a measure of a localized trembling mode's resistance to deflection of its proper-time geodesic. This resistance is not a postulated parameter but follows directly from the geometry of deviation under spacetime curvature. Recall that in the trembling framework, a localized mode follows a perturbed geodesic determined by the effective metric, given in Equation (2), where $\xi_{\mu\nu}(x)$ encodes localized, bounded trembling deviations.

To analyze the stability and inertia of this path, we consider a one-parameter family of nearby trajectories $x^{\mu}(\tau, \epsilon)$, with the deviation vector:

$$ \delta x^{\mu}(\tau) = \left. \frac{\partial x^{\mu}}{\partial \epsilon} \right|_{\epsilon=0}. \qquad (30) $$

The standard Jacobi (geodesic deviation) equation governs how small displacements evolve under curvature [19]:

$$ \frac{D^2 \delta x^{\mu}}{D \tau^2} + R^{\mu}_{\nu\alpha\beta} \dot{x}^{\nu} \delta x^{\alpha} \dot{x}^{\beta} = 0, \qquad (31) $$

where $R^{\mu}_{\nu\alpha\beta}$ is the Riemann tensor derived from the effective metric.

This equation expresses that local curvature tends to deflect the relative separation between nearby geodesics. In TSRT, the amplitude of trembling-induced deflection is finite but bounded by the coherence constraints of localized modes. Intuitively, the more resistant a mode is to deflection under curvature, the larger its effective inertia.

To make this notion quantitative, we observe that the second proper-time derivative of the worldline itself,

$$ a^{\mu} \equiv \frac{D^2 x^{\mu}}{D \tau^2}, \qquad (32) $$

represents the local four-acceleration relative to geodesic free fall in the unperturbed background metric. Projecting this acceleration onto the four-velocity and integrating over one coherence interval yields a scalar measure of inertial resistance.

Accordingly, we define the effective inertial mass as:

$$ m = \frac{1}{c^2} \langle \sqrt{a^{\mu} g_{\mu\nu} a^{\nu}} \rangle_{\tau}, \qquad (33) $$

where the angled brackets denote an average over a full trembling cycle:

$$ \langle f(\tau) \rangle_{\tau} \equiv \frac{1}{\Delta\tau} \int_{0}^{\Delta\tau} f(\tau) d\tau. \qquad (34) $$

This expression has a clear interpretation: it measures the average rate at which proper-time curvature attempts to deflect the trajectory, projected along the motion itself. The more such deflection must be opposed to maintain coherence, the larger the effective mass.
This formulation connects smoothly to the later expressions in Section 8, where the inertial mass emerges as a geometric integral over the deviation field and its induced accelerations.

Importantly, Equation (33) does not rely on arbitrary external forces or additional assumptions: it is fully determined by the trembling geometry encoded in $\xi_{\mu\nu}(x)$. It also provides a direct bridge to the canonical rest energy relation:

$$E = m c^2, \qquad (35)$$

because the energy derived from action accumulation (Section 3.1) and the inertia derived here are both consequences of the same coherent trembling dynamics.

In summary, the inertial mass in TSRT is a geometric response coefficient: it quantifies how strongly a localized trembling mode resists deviation from causal coherence under effective curvature. This geometric viewpoint unifies mass, energy, and proper-time stability without requiring field-theoretic mass terms or Higgs-like mechanisms.

Thus, mass in TSRT is a geometric invariant of the mode's resistance to curvature within a coherence interval. It reflects how tightly the internal structure is coupled to the background trembling metric. This explains why massless modes (like photons) follow null geodesics with no curvature-induced deflection, while massive modes resist deformation and accumulate internal action as rest energy. The equivalence $E = mc^2$ then follows from this curvature response, as shown in Section 3.3.

Historically, the equivalence between gravitational mass ($m_g$) and inertial mass ($m_i$) has played a central role in the development of physics. Newton, in his *Principia* (1687), already noted their proportionality based on pendulum experiments accurate to about one part in a thousand [55]. As mentioned above, this empirical observation—that all objects fall with the same acceleration in a gravitational field—later became formalized as the Weak Equivalence Principle (WEP). Einstein elevated this principle in 1907 [41] to a foundational postulate of general relativity, asserting that the exact equality $m_g = m_i$ required a geometric reformulation of gravity itself. Modern high-precision experiments, including torsion balance measurements [56, 57] and lunar laser ranging [58], have confirmed this equivalence to within one part in $10^{14}$. Yet, despite its foundational status in relativity, no existing theory has derived this equality from more basic physical principles [59].

Within TSRT, however, this equivalence finds a natural geometric origin: inertial mass and gravitational interaction both emerge from the same underlying trembling geometry. Since all mass is defined by its resistance to curvature-induced deviation along proper-time geodesics, and since spacetime curvature is also the source of gravitational effects, the distinction between $m_i$ and $m_g$ dissolves. Thus, TSRT offers not just a reinterpretation of mass, but a conceptual closure to one of physics' oldest questions.

From this perspective, inertial mass in TSRT can be interpreted as a geometric stiffness coefficient of the trembling geodesic. Just as an elastic medium resists deformation under applied stress, a localized trembling mode resists deflection of its causal trajectory under internal curvature perturbations. The larger the effective mass, the greater the proper-time curvature required to alter its motion. This analogy clarifies why mass emerges as an intrinsic property of stable trembling configurations: it quantifies the resistance of the spacetime fabric itself to
reconfiguration by localized action density. In this sense, mass is not an extrinsic parameter but a measure of the inherent stiffness of causally admissible trembling geometry.

### 3.3 Rest Energy from Coherent Geodesic Resistance

In TSRT, the rest energy of a localized trembling mode arises as the conserved quantity associated with invariance under proper-time translation of the worldline action. Unlike in conventional field theory, where energy is defined through Hamiltonian operators, here energy is an invariant scalar quantity derived from the causal geometry of geodesic evolution.

The action functional for a free localized mode with rest mass $m$ propagating along a causally admissible geodesic parametrized by proper time $\tau$ is given by

$$S = -mc \int ds = -mc^2 \int d\tau, \qquad (36)$$

where

$$ds = \sqrt{g_{\mu\nu} dx^\mu dx^\nu}$$

and the negative sign ensures that the action is minimized along timelike geodesics.

In this parametrization, the Lagrangian becomes a constant,

$$\mathcal{L} = -mc^2. \qquad (37)$$

Because the Lagrangian is independent of the coordinates $x^\mu$, the standard canonical momentum vanishes identically in this parametrization. Instead, the physical content is contained in the normalization constraint

$$g_{\mu\nu} \frac{dx^\mu}{d\tau} \frac{dx^\nu}{d\tau} = c^2, \qquad (38)$$

which encodes the invariant interval along the trajectory.

To define the rest energy consistently, one constructs the four-momentum

$$P^\mu = m \frac{dx^\mu}{ds} = m \frac{dx^\mu}{c d\tau}, \qquad (39)$$

so that

$$g_{\mu\nu} P^\mu P^\nu = m^2 c^2. \qquad (40)$$

In the rest frame of the mode, where the spatial components vanish ($dx^i/d\tau = 0$), we have

$$\frac{dx^0}{d\tau} = c,$$

and thus

$$P^0 = m \frac{c}{c} = m. \qquad (41)$$

The observable rest energy is then given by

$$E = c P^0 = mc^2. \qquad (42)$$
This derivation shows that the Einstein relation $E = mc^2$ arises in TSRT not from any quantized field operator, but as a geometric identity: it reflects the invariant normalization of the four-momentum generated by causal evolution in trembling spacetime.

Finally, the accumulation of action over a finite coherence interval $\Delta\tau$ is simply

$$S = -mc^2 \Delta\tau, \qquad (43)$$

demonstrating that the action grows linearly with the elapsed proper time but does not, by itself, define the rest energy. Instead, the rest energy is the conserved scalar quantity derived from time translation invariance and the normalization constraint (38).

In this way, TSRT provides a fully deterministic and geometrically grounded derivation of rest energy as the invariant measure of a mode's resistance to deflection and the causal integrity of its geodesic evolution.

# 4 Localized Trembling Modes and Particle Formation

In Trembling Spacetime Relativity Theory (TSRT), fundamental particles are identified with localized, symmetry-bound solutions to the trembling geodesic equation. These configurations remain causally coherent over a finite proper-time interval and exhibit internally bounded curvature oscillations. The concept of a particle thus emerges geometrically: it is not an excitation of a quantized field, but a stable trembling structure in spacetime itself.

Localized trembling modes are defined by three key properties:

(i) their stress-energy density $^{23}$ remains confined within a finite proper-time coherence interval;

(ii) their deviation field $\xi_{\mu\nu}(x)$ exhibits closed-cycle variation, yielding discrete action accumulation per coherence loop;

(iii) they respect the trembling constraints derived from variational causality and curvature admissibility.

Such configurations serve as the geometric prototypes of particles. Each coherent trembling mode corresponds to a distinct class of physical entity, with its observable properties—mass, spin, charge—derivable from its internal symmetry and deviation structure. In this view, particle formation is not a separate physical process but a classification of stable solutions to the trembling geodesic equations under symmetry constraints.

The remainder of this section formalizes the definition of localized trembling modes, explains their coherence properties, and derives their physical observables from the underlying geometry.

<footnote>$^{23}$In general relativity, the stress-energy tensor $T_{\mu\nu}$ encodes the distribution and flow of energy, momentum, and stress (pressure and shear) within spacetime. As the source term in Einstein's field equations $G_{\mu\nu} = 8\pi GT_{\mu\nu}$, it couples matter and energy to the curvature of spacetime. While its mathematical structure resembles classical stress tensors (e.g., in elasticity theory), $T_{\mu\nu}$ is fundamentally relativistic: it incorporates all forms of energy and momentum—including mass-energy, electromagnetic fields, and quantum contributions—as gravitational sources. Unlike classical analogs, it is a rank-2 tensor in 4-dimensional spacetime, with components that transform covariantly under Lorentz transformations.</footnote>
## 4.1 Definition and Properties of Localized Trembling Modes

A localized trembling mode in TSRT is defined as a curvature-bound solution to the trembling geodesic equation whose deviation field $\xi_{\mu\nu}(x)$ remains coherent and bounded over a finite proper-time interval. These solutions are characterized by their internal geometric periodicity and by their preservation of causal structure throughout the coherence cycle.

The defining feature of such a mode is its confinement to a compact support in proper time, during which the deviation field completes a full oscillatory loop and returns to its original configuration up to gauge equivalence<fnref n="24" />:

$$ \xi_{\mu\nu}(\tau + \Delta\tau) \approx \xi_{\mu\nu}(\tau), \qquad (44) $$

for some minimal coherence interval $\Delta\tau$, where $\approx$ denotes equivalence under internal symmetry holonomy.

To satisfy causal admissibility, each trembling mode must obey the bounded deviation constraint:

$$ \xi_{\mu\nu}(x)\xi^{\mu\nu}(x) < \epsilon^2, \qquad (45) $$

where $\epsilon \ll 1$ represents the maximal allowed curvature deviation compatible with geodesic closure. This condition ensures that the trembling remains in the weak-deviation regime, where the background spacetime retains a well-defined causal structure and the deviation field can be treated perturbatively without inducing large-scale metric singularities. This ensures that trembling remains within the small-deviation regime where proper-time parametrization and geodesic stability are preserved. Furthermore, a localized mode must conserve its internal action over the coherence cycle:

$$ \oint_{\text{cycle}} \mathcal{L} d\tau = n\mathcal{S}_0, \qquad (46) $$

with $n \in \mathbb{Z}$ and $\mathcal{S}_0$ the geometric action quantum derived in TSRT as the minimal unit of coherent trembling.

These criteria jointly define a localized trembling mode as a physically admissible, symmetry-bound, stable solution to the trembling geodesic equations. Such modes give rise to the particle-like configurations explored in the remainder of this paper, including their conserved charges, mass, spin, and gauge properties.

**Remark.** Although the parameter $\epsilon$ is defined locally as a bound on the squared norm of the deviation field, its existence implies that the total action accumulated over a closed trembling cycle cannot vanish. In particular, even when the trembling amplitude is minimized to the smallest causally admissible configuration, the integrated action over the coherence interval remains finite. This irreducible action per cycle defines a universal scale $\mathcal{T}$, which was identified in earlier work [19,21] as the geometric precursor of Planck's constant $\hbar$. While $\epsilon$ itself depends on the coherence geometry and is therefore not uniquely determined without specifying the

<footnote n="24">²⁴That is, the deviation field may differ by a transformation belonging to the internal symmetry group under which it transforms (e.g., a global U(1) phase rotation, an SU(2) chiral rotation, or an SU(3) color rotation), but remains physically identical in observable quantities such as conserved charges and curvature invariants.</footnote>
spatial extent and proper-time duration of the mode, its nonzero lower bound reflects the quantization of trembling action.<fnref n="25" />

## 4.2 Stability Conditions for Localized Trembling Modes

Stability in TSRT is defined geometrically: a trembling mode is stable if its deviation field $\xi_{\mu\nu}(x)$ evolves cyclically and returns to its original configuration (up to gauge equivalence) after a finite proper-time interval. This recurrence must be dynamically self-sustaining—meaning the deviation cycle must be a variational attractor of the proper-time action principle.<fnref n="26" />

Let the action accumulated over a full trembling cycle be

$$S = \oint \mathcal{L}(\xi, \dot{\xi}, x) d\tau. \qquad (47)$$

The requirement of variational stability demands that the first variation vanish<fnref n="27" />:

$$\delta S = 0, \qquad (48)$$

under all causally admissible perturbations that preserve proper-time parametrization and boundary holonomy. This ensures that the trembling configuration is a stationary point of the action functional under the TSRT variational constraint.<fnref n="28" />

Furthermore, mode stability implies that the deviation field must reside in a closed holonomy class of the internal symmetry group governing the mode (U(1), SU(2), SU(3), or curvature-

$$S_{cycle} = \int_{0}^{\Delta\tau} \mathcal{L} d\tau.$$

Since the Lagrangian contains a factor proportional to $\sqrt{1 + \delta}$, with

$$\delta \sim \xi_{\mu\nu} u^{\mu} u^{\nu},$$

a nonzero bound $\epsilon$ implies that $\delta$ cannot be made identically zero everywhere along the cycle. Consequently, the minimal nontrivial trembling configuration yields

$$S_{cycle} \geq \mathcal{T},$$

with $\mathcal{T}$ the irreducible action scale. If one further imposes that the coherence interval is of order the Compton wavelength divided by $c$, then dimensional analysis gives

$$\epsilon \sim \frac{\sqrt{\mathcal{T}}}{L^2},$$

where $L$ denotes the spatial extent of the coherence domain. This connection underlies the identification $\mathcal{T} \simeq \hbar$.

<footnote n="25">²⁵Recall that the action over a coherence interval $\Delta\tau$ is given by</footnote>

<footnote n="26">²⁶In other words, small perturbations of the deviation field away from this cyclic solution decay over time, so that the trembling mode dynamically returns to the same periodic configuration. This stability condition is analogous to a limit cycle in dynamical systems or an energy minimum in classical mechanics.</footnote>

<footnote n="27">²⁷This means that, when the deviation field $\xi_{\mu\nu}$ is perturbed infinitesimally (i.e., $\xi \to \xi + \delta\xi$), the resulting change in the action $\delta S$ must be zero to first order in $\delta\xi$. This is the geometric analog of requiring that a classical trajectory makes the action stationary, ensuring the trembling mode is an extremal (i.e., stable) solution to the variational principle.</footnote>

<footnote n="28">²⁸Being a “stationary point of the action functional” means that small, causally admissible variations of the deviation field $\xi_{\mu\nu}$ do not change the value of the action to first order. This condition ensures that the trembling mode is dynamically self-sustaining: it persists without decaying or diverging over time. The phrase “under the TSRT variational constraint” emphasizes that only variations compatible with causal structure, proper-time parametrization, and internal symmetry are allowed, in contrast to arbitrary variations in unconstrained field theories.</footnote>
bound diffeomorphisms). Each such class supports a discrete set of closed paths in configuration space whose geometric curvature matches the TSRT constraint conditions.

The result is that trembling modes exhibit topologically quantized recurrence: only a discrete subset of possible internal oscillation patterns closes coherently, satisfying both causal and variational constraints. This mechanism explains why particle-like solutions in TSRT are discrete and stable, while most arbitrary perturbations lead to dispersion, mode decay, or incoherence.

These stability conditions also justify the exclusion of transient or exotic configurations that do not satisfy action closure. In this framework, a fundamental particle is rigorously defined as a localized trembling mode whose deviation field completes a geometrically coherent loop in proper time, preserving causal admissibility and variational stability over indefinite durations. Configurations that arise spontaneously under extreme conditions—such as high-energy collisions—may exhibit temporary trembling patterns resembling particles, but lacking long-term coherence, they do not constitute true eigenmodes of the trembling geometry. As a result, the possible observation of short-lived, rapidly decaying resonances or metastable excitations does not in itself contradict TSRT, provided these configurations can be interpreted as transient, non-eigenmode deformations of the spacetime manifold.

This distinction establishes a physically precise selection principle: only those trembling modes that maintain full causal and variational stability define the finite, predictive spectrum of fundamental particles in TSRT. Observations of ephemeral or incoherent phenomena, while important for phenomenology, do not automatically falsify the theory unless such states can be demonstrated to satisfy the same coherence criteria as stable trembling eigenmodes.

# 5 Derivation of Particle Properties from Geometry

Localized trembling modes in TSRT do not merely exhibit spacetime-bound oscillations—they give rise to all observable properties of particles. Each property, from inertial mass to internal charge, arises from a distinct geometric feature of the deviation field $\xi_{\mu\nu}(x)$ and its coupling to proper-time curvature. The analysis in this section shows how the full range of physical particle properties—rest energy, spin, electric and color charge, and quantum numbers—are not postulated, but derived from the structure of causally admissible trembling configurations. These properties emerge naturally from holonomy, symmetry class, and curvature response$^{29}$, and are quantized due to coherence constraints on closed trembling cycles. Thus, TSRT recovers the observed particle phenomenology from first geometric principles, without invoking field quantization, Hilbert spaces, or operator algebras.

In TSRT [19], fundamental particles are interpreted as stable, localized, symmetry-bound trembling modes of the spacetime metric, as established in Section 4. These modes are not excitations of pre-assumed fields, but geometric structures characterized by their internal coherence, causal admissibility, and binding to one of the four fundamental interaction symmetries. It fol-

<footnote>$^{29}$“Curvature response” refers to how the trembling deviation field $\xi_{\mu\nu}(x)$ reacts to and sources spacetime curvature. In TSRT, the localized mode both deforms the geometry and adapts its own configuration in response to the resulting curvature. This mutual interaction determines properties such as effective mass and energy: a stronger curvature response typically implies higher inertial resistance and larger rest energy.</footnote>
lows that observable particle properties—mass, spin, and charge—must arise from this internal trembling geometry rather than from external imposition.

This section formalizes that idea. Each property is shown to emerge from a specific aspect of the trembling configuration, and in every case, the result is constrained by the causal and variational structure of the theory. No degrees of freedom exist beyond those allowed by the trembling classification, and therefore, the range of possible particle properties is not only finite but derivable.

The subsections that follow develop these derivations in detail and in the order presented here. Section 5.1 demonstrates how spin arises as a geometric phase accumulated by closed trembling cycles and quantizes naturally from holonomy constraints. Section 5.2 establishes that the discreteness of observable particle properties follows from coherence intervals imposing quantized action per cycle. Section 5.3 explains how particle-antiparticle duality emerges as a reversal of internal orientation while preserving causal evolution. Section 5.4 derives electric, weak, and color charges as gauge-class properties of the deviation field, including their conserved currents and confinement mechanisms. Finally, Section 5.5 formalizes the quantization of charge as a geometric consequence of coherence and internal symmetry holonomy.

Together, these results confirm that particle properties are not added to the theory—they are determined by it. The trembling spacetime geometry, once fixed by causal and variational constraints, yields a closed and predictive classification of all admissible physical modes. This classification is finite, internally consistent, and non-extendable—unlike the Standard Model, which allows additional symmetries or particles by assumption. By the end of this section, we will have established the core claim of this paper: the observable spectrum of mass, spin, and charge follows from first-principles geometric dynamics in trembling spacetime.

## 5.1 Spin as Geometric Phase of Closed Trembling Paths

Spin, in the broadest sense, refers to a property of particles that manifests as quantized angular momentum even in the absence of classical rotation [60]. In classical physics, angular momentum arises from a mass distribution spinning about an axis, as in a rotating top [61]. In quantum mechanics, spin instead describes an intrinsic, irreducible form of angular momentum without a literal spinning motion: it is defined mathematically by the transformation properties of a particle's wavefunction under rotations (for example, whether it requires a $2\pi$ or $4\pi$ rotation to return to the same state) [62]. This abstraction sometimes obscures its geometric origin [63].

In TSRT, spin is neither a label assigned by representation theory nor an ad hoc property: it emerges deterministically as a geometric phase accumulated by a localized trembling configuration transported along a closed proper-time loop [64]. This phase can be visualized as a holonomy—an accumulated angle associated with the internal oscillation geometry of spacetime itself [65]. Although this quantization reproduces the discrete spectrum familiar from quantum theory, its origin lies entirely in the causal geometry of trembling spacetime [19]. This formulation unifies the intuitive picture of rotational behavior with the abstract algebraic description, showing that both are special cases of the same geometric mechanism [66], and generalizes angular momentum within a purely geometric framework, avoiding the need for quantization rules, spinor representations, or operator formalism.
Trembling modes in TSRT are characterized by internal oscillatory structures encoded in the deviation field $\xi_{\mu\nu}(x)$. When these structures undergo closed cycles in the local rest frame, they trace out spatial loops that, due to the curvature of spacetime, accumulate a net geometric phase. This phase is path-independent for homotopically equivalent cycles and reflects the intrinsic angular momentum of the mode.<fnref n="30" /> Physically, this means that a phenomenon, spatially confined and returning to its initial position after each cycle—may experience a net rotational shift in its internal geometry, akin to the holonomy of a gyroscope transported in curved space.<fnref n="31" /> This holonomy, constrained by the causal coherence of the trembling solution, defines the spin of the particle. Quantized values of spin arise from the requirement that the trembling mode returns to a physically equivalent state after a full proper-time cycle.

To make this precise, consider a localized trembling mode centered on a proper-time-parametrized timelike worldline $x^{\mu}(\tau)$, with an internal deviation field $\xi_{\mu\nu}(x)$ whose spatial support is localized within a compact region $\Omega$ and whose temporal evolution is periodic. In the local rest frame of the mode—where the four-velocity is $u^{\mu} = dx^{\mu}/d\tau$—these internal oscillations occur in directions transverse to the mode's motion, i.e., orthogonal to $u^{\mu}$. Viewed in this rest frame, the trembling induces closed spatial loops, similar to circular or elliptical vibrations confined to a plane.

As the internal mode undergoes these closed cycles, a geometric phase is accumulated due to the curvature of spacetime along the path. This phase shift is formally captured by the holonomy [34, 71–73] of parallel transport around the loop [35]:

$$ \Delta\theta = \oint_{\gamma} \omega, \qquad (49) $$

where $\gamma$ is the closed trembling path in the three-dimensional spatial hypersurface orthogonal to $u^{\mu}$, and $\omega$ is the connection one-form<fnref n="32" /> that defines how vectors are rotated under parallel

$$ \Delta\theta^{\mu}_{\nu} = \mathcal{P} \exp \left( - \oint_{\gamma} \Gamma^{\mu}_{\nu\alpha} dx^{\alpha} \right), $$

$$ \omega = \omega^{a}_{\mu} J_{a} dx^{\mu}, $$

<footnote n="30">³⁰Two closed cycles (loops) $\gamma_1, \gamma_2$ in a space $M$ are homotopically equivalent if one can be continuously deformed into the other without tearing or leaving $M$. Formally, $\gamma_1 \sim \gamma_2$ if there exists a map $H: [0, 1] \times S^1 \to M$ such that $H(0, \cdot) = \gamma_1$ and $H(1, \cdot) = \gamma_2$. For the geometric phase, path-independence under homotopy means that the net phase shift depends only on the topology of the loop (e.g., its winding number around a singularity) and the curvature (e.g., the Berry curvature $\Omega$ in quantum systems, or spacetime curvature $R^{\mu}_{\nu\alpha\beta}$ in GR). Intrinsic angular momentum arises because non-trivial homotopy classes (e.g., non-contractible loops) encode global geometric constraints. See [39] for topological phases and [67] for spacetime applications.</footnote>

<footnote n="31">³¹The holonomy of a gyroscope (or any vector) transported along a closed path in curved spacetime is the net rotation of its orientation relative to a locally inertial frame, due to the curvature-induced parallel transport. This is a classical manifestation of the connection's holonomy group. Mathematically, for a gyroscope transported along a loop $\gamma$ in spacetime with Riemann curvature $R^{\mu}_{\nu\alpha\beta}$, its orientation undergoes a rotation described by the path-ordered exponential:</footnote>

<footnote>where $\Gamma^{\mu}_{\nu\alpha}$ are the connection coefficients. In weak-field gravity, this reduces to the Thomas precession (special relativity) and the geodetic + frame-dragging effects (general relativity). Experimentally observed via satellite missions like Gravity Probe B [68], this phenomenon is a geometric phase (akin to the Aharonov-Bohm effect for quantum states). For derivations, see [25, 69] or the modern treatment in [70].</footnote>

<footnote n="32">³²The connection one-form $\omega$ (or gauge field) is a Lie-algebra-valued differential form that defines infinitesimal parallel transport of vectors or internal states along a path. For the spatial rotation group $SO(3)$, $\omega$ takes values in the Lie algebra $so(3)$ and acts as:</footnote>

<footnote>where $J_a$ are the generators of $so(3)$ (e.g., $J_z$ generates rotations about the z-axis), and $\omega^a_\mu$ encodes the rotation rate of a vector under transport in direction $\partial_\mu$. The holonomy $\Delta\theta$ arises from integrating $\omega$ along $\gamma$, yielding a net rotation $\exp(\Delta\theta^a J_a) \in SO(3)$. In curved spacetime, $\omega$ may include contributions from the Levi-Civita</footnote>
transport. In this context, $\omega$ encodes the rotation properties associated with the local SO(3) symmetry group, which governs spatial rotations in the rest frame.

Physically, this means that even though the internal mode returns to its original spatial position after a complete trembling cycle, it may acquire a net rotational shift in phase—just as a Foucault pendulum accumulates a shift due to Earth’s rotation.<fnref n="33" /> If this phase shift is a nonzero multiple of $2\pi$, it reflects intrinsic angular momentum. Specifically, if $\Delta\theta = 2\pi s$, where s is a half-integer or integer, then the trembling mode acquires spin s. This value is intrinsic to the geometry of the trembling path and remains invariant under coordinate transformations.

If the trembling path returns to its original spatial position but does not return to its initial internal phase—i.e., if the accumulated geometric phase along the loop is

$$ \Delta\theta = 2\pi s, \qquad (50) $$

where s is a half-integer—then the mode is said to possess spin s. In this context, the phase shift is a consequence of the internal curvature structure encoded in the deviation field $\xi_{\mu\nu}(x)$. More precisely, the trembling configuration undergoes parallel transport around a closed proper-time loop, and the holonomy—the net rotation accumulated by this transport—is what determines the spin. This geometric phase cannot be removed by any smooth change of coordinates or local deformation of the path because it is a topological invariant: it depends on the global properties of the space in which the trembling evolves. In other words, even if the trembling loop is continuously deformed or reparametrized, the total phase accumulation remains fixed, reflecting the intrinsic curvature and connectivity of the underlying spacetime manifold. This is analogous to how a vector parallel transported around a closed path on a curved surface acquires a rotation angle that depends only on the enclosed curvature. Hence, in TSRT, spin is not an arbitrary label but a holonomy-induced geometric phase that is invariant under all allowed coordinate transformations and uniquely classifies the trembling mode.

In the simplest case, consider a localized trembling mode undergoing planar oscillations in the spatial hypersurface orthogonal to its worldline. Let the trembling path form a closed, approximately circular loop over a complete cycle in proper time. Such configurations model the most elementary form of internal rotation and serve as idealized representations of spin-bearing particles. In this setting, the geometric phase accumulated over one full trembling cycle can be interpreted as an angular momentum. Specifically, the internal structure of the mode generates a nonzero area in the transverse plane swept out by the oscillating trajectory. This area contributes directly to a rotational moment, much like how classical angular momentum arises from mass moving in a circular path. Mathematically, the intrinsic angular momentum associated with such internal trembling is encoded in the spatial components of the antisymmetric angular

<footnote>connection (e.g., frame-dragging) or gauge fields (e.g., non-Abelian phases). For the intrinsic angular momentum interpretation, see [71]; for the SO(3) case, see [34], §11.8. In quantum systems, $\omega$ generalizes to the Berry connection $\mathcal{A} = \langle\psi|d\psi\rangle$, with holonomy as the Berry phase [65]. Geometrically, $\omega$ is the principal so(3)-connection on the frame bundle of the spatial hypersurface.</footnote>

<footnote n="33">³³This phase shift is analogous to the Foucault pendulum’s precession, where a pendulum’s plane of oscillation rotates by an angle $\Delta\phi = 2\pi \cos\theta$ after one Earth rotation (with $\theta$ the latitude), due to parallel transport on a curved manifold (here, Earth’s surface). Similarly, the internal mode’s phase shift arises from the holonomy of the governing symmetry group (U(1), SU(2), etc.), reflecting a geometric phase (like the Berry phase in quantum mechanics). For the Foucault pendulum’s geometric interpretation, see [65]; for its mathematical equivalence to gauge theories, see [74].</footnote>
momentum tensor:

$$S^{ij} = \int_{\Sigma} (x^i \dot{x}^j - x^j \dot{x}^i) d\tau, \qquad (51)$$

where $x^i$ and $\dot{x}^i \equiv dx^i/d\tau$ are the spatial coordinates and velocities within the local rest frame of the mode, and $\Sigma$ denotes the trembling surface (the spatial region enclosed by the internal oscillation). This integral gives the effective rotational momentum of the mode in the $i-j$ plane, directly analogous to classical definitions of spin in rigid-body mechanics, but here arising from periodic geodesic deviation in spacetime.

Emergent quantization of spin in TSRT follows directly from the periodicity of the trembling cycle and the requirement of action quantization. A closed trembling loop satisfies the geometric quantization condition:

$$\oint \mathcal{L} d\tau = 2\pi s \mathcal{T}, \qquad (52)$$

where $\mathcal{L}$ is the effective Lagrangian density of the trembling mode, $\mathcal{T}$ is the fundamental unit of action derived in TSRT (linked to Planck's constant in the weak-field limit [19]), and $s \in \mathbb{Z}/2$ is the spin quantum number. This condition arises naturally from the proper-time periodicity of geodesic trembling and the requirement of phase coherence for stable, localized configurations (as detailed in Section 4.1). Only those trembling cycles whose total action is an integer multiple of $2\pi\mathcal{T}$ are physically realized.

This formulation provides a transparent geometric interpretation of discrete spin values. Spin-$\frac{1}{2}$ fermions correspond to trembling cycles that accumulate a geometric phase of $\pi$, spin-1 bosons to cycles with phase accumulation of $2\pi$, and spin-0 particles to configurations with no net holonomy in transverse directions. In this framework, spin arises not as an imposed quantum label but as an emergent, discretized geometric feature of trembling dynamics, grounded in the holonomy structure of closed paths in curved spacetime [34, 71-73]. Thus, TSRT offers a deterministic and purely geometric account of spin, replacing the conventional notion of abstract spin operators with action-integrated loops and coherent holonomies.

The restriction to these discrete spin values is not imposed but results from the geometric constraints on stable trembling configurations in causal spacetime. In TSRT, trembling modes must satisfy two key conditions to represent physical particles: (1) the total trembling pattern must be periodic in proper time to ensure localized, non-dispersive evolution, and (2) the accumulated geometric phase (holonomy [34, 71-73]) after one full cycle must return the mode to its original physical configuration, possibly up to a sign. These constraints sharply limit the allowable holonomies to those consistent with the topological structure of rotation groups in three spatial dimensions.

Specifically, in the local rest frame, the spatial symmetry group governing transverse trembling is $SO(3)$. However, to capture spinor behavior (such as for electrons), we must consider its double cover, $SU(2)$, which allows for half-integer phase shifts. In this setting, a geometric phase of $2\pi$ corresponds to a full rotation (bosonic modes), while a phase of $\pi$ corresponds to a double-valued representation (fermionic modes), where a second cycle is needed to return the system to its original state.

Trembling modes with phase accumulations different from 0, $\pi$, $2\pi$, ... would violate either periodicity or causal continuity, resulting in non-physical, unstable configurations that cannot
remain localized in spacetime. Thus, only certain discrete spin values emerge as physically realizable trembling solutions. This quantization follows from the topological and variational structure of spacetime itself—not from imposed quantum rules.

The result is a deterministic and geometric explanation for why nature exhibits only a finite, discrete spectrum of particle spins: spin-0 (scalar), spin-$\frac{1}{2}$ (fermionic), and spin-1 (vector bosonic) arise as the only stable, causally coherent trembling modes consistent with the internal geometry and action quantization of TSRT.

Higher-spin modes—such as spin-2 excitations—are not categorically excluded in TSRT, but their existence is subject to far more stringent geometric constraints. For a trembling mode to be both stable and localized, the internal geodesic deviations must form closed cycles with well-defined, periodic holonomy [34, 71–73] in the transverse directions. In the case of spin-2, this would require a quadruple-covering representation of the local rotation group, meaning that the mode must return to its original configuration only after four full cycles. While mathematically conceivable, such configurations are generically unstable under causal propagation and fail to produce localized energy densities consistent with action quantization. Consequently, no stable spin-2 trembling mode has yet been identified within the admissible solution space of TSRT. This aligns with the empirical absence of free, localized spin-2 particles in nature, despite the theoretical possibility of massless spin-2 fields such as the graviton in quantum field theory. More generally, TSRT prohibits arbitrary spin states by grounding particle identity in the topology and periodicity of spacetime itself, not in abstract field representations. The absence of higher or fractional spin states beyond the Standard Model is therefore not a gap, but a prediction: only those trembling modes that respect causal closure, internal coherence, and geometric periodicity can emerge as physically observable particles.

The TSRT interpretation of spin is therefore entirely geometric, consistent with all known spin classifications, and does not rely on spinor algebra or quantized field excitations. It further implies that the allowed spins of elementary particles are determined by the topology and curvature of their internal trembling structure. This provides a natural constraint on the particle spectrum and forbids fractional spins or arbitrary values not supported by the holonomy [34, 71–73] of trembling loops.

## 5.2 Quantized Action and Coherence Intervals

The purpose of this subsection is to formalize how the discreteness of observable particle properties arises from the internal dynamics of trembling modes. In TSRT, particles are not continuous excitations that can accumulate arbitrary action over time; instead, their evolution is segmented into coherence intervals—finite proper-time loops during which the trembling configuration maintains causal consistency and internal phase coherence. This segmentation imposes a natural quantization condition on the total action, effectively constraining the allowed trembling solutions to discrete classes. As discussed extensively in earlier work [21], this coherence quantization is the same mechanism that yields the Planck–Einstein energy relation for photons without invoking canonical quantization.

More formally, the foundational TSRT framework [19] establishes that localized trembling modes evolve over finite proper-time coherence intervals, during which the deviation field $\xi_{\mu\nu}(x)$
completes one or more full geometric cycles. Over each cycle, the action accumulated is given by:

$$ \oint \mathcal{L} d\tau = 2\pi k \mathcal{T}, \quad k \in \mathbb{Z}, \qquad (53) $$

where $\mathcal{L}$ is the effective Lagrangian of the trembling mode, and $\mathcal{T}$ is the fundamental action unit derived from the proper-time invariant measure [21]. In Equation (53), the integer $k$ counts the number of completed action cycles. This is distinct from Equation (52), where the factor $s$ indexes the quantized geometric phase accumulated in spin-like closed loops. Specifically, Equation (53) governs action quantization over coherence intervals, constraining the total internal action to integer multiples of $2\pi\mathcal{T}$, while Equation (52) governs phase holonomy quantization, associating the spin $s$ with the discrete geometric phase accumulated in the internal configuration space.

Both quantizations arise from the same principle: the requirement that trembling modes return to an equivalent geometric and causal state after a full coherence interval, but they encode complementary aspects—action closure and rotational holonomy.

This quantization condition is not a heuristic assumption; it follows rigorously from the variational closure of trembling solutions described in the foundational work on TSRT [19] and from the phase-coherence arguments developed in earlier work on the corpuscular nature of light [21] for massless modes. In the massive case, coherence intervals similarly constrain the allowed spectrum of stable trembling configurations.

As no partial cycles are dynamically admissible under causal evolution, the allowed values of accumulated action and the associated conserved quantities—such as charge, spin, and rest mass—are necessarily discrete. This geometric discreteness replaces operator-based quantization, unifying particle identity with the causal structure of spacetime.

It is thus essential to emphasize that in TSRT, a particle means a localized trembling mode satisfying both of these quantization constraints: Coherence quantization (Equation (53)), and spin holonomy quantization (Equation (52)). This dual condition ensures that only stable, causally admissible configurations can persist over macroscopic timescales, while arbitrary transient deformations remain unquantized and decay rapidly.

In summary, the discreteness of charges, masses, and spins in TSRT is not a consequence of introducing operator algebras or field quantization postulates. Rather, it emerges naturally from the requirement that the trembling deviation field complete geometrically and causally consistent cycles, with each coherence interval contributing an integer multiple of the fundamental action unit $2\pi\mathcal{T}$.

## 5.3 Particle–Antiparticle Duality from Orientation Reversal

In TSRT [19], the existence of antiparticles is not postulated but emerges as a geometric consequence of internal orientation in the trembling geodesic framework. Each localized trembling mode corresponds to a causally admissible solution of the geodesic equations in the metric of Equation (2), where the deviation field $\xi_{\mu\nu}$ remains bounded and coherent over proper time. The proper-time parameter $\tau$ progresses monotonically, ensuring that causality is preserved.

However, each mode also possesses an internal phase orientation—an intrinsic directionality
of the trembling cycle within the compact internal symmetry space. This orientation is mathematically expressed by the sign of the phase accumulation per coherence interval. Concretely, a trembling mode completing a coherence cycle accumulates action

$$S = \oint \mathcal{L} d\tau, \qquad (54)$$

and acquires a geometric phase whose sign reflects the orientation of internal oscillation relative to proper-time progression.

Importantly, the causal progression $\tau$ is never reversed. Instead, the internal phase orientation can be flipped while keeping the geodesic evolution forward in time. This corresponds to transforming

$$\xi_{\mu\nu}(x) \rightarrow -\xi_{\mu\nu}(x), \qquad (55)$$

which reverses the direction of internal curvature oscillation while preserving causal admissibility. Under this transformation, all orientation-dependent conserved quantities—such as electric charge $q$, isospin $I^a$, and color $T^a$—reverse sign, while mass and inertial coherence remain unchanged.

This yields the natural interpretation of antiparticles as internal orientation-reversed trembling modes:

$$\text{Particle} \xleftarrow{\text{Orientation Flip}} \text{Antiparticle}, \qquad (56)$$

with identical proper-time evolution but opposite internal charges and phase orientation.

Unlike standard quantum field theory, which introduces antiparticles through negative-frequency solutions and field quantization, TSRT derives particle-antiparticle duality as a geometric feature of trembling orientation. The fact that this reversal does not require time reversal ensures compatibility with the causality principle at the heart of TSRT.

This interpretation also clarifies charge conservation in pair creation and annihilation: the total internal phase accumulated across both trembling modes cancels exactly, preserving the net geometric action. Thus, TSRT provides a deterministic, causally consistent foundation for antiparticles without invoking probabilistic vacuum fluctuations or retrocausal propagation.

## 5.4 Charge and Color as Gauge-Class Properties

In the foundational formulation of TSRT [19], all conserved emergent quantum numbers—including electric charge, weak isospin, and color charge—are derived from internal geometric symmetries of localized trembling modes. These symmetries are not imposed externally via gauge fields or quantization rules, but emerge from the structure-preserving transformations of the deviation field $\xi_{\mu\nu}(x)$ under the four irreducible trembling classes.

Each symmetry class supports its own invariants, corresponding to familiar physical charges. The associated conservation laws follow from Noether's theorem applied to the proper-time action of geodesics in trembling spacetime.<fnref n="34" /> The resulting charges are discrete, stable, and geometrically determined by holonomy and phase structure of the deviation field over coherence

<footnote n="34">³⁴TSRT Foundations [19], Supplementary Material: Conservation Laws from Geometric Symmetries - Geometric Invariance and Noether-Type Correspondence.</footnote>
intervals. Further, we describe how charge, isospin, and color emerge as distinct gauge-class properties of TSRT solutions.

**Electric Charge.** The geometric origin of electric charge in Trembling Spacetime Relativity Theory (TSRT) was established in earlier work [19]. In that framework, electric charge is not imposed as a separate quantum number but arises as a conserved Noether quantity associated with the internal Abelian phase symmetry of localized trembling modes.

More concretely, in the Abelian sector, the trembling deviation can be represented in the proper-time formalism as a localized vector field:

$$ \Xi^{\mu}(\tau) = \epsilon(\tau) u^{\mu}(\tau), \qquad (57) $$

where $\epsilon(\tau)$ encodes the phase-locked oscillation amplitude, and $u^{\mu}$ is the local four-velocity along the worldline. This construction leads to an emergent electromagnetic potential:

$$ A_{\mu}(x) = \nabla_{\mu} \int \epsilon(\tau) \delta^{(4)}(x - x(\tau)) d\tau, \qquad (58) $$

as derived in the foundational work [19]. Under global phase transformations of the internal oscillation:

$$ \epsilon(\tau) \rightarrow e^{i\alpha} \epsilon(\tau), \qquad (59) $$

the action remains invariant. Noether's theorem then guarantees the existence of a conserved current:

$$ J_{\text{EM}}^{\mu} = q u^{\mu}, \quad \nabla_{\mu} J_{\text{EM}}^{\mu} = 0, \qquad (60) $$

where the coefficient $q$ is identified as the electric charge$^{35}$.

It is important to clarify that the gauge-like transformation often written heuristically as

$$ \xi_{\mu\nu}(x) \rightarrow \xi_{\mu\nu}(x) + \frac{1}{2} [\nabla_{\mu}\Lambda(x)v_{\nu} + \nabla_{\nu}\Lambda(x)v_{\mu}] \qquad (61) $$

should be understood as a symbolic expression illustrating the fact that adding an exact derivative to the potential does not affect the curvature (field strength). Strictly speaking, in the Abelian trembling formulation, it is the phase rotation of $\epsilon(\tau)$ that is the primary gauge degree of freedom [19].

Electric charge is therefore defined as the invariant integral over a spacelike hypersurface intersecting the trembling worldline:

$$ q = \int_{\Sigma} J_{\text{EM}}^{\mu} d\Sigma_{\mu}, \qquad (62) $$

where $d\Sigma_{\mu}$ is the normal volume element.$^{36}$

Regarding quantization, it is crucial to distinguish two aspects: First, the action quantization

<footnote>$^{35}$TSRT Foundations [19], Supplementary Material: Conservation Laws from Geometric Symmetries - Geometric Invariance and Noether-Type Correspondence.</footnote>

<footnote>$^{36}$This expression is identical in content to the one in TSRT Foundations [19], Supplementary Material: Conservation Laws from Geometric Symmetries - Geometric Invariance and Noether-Type Correspondence.</footnote>
condition (Equation (53)), which constrains the accumulated action per coherence interval to discrete multiples of $2\pi T$. This reflects the causal closure of trembling cycles. Second, the charge quantization, which emerges because each complete coherence cycle corresponds to a fixed increment of the conserved Noether charge $q$. In other words, the internal Abelian phase must return to its original configuration modulo $2\pi$, implying that only integer multiples of a minimal unit charge $q_0$ are dynamically admissible:

$$q = n q_0, \quad n \in \mathbb{Z}, \qquad (63)$$

where $q_0$ corresponds to the minimal nontrivial trembling configuration.

This distinction means that in TSRT, charge quantization does not derive from field operator spectra but from the topological closure of the phase in the internal Abelian cycle. Accordingly, there is indeed a fundamental unit charge, but its origin is purely geometric rather than quantum-mechanical.

Thus, discrete electric charges reflect the geometric quantization of coherence intervals and phase holonomy in the Abelian trembling sector. This perspective yields a unified explanation of electric charge, consistent with Noether's theorem, causal coherence, and the action quantization underlying the entire TSRT framework. For further derivations and examples, see also Section 5.5.

**Weak Isospin.** In the TSRT framework [19], weak isospin arises from the chiral asymmetry of localized non-Abelian trembling modes. Specifically, the deviation field $\xi_{\mu\nu}(x)$ can be decomposed into components that transform under the internal SU(2) gauge symmetry associated with weak interactions. This decomposition reflects the fact that the trembling-induced metric deviations couple differently to left-handed and right-handed components of the proper-time four-velocity $u^\mu = dx^\mu/d\tau$. The resulting asymmetry is not imposed by hand but derived from the causal geometry of the deviation field.

To see this concretely, recall that in the spinor formulation [19], the effective deviation field can be expressed schematically in the bispinor representation as:

$$\xi_{\mu\nu}(x) \sim \bar{\psi}(x) \Gamma_{\mu\nu} \psi(x), \qquad (64)$$

where $\Gamma_{\mu\nu}$ denotes the bilinear operator encoding internal symmetry and spacetime index structure. Because the weak interaction is known experimentally to be purely left-chiral, TSRT identifies the projection of this deviation field onto the left-handed subspace as the physically relevant component for weak interactions.

This projection uses the familiar left-chiral projector:

$$P_L = \frac{1}{2}(1 - \gamma^5), \qquad (65)$$

where $\gamma^5$ is the usual Dirac matrix encoding chirality. The left-chiral part of the deviation field is then defined as:

$$\xi_{\mu\nu}^{(L)}(x) = P_L \xi_{\mu\nu}(x) P_L. \qquad (66)$$
This operation simply means that, in the bispinor basis, you zero out any components corresponding to right-chiral or mixed-chirality combinations.

Under an SU(2) local gauge transformation $U(x) \in SU(2)$, the left-chiral deviation field transforms covariantly:

$$ \xi_{\mu\nu}^{(L)}(x) \rightarrow U(x) \xi_{\mu\nu}^{(L)}(x) U^{\dagger}(x). \qquad (67) $$

This relation follows from the fact that in TSRT, the trembling deviation acts as an adjoint-valued bispinor operator: applying a gauge transformation conjugates the internal indices, preserving the underlying metric symmetry but rotating the internal SU(2) orientation.

The physical interpretation is that the internal SU(2) symmetry acts only on the left-chiral projection of the trembling deviation field, in complete parallel to how weak interactions couple only to left-handed fermions in the Standard Model. Right-handed projections remain invariant under SU(2), effectively acting as gauge singlets.

This internal SU(2) symmetry implies the existence of conserved weak isospin currents. Noether's theorem [43] applied to local SU(2) gauge transformations yields:

$$ \nabla_{\mu} J_{\text{weak}}^{a\mu} = 0, \qquad (68) $$

where each $J_{\text{weak}}^{a\mu}$ is associated with one generator $T^a$ of the SU(2) Lie algebra. The integrated conserved quantities,

$$ I^a = \int_{\Sigma} J_{\text{weak}}^{a\mu} d\Sigma_{\mu}, \qquad (69) $$

define the components of the weak isospin vector. They correspond geometrically to the internal orientation of the trembling mode in SU(2) space. Because only the left-chiral part of the deviation field transforms nontrivially under SU(2), only left-handed components participate in weak isospin dynamics, yielding the familiar experimental observation of parity violation.

TSRT thus recovers the complete structure of weak isospin symmetry, including chirality dependence, purely from the causal, variational geometry of trembling spacetime—without recourse to scalar fields or spontaneous symmetry breaking. The conservation of weak isospin emerges from the topological SU(2) holonomy [34, 71–73] encoded in the internal antisymmetric structure of localized trembling modes.

**Color Charge.** In the TSRT framework [19], color charge emerges from the internal SU(3) symmetry structure of non-Abelian trembling modes. Concretely, these modes are described by a multicomponent deviation field whose internal directions correspond to the eight generators $T^a$ of the SU(3) Lie algebra.

To represent this structure, one introduces eight effective vector potentials $A_{\mu}^{a}(x)$, each encoding the local amplitude and orientation of trembling along the corresponding generator. The deviation field can then be expressed in the form:

$$ \xi_{\mu\nu}(x) = \frac{1}{2} \sum_{a=1}^{8} \left[ A_{\mu}^{a}(x) v_{\nu} + A_{\nu}^{a}(x) v_{\mu} \right], \qquad (70) $$

where $v_{\mu}$ is the local timelike reference vector defining the direction of the coherence cycle. This
ansatz ensures that the deviation field remains symmetric in $\mu \leftrightarrow \nu$ and consistent with causal admissibility constraints. It is analogous in structure to the symmetric part of a gauge-covariant derivative applied to a local internal phase vector [19].

Under a local SU(3) gauge transformation $U(x) \in SU(3)$, the combination $A_{\mu}^{a} T^{a}$ transforms according to the standard adjoint rule augmented by an inhomogeneous term:

$$A_{\mu}^{a}(x) T^{a} \rightarrow U(x) [A_{\mu}^{a}(x) T^{a}] U^{\dagger}(x) + \frac{i}{g} U(x) \partial_{\mu} U^{\dagger}(x), \qquad (71)$$

where $g$ is the effective geometric coupling constant. This transformation law follows directly from demanding that the trembling deviation field $\xi_{\mu\nu}$ transform covariantly under parallel transport in the internal bundle.

The corresponding color current is derived by applying Noether's theorem to infinitesimal internal gauge transformations of the trembling action functional:

$$\delta S = \int d\tau \frac{\delta \mathcal{L}}{\delta x^{\alpha}} \delta x^{\alpha} + \int d\tau \frac{\delta \mathcal{L}}{\delta \xi_{a}^{\mu\nu}} \delta \xi_{a}^{\mu\nu}. \qquad (72)$$

Focusing on the gauge-variation of the internal field leads to the conserved current:

$$J_{a}^{\mu} = \frac{\partial \mathcal{L}}{\partial(\partial_{\mu} \xi_{\alpha\beta})} [T_{a}, \xi_{\alpha\beta}], \qquad (73)$$

where the commutator expresses how the internal SU(3) rotation acts non-trivially on the trembling configuration. In the simpler (linearized) approximation used for clarity in this article, the current can be heuristically expressed in the form:

$$J_{a}^{\mu} \approx \frac{\partial \mathcal{L}}{\partial \dot{x}^{\nu}} \frac{\partial \xi_{a}^{\nu\mu}}{\partial x^{\alpha}} \delta x^{\alpha}, \qquad (74)$$

though in fully nonlinear treatment the commutator form is preferred.

This color current satisfies a non-Abelian gauge-covariant conservation law:

$$D_{\mu} J_{a}^{\mu} \equiv \partial_{\mu} J_{a}^{\mu} + g f_{abc} A_{\mu}^{b} J^{\mu c} = 0, \qquad (75)$$

where $f_{abc}$ are the SU(3) structure constants, and $D_{\mu}$ denotes the internal covariant derivative associated with the trembling geometry. This equation directly generalizes the Abelian conservation law and captures the self-interaction of color currents via the non-Abelian connection.

Crucially, the presence of the structure constants $f_{abc}$ implies that isolated color-charged configurations are dynamically unstable. Because the non-Abelian deviation field $\xi_{\mu\nu}$ includes self-interacting curvature components (arising from commutators of internal rotations), any attempt to separate a single color charge produces divergent contributions to the effective curvature and breaks causal coherence over large distances.

This leads naturally to confinement: only color-neutral combinations—such as triplet-singlet baryons and octet-balanced mesons—admit stable, globally coherent trembling solutions. The mechanism is fully geometric: it does not invoke quantized gluon fields or operator-valued potentials but arises from the causal structure of SU(3) trembling bundles themselves.
Thus, TSRT recovers the empirical classification of hadronic states and color confinement without relying on auxiliary quantization hypotheses or asymptotic freedom. All such features emerge as geometric constraints on the allowable trembling configurations in the internal non-Abelian symmetry space, in direct analogy to the color confinement discussed the foundational work on TSRT [19].

**Gauge-Class Constraint.** Each localized trembling mode in TSRT [19] is uniquely associated with a single geometric gauge class—U(1), SU(2), or SU(3)—defined by the internal symmetry of its deviation field $\xi_{\mu\nu}(x)$. These symmetry classes correspond to distinct types of trembling-induced curvature structures and cannot be continuously deformed into one another without breaking essential geometric constraints. Specifically, each class imposes a distinct set of boundary conditions and transformation rules under local gauge symmetry, and these are preserved only under compatible variational flows.

Cross-class transitions—such as between an SU(2) chiral mode and an SU(3) color-charged mode—are prohibited by two core principles of TSRT. First, the causal coherence condition requires that the trembling mode maintain a continuous proper-time parametrization and internal phase coherence throughout its evolution. Attempting to interpolate between symmetry classes would break this coherence, as the internal structure of the deviation field would no longer remain compatible with a single set of symmetry generators. Second, the variational principle underlying the trembling equations of motion leads to separate extremal solutions for each symmetry class. The action functional for SU(2) trembling modes, for instance, yields a distinct Euler–Lagrange structure from that of SU(3) modes, with different coupling tensors, conserved currents, and holonomy behavior [34,71–73]. These solutions exist in disjoint functional spaces and are not connected by any smooth deformation.

Thus, each particle in TSRT is bound to a single gauge class by construction. Its electric charge, weak isospin, and color are not superimposed freely but instead arise from the unique geometric symmetry of its trembling mode. This framework explains why no particle in nature simultaneously carries both SU(3) color and SU(2) weak charge in a way that allows coherent trembling propagation—there are no SU(2)–SU(3) hybrid modes in the causal solution space of TSRT. Instead, the observed properties of known particles emerge from modes that are symmetry-bound within a fixed gauge class, providing a geometric explanation for the strict charge profiles observed in the Standard Model.

## 5.5 Geometric Origin of Electric and Gauge Charge Quantization

Emergent quantization of electric charge was introduced in Section 5.4. In Trembling Spacetime Relativity Theory (TSRT) [19], the quantization of charge emerges not from algebraic assumptions or field-theoretic postulates, but as a geometric necessity. Specifically, it results from the intrinsic structure of causally admissible trembling modes and the requirement of action periodicity over coherent proper-time cycles.

A localized trembling mode evolves along a proper-time-parametrized geodesic $x^{\mu}(\tau)$, with internal deviation structure encoded by a compactly supported field $\xi_{\mu\nu}(x)$. For such a mode to remain stable and localized, the internal oscillation must repeat identically after a finite
coherence interval $\Delta\tau$, forming a closed trembling loop. This constraint implies that the action accumulated over one complete trembling cycle is quantized:

$$ \oint \mathcal{L} d\tau = n \mathcal{S}_0, \quad n \in \mathbb{Z}, \qquad (76) $$

where $\mathcal{L}$ is the proper-time Lagrangian and $\mathcal{S}_0$ is the fundamental unit of geometric action associated with a single cycle.

This quantization arises from the variational closure of the trembling solution: the mode must return to a physically indistinguishable state under proper-time evolution. Since action is an additive invariant under reparametrization, only integer multiples of the base unit $\mathcal{S}_0$ are consistent with periodic coherence.

Conserved charges emerge from internal symmetries of the trembling deviation field $\xi_{\mu\nu}(x)$. Applying the covariant form of Noether's theorem to the proper-time action under symmetry transformations of the deviation structure yields conserved currents $J^\mu$, each associated with a geometric symmetry class:

$$ U(1) \rightarrow \text{electric charge } q, \qquad (77) $$

$$ SU(2) \rightarrow \text{weak isospin } I^a, \qquad (78) $$

$$ SU(3) \rightarrow \text{color charge } T^a. \qquad (79) $$

Each current integrates to a conserved quantity over a coherence interval. However, because the underlying trembling mode is quantized in action by Equation (76), the total charge must also be quantized:

$$ q = n_q q_0, \quad I^a = n_I I_0^a, \quad T^a = n_T T_0^a, \qquad (80) $$

where $n_q, n_I, n_T \in \mathbb{Z}$, and $q_0, I_0^a, T_0^a$ are the fundamental charge units determined by the holonomy class of the associated trembling symmetry.

Crucially, this result is not an imposed quantization rule, but a structural property of geometric coherence. The trembling mode must return to a causally consistent configuration after a full cycle; fractional or continuously varying charge values would correspond to incomplete or non-closing trembling paths, which are non-physical in TSRT. Hence, discreteness of electric charge and gauge charges is enforced by the geometry of the internal cycle.

This mechanism resolves one of the long-standing puzzles in quantum theory: the origin of charge quantization. In standard quantum field theory, quantization is introduced by hand—either through operator spectra, topological arguments, or spontaneous symmetry breaking. In contrast, TSRT predicts it as an unavoidable consequence of geodesic coherence and proper-time periodicity in the structure of spacetime itself.

Furthermore, this explanation applies uniformly to all known conserved charges. Since all internal symmetries in TSRT arise from coherent trembling deviations—classified by their symmetry group, curvature class, and holonomy—the corresponding charges are automatically discretized by the geometric constraints of localized trembling modes.

Thus, the quantization of charge in TSRT is not an artifact of representation theory or group algebra, but a direct outcome of the causal geometry that governs particle formation.
It unifies action, coherence, and conservation into a single geometric principle: only whole trembling cycles are physically permissible, and each such cycle carries a discrete unit of charge determined by its internal symmetry structure.

Note, however, that while spin is also quantized in TSRT through the holonomy of closed trembling cycles (as detailed in Section 5.1), it arises from geometric phase accumulation under spacetime rotations rather than from internal gauge symmetries. Hence, spin is not interpreted as a charge, but as a separate manifestation of geometric quantization.

# 6 Fundamental Symmetries and Particle Families

In TSRT, gauge symmetries arise not as abstract postulates or internal group assignments, but as manifestations of geometric redundancies in the description of trembling paths. These symmetries are intrinsic to the causal and variational structure of localized trembling modes and reflect the freedom to reparameterize the internal deviation fields without altering physical observables.

Each localized trembling mode evolves along a proper-time geodesic $x^\mu(\tau)$, accompanied by a deviation field $\xi_{\mu\nu}(x)$ describing bounded fluctuations of the spacetime metric. The internal structure of this deviation field admits transformations that leave the overall geodesic solution invariant, provided these transformations preserve causal coherence, boundedness, and action quantization. These internal transformations form a symmetry group $G$, which acts on the class of admissible trembling configurations.

Such transformations are geometrically redundant: they relabel internal structures without changing the causal evolution or energetic content of the mode. In analogy with fiber bundles in gauge theory, the deviation field $\xi_{\mu\nu}(x)$ can be interpreted as defining a local frame of trembling directions over spacetime. Transformations that rotate or rephase this frame while preserving causal structure correspond to gauge symmetries.

Mathematically, the set of admissible trembling configurations defines an equivalence class $[\xi]$, where two deviation fields $\xi_{\mu\nu}(x)$ and $\xi'_{\mu\nu}(x)$ are considered equivalent if they are related by a local transformation that preserves: (i) Proper-time periodicity of the trembling cycle, (ii) Bounded curvature norms within the local support region $\Omega$, (iii) Quantized action integral $\int \mathcal{L} d\tau = nS_0$, (iv) Conservation of the associated Noether current $J^\mu$.

Transformations satisfying these constraints form a local symmetry group $G(x)$ acting at each point of the trembling support. The structure of this group depends on the internal topology and the degree of freedom of the trembling mode:

- $U(1)$: phase redundancy of single-cycle coherent trembling (electromagnetic symmetry),

- $SU(2)$: rotational freedom in isospin-like internal trembling (weak interaction),

- $SU(3)$: color rotations in multi-component trembling fields (strong interaction).

In this picture, gauge transformations correspond to reparametrizations of internal trembling frames that leave the observed trajectory and accumulated action invariant. The associated gauge potentials arise as connection forms that compensate for local changes in the trembling basis to preserve coherence. Thus, the gauge field is not a fundamental entity, but a geometric artifact that ensures internal trembling consistency under local frame variation.
This geometric interpretation resolves long-standing conceptual issues in gauge theory. In standard formulations, gauge symmetry is often viewed as an abstract redundancy without physical origin. In TSRT, by contrast, it is a concrete manifestation of how localized trembling modes are internally structured. The necessity of gauge potentials stems from the requirement to maintain causal closure and coherent phase across internal cycles as the trembling configuration evolves along the geodesic.

Moreover, since all gauge degrees of freedom are tied to the internal topology of trembling paths, their algebraic structure follows from the local holonomy of the deviation field. The Lie group structure of $U(1)$, $SU(2)$, or $SU(3)$ is not arbitrarily chosen, but derives from the type of geometric redundancy permitted by the internal action cycle [19].

This view has several implications. First, gauge fields do not mediate forces in the classical sense; they encode the curvature of internal trembling evolution required for coherent propagation. Second, coupling constants arise as geometric coefficients linking trembling deformation to external spacetime curvature and are derivable from the intrinsic structure of the deviation field. Finally, gauge invariance is a manifestation of the topological equivalence of trembling frames, not an imposed symmetry principle.

In summary, gauge symmetries in TSRT are emergent consequences of internal trembling path redundancies. They guarantee the consistent evolution of localized trembling modes under frame redefinition, enforce charge conservation through Noether currents, and quantize the allowed interaction structure without invoking auxiliary fields or quantum postulates. All gauge interactions thus find their origin in the causal geometry of trembling spacetime.

## 6.1 U(1)-Bound Trembling Modes: Leptons

Leptons arise in TSRT as the simplest class of stable, localized trembling excitations: those whose deviation field $\xi_{\mu\nu}(x)$ transforms under an internal U(1) symmetry group. These modes exhibit Abelian coherence and preserve a well-defined local phase under parallel transport along their trembling cycle. The internal structure of such modes geometrically encodes electric charge, as derived in Section 5.4, where U(1) holonomy corresponds to conserved electromagnetic current.

Each lepton is a stable solution whose internal trembling executes a full U(1) phase loop over a proper-time coherence interval. The minimal excitation—corresponding to one unit of geometric phase winding—generates a mode with elementary electric charge $q = \pm e$, depending on the orientation of the phase evolution. The TSRT framework thereby predicts the natural emergence of lepton-antilepton pairs as conjugate trembling configurations with opposite holonomy.

Because U(1)-bound modes do not support internal SU(3) curvature, leptons are inherently color-neutral. However, some leptonic solutions may exhibit SU(2) chiral asymmetry in their internal trembling structure, distinguishing between left- and right-handed configurations. This difference underlies the physical separation between charged leptons (electron, muon, tau) and their corresponding neutrinos, which carry no electric charge but retain chiral phase symmetry.

TSRT imposes geometric constraints on the number of causally coherent U(1) trembling configurations. Since only a finite number of distinct trembling eigenmodes can maintain in-
ternal phase closure and external stability under causal propagation, the lepton spectrum is discretely bounded. In particular, the number of dynamically allowed phase-locked solutions determines the number of distinct lepton generations. This geometric quantization mechanism yields precisely three stable lepton families, consistent with experimental observation and neutrino oscillation data.

## 6.2 SU(2)-Bound Trembling Modes: Chiral Weak Sector

TSRT predicts the existence of trembling configurations whose internal deviation fields $\xi_{\mu\nu}(x)$ exhibit local SU(2) symmetry. These correspond to chiral trembling modes—solutions whose geometric structure distinguishes between left- and right-handed components of their proper-time geodesics. Unlike U(1)-bound modes, which are governed by Abelian phase coherence, SU(2)-bound modes transform nontrivially under the non-Abelian SU(2) gauge group:

$$ \xi_{\mu\nu}(x) \rightarrow U(x) \xi_{\mu\nu}(x) U^{\dagger}(x), \quad U(x) \in \text{SU}(2). \qquad (81) $$

Such transformation behavior implies that the internal trembling structure couples only to components of the mode that respect SU(2) chiral symmetry. Specifically, left-handed configurations interact dynamically with the deviation field and give rise to conserved weak isospin currents $I^a$, as discussed in Section 5.4. Right-handed components, which do not align with the internal SU(2) geometry, remain inert and appear as singlets—explaining the experimentally observed parity violation in weak interactions.

The internal geometry of SU(2)-bound trembling modes supports exactly two orthogonal coherence directions in trembling space, forming an irreducible doublet under SU(2) transformations. This structure naturally gives rise to the pairing of weak isospin partners, such as $(e^{-}, \nu_{e})$, $(\mu^{-}, \nu_{\mu})$, and $(\tau^{-}, \nu_{\tau})$. In TSRT, these pairings are not imposed but emerge from the requirement that stable trembling solutions maintain internal coherence and causal admissibility within a two-dimensional deviation subspace.

Neutrinos, in this framework, appear as electrically neutral modes whose deviation field participates in SU(2) coherence but lacks U(1) phase coupling. Their neutrality is thus a direct geometric consequence of their internal symmetry: they traverse trembling loops with chiral coherence but accumulate no net U(1) holonomy. This geometric interpretation fully aligns with the known properties of weakly interacting neutrinos and their exclusive left-handed coupling.

## 6.3 SU(3)-Bound Trembling Modes: Quarks and Color

TSRT predicts the existence of localized trembling modes whose deviation fields transform under an internal SU(3) symmetry.<fnref n="37" /> These configurations correspond to non-Abelian trembling structures defined over an eight-dimensional internal color space. The general form of the

<footnote n="37">³⁷Here, SU(3) denotes the special unitary group of 3 × 3 unitary matrices with determinant 1, whose associated Lie algebra is $su(3)$, the set of traceless anti-Hermitian 3 × 3 matrices. Throughout this work, we refer to “SU(3) symmetry” for the group and reserve $su(3)$ for the Lie algebra whose generators define the commutation relations and structure constants.</footnote>
deviation field is given by:

$$ \xi_{\mu\nu}(x) = \sum_{a=1}^{8} [A_{\mu}^{a}(x) v_{\nu} + A_{\nu}^{a}(x) v_{\mu}], \quad A_{\mu}^{a}(x) \in \mathfrak{su}(3), \qquad (82) $$

where $v^\mu$ is the trembling four-velocity, and $A_\mu^a(x)$ are geometric gauge potentials aligned with the eight generators of the SU(3) Lie algebra.

As discussed in Section 5.4, the structure of these modes gives rise to conserved color currents $T^a$, which encode directional flow of internal SU(3) curvature. The non-Abelian nature of the deviation field implies intrinsic self-coupling—geometrically realized through the commutator structure of $\mathfrak{su}(3)$—mirroring the gluon self-interaction dynamics known from QCD, but arising here from curvature in trembling spacetime rather than from a quantized gauge field.

A central feature of SU(3)-bound trembling modes is confinement. In TSRT, the geometric coherence conditions required for long-term stability are violated by any trembling mode with a net non-zero SU(3) color. Isolated color-charged configurations generate divergent curvature at spatial infinity, thereby breaking causal admissibility. As a result, only color-neutral combinations of SU(3) modes form physically observable particles.

The minimal such configurations include quark-antiquark pairs (mesons) and three-quark bound states (baryons), whose trembling deviations form superposed holonomies that cancel net color curvature while preserving local internal structure. These bound configurations emerge dynamically as the only causally coherent SU(3) trembling solutions, providing a geometric explanation for confinement and hadronization without invoking quantized gauge fields or vacuum symmetry breaking.

TSRT thus reproduces both the observed color triplet structure of quarks and their confinement within hadrons, grounded entirely in the differential-geometric constraints on SU(3) trembling modes. Color is not an abstract label but a property of internal curvature structure in deviation space—fully consistent with the Standard Model’s color dynamics, yet derived here from first principles in causal geometry.

## 6.4 Coherent Gravitational Modes: Geometry-Bound Structures

The fourth and final class of fundamental trembling modes in TSRT comprises coherent geometric configurations that generate spacetime curvature without carrying any internal gauge symmetry. These gravitational modes are not associated with internal groups like U(1), SU(2), or SU(3), but arise instead from large-scale, proper-time-coherent fluctuations in the metric in Equation (2), where $\xi_{\mu\nu}(x)$ encodes causally admissible deviations that preserve timelike geodesic structure.

When coherently averaged over finite proper-time intervals, these trembling-induced deviations yield emergent curvature consistent with the effective Einstein field equations:

$$ \langle R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R \rangle_{\tau} = 8\pi G \langle T_{\mu\nu} \rangle_{\tau}, \qquad (83) $$

where $\langle \cdot \rangle_\tau$ denotes coarse-grained integration over a proper-time segment of the trembling geodesic. This relation arises from the TSRT action formalism and does not rely on postu-
lating Einstein's equations as axioms.

Unlike the three gauge-bound trembling classes, these gravitational modes possess no internal symmetry index and are singlets under all internal transformations; and they exhibit curvature coherence across extended regions of spacetime, rather than localized geodesic loops.

Physically, this class accounts for classical gravitational phenomena—including black holes, gravitational waves, and cosmological expansion—as manifestations of macroscopic trembling coherence. These are not quantum excitations, but large-scale emergent geometries shaped by the collective behavior of trembling deviations.

A defining property of gravitational modes is universality: all particles, regardless of their internal trembling symmetry (U(1), SU(2), SU(3)), couple identically to these curvature structures. This universality stems from the fact that all geodesic motion is modulated by the same coarse-grained curvature, and hence all matter responds identically to gravitational trembling.

In TSRT, gravitational interaction is thus not a separate force but the residual macroscopic structure of trembling spacetime itself. The emergence of Einstein-like dynamics becomes a theorem of causal geometric averaging, not a fundamental assumption.

This completes the TSRT classification of interaction modes. The four symmetry classes—U(1), SU(2), SU(3), and the curvature-bound gravitational singlet—exhaust the family of stable, causally admissible trembling solutions in curved spacetime geometry.

## 6.5 Completion of the Symmetry Classification

TSRT identifies four distinct classes of admissible trembling deviations $\xi_{\mu\nu}(x)$: the Abelian U(1), the non-Abelian SU(2) and SU(3), and a coarse curvature class that gives rise to gravitation. This classification is not imposed externally but derived from the variational structure of trembling spacetime itself. A trembling mode is physically admissible only if it preserves causal propagation, supports quantized action cycles in proper time, and remains stable under small curvature fluctuations.

These criteria restrict viable deviations to those geometrically compatible with compact Lie group symmetries acting on tangent space fibers. The only internal symmetries satisfying these constraints—while maintaining Lorentz covariance and proper-time coherence—are U(1), SU(2), and SU(3). Each defines a distinct family of stable trembling modes, corresponding to observed particle interactions.

In the long-wavelength limit, a fourth class emerges in which local trembling deviations blend smoothly with background curvature. This coarse curvature class does not correspond to an internal symmetry but instead describes metric-level geodesic perturbations with globally coherent proper-time behavior. The gravitational interaction thus arises not from a new symmetry group, but from this geometric limit.

This symmetry classification has been derived constructively in the foundational work [19], which demonstrated that no further symmetry class has been identified that satisfies all constraints of causal admissibility, variational closure, and proper-time coherence. However, a fully general proof that no additional symmetry structures can arise in broader functional spaces remains an open mathematical question to be addressed in future work.
Accordingly, the four classes identified here are conjectured to exhaust the space of permissi-
ble trembling geodesic solutions under the TSRT framework, subject to the analyzed constraints.

The resulting particle families—photons and leptons (U(1)), weak-interacting neutrinos and doublets (SU(2)), color-charged quarks (SU(3)), and gravitating mass-energy carriers (coherent curvature)—exhibit all known symmetry properties of observed matter without invoking quantum postulates.

This constrained symmetry structure implies that any further interaction types or gauge extensions would require either violation of causal admissibility or refinement of the trembling classification, making TSRT empirically falsifiable.

# 7 Interaction Mechanisms in Trembling Spacetime

## 7.1 Introduction to Geometric Interaction

In Trembling Spacetime Relativity Theory (TSRT), the origin of all fundamental interactions is purely geometric. Unlike conventional quantum field theory, which describes forces via exchange of quantized gauge bosons or virtual particles, TSRT posits that interaction arises from the coupling of localized trembling modes through their shared influence on spacetime curvature. Each trembling mode contributes a deviation field $\xi_{\mu\nu}(x)$ to the effective metric. The combined geometry then determines how other modes propagate, leading to observable interaction effects.

This framework has been demonstrated in detail for atomic systems in earlier work [20], where electromagnetic binding potentials and characteristic spectral lines were derived without invoking quantized photon exchange. Instead, electromagnetic forces emerge as the cumulative deflection of geodesics due to coherent abelian trembling modes. Similarly, color and weak interactions follow from non-abelian deviation structures, which perturb the motion of other localized modes by deforming the background curvature.

Importantly, no virtual excitations or probabilistic vacuum fluctuations are required in this approach. All forces are mediated by continuous, deterministic changes in the metric caused by the presence and coherence properties of trembling configurations. As discussed in Section 5, the quantized charges and spins of particles are themselves geometric consequences of the internal trembling symmetry; interactions therefore arise from the overlap and alignment of these internal structures.

This section formalizes the mechanism by which trembling modes couple through shared curvature. We begin by defining how individual deviation fields combine to form an effective metric, then show how coherence and phase relations determine the strength and character of the resulting forces. Finally, we illustrate how familiar potentials such as Coulomb and Yukawa forms emerge as integrals over curvature contributions in the trembling manifold, consistent with the atomic-scale predictions of TSRT [20].

## 7.2 Mode Coupling via Shared Geodesic Deformation

In TSRT, interaction between particles arises because each localized trembling mode contributes a geometric deformation of spacetime that affects the geodesics of all other modes. This process
is deterministic and does not rely on discrete force carriers. Instead, it proceeds through the superposition of deviation fields, through an equation equivalent to Equation (2):

$$g_{\mu\nu}(x) = \eta_{\mu\nu} + \sum_{i} \xi_{\mu\nu}^{(i)}(x), \qquad (84)$$

where the sum runs over all trembling modes present in the spacetime region under consideration.

Each trembling mode evolves along a geodesic determined by this composite metric. Consequently, its motion is continuously deflected by the presence of other modes:

$$\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta}(x) \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0, \qquad (85)$$

where $\Gamma^\mu_{\alpha\beta}$ is the Christoffel connection built from the total metric perturbation.

This formulation recovers the familiar concept of a force as a geodesic deviation induced by other localized modes. For example, in the electromagnetic case, an abelian trembling mode generates a metric deformation that precisely reproduces the Lorentz force term in the weak-curvature limit [20]. The deflection of the geodesic encodes the effective potential energy of interaction, but no separate field operator or quantum excitation is invoked.

Importantly, the geometric superposition principle is compatible with causal admissibility. Because each $\xi_{\mu\nu}^{(i)}(x)$ satisfies the bounded deviation constraints (Equation (45)), their sum remains within the small-curvature regime necessary for proper-time parametrization and variational stability.

This mechanism implies that interaction strength is not a universal constant but depends on the local configuration of trembling modes. Their coherence, phase alignment, and spatial overlap determine the effective curvature gradients and thus the induced deflection of geodesics. In this way, TSRT provides a unified geometric picture of all fundamental interactions as consequences of shared deformation of causal structure.

## 7.3 Coherence Overlap and Interaction Strength

In TSRT, the effective strength of interaction between trembling modes is determined by the degree of coherence overlap between their internal oscillations. Unlike conventional field theories, where coupling constants are inserted by hand, here the coupling arises dynamically from the phase relationship of deviation fields.

Consider two localized trembling modes, $P_1$ and $P_2$, with deviation fields $\xi_{\mu\nu}^{(1)}(x)$ and $\xi_{\mu\nu}^{(2)}(x)$. Their coherence overlap can be quantified by the phase-aligned inner product:

$$C = \int_{\Sigma} \xi_{\mu\nu}^{(1)}(x) \xi^{(2)\mu\nu}(x) \cos[\phi_1(x) - \phi_2(x)] d\Sigma, \qquad (86)$$

where $\phi_i(x)$ denotes the local phase of the trembling cycle, and the integration is over a spacelike hypersurface $\Sigma$.

When the phase difference $\phi_1 - \phi_2$ remains approximately constant across the support of the modes, constructive interference increases the effective curvature in the interaction region.
This leads to a stronger deflection of each mode's geodesic, manifesting macroscopically as an attractive or repulsive force proportional to $C$. Conversely, phase decoherence—random or rapidly varying phase differences—suppresses the interaction by destructive interference of deviation fields.

This coherence criterion provides an intuitive geometric explanation for the decay of interaction strength with distance or frequency mismatch. For example, in the electromagnetic case, coherence of U(1)-bound trembling modes corresponds to synchronized oscillations of the deviation field, reproducing classical Coulomb and radiation effects in the weak-curvature regime [20].

Moreover, TSRT predicts that interaction potentials are not fixed but can fluctuate dynamically as the internal phases of trembling modes evolve. This is in stark contrast with static force fields of classical theory or the instantaneous exchange diagrams of quantum field theory. Instead, coherence-based interaction is continuous, deterministic, and inherently tied to the causal structure of spacetime.

This formulation also explains why some modes—such as color-confined SU(3) trembling configurations—interact strongly only within compact domains of phase alignment. Outside these regions, their rapidly oscillating phases suppress long-range coupling, providing a natural geometric underpinning of confinement and short-range forces.

## 7.4 Effective Potentials from Curvature Accumulation

A key insight of TSRT is that the cumulative deformation of geodesics induced by interacting trembling modes gives rise to emergent effective potentials. Unlike traditional field theory, where potentials are attributed to static fields or virtual exchange particles, here they arise as integrated curvature effects along proper-time trajectories.

Consider a localized trembling mode propagating through the deviation field $\xi_{\mu\nu}^{(\text{int})}(x)$ induced by another mode. The effective potential $V_{\text{eff}}$ governing its evolution can be expressed as:

$$V_{\text{eff}}(\tau) = \int_{\gamma(\tau)} \mathcal{R}[\xi_{\mu\nu}^{(\text{int})}(x)] d\tau, \qquad (87)$$

where $\mathcal{R}$ denotes an invariant scalar curvature functional constructed from the deviation field, and $\gamma(\tau)$ is the proper-time path of the mode.

In the weak-curvature limit, this formulation reproduces familiar interaction potentials. For example, when the interacting deviation corresponds to a U(1)-bound trembling mode with quasi-static coherence, the curvature accumulation yields a Coulomb-like potential:

$$V_{\text{Coulomb}}(r) \propto \frac{q_1 q_2}{r}, \qquad (88)$$

where $r$ is the spatial separation and $q_i$ are the quantized geometric charges of the modes. This correspondence is derived rigorously in earlier work [20].

Similarly, for SU(2) trembling modes with partial coherence decay over finite intervals, the effective potential exhibits an exponential suppression, reproducing a Yukawa-type form:

$$V_{\text{Yukawa}}(r) \propto \frac{g^2}{r} \exp(-\mu r), \qquad (89)$$
where $\mu$ encodes the inverse coherence length of the trembling mode's internal phase structure. In TSRT, this decay arises not from spontaneous symmetry breaking or mass insertion but as a geometric consequence of partial phase decoherence.

These examples demonstrate that TSRT captures the correct large-scale phenomenology of known forces as emergent curvature accumulations, while remaining fully deterministic and variational. The quantization of charges and coherence intervals ensures that the effective potentials are not arbitrary but determined by the underlying geometry of trembling spacetime.

This approach eliminates the need for virtual particle exchange: interaction strength and range follow directly from the curvature integral along causally admissible geodesics. As such, TSRT offers a unified framework in which the form of interaction potentials is a derived feature of the same trembling dynamics that define particle identity.

## 7.5 Exchange Interpretation Without Virtual Particles

One of the conceptual breakthroughs of TSRT is the elimination of virtual particles as mediators of interaction. In conventional quantum field theory, forces are modeled as the exchange of off-shell quanta—virtual photons, gluons, or W/Z bosons—whose probabilistic emission and reabsorption yield effective potentials. While operationally successful, this framework relies on formal constructions (Feynman diagrams, path integrals) that do not correspond to any underlying deterministic geometry.

By contrast, in TSRT, the analogy of "exchange" arises purely as a geometric artifact: it reflects the continuous deformation of proper-time geodesics in the presence of coherent trembling curvature, not discrete quanta hopping between particles. As discussed in preceding work [20], the apparent transfer of action or momentum is the result of phase-adjusted coupling between deviation fields, leading to redistribution of geodesic deflection along interacting trajectories.

Formally, when two localized trembling modes $P_1$ and $P_2$ are causally near, their combined deviation field

$$ \xi_{\mu\nu}(x) = \xi_{\mu\nu}^{(1)}(x) + \xi_{\mu\nu}^{(2)}(x) \qquad (90) $$

induces reciprocal perturbations to the geodesic equation:

$$ \frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta}[\xi](x) \frac{dx^\alpha}{d\tau} \frac{dx^\beta}{d\tau} = 0. \qquad (91) $$

The interaction thus appears as mutual curvature entanglement rather than an exchange of discrete carriers.

This geometric viewpoint clarifies several longstanding interpretive puzzles. Indeed, the absence of virtual particles removes ambiguity about whether intermediate states "exist." Furthermore, interaction is always causal, without backward-in-time propagation or unphysical energies. Finally, no renormalization infinities arise, since trembling deviations remain bounded by coherence constraints.

In TSRT, the "exchange" language persists only as a heuristic: what field theory interprets as emission and absorption corresponds here to reconfiguration of curvature overlap between trembling modes. No quantized packets are emitted or absorbed in the spacetime manifold itself.
This deterministic account reconciles interaction with causal geometry and preserves a transparent correspondence between the internal coherence structure of each trembling mode and the observed effective potential. As such, TSRT provides a conceptually cleaner framework in which the success of QFT emerges as a limiting approximation of coherent trembling curvature, rather than as evidence of ontological virtual particles.

## 7.6 Implications for Bound States and Scattering

The geometric interaction mechanism of TSRT yields direct implications for bound-state formation, confinement phenomena, and decay processes. Because all interactions derive from coherent curvature overlap, any stable bound system must correspond to a joint trembling configuration that preserves causal admissibility and action quantization over extended intervals of proper time.

For example, in atomic structure [20], the effective Coulomb potential emerges from the integrated curvature scalar of overlapping U(1) deviation fields between electron and nucleus. The localized trembling cycles synchronize their internal phase evolution, forming a coherent compound mode whose action remains quantized. This provides a deterministic explanation for discrete energy levels and spectral lines without invoking probabilistic electron clouds or superposed states.

Similarly, in color confinement, SU(3)-bound trembling modes cannot propagate independently at macroscopic scales. Any attempt to isolate a color-charged trembling configuration results in divergence of curvature and loss of phase coherence, enforcing confinement as a geometric necessity rather than an imposed feature of QCD.

In scattering scenarios, the coherence overlap criterion predicts that cross sections are governed by phase alignment and curvature accumulation during interaction. The effective deflection angle can be computed by integrating the deviation-induced geodesic perturbation over the interaction region:

$$ \Delta\theta \propto \int_{\tau_i}^{\tau_f} \mathcal{R}[\xi_{\mu\nu}^{\text{int}}(\tau)] d\tau, \qquad (92) $$

where $\mathcal{R}$ encodes the accumulated curvature scalar of the combined trembling fields.

For weak decays, the finite coherence lifetime of SU(2)-bound trembling modes accounts for the instability of massive particles such as muons or heavy quarks. Their decay is interpreted not as a stochastic process but as deterministic decoherence: the internal trembling cycle eventually fails to close coherently, leading to geometric fragmentation into lighter trembling modes.

These insights extend and refine the framework established in earlier work [20], showing that TSRT naturally predicts the discreteness and stability of atomic bound states, the inevitability of color confinement, the causally determined lifetimes of weakly interacting particles, and the phase-dependent character of scattering interactions.

A detailed and quantitative formulation of these processes—including precise definitions of coherence overlap functionals, cross sections, and decay thresholds—is provided in Section 11.
## 7.7 Synthesis of Interaction Mechanisms

This section has established that all fundamental interactions in TSRT emerge from the causal geometry of trembling spacetime, without requiring field quantization, virtual particles, or auxiliary postulates. The coupling of localized trembling modes is mediated entirely through their combined deviation fields, which deform geodesics in a fully deterministic and variationally consistent manner.

Several core insights have been highlighted. For instance, mutual deflection arises from superposed metric deviations and preserves causal admissibility. Also, the strength of interactions is set by coherence overlap and phase synchronization, unifying long- and short-range forces within a single geometric principle. Furthermore, effective potentials—such as Coulomb, Yukawa, and confinement-like forms—result from curvature accumulation over proper time, rather than exchange of virtual quanta. Finally, bound states, decay lifetimes, and scattering processes are all explained as consequences of coherence dynamics and action quantization.

These results build directly upon the earlier treatment of atomic structure [20], extending it to encompass the full Standard Model spectrum within a single geometric framework.

Having established the geometric origin and causal mechanism of interactions, Section 8 recapitulates the mathematical structure of trembling eigenmodes, while Section 9 turns to the classification of Standard Model particles in TSRT, showing in detail how each known particle corresponds to a unique symmetry-bound trembling configuration with quantized properties. Section 10 consequently exposes how the Standard Model's fundamental particles appear in TSRT.

# 8 Mathematical Structure of Trembling Eigenmodes

In TSRT, each fundamental particle corresponds to a localized, stable trembling eigenmode—an admissible, symmetry-bound excitation of the deviation field $\xi_{\mu\nu}(x)$ in curved spacetime. These eigenmodes arise not from quantized fields or imposed boundary conditions, but from variationally stable solutions to the trembling geodesic equation under internal symmetry constraints.

The core mathematical structure begins with the perturbed metric of Equation (2), where $\xi_{\mu\nu}(x)$ encodes localized trembling deviations that remain bounded over proper-time intervals. These deviations must satisfy two distinct conditions: (i) Causal admissibility, meaning the deviation field respects the light-cone structure and remains bounded in magnitude throughout its evolution; and (ii) Periodicity in proper time, meaning the deviation configuration recurs identically after each coherence interval. Formally, this periodicity is expressed by:

$$ \xi_{\mu\nu}(x^{\alpha} + \Delta x^{\alpha}) = \xi_{\mu\nu}(x^{\alpha}), \qquad (93) $$

where

$$ \Delta x^{\alpha} = u^{\alpha} \Delta \tau. \qquad (94) $$

Here, $u^{\alpha} = \frac{dx^{\alpha}}{d\tau}$ is the local four-velocity of the trembling mode, indicating its proper-time parametrized direction of evolution, and $\Delta \tau$ is the finite proper-time interval corresponding to a complete trembling cycle (i.e., the time it takes for the internal oscillation to return to
its initial phase configuration). This relation ensures that the trembling deviation is not only bounded and causal but also closed and recurrent in proper time.

The trembling deviation must obey a variational stability condition under causal reparametriza-
tion of the proper-time path:

$$
\delta \int_{\tau_0}^{\tau_0 + \Delta\tau} \mathcal{L}[x^{\mu}(\tau), \xi_{\mu\nu}(x), \dot{x}^{\mu}(\tau)] d\tau = 0, \quad (95)
$$

where the effective Lagrangian $\mathcal{L}$ includes curvature couplings induced by $\xi_{\mu\nu}(x)$ and is con-
structed to preserve local Lorentz invariance.

Stable trembling eigenmodes are defined by four essential properties. First, localization: the
deviation field $\xi_{\mu\nu}(x)$ has compact spatial support and remains confined without dispersion over
the mode's coherence interval. Second, periodicity: the configuration completes closed cycles in
proper time and returns to an equivalent geometric state after a finite interval $\Delta\tau$. Third, holon-
omy discreteness: the net geometric phase accumulated during one complete trembling cycle is
constrained to discrete values, resulting in distinct, invariant internal angular momentum (com-
monly interpreted as spin) or conserved charges. This discreteness arises deterministically from
the topological closure of the trembling path and is not an artifact of probabilistic quantization
procedures. Finally, each mode must satisfy a symmetry constraint: it transforms covariantly
under one of the admissible internal symmetry groups —namely U(1), SU(2), or SU(3)— or
remains symmetry-free in the gravitational sector.

Each eigenmode corresponds to an extremum of the action functional over the trembling path. Its associated conserved quantities—mass, spin, and charge—arise from geometric integrals over the deviation field:

$$
m = \frac{1}{c^2} \left\langle \sqrt{\frac{d^2 x^\mu}{d\tau^2} g_{\mu\nu} \frac{d^2 x^\nu}{d\tau^2}} \right\rangle_\tau, \quad (96)
$$

as in Equation (33),

$$
S^{\mu\nu} = \int_{\Sigma} (x^{\mu} \dot{x}^{\nu} - x^{\nu} \dot{x}^{\mu}) d\tau, \quad (97)
$$

as in Equation (51), and

$$
Q = \oint J^{\mu} d\Sigma_{\mu}, \qquad (98)
$$

as in Equation (62). Here, $\Sigma$ denotes the spatial hypersurface spanned by the internal trembling,
and $J^\mu$ is the symmetry-induced current associated with Noether's theorem.

In this formulation, eigenmodes are not “particles” in a quantum field theoretic sense, but
self-consistent, conservative spacetime structures that repeat deterministically and satisfy topo-
logical and variational constraints. The quantization of their physical properties results from
the geometry and stability of these trembling cycles.

In summary, trembling eigenmodes in TSRT behave as persistent geometric structures—stable,
localized oscillatory patterns of spacetime itself. Each mode is like a self-sustained whirl or knot
in the geometry, whose mass, spin, and charges are not imposed quantities but natural outcomes
of its conserved internal motion. These properties arise from how the mode twists, cycles, and
interacts with the symmetries of spacetime over closed intervals of proper time.

With this framework in place, we are now equipped to connect these geometric entities to the
known particles of nature. The next section demonstrates how the full spectrum of Standard
Model fermions and bosons arises from specific families of such trembling eigenstates—each
uniquely classified by its symmetry-bound structure and conserved geometric features.

# 9 Overview of the Standard Model Particle Spectrum

Before comparing the Standard Model (SM) with the TSRT-based classification of particles
in Section 10, it is necessary to summarize the established particle content of the Standard
Model of particle physics. The SM successfully accounts for all known fundamental particles
and interactions (except gravity) and organizes them based on spin, mass, electric charge, and
gauge interactions. This section outlines the key particle types, grouped according to their
empirical roles and quantum numbers.

## 9.1 Fermions: Matter Constituents

Fermions are the foundational building blocks of all known matter. In the Standard Model (SM) of particle physics, there are twelve distinct fundamental fermions, each with spin-$\frac{1}{2}$ and obeying the Pauli exclusion principle [75]. These are grouped into three generations, where each generation contains four fermions: two quarks and two leptons [27, 76]. While the generations differ massively in mass, they possess identical gauge charges and interaction structures under the SM's $SU(3)_C \times SU(2)_L \times U(1)_Y$ symmetry [26].

**Quarks.** Quarks are the constituents of hadrons such as protons and neutrons [76] and carry color charge, participating in the strong interaction mediated by gluons [26]. They also carry electric charge and, for all except the top quark, form bound states due to color confinement [27, 77]. Each generation contains one up-type quark with charge $+\frac{2}{3}e$ and one down-type quark with charge $-\frac{1}{3}e$. The three generations of quarks are:

• First generation: up ($u$), down ($d$)

• Second generation: charm ($c$), strange ($s$)

• Third generation: top ($t$), bottom ($b$)

**Leptons.** Leptons do not experience the strong interaction [76] and consist of charged and neutral species [27]. Each generation includes one charged lepton with charge $-e$ and one electrically neutral neutrino [26]. The three generations of leptons are:

• First generation: electron ($e^-$), electron neutrino ($\nu_e$) [78]

• Second generation: muon ($\mu^-$), muon neutrino ($\nu_\mu$) [79]

• Third generation: tau ($\tau^-$), tau neutrino ($\nu_\tau$) [78]
**Mass Hierarchy and Mixing.** The fermion masses span several orders of magnitude [76]. For example, the top quark has a mass of approximately 173 GeV/c², while the electron mass is only about 0.511 MeV/c² [76]. Although each generation follows the same charge and interaction structure [27], heavier generations are unstable and decay to lighter ones via the weak interaction [26]. Neutrinos are nearly massless and can oscillate between generations [80], a phenomenon confirmed experimentally and requiring extensions beyond the minimal SM [81].

**Interaction Structure.** All fermions interact with the electroweak force: charged fermions couple to the electromagnetic field (via $U(1)_{EM}$), while both charged fermions and neutrinos couple to the weak force (via $SU(2)_L$). Only quarks participate in the strong interaction (via $SU(3)_C$) through their color charge.

To summarize, these twelve fermions provide the full matter content of the SM. The classification into generations is based on empirical mass and decay structure rather than symmetry requirements, and the reason for three—and only three—generations remains an open question within the Standard Model framework. In contrast, TSRT will provide a geometric classification in Section 10 in which the emergence, stability, and interaction properties of fermions follow from the underlying structure of trembling spacetime.

## 9.2 Gauge Bosons: Force Mediators

The Standard Model describes three of the four known fundamental interactions—electromagnetic, weak, and strong—through the framework of gauge theory [27]. Each of these interactions is mediated by vector bosons, which are spin-1 particles corresponding to quanta of underlying gauge fields [26]. These gauge bosons are not matter particles but carriers of force, arising from the requirement of local symmetry invariance under the gauge groups $U(1)$, $SU(2)$, and $SU(3)$ [7, 82].

**Electromagnetic Interaction (U(1)).** The photon ($\gamma$) is the massless gauge boson of the $U(1)$ symmetry group associated with electromagnetism. It mediates interactions between electrically charged particles. Because it is massless and uncharged, the photon enables long-range electromagnetic forces and does not self-interact. Its properties are fully described by quantum electrodynamics (QED), the $U(1)$ gauge theory.

**Weak Interaction (SU(2)).** The weak force is mediated by three massive vector bosons: the $W^+$, $W^-$, and $Z^0$. These arise from the $SU(2)_L$ gauge symmetry associated with left-handed weak isospin. Unlike the photon, the weak bosons are massive, which limits the range of the weak interaction. The charged $W^\pm$ bosons mediate flavor-changing processes such as beta decay, while the neutral $Z^0$ boson mediates neutral current processes. The weak force uniquely violates parity symmetry, acting only on left-handed fermions.

**Electroweak Unification.** At high energies, the electromagnetic and weak interactions unify into a single electroweak force described by the $SU(2)_L \times U(1)_Y$ gauge group. The photon
and the weak bosons arise as linear combinations of the original gauge fields via spontaneous symmetry breaking in the Higgs sector. This mixing is parametrized by the weak mixing angle $\theta_W$, defining the photon as the massless mode and the $Z^0$ as the massive neutral counterpart.

**Strong Interaction (SU(3)).** The strong force is governed by quantum chromodynamics (QCD), the SU(3)$_C$ gauge theory.<fnref n="38" /> It is mediated by eight massless gluons, $g^a$ with $a = 1, \dots, 8$, which carry color charge and interact with one another due to the non-Abelian nature of the SU(3) group. This self-interaction leads to two essential properties: confinement (quarks are never observed in isolation) and asymptotic freedom (interactions become weaker at shorter distances).

In summary, all SM gauge bosons are spin-1 and arise directly from the imposition of local gauge symmetry. Their properties—mass, charge, and interaction structure—are consequences of the symmetry group to which they belong. In TSRT, these vector bosons are reinterpreted not as fundamental particles, but as specific trembling modes of the spacetime metric, a reinterpretation that maintains empirical agreement while eliminating field quantization assumptions.

## 9.3 Scalar Boson: Higgs Particle

The Higgs boson is the only fundamental scalar (spin-0) particle in the Standard Model (SM). It plays a central role in the mechanism of electroweak symmetry breaking and the generation of mass for both gauge bosons and fermions. This mechanism is known as the Brout-Englert-Higgs mechanism, first proposed in 1964 [11, 52, 83] and incorporated into the unified electroweak theory developed by Weinberg and Salam [9].

**The Higgs Field and Symmetry Breaking.** In the SM, a complex scalar field $\phi$ is introduced, transforming as a doublet under<fnref n="39" /> SU(2)$_L$ and carrying hypercharge under<fnref n="40" /> U(1)$_Y$. The Higgs potential has the form:

$$V(\phi) = \mu^2 \phi^\dagger \phi + \lambda (\phi^\dagger \phi)^2, \qquad (99)$$

where $\mu^2 < 0$ and $\lambda > 0$, producing a "Mexican-hat" potential. This causes the field to acquire a nonzero vacuum expectation value (VEV) $v \approx 246$ GeV, spontaneously breaking the SU(2)$_L \times U(1)_Y$ symmetry down to U(1)$_{EM}$, the symmetry group of electromagnetism.

<footnote n="38">³⁸The subscript $C$ in SU(3)$_C$ denotes color charge, the fundamental degree of freedom in QCD. Unlike the flavor symmetry SU(3)$_F$ of quark types $(u, d, s, \dots)$, SU(3)$_C$ acts on the three color states (red, green, blue) of quarks and gluons. The $C$ emphasizes that this is a local gauge symmetry mediating the strong force, with gluons (the gauge bosons) transforming under the adjoint representation of SU(3)$_C$.</footnote>

<footnote n="39">³⁹The subscript $L$ in SU(2)$_L$ denotes that this group acts exclusively on left-handed fermion fields (and the Higgs doublet) in the Standard Model. It distinguishes the weak isospin group from a hypothetical SU(2)$_R$ (absent in the SM). The $L$ reflects the chiral structure of the electroweak theory, where only left-handed particles couple to $W^\pm/Z$ bosons.</footnote>

<footnote n="40">⁴⁰The subscript $Y$ in U(1)$_Y$ indicates the hypercharge gauge group, whose charge $Y$ is related to electric charge $Q$ and weak isospin $I_3$ by $Q = I_3 + \frac{Y}{2}$. Unlike U(1)$_{em}$, it couples to both left- and right-handed fields. The $Y$ emphasizes its role as the unbroken subgroup before electroweak symmetry breaking (EWSB), after which it combines with SU(2)$_L$ to give U(1)$_{em}$.</footnote>
**Mass Generation.** The nonzero VEV of the Higgs field gives mass to the weak bosons through gauge-invariant kinetic terms. The photon remains massless, consistent with long-range electromagnetic forces, while the $W^{\pm}$ and $Z^0$ acquire mass in a ratio determined by the weak mixing angle. Fermions also acquire mass via Yukawa couplings, which couple the Higgs field to left- and right-handed fermion fields. Each fermion's mass is proportional to its Yukawa coupling constant.

**Experimental Discovery.** The Higgs boson was discovered in 2012 by the ATLAS and CMS collaborations at CERN's Large Hadron Collider (LHC), through its decay into multiple final states including photon pairs $(\gamma\gamma)$, $ZZ^*$ → $4\ell$, and bottom quark-antiquark pairs [54, 84]. It was observed as a resonance peak at 125 GeV/c$^2$, consistent with SM predictions. Because the Higgs boson decays rapidly (with a lifetime on the order of $10^{-22}$ s), its presence is inferred statistically from decay products.

In summary, the Higgs boson is central to the SM's explanation of why particles have mass. It provides a unifying mechanism through spontaneous symmetry breaking, linking gauge invariance to particle masses via its scalar field dynamics. In TSRT, this same phenomenon is reinterpreted geometrically, as arising from curvature-induced trembling modes of spacetime, thereby eliminating the need for a fundamental scalar field while preserving all experimental observations.

## 9.4 Particle Properties and Quantum Numbers

Each particle in the Standard Model (SM) is defined by a unique set of intrinsic properties known as quantum numbers. These numbers classify particles according to their behavior under the SM's gauge symmetries—$U(1)_Y$, $SU(2)_L$, and $SU(3)_C$—and determine how they interact via the fundamental forces. The principal quantum numbers include:

**Electric Charge ($Q$).** This determines a particle's coupling to the electromagnetic field. It is related to weak isospin and hypercharge via the Gell-Mann–Nishijima relation:

$$Q = T_3 + \frac{Y}{2}, \qquad (100)$$

where $T_3$ is the third component of weak isospin and $Y$ is the weak hypercharge. Electric charge is conserved in all SM interactions.

**Weak Isospin ($T$).** Particles that participate in the weak interaction belong to $SU(2)_L$ multiplets. Left-handed fermions form weak isospin doublets, while right-handed fermions are singlets. The $W^{\pm}$ and $Z^0$ bosons couple to this degree of freedom.

**Hypercharge ($Y$).** Associated with the $U(1)_Y$ gauge symmetry, hypercharge determines how particles mix under electroweak unification. It plays a role in determining electric charge via the above relation.
**Color Charge.** Quarks carry one of three color charges—red, green, or blue—under SU(3)$_C$, and gluons carry combinations of color and anticolor. Leptons and the photon are color-neutral. Color charge determines a particle's participation in the strong interaction. The confinement of color ensures that only color-neutral combinations (hadrons) are observed in isolation.

**Spin.** All SM particles have a well-defined intrinsic angular momentum. Fermions have spin-$\frac{1}{2}$, bosons mediating interactions have spin-1, and the Higgs boson has spin-0. Spin determines quantum statistics: fermions obey Fermi–Dirac statistics (Pauli exclusion principle), while bosons obey Bose–Einstein statistics.

**Mass.** In the SM, mass arises from the coupling to the Higgs field. Gauge bosons acquire mass through spontaneous symmetry breaking, while fermions gain mass via Yukawa couplings. Neutrino masses, not originally included in the minimal SM, require extensions such as the seesaw mechanism.<fnref n="41" />

**Other Quantum Numbers.** Additional classification numbers include lepton number (conserved in most processes), baryon number, and generation number. Some, such as lepton and baryon number, are conserved empirically but not strictly required by the SM's gauge symmetries, suggesting possible violation in physics beyond the SM.

**Conservation Laws.** The quantum numbers above define various conservation laws that govern all known interactions. These include:

* **Electric charge conservation:** Exact symmetry of $U(1)_{EM}$.

* **Color confinement:** Consequence of $SU(3)_C$ non-Abelian gauge symmetry.

* **Lepton and baryon number conservation:** Empirically valid at low energies, but not enforced by any geometric symmetry in TSRT; small violations may arise from nontrivial trembling transitions, analogous to nonperturbative effects.

* **Spin and angular momentum conservation:** Always preserved under Lorentz invariance.

Together, these quantum numbers and their associated symmetries fully determine the behavior and classification of all SM particles. They also provide a critical benchmark for comparing with alternative theories such as TSRT, which seeks to recover these properties from geometric principles alone.

<footnote n="41">⁴¹The seesaw mechanism explains the smallness of observed neutrino masses by introducing heavy right-handed neutrinos $N_R$ (sterile under $SU(2)_L \times U(1)_Y$). These couple to left-handed neutrinos $\nu_L$ via the Yukawa interaction $y_\nu \bar{\nu}_L \tilde{\phi} N_R + \text{h.c.}$, and possess a Majorana mass term $M_N \bar{N}_R^c N_R$. Diagonalizing the mass matrix yields light neutrinos $m_\nu \sim y_\nu^2 v^2 / M_N$ (with $v = 246$ GeV the Higgs vacuum expectation value) and heavy partners $m_N \sim M_N$. For $M_N \gg v$, this naturally suppresses $m_\nu$ below eV scales. See [85] for the original proposal and [86, 87] for early formulations.</footnote>
## 9.5 Summary Table of SM Particles

For clarity and reference, we summarize the full spectrum of Standard Model particles, organized by type and generation. The table below lists each particle's name, symbol, mass, electric charge, spin, and participation in the fundamental interactions: electromagnetic (EM), weak (W), and strong (S).

All masses are given in GeV/c²; electric charges are in units of the elementary charge e; and spins are dimensionless. A checkmark (✓) indicates that the particle participates in the corresponding interaction. The given values are indicative; note that neutrino masses are upper limits. Refined values can be found in the literature.

<table><thead><tr><th>Particle</th><th>Symbol</th><th>Mass</th><th>Charge</th><th>Spin</th><th>Gen.</th><th>EM</th><th>W</th><th>S</th></tr></thead><tbody><tr><td colspan="9"><strong>Leptons</strong></td></tr><tr><td>Electron</td><td>e<sup>-</sup></td><td>0.000511</td><td>-1</td><td>1/2</td><td>1</td><td>✓</td><td>✓</td><td></td></tr><tr><td>Muon</td><td>&mu;<sup>-</sup></td><td>0.1057</td><td>-1</td><td>1/2</td><td>2</td><td>✓</td><td>✓</td><td></td></tr><tr><td>Tau</td><td>&tau;<sup>-</sup></td><td>1.776</td><td>-1</td><td>1/2</td><td>3</td><td>✓</td><td>✓</td><td></td></tr><tr><td>Electron Neutrino</td><td>&nu;<sub>e</sub></td><td>&lt; 10<sup>-9</sup></td><td>0</td><td>1/2</td><td>1</td><td></td><td>✓</td><td></td></tr><tr><td>Muon Neutrino</td><td>&nu;<sub>&mu;</sub></td><td>&lt; 10<sup>-9</sup></td><td>0</td><td>1/2</td><td>2</td><td></td><td>✓</td><td></td></tr><tr><td>Tau Neutrino</td><td>&nu;<sub>&tau;</sub></td><td>&lt; 10<sup>-9</sup></td><td>0</td><td>1/2</td><td>3</td><td></td><td>✓</td><td></td></tr><tr><td colspan="9"><strong>Quarks</strong></td></tr><tr><td>Up</td><td>u</td><td>0.0022</td><td>+2/3</td><td>1/2</td><td>1</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>Down</td><td>d</td><td>0.0047</td><td>-1/3</td><td>1/2</td><td>1</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>Charm</td><td>c</td><td>1.27</td><td>+2/3</td><td>1/2</td><td>2</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>Strange</td><td>s</td><td>0.096</td><td>-1/3</td><td>1/2</td><td>2</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>Top</td><td>t</td><td>172.76</td><td>+2/3</td><td>1/2</td><td>3</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td>Bottom</td><td>b</td><td>4.18</td><td>-1/3</td><td>1/2</td><td>3</td><td>✓</td><td>✓</td><td>✓</td></tr><tr><td colspan="9"><strong>Gauge Bosons</strong></td></tr><tr><td>Photon</td><td>&gamma;</td><td>0</td><td>0</td><td>1</td><td>-</td><td></td><td>✓</td><td></td></tr><tr><td>W Boson</td><td>W<sup>&pm;</sup></td><td>80.379</td><td>&pm;1</td><td>1</td><td>-</td><td>✓</td><td>✓</td><td></td></tr><tr><td>Z Boson</td><td>Z<sup>0</sup></td><td>91.1876</td><td>0</td><td>1</td><td>-</td><td></td><td>✓</td><td></td></tr><tr><td>Gluon</td><td>g</td><td>0</td><td>0</td><td>1</td><td>-</td><td></td><td></td><td>✓</td></tr><tr><td colspan="9"><strong>Scalar Boson</strong></td></tr><tr><td>Higgs Boson</td><td>H</td><td>125.25</td><td>0</td><td>0</td><td>-</td><td></td><td></td><td></td></tr></tbody></table>

This complete listing provides a reference benchmark for the detailed classification and interpretation that follows in the next sections, where each of these particles is derived as a symmetry-bound mode of trembling spacetime within TSRT.
# 10 Derivation of the Standard Model Particle Spectrum from Trembling Geometry

In conventional physics, the Standard Model (SM), briefly summarized in Section 9, organizes particles—including fermions, gauge bosons, and the Higgs boson—as quantized excitations of background fields, each endowed with mass, spin, electric charge, isospin, and color charge. These properties are typically assigned by constructing Lagrangians that impose internal symmetry groups by fiat.

By contrast, Trembling Spacetime Relativity Theory (TSRT) [19] derives these same structures and properties as emergent, deterministic consequences of spacetime geometry itself. In this framework, a fundamental particle is not a pointlike object nor a probabilistic wavepacket, but a stable, causally coherent trembling eigenmode of the deviation field $\xi_{\mu\nu}(x)$ defined over the effective metric in Equation (2). Such eigenmodes arise as variationally stable, symmetry-bound solutions to the trembling geodesic equations derived from the action principle, as in Equation (47) and Equation (54):

$$S = \oint \mathcal{L}(x^{\mu}, \dot{x}^{\mu}, \xi_{\mu\nu}) d\tau, \qquad (101)$$

where coherence and closure conditions quantize the allowed configurations into discrete, physically admissible modes.

This section establishes that every known Standard Model particle corresponds to one of these geometrically quantized trembling solutions. Rather than postulating arbitrary fields or symmetry assignments, TSRT sticks to the actual spacetime in which particles live and explains the discrete and finite spectrum of stable particles by showing:

1. how each mode's symmetry class (Abelian, non-Abelian chiral, non-Abelian color, or coarse-grained curvature) fixes its gauge charges;

2. how internal phase holonomy yields spin and charge quantization without operator algebras (cf. [21]);

3. how stability under the trembling evolution equations determines which configurations can persist as observable particles.

Unlike in conventional quantum field theory, the existence of each particle here is not a stipulation but a consequence of the causal structure of the metric and the action quantization constraint, as in Equation (53):

$$\oint \mathcal{L} d\tau = 2\pi n T, \quad n \in \mathbb{Z}, \qquad (102)$$

established rigorously in Refs. [19,21].

To make this derivation explicit and transparent, we proceed systematically:

* *Gauge bosons:* shown to arise as transverse trembling modes whose internal symmetries and curvature signatures reproduce the electromagnetic, weak, and strong interactions.
* Fermions: demonstrated to be localized, phase-coherent configurations with spin-1/2 holonomy, yielding the observed generations and charges.

* The Higgs-like scalar: reinterpreted as a metastable trembling solution whose coherence constraints differ from those of stable gauge and fermionic modes.

Throughout, we explicitly derive:

1. why each symmetry class yields exactly the known particles—no more, no fewer;

2. how mass and charge are linked quantitatively to action accumulation and holonomy;

3. why no additional stable trembling modes exist beyond those corresponding to the Standard Model.

This geometric classification therefore constitutes a constructive proof: each particle arises as a unique, causally admissible solution, determined from first principles in trembling spacetime geometry.

# 10.1 Formal Statement of Spectrum Exhaustiveness

Under the following conditions—(i) the deviation field $\xi_{\mu\nu}(x)$ is smooth, causal, and bounded in curvature norm, (ii) the proper-time action integral satisfies the quantization condition of Equation (25), and (iii) the trembling configuration remains variationally stable under infinitesimal perturbations—the set of causally admissible trembling eigenmodes is conjectured to coincide exactly with the known Standard Model particle content: three generations of chiral fermions, gauge bosons corresponding to U(1), SU(2), SU(3) symmetries, and the curvature-bound Higgs mode.

This conjecture is supported by constructive derivations in Sections 4.1, 4.2, and 10.6, where each symmetry class is shown to yield a finite discrete spectrum under the specified constraints. To date, no additional stable solutions satisfying all criteria have been identified. Any hypothetical mode beyond this classification would necessarily violate at least one of the defining conditions—either by lacking proper-time coherence, exceeding bounded curvature norms, or failing variational stability. A rigorous existence and uniqueness proof remains an open problem and is deferred to future work.

# 10.2 Fermions: Localized Chiral Trembling Modes

In TSRT, fermions arise not as quantized field excitations but as stable, localized trembling eigenmodes whose deviation fields are constrained by SU(2) chiral symmetry. These modes possess three defining features:

1. *Chiral asymmetry*, encoded in the antisymmetric component of the deviation tensor, leading to parity violation.

2. *Spin-$\frac{1}{2}$ holonomy*, resulting in the characteristic sign inversion under $2\pi$ rotations.

3. *Localized, causally coherent support*, ensuring their stability as particle-like excitations.
This subsection establishes these properties constructively, showing how they yield the known leptons and neutrinos without auxiliary quantization rules.

**Chiral Deviation Fields and Parity Violation** Chiral trembling modes are built from deviation fields that couple asymmetrically to the proper-time velocity:

$$ \xi_{\mu\nu}(x) = -\xi_{\nu\mu}(x) = \sum_{a=1}^{3} W_{\mu}^{a}(x) (T_{L}^{a})_{\nu}, \qquad (103) $$

where $W_{\mu}^{a}(x)$ are real-valued fluctuation components, and $(T_{L}^{a})$ are SU(2) generators acting exclusively on left-handed subspaces. Unlike in quantum field theory, where parity violation is imposed via projection operators, here it arises directly from the antisymmetric structure of $\xi_{\mu\nu}$. This configuration is not invariant under spatial inversion $\vec{x} \to -\vec{x}$, providing a geometric explanation for the observed left-handedness of weak interactions.

**Spin-$\frac{1}{2}$ Holonomy from Closed Trembling Cycles** As developed in Section 5.1, the spin of a trembling mode is determined by the net geometric phase (holonomy) accumulated along a closed coherence interval. For fermions, this holonomy satisfies:

$$ \oint \mathcal{L} d\tau = \pi \mathcal{T}, \qquad (104) $$

implying that the internal configuration returns to its original state only after two cycles. This phase condition yields spin-$\frac{1}{2}$ behavior, including the sign inversion under $2\pi$ rotations, without requiring spinor fields or anticommutation relations. This is the precise geometric counterpart of the spin-statistics connection, but derived here as a consequence of trembling periodicity. It is important to note that the geometric holonomy phase of Equation (104) corresponds formally to the projective representation of the rotation group SO(3) that yields spin-$\frac{1}{2}$ behavior. Specifically, the double-valuedness under $2\pi$ rotations recovers the same transformation properties as the fundamental representation of SL(2,C), ensuring consistency with observed spinor behavior.

**Localization and Mass from Geodesic Resistance** The proper-time evolution of the deviation field is governed by the trembling geodesic equation:

$$ \frac{D^2 x^\mu}{D\tau^2} = -\mathcal{R}^\mu_\nu \delta x^\nu, \qquad (105) $$

where $\mathcal{R}^\mu_\nu$ encodes curvature response from both SU(2) and U(1) trembling components. The mass of the mode arises as the resistance to deflection in this effective geometry:

$$ m = \frac{1}{c^2} \left\langle -\frac{d^2 x^\mu}{d\tau^2} g_{\mu\nu} \dot{x}^\nu \right\rangle_\tau, \qquad (106) $$

establishing inertial mass as a measurable consequence of internal trembling resistance. In neutrinos, which lack significant U(1) coupling, this resistance is minimal, explaining their near-null propagation and small mass.
**Hierarchy and Charge Structure** Charged leptons (electron, muon, tau) arise when the chiral SU(2) trembling mode is coherently coupled to an additional Abelian U(1) component:

$$ \xi_{\mu\nu}(\text{charged}) = \xi_{\mu\nu}^{\text{SU(2)}} + \xi_{\mu\nu}^{\text{U(1)}}, \qquad (107) $$

while neutrinos are purely chiral SU(2) modes. This construction yields integer-multiple quantization of electric charge:

$$ q = n q_0, \quad n \in \mathbb{Z}, \qquad (108) $$

where $q_0$ is the fundamental charge quantum associated with one cycle of U(1) trembling holonomy (see Section 5.5).

**Flavors and Neutrino Species** The existence of exactly three neutrino species emerges as the minimal degeneracy allowed by the internal trembling configuration: each neutrino corresponds to a distinct stable coherence mode differing in phase topology or internal frequency. This classification requires no arbitrary symmetry enlargement beyond the SU(2) sector and is consistent with the minimal particle content observed experimentally.

To summarize, in TSRT, fermions are deterministic solutions satisfying:

* Chiral coupling to left-handed SU(2) deviations (explaining weak parity violation).

* Spin-$\frac{1}{2}$ phase holonomy (explaining spin and statistics).

* Localization and mass from curvature resistance (explaining inertial properties).

* Charge from combined SU(2)-U(1) coherence (explaining electric charge).

Their existence follows from the same variational and causal coherence principles that define all trembling modes, not from quantized fields or imposed spinor representations. This provides a closed, predictive derivation of the fermion sector in the Standard Model.

We now proceed to classify the gauge bosons, the symmetry-bound trembling modes mediating interactions.

### 10.2.1 Leptons

In TSRT, leptons arise as localized, chiral trembling eigenmodes bound by SU(2) symmetry but without SU(3) color charge. Their defining features are:

* Spin-$\frac{1}{2}$ holonomy, resulting from half-cycle phase accumulation ($\pi$) per trembling loop, as established in Equation (104).

* Electric charge, emerging from coherent U(1) phase tension superposed on the SU(2) deviation field.

* Localization, maintained through causal coherence and bounded spatial support of the deviation tensor $\xi_{\mu\nu}(x)$.
**Charged Leptons** The three charged leptons—the electron, muon, and tau—correspond to distinct stable solutions of the trembling geodesic equations in which the SU(2) chiral deviation is coherently coupled to a U(1) Abelian component:

$$ \xi_{\mu\nu}(x) = \xi_{\mu\nu}^{\text{SU(2)}}(x) + \xi_{\mu\nu}^{\text{U(1)}}(x). \qquad (109) $$

This combined configuration yields a conserved electric charge $q = -1$, derived geometrically via the Noether current integral

$$ q = \int_{\Sigma} J_{\text{EM}}^{\mu} d\Sigma_{\mu}, \qquad (110) $$

as developed in Section 5.4. Also, spin-$\frac{1}{2}$ behavior, through holonomy quantization, and a rest mass determined by the magnitude and frequency of internal trembling, via:

$$ m = \frac{1}{c^2} \left\langle -\frac{d^2 x^{\mu}}{d\tau^2} g_{\mu\nu} \dot{x}^{\nu} \right\rangle_{\tau}. \qquad (111) $$

The distinct masses of the electron, muon, and tau result from differences in the internal trembling amplitude and proper-time periodicity, rather than from external Yukawa couplings. This provides a geometric explanation for the lepton mass hierarchy as a spectrum of energetically stable coherence intervals.

**Neutrinos** Neutrinos in TSRT are complementary trembling modes:

$$ \xi_{\mu\nu}(x) = \xi_{\mu\nu}^{\text{SU(2)}}(x), \qquad (112) $$

with no U(1) component. This implies $q = 0$ electric charge, pure SU(2) chiral coupling, and weak localization and near-null geodesic propagation, leading to extremely small rest masses.

Importantly, their masses are not introduced by hand but arise from residual coupling to long-wavelength gravitational trembling modes (Section 3.2). Because this coupling is much weaker than the electromagnetic contribution in charged leptons, neutrino mass remains many orders of magnitude smaller.

**Generational Structure** The geometric classification of SU(2)-bound trembling modes yields exactly three discrete, causally admissible coherence configurations satisfying:

$$ \oint \mathcal{L} d\tau = 2\pi n T, \quad n = 1, 2, 3, \qquad (113) $$

with each integer $n$ corresponding to a stable generation. This count is not arbitrary: it follows from the quantization of action accumulation and the requirement that the trembling mode remain localized without dispersion. As a result, TSRT naturally predicts three generations of charged leptons and three neutrinos, matching observation.

To summarize, in this framework, the lepton sector emerges as through spin-$\frac{1}{2}$ holonomy from chiral trembling, electric charge from U(1) phase tension, mass from curvature resistance of coherent deviations, and generational structure from the discrete action spectrum.
No quantization postulates, spontaneous symmetry breaking, or auxiliary Higgs mechanisms are required. All lepton properties derive directly from the geometry of trembling spacetime.

### 10.2.2 Quarks

In TSRT, quarks appear as localized spin-$\frac{1}{2}$ trembling eigenmodes characterized by non-Abelian SU(3) internal symmetry. Unlike leptons, which are SU(3)-neutral, quarks possess a triplet-valued internal structure corresponding to color charge. Geometrically, these modes are defined by entangled trembling deviations:

$$ \xi_{\mu\nu}(x) = \sum_{a=1}^{8} A_{\mu}^{a}(x) (T^{a})_{\nu}, \qquad (114) $$

where $A_{\mu}^{a}(x)$ are vector potentials associated with the eight SU(3) generators $T^{a}$. This configuration supports color exchange and non-trivial curvature feedback, matching the known gluon degrees of freedom in QCD (Section 5.4).

**Fractional Electric Charge** Quark electric charges arise from the interplay between SU(3) color symmetry and the Abelian U(1) phase winding of the deviation field. Unlike leptons—which complete full U(1) trembling cycles ($2\pi$ holonomy) to yield integer charges—quark modes maintain coherence over fractional phase intervals. Specifically, TSRT identifies that:

$$ \oint \mathcal{L} d\tau = \frac{2\pi}{3} n \mathcal{T}, \quad n \in \mathbb{Z}, \qquad (115) $$

with $n = 1, 2$ corresponding to $+\frac{1}{3}$ and $+\frac{2}{3}$ charges respectively (or their negative counterparts depending on phase alignment). This quantized partial holonomy is a geometric property of the SU(3)-bound trembling configuration rather than a phenomenological assignment (see Section 5.2). A calculation is provided for the down quark electric charge from partial holonomy in Section 10.9.

**Spin and Chirality** Quarks retain spin-$\frac{1}{2}$ holonomy because their trembling cycles accumulate a $\pi$ geometric phase per coherence interval (Section 5.1). Like leptons, their deviation fields couple asymmetrically to the left-handed components of the proper-time velocity:

$$ \xi_{\mu\nu}^{(L)} = P_L \xi_{\mu\nu} P_L, \qquad (116) $$

explaining parity violation in weak interactions without invoking projection operators or spinor fields. This chiral coupling differentiates quark flavors through variations in internal phase topology and coherence scale.

**Flavor Structure and Mass Hierarchy** TSRT predicts six distinct stable trembling eigenmodes—each corresponding to an observed quark flavor—through discrete quantization of in-
ternal action and curvature tension:

$$ \oint \mathcal{L} d\tau = 2\pi n T, \quad n = 1 \dots 6. \qquad (117) $$

Heavier quark modes exhibit tighter spatial confinement and higher internal curvature, directly producing larger inertial masses as derived in Section 3.2. This mechanism replaces arbitrary Yukawa couplings with a geometric tension–curvature balance.

**Color Confinement** A central prediction of TSRT is that isolated SU(3)-charged trembling modes cannot remain causally coherent over macroscopic scales. Specifically, when the deviation field extends beyond a critical separation, the accumulated curvature prevents closure of the internal trembling cycle:

$$ \xi_{\mu\nu}(x) \text{ fails to satisfy } \xi_{\mu\nu}(x + \Delta x) = \xi_{\mu\nu}(x). \qquad (118) $$

This breakdown invalidates the coherence condition necessary for stable particle-like behavior (Section 4.2). Consequently, quarks are permanently confined within color-neutral combinations—baryons and mesons—without requiring additional assumptions such as potential walls or running couplings. The non-linear feedback of SU(3) curvature self-interactions suffices to enforce confinement.

To summarize, TSRT therefore yields a fully geometric derivation of quark properties:

* Spin-$\frac{1}{2}$ holonomy and parity-violating chiral coupling.

* Fractional electric charge from partial U(1) phase winding.

* Six discrete flavor modes with mass hierarchy determined by curvature tension.

* Color confinement as a dynamical consequence of non-Abelian coherence breakdown.

No operator algebra, gauge quantization, or external postulates are required. These features emerge as inevitable consequences of causally admissible SU(3)-bound trembling configurations in curved spacetime.

## 10.3 Gauge Bosons: Extended Coherent Trembling Modes

Having established that fermions in TSRT arise as spatially localized, chiral trembling modes with discrete phase holonomy, we now turn to the second major class of fundamental particles: the gauge bosons. In the TSRT framework, these modes do not serve as matter constituents but as extended, coherence-bound configurations whose geometric role is to mediate interactions between localized trembling regions.

Unlike fermions, gauge bosons are not confined within compact spatial domains. Instead, they propagate as globally phase-consistent trembling deviations that remain coherent under proper-time evolution across macroscopic distances. This property allows them to maintain phase alignment between distant localized modes and thereby encode interaction potentials and conservation laws.
Formally, gauge bosons in TSRT correspond to deviation fields

$$ \xi_{\mu\nu}(x) \quad \text{such that} \quad \xi_{\mu\nu}(x + \Delta x) = \xi_{\mu\nu}(x) \qquad (119) $$

with spatially non-compact support and internal alignment to a symmetry group generator. In contrast to Standard Model treatments—where gauge bosons are quantized excitations of gauge fields—TSRT derives them deterministically as the minimal-energy solutions that satisfy:

1. Coherence: the internal trembling phase remains constant along the propagation direction.

2. Symmetry alignment: the deviation field belongs strictly to one of the three geometric symmetry classes—U(1), SU(2), or SU(3).

3. Variational admissibility: the configuration minimizes the action integral under proper-time evolution (see Section 5.2).

The distinction between massless and massive gauge bosons in TSRT is not imposed by hand but arises from the extent of spatial coherence and the curvature coupling of the trembling configuration. Massless gauge bosons (e.g., photon and gluons) correspond to trembling deviations with perfect global coherence and no intrinsic curvature-induced phase decay. Because the coherence interval can be extended indefinitely, their effective inertial mass vanishes (Section 3.1). Massive gauge bosons (e.g., $W^{\pm}$ and $Z$) correspond to modes whose internal coherence decays over finite spatial intervals due to curvature tension in the SU(2) sector. This decay imposes a minimal action per coherence interval, which translates to a finite rest mass (see Section 3.2).

This mechanism replaces the need for spontaneous symmetry breaking or external Higgs fields: the observed mass spectrum emerges naturally from the geometric balance between phase coherence and curvature feedback.

Gauge bosons exhibit integer spin because their trembling cycles complete full $2\pi$ holonomy per coherence interval. Specifically:

$$ \oint \mathcal{L} d\tau = 2\pi s \mathcal{T}, \quad s \in \mathbb{Z}, \qquad (120) $$

with $s = 1$ corresponding to vector bosons. This contrasts with fermions, where half-integer spin arises from partial phase accumulation (Section 5.1).

TSRT thereby recovers all known gauge bosons as extended trembling solutions. Indeed, the photon emerges as the unique globally coherent U(1)-aligned trembling mode with vanishing curvature coupling, hence massless. The eight gluons arise as SU(3)-aligned, color-charged trembling deviations with perfect coherence and zero mass but self-interacting curvature feedback (see Section 5.4). Finally, the $W^{\pm}$ and $Z$ bosons correspond to SU(2)-aligned modes whose coherence decays over finite intervals due to nontrivial internal curvature, resulting in finite rest mass.

This classification requires no auxiliary gauge fields, operator quantization, or spontaneous symmetry breaking. All properties derive directly from geometric constraints on causally admissible trembling configurations.