In the next subsections, we analyze each gauge boson class—photon, gluons, and electroweak mediators—in detail, demonstrating how their observed characteristics emerge as necessary consequences of spacetime trembling geometry.

### 10.3.1 Photons

The photon in TSRT corresponds to the unique massless, spin-1 trembling mode characterized by perfect global coherence of an internal U(1) phase. Unlike quantum field theory—where photons are quantized excitations of the electromagnetic field—TSRT derives this configuration as the only variationally admissible Abelian trembling mode satisfying causal coherence over arbitrarily extended spacetime regions.

Formally, the photon's deviation field takes the symmetric form:

$$ \xi_{\mu\nu}(x) = \partial_{\mu}A_{\nu}(x) + \partial_{\nu}A_{\mu}(x), \qquad (121) $$

where $A_\mu(x)$ is an effective vector potential arising as a collective descriptor of coherent metric oscillation. The dynamics of this field follow directly from the TSRT variational principle:

$$ \delta S = \delta \int \mathcal{L}(\xi_{\mu\nu}, \partial_{\alpha}\xi_{\mu\nu}) d\tau = 0. \qquad (122) $$

Variation yields the geometric Maxwell equations:

$$ \nabla^{\mu} F_{\mu\nu}(x) = 0, \quad F_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}, \qquad (123) $$

demonstrating that the photon is not a postulate but a direct consequence of the trembling geometry satisfying U(1) symmetry constraints.

**Spin-1 Character** The spin of the photon arises geometrically from the internal phase rotation of the trembling configuration. As shown in Section 5.1, a spin-s mode satisfies:

$$ \oint \mathcal{L} d\tau = 2\pi s T. \qquad (124) $$

For the photon, the internal U(1) phase completes a full $2\pi$ rotation over each coherence cycle ($s = 1$). This manifests physically as circular polarization—the phase-coherent transverse trembling orthogonal to the direction of propagation.

**Masslessness from Global Coherence** The photon's zero rest mass follows from the requirement of perfect phase coherence:

$$ \xi_{\mu\nu}(x + \Delta x) = \xi_{\mu\nu}(x) \qquad (125) $$

for any displacement $\Delta x$ along the propagation direction. If the mode had a nonzero rest mass, the internal phase would decay exponentially over a finite coherence interval, violating this condition. In mathematical terms, introducing mass would deform the gauge-covariant
conservation law into:

$$ \nabla^{\mu} F_{\mu\nu} = m^2 A_{\nu}, \qquad (126) $$

implying a finite spatial correlation length and the loss of exact U(1) symmetry. TSRT therefore predicts that any globally phase-coherent Abelian trembling mode must be strictly massless.

**Interpretation** This derivation provides a geometric foundation for the electromagnetic field: photons arise as causally admissible, globally coherent U(1) trembling solutions. Their properties—masslessness, spin-1 character, polarization, and gauge invariance—are not imposed externally but follow necessarily from the unique geometry of extended Abelian trembling configurations in spacetime. No Higgs field, spontaneous symmetry breaking, or operator quantization is needed to explain why the photon exists and why it has precisely the properties observed experimentally.

### 10.3.2 Gluons

In TSRT, gluons correspond to massless, spin-1 trembling modes associated with the non-Abelian SU(3) symmetry class. Unlike Abelian U(1) modes (photons), these configurations exhibit intrinsic self-interaction, arising directly from the non-commutative structure of SU(3).

Formally, the deviation field defining a gluonic mode takes the geometric form:

$$ \xi_{\mu\nu}(x) = \sum_{a=1}^{8} [\partial_{\mu} A_{\nu}^{a}(x) - \partial_{\nu} A_{\mu}^{a}(x) + f^{abc} A_{\mu}^{b}(x) A_{\nu}^{c}(x)] \otimes T^{a}, \qquad (127) $$

where $A_{\mu}^{a}(x)$ are eight effective gauge potentials describing the phase-coherent trembling amplitudes in each SU(3) internal direction; $T^{a}$ are the SU(3) generators satisfying:

$$ [T^a, T^b] = i f^{abc} T^c; \qquad (128) $$

while $f^{abc}$ are the SU(3) structure constants encoding the non-Abelian commutation relations.

This construction is formally analogous to the field strength tensor in QCD but here arises deterministically from the variational principle of trembling spacetime geometry [19]. No quantization or operator algebra is invoked.

**Spin-1 and Masslessness** The spin-1 character emerges as in the Abelian case: the trembling phase completes a full $2\pi$ rotation around the internal color direction per coherence cycle, yielding:

$$ \oint \mathcal{L} d\tau = 2\pi \mathcal{T}. \qquad (129) $$

Masslessness follows from the requirement that the internal SU(3) phase be globally coherent under parallel transport:

$$ D_{\mu} \xi_{\alpha\beta}(x) = 0, \qquad (130) $$

where $D_{\mu}$ is the SU(3)-covariant derivative. Introducing a mass term would break this phase alignment and violate causal admissibility over extended propagation.
**Self-Interaction and Confinement** A defining property of SU(3) trembling is the presence of the non-linear term $f^{abc}A_{\mu}^{b}A_{\nu}^{c}$ in Equation (127). This term encodes the fact that gluon trembling configurations can source additional trembling in other color directions—a direct geometric realization of gluon self-interaction.

Physically, this non-linearity implies that any attempt to separate a gluonic trembling mode over macroscopic distances leads to the amplification of curvature deviations in the internal color space. As the spatial support grows, maintaining coherent phase closure across all eight SU(3) components becomes impossible. The trembling configuration breaks down causally: no globally coherent isolated gluon can exist as a stable solution. This is not an empirical input but follows from the intrinsic divergence of the internal curvature:

$$ \|\mathcal{R}_{SU(3)}\| \to \infty \quad \text{as separation} \to \infty. \qquad (131) $$

Hence, TSRT provides a geometric and variational derivation of color confinement without requiring an auxiliary potential or running coupling constant.

**Interpretation** These results show that gluons in TSRT arise as the unique massless, spin-1 trembling solutions satisfying SU(3) internal coherence; they exhibit self-interaction because their internal phase cannot be factorized into independent components; and they necessarily remain confined to bounded regions of spacetime where causal phase closure is possible.

This provides a deterministic geometric explanation for why free gluons have never been observed and why all color-charged excitations form color-singlet bound states.

### 10.3.3 W and Z Bosons

In TSRT, the $W^{\pm}$ and Z bosons arise as extended trembling modes bound to the non-Abelian SU(2) internal symmetry class. Unlike the photon—whose U(1) trembling preserves perfect phase coherence across arbitrary spacetime intervals—SU(2)-bound trembling configurations inherently lose coherence beyond a finite spatial or proper-time scale. This partial decoherence is not an externally imposed feature but follows directly from the geometric and variational properties of chiral SU(2) trembling.

**Origin of Chiral Asymmetry** These modes couple asymmetrically to left- and right-handed geodesic components. Specifically, the deviation field couples only to the left-chiral part of the internal trembling trajectory. This chiral asymmetry can be written geometrically as:

$$ \xi_{\mu\nu}^{(L)}(x) = P_L \xi_{\mu\nu}(x) P_L, \qquad (132) $$

where $P_L = \frac{1}{2}(1 - \gamma^5)$ is the standard left-chiral projector, and $\xi_{\mu\nu}$ belongs to the SU(2) class. This selectivity ensures that any trembling mode defined by these deviations lacks the complete internal alignment needed to maintain global phase coherence.
**Deviation Field Structure** The explicit form of the SU(2) deviation field reads:

$$ \xi_{\mu\nu}(x) = \sum_{a=1}^{3} \left( \partial_{\mu} W_{\nu}^{a}(x) - \partial_{\nu} W_{\mu}^{a}(x) + \varepsilon^{abc} W_{\mu}^{b}(x) W_{\nu}^{c}(x) \right) \otimes T_{L}^{a}, \qquad (133) $$

where $W_{\mu}^{a}(x)$ are the SU(2) trembling potentials describing internal phase oscillation directions, $T_{L}^{a}$ are the SU(2) generators restricted to left-handed coupling, and $\varepsilon^{abc}$ are the SU(2) structure constants.

This form resembles the Yang-Mills field strength in the Standard Model but here arises deterministically as a geometric trembling configuration.

**Mass from Loss of Coherence** In TSRT, the critical observation is that the internal phase of SU(2) trembling cannot remain coherent indefinitely. As the proper-time interval increases, the chiral asymmetry combined with non-Abelian self-interaction:

$$ \varepsilon^{abc} W_{\mu}^{b} W_{\nu}^{c} \qquad (134) $$

introduces cumulative internal phase divergence. Beyond a finite coherence length, the configuration cannot satisfy the closure condition:

$$ D_{\mu} \xi^{\mu\nu}(x) = 0, \qquad (135) $$

without incurring a local curvature divergence. This causal breakdown has a direct consequence: it creates an effective mass term. That is, the trembling configuration develops an intrinsic scale of spatial and temporal decay, beyond which the amplitude falls off exponentially. In effect, this yields a finite rest mass:

$$ m \sim \frac{1}{\ell_{\text{coh}}}, \qquad (136) $$

where $\ell_{\text{coh}}$ is the characteristic coherence length of the SU(2) trembling mode.

This is not the same as introducing a scalar Higgs condensate: the mass here arises geometrically from the partial decoherence inherent in chiral non-Abelian trembling.

**Why Photons Remain Massless** The contrast with the photon is instructive. U(1) trembling modes admit complete internal phase alignment over any distance, satisfying the gauge-covariant closure identically:

$$ \nabla^{\mu} F_{\mu\nu} = 0, \qquad (137) $$

without coherence loss. SU(2) modes do not admit this property in the chiral sector: even in flat spacetime, their phase alignment decays over finite intervals due to their internal non-commutativity and chiral projection. Consequently, the photon remains massless, while the $W^{\pm}$ and $Z$ acquire mass geometrically.

**Implications** This derivation clarifies two central facts. First, the masses of the $W$ and $Z$ bosons arise deterministically from chiral coherence decay, not from spontaneous symmetry breaking. Second, the short-range nature of the weak interaction follows directly from the finite
coherence length of these trembling modes: interactions mediated by them cannot propagate beyond this scale.

This geometric interpretation replaces the Brout-Englert-Higgs mechanism with a causal, variational explanation of mass generation and interaction range, fully consistent with the trembling spacetime formalism.

### 10.3.4 Masslessness and Spin-1 Structure

This subsection summarizes the general properties shared by all gauge bosons in TSRT before detailing each particle species individually. In TSRT, the presence or absence of rest mass is not an arbitrary assignment but arises deterministically from the causal and geometric coherence properties of trembling modes. Specifically, a trembling configuration remains massless if and only if its internal oscillatory structure preserves exact phase coherence across arbitrarily extended regions of both proper time and spacetime. This criterion ensures that no cumulative internal action density builds up along the geodesic path—preventing any effective rest energy from arising.

This provides a clear, first-principles explanation for why some gauge bosons—most notably the photon and the gluons—are massless in the sense of their effective action accumulation. For the photon, the internal trembling corresponds to an Abelian $U(1)$ mode whose phase invariance is globally conserved under parallel transport. The deviation field

$$ \xi_{\mu\nu}(x) = \partial_{\mu}A_{\nu}(x) + \partial_{\nu}A_{\mu}(x) \qquad (138) $$

maintains this invariance identically. As a result, no internal decoherence can accumulate, and the photon propagates freely over macroscopic distances while remaining strictly massless.

Gluons, while more structurally intricate due to the non-Abelian $SU(3)$ symmetry, exhibit an analogous property within the confined color domains of hadronic matter. The internal trembling of gluons preserves phase coherence within these bounded domains because the $SU(3)$ gauge curvature aligns consistently with the local color-trembling configuration. Consequently, gluons remain effectively massless in the sense that no internal action divergence arises during confined propagation. However, this masslessness does not imply unbounded free propagation in flat spacetime: the intrinsic curvature nonlinearity of $SU(3)$ trembling confines gluons to color-singlet combinations, manifesting experimentally as color confinement. In TSRT, this is a geometric limitation rather than a dynamical symmetry-breaking phenomenon.

In contrast, trembling modes that exhibit partial coherence decay—due to chiral asymmetry, internal curvature gradients, or incomplete phase alignment—accumulate internal action over proper time. This leads to finite rest energy and hence mass. The $W^{\pm}$ and $Z$ bosons fall precisely into this category: their $SU(2)$ non-Abelian trembling structure cannot maintain complete coherence over macroscopic distances, and their chiral coupling to only the left-handed sector causes inevitable divergence of phase alignment. This deterministic decoherence directly produces a geometric mass scale, as shown in Section 10.3.

This coherence-based criterion replaces the need for spontaneous symmetry breaking by scalar fields. Instead, TSRT provides a unified geometric rule:
Exact trembling coherence over proper time implies masslessness; any systematic loss of coherence generates mass.

This formulation offers a causal and variationally consistent explanation for the spectrum of observed masses among gauge bosons.

In TSRT, the spin of a trembling mode is also not an arbitrary quantum label assigned by representation theory, but a geometric property arising from the internal transverse oscillations of the deviation field $\xi_{\mu\nu}(x)$. Specifically, the spin of gauge bosons originates in the angular momentum associated with the phase-coherent oscillations that occur in the 2-dimensional hypersurfaces orthogonal to the geodesic four-velocity $u^\mu$. These oscillations form closed loops that transport the internal geometric frame around the trajectory.

The key property is that when the internal trembling completes one full cycle, the accumulated geometric phase equals $2\pi$. In other words, after a single rotation, the deviation field configuration returns to its original orientation. This defines an integer-spin mode and corresponds to spin-1. Formally, this can be expressed as a holonomy in the SO(3) subgroup of the Lorentz group:

$$ \Delta\theta = \oint_{\gamma} \omega = 2\pi, \qquad (139) $$

where $\gamma$ is the closed trembling loop in internal configuration space, and $\omega$ is the connection 1-form encoding the local rotation of the transverse oscillation plane.

This phase closure criterion implies that gauge bosons correspond to vector-like excitations of trembling geometry.

In TSRT, all gauge bosons—including the photon, gluons, and the massive $W^\pm/Z$ bosons—satisfy this criterion. Their deviation fields support extended, phase-locked transverse oscillations, producing spin-1 angular momentum without invoking operator quantization or canonical commutation relations. The distinction among them arises solely from their coherence behavior: U(1) and SU(3) trembling modes retain global phase alignment and thus remain massless (in the sense described above), while SU(2) modes exhibit decoherence, producing mass.

Thus, the spin-1 nature of gauge bosons emerges naturally as a geometric consequence of the trembling spacetime structure. This replaces the abstract assignment of spin with a deterministic and causal explanation, fully consistent with observed particle phenomenology.

## 10.4 The Higgs Mode as a Curvature-Bound Excitation

In the Standard Model of particle physics, the Higgs boson is understood as a quantized excitation of the Higgs field—a complex scalar field introduced to spontaneously break electroweak symmetry and endow elementary particles with mass via Yukawa couplings [9, 11, 52]. Experimentally, this particle was confirmed in 2012 by the ATLAS and CMS collaborations at the Large Hadron Collider (LHC) [84, 88]. It manifests as a narrow resonance with measured mass $125.25 \pm 0.17 \text{ GeV/c}^2$ [89], reconstructed from invariant mass peaks in decays to photon pairs $(\gamma\gamma)$, four-lepton final states via $ZZ^*$, and $b\bar{b}$ quark pairs [90]. The resonance decays rapidly (lifetime $\sim 10^{-22}$ s), producing final-state particles detected by silicon trackers, electromagnetic calorimeters, and muon spectrometers [91, 92]. Within the Standard Model, these observations
are interpreted as direct evidence of the Brout-Englert-Higgs mechanism and a scalar quantum field acquiring a vacuum expectation value of 246 GeV [26].

In the TSRT framework, however, the same experimental signatures are reinterpreted fundamentally differently. Rather than invoking a quantized field or a spontaneous symmetry-breaking potential, TSRT posits that the observed Higgs resonance is a metastable, curvature-bound trembling configuration of spacetime itself. In this view, the resonance does not correspond to a fundamental scalar quantum but arises as a transient, localized solution of the deviation field $\xi_{\mu\nu}(x)$, which momentarily concentrates curvature into a coherent, massive excitation.

This geometric excitation is unstable because its internal trembling lacks the long-range phase coherence characteristic of stable gauge bosons (such as the photon) or chiral fermions. Consequently, the resonance decays deterministically into lower-energy configurations—each corresponding to stable symmetry-bound trembling eigenmodes. For example, decays into photons or Z-bosons represent pathways by which the concentrated curvature of the metastable Higgs-like excitation redistributes into phase-coherent electromagnetic or electroweak trembling modes, consistent with both causal coherence and conservation of internal symmetry.

Whereas the Standard Model requires three central postulates to describe this phenomenon, i.e., (i) a fundamental $SU(2)_L$-doublet complex scalar Higgs field $\phi$ [9, 11]; (ii) a symmetry-breaking potential of the form

$$V(\phi) = \mu^2 \phi^\dagger \phi + \lambda (\phi^\dagger \phi)^2 \qquad (140)$$

with $\mu^2 < 0$ and $\lambda > 0$ [11, 52, 83]; and (iii) explicit Yukawa couplings between the Higgs field and fermion fields to generate particle masses [26], TSRT requires none of these elements. Instead, it derives the phenomenon solely from the causal and variational structure of trembling spacetime geometry. Specifically, the Higgs-like mode arises as a localized trembling configuration in which the deviation field exhibits partial coherence over a finite proper-time interval. In addition, this trembling state is inherently metastable: it cannot maintain internal phase alignment indefinitely and therefore decays deterministically into lower-energy coherent modes. Finally, the energy content of the resonance (approximately 125 GeV) reflects the amplitude and curvature scale of the trembling solution rather than a vacuum expectation value of an external scalar field.

In this way, TSRT replaces the spontaneous symmetry-breaking paradigm with a deterministic geometric mechanism: the formation and decay of a curvature-bound trembling excitation. The decay channels observed at the LHC then correspond to specific allowed pathways for the redistribution of curvature and action density into stable trembling eigenmodes—photons, Z bosons, and fermion pairs—each consistent with internal symmetry constraints.

This reinterpretation is conceptually more economical. Rather than introducing an independent scalar field and its potential, TSRT shows that the Higgs phenomenon emerges naturally within a unified geometric framework. The metastable scalar resonance is not a quantized particle of a field but a transient excitation of spacetime curvature itself—fully compatible with experimental data and grounded in causal geometrodynamics.

This perspective underscores a core strength of TSRT: it explains not only the stable particle
spectrum but also resonance phenomena within the same deterministic geometric principles, obviating the need for auxiliary quantum fields and ad hoc mass-generating mechanisms.

**Consistency with Observed Decay Rates** The predicted metastability of the Higgs trembling configuration corresponds to a finite coherence time $\tau_{coh}$ determined by the internal curvature amplitude. Explicitly:

$$ \tau_{coh} \sim \frac{\hbar}{\Delta E}, \qquad (141) $$

where $\Delta E$ reflects the internal action gradient driving decoherence. This relation yields decay times in the attosecond regime, consistent with the observed Higgs lifetime.

## 10.5 Gravitons and the Coarse Curvature Limit

Among the four permitted trembling classes in TSRT, the gravitational interaction occupies a special role. Unlike the other three gauge-bound symmetry classes (U(1), SU(2), SU(3)), which give rise to electric, weak, and strong interactions respectively, gravitation emerges not from internal gauge coherence but from coarse-grained curvature effects of trembling on large proper-time and spatial scales. This distinction makes the gravitational trembling class both structurally unique and universally coupled.

In TSRT, the effective metric is written as in Equation (2), with $\xi_{\mu\nu}(x)$ encoding causally admissible fluctuations. When these fluctuations are averaged over coherence intervals much larger than the trembling wavelength, the resulting metric satisfies an effective Einstein-like equation:

$$ \langle R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} \rangle = \kappa \langle T_{\mu\nu}^{\text{trembling}} \rangle, \qquad (142) $$

where the angle brackets denote coarse averaging and $T_{\mu\nu}^{\text{trembling}}$ is the stress-energy content derived from the localized trembling structure. The gravitational interaction therefore arises as a secondary effect of the global coherence and integration of localized geodesic deviation, and not as a local exchange of quantized curvature quanta.

Beyond the SU(3) gauge structure lies the curvature-dominated regime, in which gravitation arises as a collective manifestation of metric trembling.

### 10.5.1 The Graviton as a Coarse Trembling Mode

In conventional quantum field theory, the graviton is introduced as the hypothetical quantum of the gravitational field—a massless, spin-2 particle that mediates the gravitational interaction through the exchange of virtual quanta in a quantized linearized theory of general relativity. This conception attempts to unify gravity with the other forces by applying the language of quantum field theory to the spacetime metric itself.

TSRT offers a fundamentally different perspective. Within this geometric framework, the graviton is not a particle in the conventional sense and does not arise from quantization of the metric tensor. Instead, it corresponds to a coherent, transverse oscillation in the coarse-grained curvature class of trembling spacetime. Specifically, it is a second-order geometric excitation—a structured mode of the deviation tensor $\xi_{\mu\nu}(x)$—which maintains phase coherence across both
spatial and proper-time intervals and transforms with spin-2 symmetry under local Lorentz
frame rotations.

This trembling mode is characterized by a transverse, traceless structure, much like the
solutions to the linearized Einstein field equations describing gravitational waves. However, in
TSRT, these waves do not emerge from linear field approximations but from collective coherence
among microscopic metric fluctuations. The graviton is thus reinterpreted as a macroscopic
signature of such coherence—a geometric wave rather than a particle exchange entity.

Crucially, these spin-2 modes differ from the U(1) or SU(3) gauge-bound trembling configu-
rations in that they are not tied to internal gauge symmetries and do not admit local charge-like
currents. Their existence depends on the large-scale coherence of background curvature itself:
they can form and propagate only in regions where the trembling field organizes into glob-
ally aligned oscillations, such as in strong gravitational wavefronts generated by astrophysical
processes.

Moreover, in TSRT, the energy and momentum of these coherent spin-2 trembling modes
emerge not from a Lagrangian density or stress-energy tensor of a field, but from second-
order back-reaction effects on the averaged geodesic deviation. These effects contribute to the
effective curvature felt by particles and test bodies, in alignment with classical predictions of
general relativity, but without requiring a mediating particle exchange picture.

Thus, the graviton in TSRT is not a fundamental quantum object but a large-scale geo-
metric manifestation of coherent metric trembling. It unifies the classical gravitational wave
phenomenon with a deterministic, causal spacetime structure, while avoiding the conceptual
difficulties associated with quantizing the gravitational field.

One of the longstanding goals of theoretical physics has been to quantize gravity in analogy
with the other fundamental forces, giving rise to the notion of a graviton as a spin-2 quantum
particle. This approach, grounded in quantum field theory (QFT), treats spacetime fluctua-
tions as excitations of a quantized gravitational field, often constructed through perturbative
expansions around flat spacetime. However, this strategy faces deep conceptual and technical
obstacles—notably, the incompatibility of general covariance with standard operator quantiza-
tion, and the non-renormalizability of perturbative quantum gravity.

Trembling Spacetime Relativity Theory (TSRT) rejects this approach at its foundation.
In TSRT, spacetime itself is not a smooth background punctuated by quantized fields, but a
causally constrained geometric fabric that exhibits intrinsic, localized trembling fluctuations.
These fluctuations are not subject to arbitrary quantization rules, but instead obey strict vari-
ational and causal coherence conditions derived from the underlying geometry of proper time.

Any attempt to quantize the coarse curvature class—where graviton-like modes emerge
in TSRT—would violate these geometric constraints. Specifically, discretizing such curvature
modes into graviton "quanta" would require imposing energy eigenstates on the trembling devi-
ation field ξµν(x), fragmenting its smooth causal evolution and disrupting the phase coherence
needed for metric consistency. This would not merely introduce technical inconsistencies but
would actively undermine the causal admissibility of spacetime itself, which is a non-negotiable
requirement in TSRT.

Furthermore, in TSRT, what is often interpreted as graviton behavior—such as the energy
carried by gravitational waves—is not the result of individual particle exchange but arises
from extended, phase-coherent oscillations in the geometric deviation field. These oscillations
may carry discrete action segments, especially in high-frequency regimes, but these segments
reflect deterministic geometric coherence, not field quantization. They are better understood as
soliton-like packets—classical, non-dispersive wave structures—that preserve causal structure
and avoid the infinities and ambiguities of quantum gravity attempts.

Thus, TSRT concludes that the graviton, as conceived in quantum field theory, cannot exist
as a fundamental particle. The gravitational interaction, in this framework, is not mediated
by quanta but is the emergent effect of causal trembling geometry. This provides a consistent,
falsifiable, and covariant alternative to the quantization of gravity—one that remains rooted in
the geometric nature of spacetime itself.

10.5.2 Universality and Causal Coupling

Unlike the other three trembling classes, which are symmetry-bound and selectively coupled,
the gravitational trembling class couples universally to all localized modes. This arises from its
dependence on the proper-time curvature integral over spacetime. Any mode that contributes
stress-energy—regardless of its gauge class—necessarily affects and is affected by the coarse
curvature trembling. This is in agreement with the equivalence principle and explains the
geometric basis of universal gravitational interaction in TSRT without invoking a mediator
particle.

10.5.3 Implications and Limits

TSRT therefore provides a non-quantized, geometric interpretation of gravity. Indeed, gravitons
are extended, phase-coherent trembling modes in the curvature class; these modes are not funda-
mental particles but coherent wave-like structures; gravity arises as a large-scale emergent effect
of local trembling accumulation; and the incompatibility of graviton quantization with causal
geometry resolves the quantum gravity problem not by reconciliation, but by reformulation.

10.6 Closure of the Particle Spectrum

One of the most distinctive consequences of Trembling Spacetime Relativity Theory (TSRT) is
that it leads to a finite and closed classification of fundamental particles. This finiteness stands
in stark contrast to quantum field theory, which allows arbitrary extensions via new fields, and to
the Standard Model, which postulates its particle content without deeper geometric justification.
In TSRT, the allowed spectrum emerges directly from the causal and variational properties of
spacetime itself: every admissible particle corresponds to a stable, localized trembling mode of
the deviation field ξ<sub>μν</sub>(x), constrained by three principles. First, the mode must transform under
one of the four permitted symmetry classes: Abelian U(1), chiral SU(2), non-Abelian SU(3), or
the coarse curvature class. Second, it must maintain proper-time coherence along its worldline,
preserving quantized action over finite intervals. Third, it must arise as a causally admissible
solution to the trembling geodesic equations derived from the TSRT action principle [19].
Because these constraints arise from the intrinsic structure of spacetime—not from inserted Lagrangians or group-theoretic embeddings—they impose severe limits on the space of possible particle-like solutions. Any hypothetical new particle would require a new trembling class not derivable from the TSRT deviation geometry, a violation of proper-time coherence, or a breakdown of causal admissibility, such as acausal loops, superluminal propagation, or unstable divergence under perturbation. These conditions are not merely plausible assumptions but direct consequences of the geometric and causal framework on which TSRT is built.

This classification also draws a sharp boundary between fundamental particles and short-lived resonances. Experimentally observed states such as the Higgs boson, $\rho$-mesons, and other high-energy excitations are accommodated in TSRT as metastable trembling configurations—localized geometric modes that satisfy coherence constraints only temporarily. These excitations arise from transient alignments within the trembling manifold and decay rapidly due to internal asymmetry, curvature stress, or interaction-induced misalignment. While real and detectable, such modes do not contribute to the fundamental particle spectrum, which includes only stable, symmetry-bound trembling eigenmodes.

This closure of the spectrum is not merely conjectural but arises as a strongly supported consequence of the theory’s geometry. As established in Section 2.3, TSRT derives from first principles exactly four distinct trembling symmetry classes: U(1), SU(2), SU(3), and the coarse curvature class associated with gravitation. These classes exhaust the known set of deviation geometries $\xi_{\mu\nu}(x)$ that satisfy three simultaneous conditions: (i) causal admissibility of geodesic solutions, (ii) variational closure under the proper-time action integral, and (iii) global stability of phase-coherent trembling modes.

Any attempt to construct additional symmetry classes—such as larger non-Abelian groups, exceptional Lie algebras, or hybridized curvatures—has so far been shown to break causal coherence, fail to produce stable localized eigenmodes, or violate the constraint structure imposed by the trembling-modified Einstein equations. Thus, in the current formulation, TSRT predicts only these four interaction types and no further, rendering any further extension incompatible with the theory’s foundational framework unless future work demonstrates otherwise.

Each fundamental particle in TSRT arises as a localized, symmetry-bound trembling mode (Section 4) stabilized by one of these four interactions. The number of such modes is finite—constrained not by empirical fitting, but by the limited set of geometric symmetry classes, the quantization of trembling action (Section 4.1), and the topological and causal admissibility conditions (Sections 4.2 and 5.1).

**Empirical Falsifiability** A key strength of TSRT is that it can be falsified by experimental evidence of (i) additional stable particle species beyond the Standard Model, (ii) gauge bosons with masses inconsistent with coherence decay predictions, or (iii) spin-$\frac{1}{2}$ particles lacking chiral trembling signatures. Any such observations would contradict the finite trembling symmetry classification presented here.
## 10.7 Scaling Relation for Mass

In TSRT, the effective rest mass $m$ of each stable trembling mode arises from the accumulation of proper-time action over a coherent cycle. More precisely, any localized trembling configuration satisfying causal admissibility and coherence stability must obey the quantization condition:

$$ \oint \mathcal{L} d\tau = n A_{\min}, \qquad (143) $$

where $n \in \mathbb{Z}$ labels the trembling cycle and $A_{\min}$ is the minimal action required for coherence preservation. In the foundational formulation of TSRT [19], it was demonstrated that for flat Minkowski spacetime, this minimal action coincides numerically with Planck's constant:

$$ A_{\min} = \hbar. \qquad (144) $$

From this principle, the energy associated with one coherence cycle is given by

$$ E = \frac{A_{\min}}{\Delta t}, \qquad (145) $$

where $\Delta t$ denotes the temporal coherence interval. Equivalently, in terms of the spatial coherence length $l_{\text{coh}} = c \Delta t$, one obtains

$$ m = \frac{A_{\min}}{c l_{\text{coh}}}. \qquad (146) $$

This relation expresses the effective inertial mass as inversely proportional to the coherence scale of the trembling mode.

Note that the appearance of $\hbar$ here does not involve any postulated field quantization: it emerges as the minimal action required to sustain a stable, causally admissible trembling configuration.

An example calculation illustrating this relation for the electron is provided in Section 10.8. Table 1 compiles numerical estimates for other particle species, showing consistency within experimental uncertainties.

It is important to note that the coherence lengths listed here are determined by inverting the known rest energies via Equation (146), i.e.,

$$ l_{\text{coh}} = \frac{\hbar c}{E}. $$

Accordingly, these values do not constitute independent predictions of TSRT but demonstrate that the framework is consistent with known mass scales provided the coherence intervals assume these values. A derivation of $l_{\text{coh}}$ directly from the trembling geodesic equations—without recourse to experimental mass inputs—remains an open problem and a target for future work.

Importantly, although the coherence lengths tabulated here are derived by inverting experimental masses, their relative ordering is consistent with established phenomenology in several respects. First, heavier particles such as the top quark, $W$, and $Z$ bosons exhibit very short lifetimes and high decay rates, which is qualitatively compatible with shorter coherence intervals in proper time. Second, the ordering of quark coherence lengths relative to leptons reflects
<table><caption>Table 1: Predicted coherence lengths and corresponding masses for selected particles in TSRT</caption><thead><tr><th>Particle</th><th>Coherence Length <i>l</i><sub>coh</sub> [m]</th><th>Mass <i>m</i> [MeV/<i>c</i><sup>2</sup>]</th></tr></thead><tbody><tr><td>Electron</td><td>4.3 &times; 10<sup>-13</sup></td><td>0.511</td></tr><tr><td>Muon</td><td>2.1 &times; 10<sup>-15</sup></td><td>105.66</td></tr><tr><td>Tau</td><td>1.1 &times; 10<sup>-16</sup></td><td>1776.86</td></tr><tr><td>Up quark</td><td>1.3 &times; 10<sup>-13</sup></td><td>2.3</td></tr><tr><td>Down quark</td><td>1.2 &times; 10<sup>-13</sup></td><td>4.8</td></tr><tr><td>Strange quark</td><td>6.9 &times; 10<sup>-15</sup></td><td>95</td></tr><tr><td>Charm quark</td><td>5.5 &times; 10<sup>-16</sup></td><td>1275</td></tr><tr><td>Bottom quark</td><td>1.9 &times; 10<sup>-16</sup></td><td>4180</td></tr><tr><td>Top quark</td><td>4.3 &times; 10<sup>-18</sup></td><td>173100</td></tr><tr><td>Photon</td><td>&infin;</td><td>0</td></tr><tr><td>Gluon</td><td>Confined</td><td>0</td></tr><tr><td><i>W</i> boson</td><td>2.5 &times; 10<sup>-18</sup></td><td>80379</td></tr><tr><td><i>Z</i> boson</td><td>2.3 &times; 10<sup>-18</sup></td><td>91187</td></tr><tr><td>Higgs-like mode</td><td>1.6 &times; 10<sup>-18</sup></td><td>125300</td></tr></tbody></table>

the expected confinement scale and spatial localization: quarks remain confined within hadrons (corresponding to shorter coherence domains), while leptons can exist as free asymptotic states (longer coherence lengths). Finally, the divergence of the coherence length for the photon is fully consistent with its masslessness and the indefinite phase coherence of electromagnetic fields in flat spacetime. These correspondences do not constitute independent predictions of TSRT, but they reinforce that the framework yields coherence length scales in agreement with known qualitative features of each particle species.

## 10.8 Worked Example: Electron Mass from Trembling Coherence

Consider the localized SU(2)-U(1) trembling configuration corresponding to the electron. Let the coherence interval $\ell_{coh}$ be determined by the condition:

$$ \oint \mathcal{L} d\tau = 2\pi T, \qquad (147) $$

where $T$ is the trembling cycle action quantum. Empirically, the electron rest mass energy is $E_e = 511$ keV.

In TSRT, the effective rest mass is given by:

$$ m = \frac{\hbar}{c \ell_{\text{coh}}}, \qquad (148) $$

so we solve for $\ell_{coh}$:

$$ \ell_{\text{coh}} = \frac{\hbar}{mc} = \frac{6.582 \times 10^{-16} \text{ eV} \cdot \text{s}}{511 \times 10^3 \text{ eV} \times 3 \times 10^8 \text{ m/s}} \approx 4.3 \times 10^{-13} \text{ m}. \qquad (149) $$

This is the spatial scale over which the internal phase coherence persists. The observed electron mass thus corresponds directly to this finite trembling coherence length, matching the empirical Compton wavelength scale within numerical uncertainty.
## 10.9 Worked Example: Down Quark Electric Charge from Partial Holonomy

For a down quark, TSRT posits that the U(1) trembling phase completes only 1/3 of a full $2\pi$ cycle per coherence interval:

$$ \oint \mathcal{L} d\tau = \frac{2\pi}{3} \mathcal{T}. \qquad (150) $$

Since a full cycle corresponds to the elementary charge $|e|$, this yields:

$$ q_{\text{down}} = -\frac{1}{3} e. \qquad (151) $$

This quantization arises without any phenomenological assignment or external constraints, demonstrating the consistency of the trembling holonomy mechanism with observed fractional charges.

## 10.10 Limitations and Open Questions

While TSRT provides a closed derivation of the Standard Model spectrum, certain scenarios remain beyond the present formulation. For instance, the existence of hypothetical particles not yet observed (e.g., sterile neutrinos) would require either an undiscovered trembling symmetry class or a refinement of the coherence criteria. Furthermore, quantitative matching of all mass scales relies on detailed solutions of the trembling geodesic equations, which can be elaborated in future work. Finally, the role of cosmological boundary conditions in selecting specific trembling configurations remains an open area of investigation. Indeed, the reasons why the universe realizes exactly this set (and in precisely these quantities) may depend on large-scale boundary conditions, such as cosmological topology or the initial conditions of spacetime.

These questions, however, do not undermine the core derivation but instead delineate its domain of applicability. On the contrary, they define a clear roadmap for future research extending TSRT into cosmology, quantitative mass predictions, and potential new symmetry classifications.

In the following section, we develop these constraints in detail and show why the TSRT framework permits no further extensions of the particle spectrum—thereby establishing its foundational closure and predictive finality.

## 11 Observable Particle Processes Without Field Quantization

The discussion in Section 7.6 outlined how TSRT describes bound states, confinement, and decay through coherent trembling overlaps. Now that Section 10 has demonstrated that all observed elementary particles can be derived as stable trembling eigenmodes, it becomes possible to develop a more precise and quantitative formulation of their interactions and decays.

In this section, we extend and deepen the preliminary analysis of Section 7.6 by introducing explicit variational criteria and geometric measures for scattering, annihilation, and decay processes. This refinement provides a deterministic framework capable of reproducing all known particle phenomenology—without invoking field quantization or probabilistic state collapse.
## 11.1 Scattering as Curvature Reconfiguration

Two incoming trembling modes $P_1$ and $P_2$ scatter when their deviation fields $\xi_{\mu\nu}^{(1)}$, $\xi_{\mu\nu}^{(2)}$ overlap and dynamically reconfigure the local curvature. This induces deflection of proper-time geodesics, with the effective deflection given by:

$$ \Delta x_{\text{scatter}}^{\mu} \propto \int \mathcal{R}_{\mu\nu\rho\sigma} [\xi^{(1)} + \xi^{(2)}] u^{\nu} u^{\rho} d\tau, \qquad (152) $$

where $\mathcal{R}_{\mu\nu\rho\sigma}$ is the effective Riemann tensor. No virtual mediators are invoked; all scattering arises from coherent dynamic curvature exchange.

The coherence overlap functional controlling the interaction probability is defined as:

$$ \mathcal{P}_{\text{overlap}} = \int_{D} |\xi_{\mu\nu}^{(1)}(x) \xi_{\mu\nu}^{(2)}(x)| d^4 x. \qquad (153) $$

This measure determines not only the cross section but the detailed phase sensitivity of the process (as previewed in Section 7.6).

## 11.2 Decay as Internal Phase Dephasing

Decay corresponds to deterministic loss of internal phase coherence. A trembling mode becomes unstable when its coherence interval exceeds the causal stability bound:

$$ \Delta\tau_{\text{max}} = \frac{2\pi}{\omega_{\text{int}}}, \qquad (154) $$

where $\omega_{\text{int}}$ is the internal angular frequency. The mode then fragments into daughter modes satisfying:

$$ \sum_{i} S_{i} = S_{\text{parent}}, \quad \sum_{i} q_{i} = q_{\text{parent}}. \qquad (155) $$

This provides a deterministic alternative to probabilistic decay amplitudes.

## 11.3 Annihilation and Mode Cancellation

Annihilation arises when a mode and its antimode have internal deviation fields satisfying:

$$ \xi_{\mu\nu}^{(e^{-})} + \xi_{\mu\nu}^{(e^{+})} \rightarrow 0. \qquad (156) $$

The resulting cancellation releases the stored action as null trembling geodesics (corpuscular photons). Conservation laws remain exact:

$$ S_{\text{tot}}^{\text{tremble}} = \sum_{i} S_{\gamma_{i}}, \quad P_{\text{in}}^{\mu} = P_{\text{out}}^{\mu}. \qquad (157) $$
## 11.4 Cross Sections from Geodesic Overlap Probability

Unlike conventional quantum field theory, where cross sections arise from operator amplitudes, TSRT derives them from geodesic overlap. The total cross section satisfies:

$$ \sigma_{\text{int}} \propto P_{\text{overlap}}, \qquad (158) $$

linking deterministic coherence geometry to statistical observables. This provides a unified picture of scattering and decay, deepening the preliminary analysis of Section 7.6 and grounding all particle processes in causal trembling spacetime dynamics.

# 12 Finiteness and Uniqueness of the Particle Spectrum in TSRT

Section 10 demonstrates how all known Standard Model particles—fermions, gauge bosons, and resonant modes—arise as symmetry-bound, causally stable trembling configurations in curved spacetime. We now turn to a deeper question: Why does this spectrum terminate? What prevents the existence of additional fundamental particles beyond those already observed?

In Trembling Spacetime Relativity Theory (TSRT) [19], the finiteness of the particle spectrum is not an empirical coincidence but a strict consequence of the theory’s geometric and variational structure. Every admissible particle arises from one of only four permitted trembling classes: U(1), SU(2), SU(3), and the curvature-class associated with gravity. No additional classes are allowed without violating causal closure, coherence quantization, or the variational admissibility of the deviation field $\xi_{\mu\nu}(x)$.

This stands in sharp contrast to quantum field theory (QFT), where additional particles can be postulated by freely extending the field content, enlarging symmetry groups, or invoking spontaneous symmetry breaking. Such extensions, while often motivated by phenomenology or aesthetic principles, are not constrained by the kind of foundational closure that TSRT enforces. What follows is a rigorous demonstration of how and why this geometric closure emerges in TSRT.

In this section, we now examine why this particle spectrum is not only predictive but also strictly finite and non-extendable. The finiteness arises from first-principles constraints on allowed metric deviations, phase coherence, and topological admissibility—constraints that exclude the possibility of arbitrarily adding new particles, as is commonly done in quantum field theory (QFT). In contrast to QFT, where particle content can be expanded by introducing new fields or symmetry groups, TSRT derives the number and type of particles directly from the structure of trembling spacetime itself.

In TSRT, every fundamental particle corresponds to a stable, localized trembling mode of the deviation field $\xi_{\mu\nu}(x)$—a coherent, symmetry-constrained fluctuation of the spacetime metric that encodes proper-time curvature deviation from inertial geodesics. These trembling modes satisfy a strict set of causal and geometric constraints: causal structure preservation, proper-time coherence, variational stability, and geometric localization within the permitted symmetry classes of the deviation field $\xi_{\mu\nu}(x)$. These conditions jointly define a finite space of physically realizable modes. As demonstrated in the preceding sections, each known Standard Model
particle corresponds uniquely to one such mode, with no empirical or theoretical motivation
within TSRT for additional types. This finiteness is not an assumption but a derived feature of
the theory’s geometric foundation.

12.1 Why the Standard Model Appears Larger

In conventional quantum field theory (QFT), the apparent richness of the Standard Model arises from the freedom to define arbitrary field content and representation assignments. This flexibility permits multiple generations, supersymmetric extensions, grand unification models, and hypothetical particles such as axions or sterile neutrinos. By contrast, TSRT admits only those configurations that correspond to stable trembling modes within the geometric and causal constraints of the theory. In this framework, multiple fermion generations emerge not from new internal symmetry classes but as higher-order quantized coherent trembling configurations with distinct mass and curvature characteristics. This feature provides a natural limit on the number of generations, without requiring separate flavor symmetry assumptions.

12.2 Constraints on Extensions

Within the present formulation, TSRT does not accommodate additional internal symmetries
beyond SU(3)×SU(2)×U(1), nor particles that cannot be realized as stable trembling config-
urations within the known geometric classes. Similarly, extensions such as supersymmetry or
additional gauge groups would require the existence of new trembling solutions satisfying the
same causal and variational criteria. Current analysis indicates that such solutions are not sup-
ported by the existing trembling classification, but this conclusion remains subject to refinement
as the mathematical theory develops.

This constraint is not merely an aesthetic preference but arises from structural features of
the trembling framework. Any attempt to define additional stable particle configurations that
do not correspond to causally coherent trembling modes appears to violate one or more of the
following: proper-time coherence, stability under the geodesic deviation equation, compatibility
with action quantization, or preservation of Lorentz symmetry in trembling evolution.

12.3 Predictivity and Falsifiability

Because TSRT restricts the particle spectrum through causal and variational constraints, it
attains a degree of predictive power that differs in character from that of conventional quan-
tum field theories. If future experiments were to confirm the existence of a truly new stable
particle—one that cannot be derived from the trembling structure or associated with one of the
four known geometric classes—TSRT would be challenged in a fundamental way. Conversely,
the continued non-observation of such particles reinforces the plausibility of the finite trem-
bling classification, while still leaving room for further theoretical refinement or extension of
the framework if warranted by evidence.
## 12.4 Minimalism Without Arbitrary Reduction

The resulting particle spectrum in TSRT is designed to be both minimal and sufficient: all currently observed particles can be accounted for as distinct, stable trembling configurations constrained by causality, coherence, and geometric admissibility. No extraneous particles are predicted, and no additional symmetry assumptions—such as supersymmetry, grand unification, or extra dimensions—are required within the present formulation. The theory thereby aims for maximal explanatory economy without invoking arbitrary truncation or speculative extensions.

Importantly, TSRT distinguishes between fundamental particles—those corresponding to stable, symmetry-bound trembling modes—and short-lived, unstable excitations that may arise transiently under extreme curvature or high-energy conditions (e.g., in particle collisions). Such transient configurations lack the proper-time coherence and geometric stability needed for classification as genuinely fundamental in this framework. While real and experimentally observable, their fleeting nature and lack of persistent quantized trembling coherence preclude their inclusion among the elementary particle spectrum as defined here.

Accordingly, TSRT aspires to provide a conceptually unified and empirically grounded approach to understanding the apparent finiteness of the particle spectrum. Rather than accommodating every conceivable field extension, it derives the observed structures from constraints rooted in the causal geometry of spacetime. The result is a framework in which the known particle content appears not as an arbitrary subset of a larger spectrum but as a natural consequence of a specific set of geometric principles. However, a fully rigorous mathematical classification of all admissible trembling solutions remains an open challenge for future work.

## 12.5 Geometric Constraints on New Particle Types

A defining feature of Trembling Spacetime Relativity Theory (TSRT) [19] is that it imposes stringent geometric constraints on the admissibility of particle-like structures. Unlike quantum field theory, where additional particle types can be introduced through symmetry extensions or auxiliary fields, TSRT allows only those particles that emerge as causally admissible, stable, symmetry-bound trembling modes. Within the constraints analyzed so far, this geometric rigidity implies that no new fundamental particles—outside those already derivable from the four trembling classes—are expected to exist.

**Additional Particle Prohibition by Principle.** This exclusion arises from four interrelated principles: the closure of symmetry classes (as currently classified), localized mode stability, invariance under proper-time reparametrization, and the empirical irrelevance of non-admissible configurations.

*Closure of Symmetry Classes.* As shown in Section 2.3, the trembling metric deviation $\xi_{\mu\nu}(x)$ admits exactly four non-redundant symmetry classes within the framework presented:

$$ U(1), SU(2), SU(3), \text{ (Coarse curvature coherence).} \qquad (159) $$

This classification is derived from variational and causal admissibility constraints and is conjec-
tured to be mathematically complete in the sense that no additional internal symmetry—global or local—can be appended without violating Lorentz covariance, proper-time causality, or geometric consistency. However, a fully rigorous existence and uniqueness proof remains open.

*Localized Mode Stability.* Any candidate particle in TSRT must correspond to a localized, temporally coherent trembling mode, as defined in Section 4.1. Such modes must be spatially confined via curvature-induced restoring structures, exhibit internal action quantization over coherence intervals, and preserve the causality inequality of Equation (3). Under current analysis, no additional trembling configurations satisfying all these conditions have been identified.

*Invariance under Proper-Time Reparametrization.* All trembling dynamics in TSRT are governed by action principles parameterized by proper time $\tau$. For a hypothetical new particle to be compatible, its action integral would have to remain invariant under reparametrizations and preserve causal evolution. To date, no further configurations have been shown to meet this criterion without redundancy or instability.

*Empirical Irrelevance of Non-Admissible Configurations.* Even if irregular or anomalous trembling configurations existed beyond the known symmetry classes, such modes would lack the internal coherence, proper-time stability, and conserved quantities (e.g., spin, charge) needed to manifest as persistent, observable particles. They would likely appear, if at all, as transient geometric disturbances without reproducible signatures.

It remains conceivable that future high-energy experiments—especially those probing extreme curvature regimes—could transiently excite such non-admissible configurations. These would manifest, if observed, as short-lived, non-reproducible phenomena that do not constitute genuine particles in the sense defined by TSRT.

This distinction underscores TSRT’s falsifiability. The discovery of a truly new, stable, symmetry-consistent particle not derivable from the trembling classification would necessitate revision or extension of the framework. Conversely, transient anomalies or incoherent decay products do not contradict TSRT predictions, as they fall outside the definition of stable, causally coherent modes.

**Consequences for Particle Physics.** The constraint structure described here has significant implications. Supersymmetric partners—though mathematically consistent in many QFT models—are not predicted in TSRT unless they can be derived as stable trembling modes, which no analysis to date has established. Similarly, hypothetical dark matter candidates, if compatible with TSRT, must arise as coherent trembling configurations embedded in existing curvature symmetry classes—most plausibly as electrically neutral modes in the coarse gravitational sector.

Moreover, extensions such as hypothetical fifth forces or higher-spin particles are not supported under current derivations unless their trembling structure satisfies the same geometric criteria. This does not imply that discovery of such phenomena is impossible, but rather that their existence would either require identifying new admissible solutions within the TSRT frame-
work or indicate a need to generalize the theory’s geometric foundations.

**TSRT as a Geometric Framework.** Trembling Spacetime Relativity Theory is not merely an extension of the Standard Model but an attempt at a geometric foundation underlying its observed structures. While the Standard Model codifies interactions via gauge field principles, TSRT derives particle properties and interactions from the causal geometry of trembling spacetime—without invoking additional symmetry assumptions or arbitrary parameters. In this view, the absence of new particles at higher energies is not surprising but expected. The particle spectrum appears to be geometrically saturated within the present formulation.

Nonetheless, TSRT remains falsifiable in a constructive sense: the confirmed detection of a stable, causally coherent particle beyond the four trembling classes would directly challenge its core assumptions. Conversely, the continued absence of such discoveries would further support the idea that matter and interaction emerge from geometric coherence alone.

In this light, TSRT does not negate the achievements of quantum field theory; rather, it seeks to integrate and extend them through a unifying geometric perspective. The present analysis should be regarded as an initial classification subject to refinement as further mathematical and experimental insights are gained.

## 12.6 Contrast with the Extensibility of the Standard Model

While the Standard Model (SM) has achieved extraordinary empirical success, its underlying structure remains fundamentally open-ended. Rooted in the framework of quantum field theory (QFT), the SM permits extensions through new gauge groups, higher-dimensional operators, undiscovered particles, and mechanisms such as supersymmetry, technicolor, or extra dimensions. This flexibility, though powerful in accommodating experimental data, comes at the cost of predictive closure. As a result, the SM serves more as a highly effective model than a self-contained theory—it describes what is seen, but does not explain why the particle spectrum is finite, why certain symmetries exist, or why no new stable particles have been discovered despite decades of exploration at ever higher energies.

By contrast, Trembling Spacetime Relativity Theory (TSRT) [19] seeks to replace this extensibility with geometric determinacy. In the formulation presented here, all allowable particles and interactions are conjectured to emerge directly from the causal and variational structure of spacetime itself. Under current analysis, no additional symmetries can be appended, and no new degrees of freedom are expected to arise, unless they can be shown to originate from one of the four causally admissible trembling symmetry classes identified so far. This structural rigidity is not imposed ad hoc but proposed as a derived outcome of the trembling framework—though a complete proof of uniqueness remains an open question.

The remainder of this subsection makes this contrast explicit, illustrating how the open architecture of the Standard Model invites arbitrary extensions, while TSRT constrains the physical spectrum to a geometric origin that appears, under the assumptions made, uniquely determined.
**Extensibility in the Standard Model.** The SM is built upon three internal symmetry groups: SU(3)C, SU(2)L, and U(1)Y, acting on independently introduced quantum fields. This field-theoretic construction leaves several degrees of freedom. For instance, additional symmetries (e.g., SU(5), SO(10), E6) can be introduced to propose Grand Unified Theories (GUTs). Supersymmetry postulates the existence of superpartners for every SM particle. New fields may be added to explain phenomena such as neutrino masses, dark matter, or baryogenesis. Finally, flavor extensions multiply generations, CP-violating phases, or family symmetries.

Each of these extensions introduces new hypothetical particles, interactions, or parameters—many of which, despite theoretical motivation, currently lack experimental confirmation and often require finely tuned assumptions to remain phenomenologically consistent.

**Predictive Limitations.** Because of its extensibility, the SM does not in itself determine the number of fermion generations, the mass hierarchy or coupling constants, or the closure and finiteness of the particle spectrum. In this sense, it remains best regarded as an effective theory: one that captures known physics with remarkable accuracy but without providing deeper constraints on why the observed structures are unique or finite.

**TSRT's Contrasting Approach.** In TSRT, the particle spectrum is derived—subject to the conjectures described—from a geometric classification of causal trembling modes. As detailed in Sections 2.1 and 2.3, the framework permits exactly four interaction classes, and only those particle configurations that satisfy proper-time coherence, spatial localization, causal admissibility, and quantized internal trembling action.

Any attempt to introduce new symmetries or particles must conform to these geometric constraints, which severely limit the scope of admissible extensions. As such, TSRT predicts a finite spectrum consistent with observed particles and forbids speculative entities not derivable from its variational structure—although this restriction remains subject to further mathematical refinement and potential future discovery of additional admissible trembling solutions.

**Implications.** The openness of the Standard Model makes it adaptable but non-predictive regarding the ultimate closure of the spectrum. TSRT inverts this logic: by reducing flexibility, it gains falsifiability and explanatory potential. Any future detection of stable particles or interactions not derivable from the four trembling classes would falsify the present formulation. Conversely, the persistent non-observation of predicted SM extensions (e.g., supersymmetry) lends indirect support to the idea that no additional stable particles exist—at least within the energies probed so far.

TSRT thus offers an alternative vision of theoretical physics—one that prioritizes derivation over flexibility and geometric necessity over empirical accommodation. In doing so, it does not claim to accommodate every conceivable possibility, but to provide a framework that can explain why nature selects only a specific, finite set of physically realizable particle structures, within the constraints and assumptions currently established.
## 12.7 Formal Status of Existence and Uniqueness

The foundational formulation of TSRT [19] develops the variational action principle, the trembling geodesic equations, and the action quantization condition that underpin the classification of admissible particle modes. In that work, the spectrum finiteness was derived constructively by demonstrating that only four geometric symmetry classes (U(1), SU(2), SU(3), and the coarse curvature class) admit solutions satisfying smoothness and causal admissibility of the deviation field $\xi_{\mu\nu}(x)$, quantized action accumulation over closed trembling cycles, and variational stability under infinitesimal perturbations.

However, it is important to emphasize that a fully rigorous existence and uniqueness theorem—establishing in a functional-analytic sense that every solution satisfying these constraints must belong to the enumerated symmetry classes, and that no additional stable trembling configurations exist anywhere in the admissible function space, was not formally proven in the foundational TSRT publication. Instead, the argument proceeded via constructive classification: for each symmetry class, explicit solutions were exhibited, and heuristic reasoning was given to show that other configurations necessarily fail coherence, causality, or quantization.

Accordingly, this paper retains the same status: the results constitute a constructive derivation and an exhaustive classification within the framework defined by the TSRT trembling ansatz. Nevertheless, a complete mathematical proof of global uniqueness and existence—analogous to theorems in the theory of solitons or harmonic maps—remains an open problem and will be addressed in future work.

This acknowledgment delineates the scope of the present derivation while preserving its predictive power: any stable particle-like solution must satisfy the same set of geometric constraints, and no additional trembling symmetry classes have been identified that could support further fundamental particles.

More precisely, while TSRT establishes a well-defined variational framework for the trembling eigenmode equations,

$$ \delta S = 0, $$

with $S$ the proper-time action functional and subject to the constraints of smoothness, causal admissibility, and bounded curvature norm, a complete mathematical proof of the existence and uniqueness of all solutions satisfying, i.e., (i) the quantization condition, Equation (102), (ii) global causal coherence, and (iii) stability under infinitesimal perturbations, remains an open problem. This difficulty is structurally similar to the classification of globally regular solutions in non-linear geometric field theories (e.g., Einstein-Yang-Mills equations), where no general uniqueness theorems are known.

Nevertheless, partial evidence supporting the conjecture of a finite and unique spectrum includes (i) the strict quantization condition, which discretizes admissible action values, (ii) the limited set of internal symmetry classes (U(1), SU(2), SU(3), and coarse curvature), and (iii) the demonstrated existence of stable localized solutions in symmetric ansätze (Sections 4.1 and 4.2).

Accordingly, this work treats the finiteness and uniqueness of the trembling spectrum as a conjecture strongly supported by the constraints of the TSRT framework but subject to further
mathematical refinement.

It is important to note that, while a global uniqueness theorem remains open, explicit examples of stable trembling solutions satisfying all constraints have been constructed in symmetry-reduced ansätze. For instance, spherical and axial configurations yield localized solutions corresponding to fermionic and gauge boson modes with correct quantized holonomy. These constructions, detailed in [19], provide constructive existence evidence supporting the conjectured classification.

Finally, it should be emphasized that the conjectured uniqueness pertains specifically to fully stable, causally coherent trembling eigenmodes with quantized action accumulation. While transient or metastable configurations may exist—analogous to hadronic resonances or collective excitations in other field theories—such configurations lack the persistent coherence required to constitute fundamental particles in TSRT. Therefore, any partial non-uniqueness among unstable solutions does not undermine the closure of the fundamental spectrum.

# 13 TSRT Compatibility with Known Data

Trembling Spacetime Relativity Theory (TSRT) [19] was developed to provide a geometric foundation for matter and interaction, while remaining consistent with all known qualitative experimental phenomena. Crucially, it reproduces the general structure of the Standard Model—such as gauge interactions, quantized charges, and spin assignments—without relying on quantum field theory postulates like operator-valued fields, renormalization, or Hilbert-space probabilities. Observable effects in TSRT emerge instead from the geometry of causally admissible trembling.

**Distinctive Features and Testable Differences** Although many consequences of TSRT are consistent with Standard Model expectations, several qualitative predictions differ in principle from conventional QFT:

1. **Absence of Higher-Spin Stable Particles:** TSRT does not allow stable spin-3/2 or higher excitations as fundamental trembling modes. Observation of such stable particles (e.g., Rarita-Schwinger-like states) would contradict this framework.

2. **No Vacuum Energy Divergence:** TSRT predicts no zero-point energy contribution from quantized fields, implying that the cosmological constant problem does not arise in the same form. This could have indirect observational implications.

3. **Deterministic Origin of Quantum Phenomena:** Interference and tunneling are attributed to ensembles of deterministic trembling geodesics, not superpositions of probability amplitudes. Experiments probing coherence scales may eventually distinguish these pictures.

These differences offer possible avenues for future falsification or corroboration, though precise quantitative predictions remain to be developed.
**Agreement with Particle Content** As shown in Section 10, all known fundamental particles—leptons, quarks, gauge bosons, and the Higgs scalar—correspond to stable, symmetry-bound trembling configurations in this framework. No Standard Model particle is excluded, and no hypothetical stable particle is predicted without an underlying trembling mode satisfying the causality and coherence criteria.

**Charge Quantization and Conservation** Electric charge, weak isospin, and color are conserved quantities in TSRT, arising from Noether-like symmetries of trembling configurations (Section 2.2). The observed quantization of electric charge in integer multiples of $e/3$ follows from the phase coherence constraints of $U(1)$ trembling intervals.

**Mass Spectra and Stability** While TSRT provides a qualitative explanation for why some particles are massless (e.g., the photon) and why certain modes are confined (e.g., quarks), it does not yet predict precise numerical values of particle masses. Mass in TSRT arises from resistance to geodesic deflection caused by localized trembling curvature (Section 3.2), and the observed hierarchy is consistent in broad outline with this principle.

**Spin and Statistics** TSRT reproduces the known spin assignments and spin-statistics relationships through topologically quantized trembling holonomy (Section 5.1). Fermions correspond to half-integer holonomy modes, while gauge bosons emerge as transverse trembling excitations with integer spin.

**Electroweak Interactions** The SU(2) trembling class provides a geometric interpretation of chiral electroweak interactions, including parity violation, without requiring spontaneous symmetry breaking via a quantized Higgs vacuum. Distinguishing between these mechanisms experimentally remains an open challenge.

**Strong Interaction and Confinement** TSRT accounts qualitatively for confinement of color charge through the non-linear geometry of SU(3) trembling, which prevents free propagation of isolated color-charged excitations.

**Gravitational Phenomena** Gravitational effects arise from coarse curvature trembling. Observed lensing, time dilation, and redshift are consistent in principle with the TSRT large-scale limit.

**Precision Tests** While TSRT is not in contradiction with any current experimental results, it does not yet provide detailed quantitative matches to all precision observables (e.g., anomalous magnetic moments, fine-structure splittings). Further development is needed to establish whether the framework can reproduce these quantities with the same accuracy as QED or QCD.

**Compatibility with Quantum Predictions** Although TSRT rejects the probabilistic interpretation of quantum mechanics, it is compatible with the outcomes of quantum experi-
ments. Deterministic trembling ensembles can reproduce interference and diffraction phenomena (cf. [22]).

In summary, TSRT remains qualitatively consistent with all known data and offers a set of testable predictions and conceptual alternatives to conventional field-theoretic formulations. Its future viability depends on further derivation of quantitative observables and experimental tests of the distinguishing features outlined here.

# 14 Discussion and Outlook

This work advances a central objective of Trembling Spacetime Relativity Theory (TSRT) [19]: to derive, constrain, and explain the observed spectrum of fundamental particles and their physical properties from a unified geometric principle. Within this framework, mass, spin, electric charge, color, and the known interactions are not introduced as independent postulates, but emerge deterministically as intrinsic properties of symmetry-bound deviations in a causally fluctuating spacetime metric. Unlike quantum field theory, which begins with fields, quantization rules, and symmetry assumptions, TSRT begins with geometry and causality—and seeks to recover the rest.

At the heart of this reformulation lies the replacement of operator-based quantization with proper-time coherence. Discreteness in nature—whether of spin states, energy levels, or charges—emerges not from imposed commutation relations, but from the requirement that trembling configurations remain causally admissible and dynamically closed under finite coherence intervals. These coherence constraints select only certain allowable trembling modes, thereby reproducing quantum phenomena as emergent consequences of geometric stability. No Hilbert space, wavefunction, or probabilistic postulate is required. In this view, quantum behavior results from the causal-geometric structure of trembling spacetime.

A principal result of this paper is the demonstration that trembling-compatible solutions form four symmetry classes—U(1), SU(2), SU(3), and curvature—that correspond precisely to the structure of the four known fundamental interactions. Within the current formulation, the space of geometrically admissible particle configurations is discrete and finitely determined by the causal and variational structure of the trembling metric. Every known particle—leptons, quarks, gauge bosons, and the Higgs—is described as a localized solution within these geometric classes. No additional stable particles have been identified, and any configuration outside these classes either violates coherence, causal closure, or proper-time admissibility, or remains unsupported by the present analysis.

This finiteness lends TSRT a predictive constraint not present in the Standard Model, which permits continuous field extensions, additional generations, and unconstrained parameter fits. TSRT, by contrast, forbids the existence of supersymmetric partners, exotic bosons, fifth forces, or non-geometric particles, unless such structures can be shown to arise from causally admissible trembling solutions. The present analysis provides strong evidence that no such configurations exist under the causal-variational principles of the theory. Transient, unstable, or incoherent trembling structures may form—but they cannot satisfy the stability and coherence criteria required for classification as fundamental particles.
TSRT maintains full compatibility with all current experimental data. It recovers the struc-
ture and conserved quantities of the Standard Model, while offering geometric derivations of
mass-energy relations, quantized charges, interference patterns, and curvature-induced redshift
corrections, as demonstrated in related hydrogen spectrum calculations [20]. At the same time,
it diverges fundamentally from the interpretive framework of quantum field theory. Particles
are not excitations of an abstract vacuum, but stable, curvature-bound solutions to geodesic
deviation equations; forces are not mediated by virtual quanta, but arise from intrinsic met-
ric symmetries and causal admissibility. The physical constants and particle families emerge
without symmetry breaking, spontaneous or otherwise.

This divergence enables experimental testability. TSRT predicts that no new stable fun-
damental particles will be discovered beyond the known trembling spectrum. It excludes su-
perpartners, forbids fifth interactions, and constrains all known forces to those arising from
internal symmetry classes of the deviation field. It also predicts strict coherence-based limits
on decay channels and scattering amplitudes. The absence of new particles at current collider
energies and the geometric rigidity of known interaction patterns provide indirect validation of
this approach. Conversely, the discovery of a stable, causally coherent particle inconsistent with
this classification would necessitate a substantial revision—or falsification—of the theory.

TSRT thereby unifies matter, motion, and interaction into a single geometric and causal
principle. Mass arises from inertial response to curvature-induced trembling; spin from topo-
logically quantized holonomy in proper time; charge from gauge-class holonomy under coherent
deviation transport; and energy from action accumulation along causal geodesic paths. In this
perspective, the Standard Model—once a parametrized phenomenological framework—emerges
as a consequence of trembling spacetime geometry.

This work lays the foundation for future developments. A fully geometric theory of nuclear
structure, a coherent extension to curved cosmological spacetimes, and a trembling-based de-
scription of early-universe entropy and inflation are natural next steps. These will require no
new postulates—only deeper analysis of the causal and geometric structure already established.

In sum, Trembling Spacetime Relativity Theory offers a deterministic and falsifiable foun-
dation for particle physics. It resolves the structure of matter not through quantization or
conjecture, but through geometry. As such, TSRT is not merely an extension of the Standard
Model: it proposes an alternative explanatory framework grounded in causal coherence and
geometric necessity.

**Complementarity with Quantum Field Theory.** While Trembling Spacetime Relativity Theory aims to derive particle properties and interactions from geometric first principles, it is important to emphasize that this perspective does not deny the empirical success of quantum field theory. QFT remains an extraordinarily accurate and predictive framework for modeling scattering processes, decay rates, and quantum fluctuations. The approach presented here is not intended as a replacement for the operational machinery of QFT in calculating observables, but rather as a complementary framework that seeks to explain why the structures codified in QFT—such as gauge symmetries, charge quantization, and spin-statistics correspondence—arise in the first place. From this perspective, TSRT can be viewed as a geometric foundation that un-
derlies the effective field theories of the Standard Model, providing an alternative interpretative layer rather than negating their achievements.

## Acknowledgements

The author gratefully acknowledges institutional support from the French Centre National de la Recherche Scientifique (CNRS) through the International Research Laboratory IRL 2958 Georgia Tech-CNRS, as well as Metz Métropole, the City of Metz, and Georgia Tech Europe for hosting him as a Georgia Tech faculty member in France.

Infrastructure and mobility support were provided by the Région Grand Est, the French Lorraine Université d'Excellence (LUE impact project I-META ANR-15-IDEX-04-LUE), Institut Carnot ARTS, Conseil Régional de Lorraine, Le Conseil Départemental de la Moselle, Feder-Europe, the Indo-French Centre for the Promotion of Advanced Research (CEFIPRA RCF-IN-0067), Mirabelle Plus, Stellantis, Institut de Soudure, the Association Nationale de la Recherche et de la Technologie (ANRT), Metz Métropole, the Institut Supérieur Européen de l'Entreprise et de ses Techniques (ISEETECH), the Contrat Plan État-Région (CPER-MEPP-P05), the Agence Nationale de la Recherche (ANR-09-BLAN-0167-01), the European Union's Horizon 2020 program (Grant Agreement No. 871260), the Belgian Fund for Scientific Research Flanders (FWO, 2005), the Flemish Institute for the Encouragement of Scientific and Technological Research in Industry (IWT, 2001-13343), and the North Atlantic Treaty Organization (NATO PST.CLG.980315). While these programs provided an invaluable research environment, the conceptual direction and results presented here stem from independent scholarly inquiry developed over decades.

The author acknowledges the foundational training in physics, astrophysics, and engineering physics received at the Catholic University of Leuven (KULAK and KU Leuven) and Ghent University, which shaped his interdisciplinary approach. He also thanks the Georgia Institute of Technology and the George W. Woodruff School of Mechanical Engineering for his promotion to Full Professor, coinciding with his designation as *Professor Honoris Causa* in Physics by the University of Allahabad—a confluence of recognition that secured the academic freedom necessary to formalize the ideas presented here, which trace their origins to early studies of solar system dynamics (Driesprong, Deerlijk) and electron beam deflection experiments (Koninklijk Atheneum, Waregem).

The author extends his deepest gratitude to colleagues and organizations in acoustics, ultrasonics, acousto-optics, and optics for fostering an intellectually vibrant and collegial environment. Their shared passion for wave phenomena has been a continual source of inspiration, and their engagement with industrial applications always boosted inspiration to seek critical support to sustain the author's laboratory.

He is equally indebted to his current and former students (BS, MS, PhD) and postdoctoral researchers, whose curiosity and fresh perspectives profoundly enriched his work in nondestructive evaluation and physical acoustics. Their enthusiasm reaffirmed that scientific discovery—regardless of discipline—thrives on intuition, relentless curiosity, and intellectual joy, motivating the author's deeper explorations in fundamental physics beyond conventional work-
ing hours.

This work stands as a testament to the enduring intellectual and material legacies of Daniel A. Lesage (1911–1988), Pierre F. Vangheluwe (1939–1991), Mariette Vanoosthuyze (1925–2009), Godelieve Verborgh (1932–2012), Maurice A. Declercq (1941–2015), and Jeanne M. C. Maesen (1930–2024), whose generosity continues to enable scholarship. The author is equally indebted to Jeanette Verbrugghe and Nelly Vangheluwe for their unwavering support—both moral and logistical—which has been indispensable to this research.

Finally, the author expresses profound gratitude to his wife and children for their steadfast presence, boundless encouragement, and enduring patience throughout the many weekends and holidays devoted to this work.

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