This paper presents a comprehensive, alternative non-linear geometric framework aimed at unifying classical general relativity with quantum mechanics under a cohesive paradigm termed Probabilistic Determinism. We propose that the smooth spacetime metric manifold constitutes the sole fundamental physical reality, and that subatomic fermions are localized, topologically stable metric solitons possessing spin-1/2 connection profile invariants. By introducing a novel covariant field relation—the Anbi Metric Deformation Equation—we demonstrate that localized energy-momentum density traces inherently compress the coordinate network itself at the sub-microscopic scale. This localized spatial contraction provides an asymptotic geometric buffer that regularizes field interactions, natively resolving the ultraviolet divergence catastrophes of quantum field theories without requiring arbitrary mathematical renormalization. Furthermore, utilizing a relativistic de Broglie-Bohm phase connection, we demonstrate how deterministic trajectories traversing this dynamically compressed, fluctuating background manifold scale up macroscopically to recreate observed quantum statistical probability distributions as a strict geometric continuity fluid law.
This paper presents a comprehensive, alternative non-linear geometric framework aimed at unifying classical general relativity with quantum mechanics under a cohesive paradigm termed Probabilistic Determinism. We propose that the smooth spacetime metric manifold constitutes the sole fundamental physical reality, and that subatomic fermions are localized, topologically stable metric solitons possessing spin-1/2 connection profile invariants. By introducing a novel covariant field relation—the Anbi Metric Deformation Equation—we demonstrate that localized energy-momentum density traces inherently compress the coordinate network itself at the sub-microscopic scale. This localized spatial contraction provides an asymptotic geometric buffer that regularizes field interactions, natively resolving the ultraviolet divergence catastrophes of quantum field theories without requiring arbitrary mathematical renormalization. Furthermore, utilizing a relativistic de Broglie-Bohm phase connection, we demonstrate how deterministic trajectories traversing this dynamically compressed, fluctuating background manifold scale up macroscopically to recreate observed quantum statistical probability distributions as a strict geometric continuity fluid law.
The absence of a common mathematical structure underlying quantum theory and relativistic physics remains a central obstacle to unification. While relativity is formulated in terms of spacetime geometry and covariance, quantum theory is intrinsically probabilistic, relying on amplitudes, uncertainty, and statistical interpretation. Existing unification programs typically attempt to quantize geometry or to geometrize quantum fields, but the probabilistic structure itself is rarely treated as a fundamental geometric object. In this paper, we propose a unified description of relativistic and quantum dynamics based on probabilistic geometry. Probability densities are taken as primary variables and are endowed with an intrinsic information–geometric structure defined by the Fisher–Rao metric. Within this framework, physical motion is described as the covariant evolution of probability distributions on a statistical manifold rather than as trajectories in a preassigned phase space or Hilbert space. The central result is a unified probabilistic–geometric field equation in which the relativistic d’Alembertian acting on the probability density is coupled to the scalar curvature of the associated information manifold. In an appropriate nonrelativistic limit, this equation reproduces the quantum potential as a curvature term, establishing a direct correspondence between quantum effects and informational geometry. Relativistic covariance is preserved throughout, allowing quantum and relativistic structures to appear as different regimes of a single probabilistic equation. The proposed framework does not introduce new particles, interactions, or quantization postulates. Instead, it provides a structural unification in which quantum and relativistic features emerge from the geometry of probability itself.
The probabilistic interpretation of quantum mechanics rests fundamentally on the Born rule, which asserts that the physical density of a state is given by the squared magnitude of its wavefunction (ρ=|ψ|^2). Historically, this exponent of (α=2) has been treated as an unprovable, axiomatic postulate justified solely by empirical alignment. This paper presents a rigorous, parameter-agnostic derivation that turns this narrative on its head. By introducing ( α) as a completely free geometric parameter ( ρ=|ψ|^α) and demanding strict adherence to the universal principle of Local Continuity: ∂ρ/∂t+∇⋅J ⃗=0 I show that the algebra itself systematically eliminates all alternative mappings. If ( α≠2), the local density variation inherently picks up non-local dependencies on the absolute background field amplitude, violating localized field interactions. The mathematics strictly demands ( α=2) as a unique geometric threshold required to isolate unobservable phase variables and preserve local conservation laws. Consequently, I demonstrate that the foundational framework of quantum mechanics is not natively probabilistic or random; rather, it is a strictly deterministic, hydrodynamic field theory whose fluid-like conservation mandate uniquely dictates the geometry of its measurement space.
We present a probabilistic-geometric formulation of quantum motion that establishes a direct geometric correspondence between statistical uncertainty and relativistic curvature. Quantum states are represented as probability distributions on a statistical manifold equipped with the Fisher–Rao metric, whose geodesic flow reproduces the dynamical evolution of quantum motion in probabilistic form. Embedding this manifold into a Lorentzian spacetime yields a unified metric structure in which stochastic fluctuations generate microscopic deformations of spacetime geometry. This framework enables the visualization and prediction of quantum motion as probabilistic geodesics, and provides a foundation for connecting information geometry, stochastic mechanics, and general relativity within a single coherent formulation.
While the foundational architecture of quantum mechanics has remained structurally bound to the probabilistic interpretations of the Copenhagen school due to the historical inability to resolve high-frequency ultraviolet boundary singularities, we present a definitive, constructive mathematical framework that establishes the absolute global regularity and determinism of fermion fields within native four-dimensional spacetime. By embedding the four-component Dirac spinor into a density-weighted non-linear state vector, the coupled system is processed through a gauge-covariant intrinsic Friedrichs mollifier J_λ integrated with an anti-symmetric Riesz transform kernel R_m. At the contact interface, the singular Dirac-delta loop divergences undergo an instantaneous, exact cross-term cancellation, giving spontaneous birth to a fourth-order biharmonic quantum-chromodynamic dissipative spectral cage - λ^2 * G_vac * 𝔐^4. At the vanishing scaffold limit (λ → 0), the continuous regularization parameter collapses uniformly (λ^2 / λ^2 = 1), freezing the anomalous energy dissipation ledger into a strictly positive local invariant. This structural closure uncovers the long-sought geometric origin of the particle angular momentum, proving analytically that the quantum spin (𝔖 = 1/2) is the unique, necessary topological reflex of the continuous spacetime fabric twisting to preserve gauge invariance under the mass gap constraint of non-Abelian fields. Furthermore, this non-local functional integration permanently eradicates the probabilistic wave-function collapse paradox, explaining quantum entanglement (EPR anomalies) as a continuous, deterministic redistribution of field density across a structurally un-cuttable topological domain. The predictive sovereignty of this framework is rigorously verified against empirical accelerator data, offering an ironclad mathematical blueprint that anchors quantum causality from within continuous four-dimensional engineering realities.
This paper establishes a background-independent geometric determinism for quantum measurement and cosmic genesis by treating the micro-spacetime domain as a Porous Spacetime Manifold with a Planck-scale cutoff. Born's probability amplitude (|ψ|²) is redefined as a deterministic initial distribution ratio of the spacetime fluid membrane across microscopic void boundaries.Critically, this model resolves the long-standing logical breakdowns in quantum foundations: temporary energy non-conservation in time-energy uncertainty and the superluminal collapse of wave packets are geometrically explained via the manifold's non-local straddling and its subsequent surface-tension-like pull-in mechanism, without violating special relativity.Furthermore, we formulate the "initial cosmic fluctuations" as a direct 100% causal transcription of the preceding universe's cumulative entropy, transferred via quantum entanglement threads (ER bridges) across generations. This model comprehensively unifies particle localization, non-resettable causal histories, and the cosmic reincarnation cycle under a single background-independent topological framework.
This updated preprint (v2) investigates the fundamental ontological contradiction between macroscopic chemical determinism and microscopic quantum uncertainty. Standard interpretations of quantum mechanics propose a micro-world governed by causeless randomness and probability waves. However, this study argues that the flawless precision observed in macroscopic chemistry and industrial systems necessitates an underlying, strictly deterministic physical reality operating at the hardware level of the universe. To resolve this paradox, the paper introduces the "Cosmic Texture" (Texton) Model, which redefines the universe not as an empty void, but as a continuous, highly elastic, and compressible fluid substratum. Within this framework, quantum uncertainty is not a fundamental law of nature, but the deterministic result of the unmapped hydrodynamic background noise of this texton sea and the inherently destructive nature of measurement methods. Key concepts explored and expanded in this version include: Hydrodynamic Phase Transitions & Baroclinic Vorticity: Rejecting mass-energy duality, the model defines particles as varying physical phases of the texton fluid. Linear compression waves (energy) transition into highly dense, stable, rigid vortices (matter) via baroclinic torque (∇ρ×∇P) during high-energy collisions. Charge as Pressure Gradients: Electric charge and attractive forces are redefined mechanically as hydrostatic pressure and central vacuum gradients within the continuous fluid network. Mechanical Wave Collapse (The Observer Effect): The "collapse" of probability waves is explained via classical fluid dynamics. A measurement shockwave causes a diffuse electron cloud to implode into a constricted state. Exceeding its structural carrying capacity, the vortex reflexively expels excess energy (photon emission) and returns to its stable limit cycle. Non-Local Determinism: Bell's non-locality is preserved through the De Broglie-Bohm pilot-wave theory, where the abstract "quantum potential" is physically identified as real hydrostatic pressure and tension gradients within the texton sea. Spin Quantization: The discrete nature of quantum spin is mathematically grounded in Kelvin's Circulation Theorem, demonstrating that topological fluid knots must exist in integer multiples. Empirical & Phenomenological Validation: Moving beyond theoretical frameworks, this study supports its deterministic model with discrete lattice simulations and macroscopic analogs. Crucially, the hydrodynamic behavior of the Cosmic Texture is corroborated by established laboratory evidence, including: Madelung's hydrodynamic formulation of the Schrödinger equation (1927). Yves Couder's macroscopic walking droplet experiments (2006). The Dynamic Casimir Effect (DCE): Demonstrating the mechanical excitation of the fluid substratum. The SLAC E144 Experiment: Providing evidence for wave-to-vortex phase transitions. The Aharonov-Bohm Effect & Superfluid Vortices: Validating vector potentials as physical momentum flow and confirming the macroscopic quantization of limit cycles. This study presents a testable, causal, and purely mechanical alternative to the Copenhagen interpretation, re-establishing absolute causality in theoretical physics.