We develop a mathematically rigorous theory of quantum measurement within Entanglement-Weighted Operator Gravity (EWOG), in which spacetime geometry is an operator-valued entity weighted by quantum entanglement. We show that quantum measurement corresponds to a localized, transient fluctuation of spacetime curvature induced by rapid redistribution of entanglement. We prove that competing measurement outcomes correspond to competing semiclassical geometries whose stability is governed by curvature variance. Minimization of curvature fluctuations leads uniquely to probability weights proportional to $|\psi|^2$, yielding the Born rule as an emergent geometric law rather than an axiom. This framework unifies quantum measurement, decoherence, and classical spacetime emergence within a single gravitational-informational principle. We further derive the spectral density of measurement-induced curvature fluctuations and establish experimental bounds on their detectability.
We develop a mathematically rigorous theory of quantum measurement within Entanglement-Weighted Operator Gravity (EWOG), in which spacetime geometry is an operator-valued entity weighted by quantum entanglement. We show that quantum measurement corresponds to a localized, transient fluctuation of spacetime curvature induced by rapid redistribution of entanglement. We prove that competing measurement outcomes correspond to competing semiclassical geometries whose stability is governed by curvature variance. Minimization of curvature fluctuations leads uniquely to probability weights proportional to ∣ψ∣2, yielding the Born rule as an emergent geometric law rather than an axiom. This framework unifies quantum measurement, decoherence, and classical spacetime emergence within a single gravitational-informational principle. We further derive the spectral density of measurement-induced curvature fluctuations and establish experimental bounds on their detectability.
We present a first-principles derivation of the Born rule using a curvature-filtered gravitational decoherence model combined with standard environmental decoherence.The derivation focuses on filtered density matrices and geometric projection operators to explain how outcome probabilities emerge as curvature-weighted classicalizations of quantum states. The curvature filter, motivated by renormalization group flow, regulates when vacuum fluctuations become gravitationally classical. We show that the Born rule emerges as a special case of a normalized curvature-weighted density matrix and as the limit of curvature-saturated operator projections. We briefly supplement this with a path-integral formulation and discuss competing theories. This approach offers a novel, observer-independent foundation for quantum measurement.
This work analyzes the stability of probabilistic manifolds embedded in curved spacetime, extending earlier formulations in which gravity emerges from the localization of probability. By introducing an affine probabilistic connection that preserves normalization, the framework shows that curvature induces a natural correction to coherence transport on the underlying state manifold. The resulting curvature‑dependent coherence equation yields a stability criterion in which positive curvature enhances localization and suppresses divergence between probability configurations, while negative curvature drives decoherence growth. A generalized divergence metric and curvature‑modulated coherence gate unify gravitational localization, quantum coherence limits, and vacuum stability within a single geometric principle: the self‑consistency of probability under curvature. The analysis predicts curvature‑dependent decoherence rates, suppression of false‑vacuum tunneling in regions of positive curvature, and cosmological thresholds associated with coherence instability. This establishes a dynamic feedback loop between probability, coherence, and geometry, offering a unified interpretation of gravitational behavior and quantum stability.
Abstract We develop a novel framework connecting quantum mechanics with gravitational geometry through a deformation of probability amplitudes. Inspired by Rohit Dhormare’s geometric formulation of probability under gravitational fields, we reinterpret the influence of curvature on probability not within classical settings, but in the domain of quantum amplitudes. We propose a quantum probabilistic manifold where space-time curvature induces topological and metric deformations on Hilbert space projections, affecting interference, measurement outcomes, and entropy. This formulation offers new pathways for understanding quantum gravity, decoherence, and information localization.
Abstract We develop a geometric framework in which quantum amplitudes, superposition, and probabilistic interpretation emerge from phase–channel geometry rather than from axiomatic postulates. The theory is formulated using generalized phase algebras Cₙ, with particular emphasis on the reduction chain C₆ → C₃ → C₂. At the C₆ level, the adjoint-defined norm selects a two-dimensional visible subspace spanned by antisymmetric phase channels, enforcing superposition as a direct consequence of higher phase symmetry. Projection to lower phase geometries collapses this superposition to a single effective channel, not through an external measurement postulate but via a curvature-driven stability mechanism. We show that distinct visible phase directions in C₆ project onto the same visible direction in C₃, with their geometric differences encoded solely in scalar amplitudes. These amplitude weights are fixed by phase geometry and lead naturally to a squared-norm probability rule. In this sense, the Born rule is obtained as a geometric projection law rather than assumed as a foundational axiom. The framework thus provides a unified geometric account of superposition, collapse, and probabilistic interpretation, and suggests a reinterpretation of quantum amplitudes as remnants of higher-dimensional phase geometry.
We develop a geometric formulation of stochastic dynamics in which noise, diffusion, path probabilities, fluctuation theorems, and entropy production arise from the intrinsic geometry of an evolving manifold rather than from externally imposed randomness. Within the theory of moving manifolds, we establish a curvature-noise correspondence: fluctuations are governed by the inverse curvature tensor, while entropy production is controlled by curvature deformation. The invariant continuity law on a moving hypersurface yields a geometric Fokker-Planck equation, and curvature-velocity coupling generates a quadratic Onsager-Machlup functional determining path weights. The resulting entropy functional satisfies a curvature-driven monotonicity law, providing a geometric derivation of the Second Law. In two dimensions, the curvature invariant reduces to Gaussian curvature and encodes topology, so topological transitions produce discrete entropy jumps. When the ambient space carries a Minkowskian signature, the same curvature-kinetic quadratic form that generates dissipative thermal weights produces oscillatory phase weights, and the Laplace-Beltrami operator governing entropy evolution acquires a Schrödinger-type structure. This provides a geometric resolution of the apparent distinction between classical stochastic behaviour and quantum dynamics. These results show that stochastic behaviour, thermodynamic irreversibility, and quantum transition amplitudes are unified within the moving manifold framework. Geometry does not merely accommodate stochasticity; stochastic behaviour arises as a consequence of deterministic geometric evolution. The theory predicts curvature-controlled anisotropic diffusion, entropy jumps at topology-changing events, and a geometric thermal-quantum crossover in which classical stochastic weights and quantum amplitudes are generated by the same curvature-kinetic action.
Quantum measurement lies at the intersection of precision sensing, entanglement theory, and the foundational questions of quantum mechanics. Recent advances in cold-atom interferometry have enabled unprecedented sensitivity in gravity measurements, while simultaneously raising profound questions about the interface between gravity and quantum theory [1]. Entropy plays a multifaceted role in this context: it quantifies quantum entanglement as a resource for quantum-enhanced measurements, provides a framework for evaluating noise sources in quantum sensors, and serves as a key signature for investigating whether gravity itself induces quantum decoherence [1]. This paper introduces Spectral Nod Gravity (SNG)—a unified framework in which spacetime emerges from discrete Planck-scale ``nod'' networks governed by four dynamic relational operators—to address these interconnected themes. We demonstrate how the Fluctuating Equivalence Operator (\(\fleq\)) dynamically modifies effective Lindblad rates to suppress decoherence from two-level fluctuators (TLFs) in quantum sensors. The Cyclic Equivalence Operator (\(\cyceq\)) enables controlled reset mechanisms that extend coherence times in precision measurements. The Phase Nexter Operator (\(\nexteq\)) facilitates error-free state preparation by bypassing unstable intermediate configurations. The Phase Reverser Operator (\(\reveq\)) naturally realizes quantum echo mechanisms for non-local coherence protection. We derive the operator algebra, present QuTiP simulations validating the predicted coherence improvements, and discuss implications for gravitational entropy bounds and the holographic principle. This work unifies quantum measurement theory with gravitational thermodynamics, offering testable predictions for next-generation quantum sensors.