The Unified Curvature Principle

The Unified Curvature Principle: Fisher Geometry as the Common Origin of Quantum Potential and Spacetime

The sPaceNPilottime (sPNP) framework proposes a radical yet parsimonious unification of quantum and gravitational phenomena by positing that the fundamental geometry of the universe is not spacetime per se, but the curvature of a physically real information manifold: the configuration space Fisher Information Metric. While traditional quantum theory introduces probability epistemically, and general relativity treats curvature as arising from stress-energy in a Riemannian spacetime, sPNP argues that both quantum potential and spacetime curvature are projections of a deeper geometric structure: the curvature tensor derived from the Fisher metric. sPNP employs either the full Fisher–Rao metric on configuration distributions or a regularized local form that ensures a non-degenerate, well-posed geometry. These constructions preserve the curvature–quantum potential link and support the unified dynamics without requiring inverse-metric-dependent Lagrangians. sPNP requires the Fisher metric (and its inverse as needed to form Δ𝑔) but does not posit any extra independent metric for spacetime, spacetime geometry is emergent via the projection kernel.

In Bohmian mechanics, the quantum potential (QP) governs particle guidance through configuration space and is derived from the wavefunction's amplitude via a Laplacian expression. While Bohmian mechanics retains determinism, it explains the probabilistic Born rule as arising from ignorance of initial conditions. In sPNP, probability is no longer about ignorance, it is the fabric of the universe’s capacity to distinguish between alternative configurations. Epistemic ignorance is a coarse-grained shadow of the universe’s intrinsic distinguishability structure.The origin of the QP's curvature structure is unexplained in naive Bohmian Mechanics. sPNP resolves this with the Fisher Information Metric as the natural geometric structure encoding its gradients and curvatures. The quantum potential is not an ad hoc extra term; it is the Laplace–Beltrami of the wave amplitude built from the Fisher metric.

Explicitly, Q(X) = - ℏ² / (2 m) * (Δ_g R(X)) / R(X), where Δ_g is the Laplace–Beltrami operator constructed from the Fisher–Rao metric g_IJ(X): Δ_g f = (1 / sqrt(det g)) * ∂_I [ sqrt(det g) * g^IJ * ∂_J f ]​

Thus the QP is a geometric scalar constructed from amplitude gradients and second derivatives and is tightly linked to Fisher curvature. In regimes where the Fisher metric itself is dominated by amplitude-gradients, the QP is numerically controlled by the same invariants that determine the manifold’s curvature; in that sense the QP is a manifestation of Fisher curvature.

Crucially, this configuration-space geometry is not an abstract computational tool but a real ontological structure. In sPNP, the Fisher metric defines geodesics, curvature, and stress in a generalized sense, leading to dynamical evolution not via external potentials but through intrinsic constraint geometry. The result is a unified picture where geodesic deviation in Fisher space explains both quantum guidance and gravitational interaction. When this curved geometry is projected onto coarse, relational observables (e.g., clock readings, rods), it manifests as spacetime curvature. Therefore, spacetime itself emerges as a limit or projection of a deeper, information-based configuration geometry. Non-locality, in sPNP, remains a natural consequence of configuration space structure; a feature, not a bug.

This formulation answers a longstanding incompleteness in both quantum and gravitational theory. Bohmian mechanics lacks an ontological basis for the wavefunction’s curvature and treats quantum equilibrium as a postulate. General relativity lacks a quantum-consistent explanation for stress-energy and assumes a classical metric background. By rooting both in a single ontological curvature, the Fisher curvature, sPNP eliminates the distinction. The quantum potential becomes the informational curvature of the configuration manifold; the Einstein tensor is understood as an effective projection of geodesic deviation in the Fisher-curved configuration manifold.

The implications extend beyond interpretation. In this unified framework (Jacobi-Fisher Metric), each operator is natural to its geometry: the Laplace–Beltrami operator constructed from the Fisher metric curves the flat Laplacian of QP just as the covariant derivative defines curvature in spacetime. The geodesic principle underlying classical mechanics is retained but recast through Jacobi-type evolution in configuration space. Pilot time replaces Newtonian and coordinate time with an emergent intrinsic parameter governing motion through Fisher-curved geometry. Pilot time t(X) in the kernel is an internally constructed scalar (e.g., the phase of a heavy semiclassical subsystem), so no external foliation appears.

In sum, the sPNP approach reframes the deepest physical laws as consequences of information geometry. The wavefunction does not merely guide; it curves. The metric does not merely constrain; it distinguishes. The universe evolves not against a passive background, but through the intrinsic dynamics of its own informational distinguishability. In this view, Fisher Information is not a tool of inference, it is the medium of reality.