From Bits to Landscapes Wheeler’s “It from Bit” treats a yes/no distinction as the atom of reality. sPNP goes further:
Distinction = Geometry.
Sharp changes in the wave amplitude makes neighboring configurations more distinguishable. The gradient of that sharpness defines the Fisher Information metric—a real curvature on the full 3 N-dimensional configuration space (or on the 3 N − 6 shape-plus-scale sub-manifold when centre-of-mass and rotations are factored out).
G_IJ(X) = (1 / Q0²) · ∂I ln R · ∂J ln R + Σ_k m_k · δ_IJ
The first term is pure information curvature; the second is the classical Jacobi mass term. Together they form the Fisher–Jacobi metric that sculpts motion.
Clarification: (Q0). Q0 has units of length. Fixing Q0² = ħ / m_ref makes the first term reproduce the standard Bohm quantum potential. Any fixed value simply sets the information length scale; it could run with scale in an RG treatment.
Entanglement: Curvature That Spans Systems When two subsystems entangle, distinguishability is no longer local. Off-diagonal components appear in G_IJ, so curvature is literally shared across the composite system. In sPNP, entanglement is a geometric agent: it reshapes the metric rather than just correlating measurement statistics.
Clarification: No preferred foliation. Curvature lives on timeless configuration space. Lorentz symmetry emerges only after projecting into 4-D spacetime, so no spatial slice is privileged.
Projection: From Configuration Curvature to Spacetime Gravity
sPNP projects curvature into spacetime with a Gaussian kernel. K(X,x) = (4πQ₀²)^(3N/2) e^(-D²(X,x)/4Q₀²) [1+O(Q₀²RF)]
sPNP introduces a geometric projection tensor F_μν(X, x) that folds the high-dimensional down to a 4-D one:
g_μν(x) = η_μν + κ ∫ dμ(X) • P(X) • F_μν(X, x)
The kernel K(X,x) localizes the projection, while F_μν(X,x) extracts the specific geometric contribution to the emergent spacetime metric. Gravity is therefore a shadow of information curvature. Large Fisher gradients project to large spacetime curvature; weak gradients recover near-flat Minkowski space.
Clarification:Test-able consequences • Tiny curvature corrections predict O(10⁻²⁸) shifts in the Lamb shift for hydrogen-like ions. • In entangled macroscopic masses, sPNP forecasts ~ femtometer-scale deviations from Newtonian trajectories—one experimental target for next-generation cold-atom interferometers.
Jacobi Dynamics: Motion as Shortest Distinctive Path Instead of “force = mass × acceleration,” sPNP uses an action equal to the arc-length in the Fisher–Jacobi metric. The system moves along geodesics that minimise that arc-length. Free particles coast (uniform motion) where the amplitude is flat; they accelerate where the information landscape steepens (steep gradients).
Because the action is reparametrisation-invariant, there is no absolute time variable—only a relational parameter. This automatically avoids the preferred-time problem that plagues non-relativistic pilot-wave models.
Entropy and the Arrow of Time Decoherence spreads amplitude gradients into the environment, flattening the local Fisher curvature. That flattening is what we call entropy increase. In sPNP the thermodynamic arrow of time is the one-way diffusion of information curvature from subsystem to bath.
Field Theory Re-imagined If curvature is the basic substance, then:
Gauge fields = infinitesimal changes in the Fisher–Jacobi metric.
Fermions = topological obstructions in that metric (non-trivial spinor bundles over configuration space).
Conservation laws = balance equations for total Fisher curvature flux.
In this Energy-momentum picture, energy is secondary—a bookkeeping device for how much curvature is stored or transferred; total Fisher curvature.
What Stays Real When No One Looks? Because R(X) is ontic, the curvature exists even when no agent queries it. The Moon follows a geodesic in configuration space whether or not we observe it; observation merely updates our Shannon entropy, not the underlying Fisher geometry.
Distinction: Take-aways Fisher Information is not a statistical afterthought. It is the curvature that structures reality.
Entanglement = shared curvature, neatly explaining non-local correlations without superluminal signals in spacetime.
Jacobi geodesics encode dynamics, restoring determinism while remaining Lorentz-compatible after projection.
Entropy is curvature flattening, providing an informational account of the arrow of time.
sPNP unifies motion, gravity, and thermodynamics in one information geometric language.
