Fisher Information as Relational Geometry
Fisher Information is relational at a deeper level than the statement that it measures distinguishability. Something is distinguishable only relative to something else. The relation is already built into the definition of the Fisher metric. A state is not geometrically specified merely by “where it is” in some pre-existing background. What matters is how its probability structure changes relative to neighboring states. Geometry is generated by those comparisons.
The deeper point is that Fisher does not merely measure relations; it makes the space of relations geometric. Once the local distinguishability of neighboring states is specified, one can define lengths, angles, geodesics, parallel transport and curvature. The resulting geometry is therefore a geometry of relational variation.
Curvature enters one level above ordinary distinguishability. Fisher Information tells us how strongly one state differs from another. The Riemann curvature tells us how that rule of comparison itself changes across the manifold. In this sense,
distinction → Fisher metric → variation of distinguishability → curvature.
Curvature is therefore not simply an additional property placed on top of Fisher Information. It records the failure of the network of local distinguishability relations to remain globally Euclidean. Moving from one region of the manifold to another changes the way nearby states can be compared. The geometry contains memory of how distinctions are organized around other distinctions.
This provides a precise meaning to the statement that distinctions become curvature. A single distinction gives a local comparison. A field of distinctions gives a metric. A non-uniform field of distinctions gives a metric whose structure changes from place to place. Curvature is the geometrical expression of that relational non-uniformity.
The relational character becomes especially clear under coarse-graining. Fisher geometry is not tied to a particular coordinate representation of the probability distribution. Under an admissible stochastic map, distinguishability cannot increase. The geometry therefore expresses a relation that is preserved in a precise sense even when microscopic descriptions are compressed.
The relevant geometric question is not whether configuration X possesses some intrinsic informational magnitude independent of everything else. The physical question is how the universal state changes under relational displacements of X, and how sharply those changes can be distinguished. The Fisher metric provides the ruler for those changes without introducing an external ruler.
R defines the relational geometry of distinguishability, while S defines the relational flow through that geometry: dS supplies the momentum 1-form whose Fisher-metric dual determines how the state moves between distinguished configurations.
The same logic extends from configuration space to the geometry itself. In ordinary relationalism, objects are characterized through their relations. In sPNP, the wavefunctional determines the geometry of distinguishability among relational configurations, and that geometry in turn determines geodesic structure and dynamical propagation. The relation is therefore reflexive:
Ψ → distinguishability → Gᴵᴶ → curvature → geodesic structure.
But the important point is not merely that information happens to be represented geometrically. The geometry is itself a structured statement about relations between possible states. Fisher Information is consequently relational in both directions: it measures how states differ, while its induced geometry determines how those differences are organized.
This suggests a stronger formulation of the sPNP principle:
Reality is not composed of intrinsically specified states to which relations are subsequently added. Physical structure consists of states together with the invariant geometry of their distinguishability.
On this view, Fisher Information is especially natural because it does not require an independent background notion of distance between probability distributions. The metric is extracted from the distributions themselves. The geometry is therefore endogenous to the relational structure rather than imposed from outside.
In sPNP, curvature is therefore not merely a geometrical representation of relations. It is the geometric organization of the relations themselves.
