Information-curvature conservation law

Abstract

This paper shows that Fisher-information geometry underlying divergence minimization obeys a “information-curvature conservation law” across coupled systems. For any collection of informational manifolds related by mutual-information-preserving maps, the sum of their Ricci curvatures equals the Laplacian of the total stability functional described in Information-Theoretic Stability as Reward Function. (emsenn, 2025) This result establishes a general constraint linking local learning dynamics to informational geometric invariants.

1. Introduction

Information geometry relates the curvature of probability manifolds to statistical inference.[@amari_InformationGeometryIts_2016] Systems that minimize divergence between successive distributions trace geodesics under the Fisher metric. In Information-Theoretic Stability as Reward Function, I defined a “stability functional” whose maximation describes local divergence minimization.

$$R_s(t)=-\frac{1}{\delta t}{D}_{\mathrm{KL}}(p_{t+\delta }||p_t),$$

In this paper, we’ll look at how when multiple such systems are coupled through mutual information, the Ricci curvature of their respective Fisher manifolds satisfies a conservation law linking geometry and stability.

2. Preliminaries

2.1 Informational Manifolds

Let each system $i$ be represented by a Riemannian manifold $({\mathcal{P}}_i,g^{(i)})$, where $g^{(i)}$ is the Fisher-Rao metric, and each $p_i(x,t)$ is smooth with finite entropy $H_i(t)$:

$$g_{ab}^{(i)}={\mathbb{E}}_{p_i}\left[{\partial }_a\log p_i{\partial }_b\log p_i\right]$$

2.2 Coupling Maps

Two systems $i,j$ are informationally coupled if there exists a smooth map ${\Phi }_{ij}:({\mathcal{P}}_i,g^{(i)})\to ({\mathcal{P}}_j,g^{(j)})$ satisfying $I(i;j)=I({\Phi }_{ij}(p_i);p_j),$ up to small curvature correction ${\kappa }_{ij}$.

3. Curvature—Stability Relation

3.1 Local Relation

For each manifold, the Ricci curvature satisfies $$ \operatorname{Ric}(g^{(i)})_{ab} = -,\nabla_a\nabla_b \log p_i(x,t)

• \mathcal{O}(\partial^2 D_{\mathrm{KL}}),

## 3.2 Conservation Lemma Let the total stability functional of $N$ coupled systems be $$ R_s^{\mathrm{tot}} = \sum_{i=1}^N R_s^{(i)}. $$ Assuming bounded entropy and differentiable couplings $\Phi_{ij}$, the following holds: $$ \boxed{ \sum_{i=1}^N \operatorname{Ric}(g^{(i)}) = \nabla^2 R_s^{\mathrm{tot}}. } $$ **Proof sketch.** Starting from the Bianchi identity $\nabla^\mu G_{\mu\nu}=0$ on each manifold and substituting the Fisher metric's expression for the information potential $\psi_i = -\log p_i$, we obtain $$ \nabla^2 \psi_i = \operatorname{Tr}\operatorname{Ric}(g^{(i)}). $$ Since $R_s^{(i)} = -\partial_t D_{\mathrm{KL}}(p_{t+\delta}||p_t)$ depends on $\nabla^2\psi_i$ through $\partial_t g^{(i)}_{ab}$, summing over coupled manifolds and using $\sum_i\nabla^\mu G_{\mu\nu}^{(i)}=0$ yields the stated conservation law. # 4. Consequences 1. ****Geometric invariance.**** Divergence minimization across coupled information manifolds preserves the total informational curvature. 2. ****Bounded stability.**** Local increases in $R_s^{(i)}$ require compensating decreases in curvature elsewhere, ensuring finite total informational variance. 3. ****Model scope.**** The result applies to any differentiable system modeled by Fisher geometry---statistical, algorithmic, biological, or physical---without further assumption. # 5. Conclusion Under general information-geometric conditions, the total Ricci curvature of coupled Fisher manifolds equals the Laplacian of the joint stability functional. This curvature conservation principle provides a minimal geometric constraint governing divergence-minimizing dynamics across heterogeneous systems, linking local learning behavior to a global invariant of informational geometry.

emsenn. (2025). Information-Theoretic Stability as Reward Function.