Home / Math / Recurgent Field Equations and Global Dynamics / Core Field Equations

Overview

This section introduces the core field equations that govern the global evolution of the coherence field $C(p, t)$ on the semantic manifold $\mathcal{M}$. Drawing on the constructs of semantic mass, recursive tensors, autopoietic ignition, and regulatory dynamics, these equations articulate how meaning propagates, deforms, and stabilizes within Recurgent Field Theory (RFT).

Key mathematical structures:

  1. Recurgent Field Equation — describes the dynamics of coherence propagation under recursive stress.
  2. Semantic Geodesics — define the natural trajectories of meaning shaped by manifold curvature.
  3. Metric Evolution Equation — captures the feedback loop by which recursion reshapes semantic geometry.

Taken together, these equations form a closed dynamical system for recursive cognition and the emergence of meaning, weaving together local field interactions and the evolution of global geometry.

9.2 Recurgent Field Equation

The evolution of the coherence field $C_i(p, t)$ under recursive stress is governed by the recurgent field equation: \(\Box C_i = T^{\text{rec}}_{ij} \, g^{jk} C_k\) where

  • $\Box = \nabla^a \nabla_a$ denotes the covariant d’Alembertian operator on the semantic manifold $\mathcal{M}$,
  • $T^{\text{rec}}_{ij}$ is the recursive stress-energy tensor (see Section 2.3),
  • $g^{jk}$ is the inverse metric tensor.

Interpretation:

This equation expresses that the acceleration of coherence, both spatially and temporally, is shaped by the local recursive stress and the geometry of semantic constraints. In regions of elevated semantic mass or pronounced recursive torsion, the coherence field bends and may collapse into attractor basins. In contrast, where density is low, coherence spreads more diffusively.

9.3 Semantic Geodesics

The natural trajectory of a semantic point $p \in \mathcal{M}$ under recursive evolution is described by the geodesic equation: \(\frac{d^2 p^i}{ds^2} + \Gamma^i_{jk} \frac{dp^j}{ds} \frac{dp^k}{ds} = 0\) where

  • $s$ is a parameter along the curve (e.g., time or recursive depth),
  • $\Gamma^i_{jk}$ are the Christoffel symbols associated with the metric $g_{ij}$,
  • $p^i(s)$ are the coordinates of the evolving semantic state.

Interpretation:

Geodesics trace the extremal (least-resistance) paths of recursive transformation on the manifold. The curvature encoded by $\Gamma^i_{jk}$ bends these paths, giving rise to semantic attractors. In this way, the geodesic structure guides the spontaneous alignment of evolving meaning with established semantic trajectories.

9.4 Metric Evolution

The geometry of the semantic manifold is itself dynamic, evolving in response to recursive flows and the accumulation of semantic mass. The evolution of the metric tensor is given by: \(\frac{\partial g_{ij}}{\partial t} = -2 R_{ij} + F_{ij}(R, D, A)\) where

  • $R_{ij}$ is the Ricci curvature tensor, encoding the torsion induced by recursion,
  • $F_{ij}$ is a forcing term dependent on the recursive coupling $R$, recursive depth $D$, and attractor stability $A$.

Implication:

Recursive processes generate curvature in the semantic geometry, which then modulates subsequent recursive flows. This interplay creates a closed feedback system: the manifold’s structure is continually reshaped by the propagation of coherence, even as it shapes that propagation in turn.

9.5 Recursive Dynamical Structure

The interdependence of the primary fields and their governing equations can be schematically represented as follows:

Next: Bidirectional Temporal Flow