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Overview

This section introduces the semantic manifold $\mathcal{M}$ as the foundational geometric substrate of Recurgent Field Theory (RFT). Meaning is formalized as a differentiable manifold equipped with a dynamic metric tensor $g_{ij}(p, t)$, which encodes both local and global constraint structures. The geometry of $\mathcal{M}$ sets the configuration space for semantic states and shapes their evolution under recursive processes.

Key mathematical structures:

  1. Metric Tensor $g_{ij}(p, t)$ — defines semantic distances and constraint geometry.
  2. Constraint Density $\rho(p, t)$ — quantifies the rigidity or plasticity of semantic regions.
  3. Coherence Field $C_i(p, t)$ — represents local alignment and self-consistency.
  4. Recursive Depth $D(p, t)$ and Attractor Stability $A(p, t)$ — modulate the manifold’s structure and resilience.
  5. Semantic Mass $M(p, t)$ — integrates depth, density, and stability to curve the manifold and generate attractor basins.

The metric evolves according to a flow equation that couples intrinsic curvature with recursive feedback, supporting both stable and fluid semantic regimes. High constraint regions produce sharply defined geodesics and semantic rigidity; low constraint regions open space for flexible, innovative transitions. The interplay of these fields provides the geometric and dynamical context for recursive coupling and semantic evolution developed in subsequent sections.

3.1 The Metric Tensor and Semantic Distance

The semantic manifold $\mathcal{M}$ is equipped with a time-dependent metric tensor $g_{ij}(p, t)$, which defines the infinitesimal squared distance between neighboring points:

\[ds^2 = g_{ij}(p, t) \, dp^i \, dp^j\]

where $dp^i$ is an infinitesimal displacement in the $i$-th semantic dimension. The metric $g_{ij}$ encodes the local constraint structure, modulating the cost of semantic displacement along and between dimensions.

Interpretation:

  • High constraint: Large $g_{ij}$ components correspond to regions where semantic distinctions are rigid and transitions are energetically costly.
  • Low constraint: Small $g_{ij}$ components correspond to regions of semantic fluidity, where transitions are facile.

3.2 Evolution Equation for the Semantic Metric

The evolution of the metric tensor is governed by a flow equation analogous to Ricci flow, but with additional forcing terms reflecting recursive structure:

\[\frac{\partial g_{ij}}{\partial t} = -2 R_{ij} + F_{ij}(R, D, A)\]

where:

  • $R_{ij}$ is the Ricci curvature tensor associated with $g_{ij}$, encoding the intrinsic curvature induced by constraint density.
  • $F_{ij}(R, D, A)$ is a symmetric tensor-valued functional incorporating:
    • $R$: the recursive coupling tensor (quantifying nonlocal feedback, see Section 4),
    • $D$: the recursive depth field (maximal sustainable recursion at $p$),
    • $A$: the attractor stability field (local resilience to perturbation).

This flow describes how semantic geometry deforms under the combined influence of intrinsic curvature and recursive feedback mechanisms.

3.3 Constraint Density

The metric tensor gives rise to the constraint density $\rho(p, t)$ at each point via:

\[\rho(p, t) = \frac{1}{\det(g_{ij}(p, t))}\]

Regions of high constraint density ($\rho \gg 1$) correspond to tightly packed semantic states, where transitions are suppressed. Low constraint density ($\rho \ll 1$) marks regions of semantic flexibility, where boundaries are diffuse and transitions are energetically favorable. The geometry of $\mathcal{M}$ thus encodes both rigidity and plasticity, modulating the propagation of coherence and the formation of recursive structures.

3.4 The Coherence Field

The coherence field $C_i(p, t)$ is a vector field on $\mathcal{M}$, representing the local alignment and self-consistency of semantic structure. The metric $g_{ij}$ is used to raise and lower indices, compute gradients, and define the norm of coherence:

\[C_{\mathrm{mag}}(p, t) = \sqrt{g^{ij}(p, t) C_i(p, t) C_j(p, t)}\]

where $g^{ij}$ is the inverse metric. $C_{\mathrm{mag}}$ quantifies the scalar magnitude of coherence at $p$, independent of direction, and serves as the basis for defining attractor potentials and autopoietic capacity in subsequent sections.

3.5 Recursive Depth, Attractor Stability, and Semantic Mass

The geometry of $\mathcal{M}$ is further modulated by the recursive depth field $D(p, t)$ and the attractor stability field $A(p, t)$. $D(p, t)$ quantifies the maximal recursion depth sustainable at $p$ before coherence degrades, while $A(p, t)$ measures the local tendency of a semantic state to return after perturbation. Together with constraint density, these fields define the semantic mass:

\[M(p, t) = D(p, t) \cdot \rho(p, t) \cdot A(p, t)\]

Semantic mass $M(p, t)$ curves the manifold, generating attractor basins and shaping the flow of coherence. High-mass regions function as stable attractors, anchoring interpretation and resisting transformation; low-mass regions are more open to innovation and recursive branching.

Next: Recursive Coupling and Depth Fields