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Overview

This section formalizes semantic mass as the principal source of curvature in Recurgent Field Theory, shaping the persistence and stability of meaning structures under recursive pressure. The mass distribution $M(p,t)$ arises from the triple product of recursive depth $D$, constraint density $\rho$, and attractor stability $A$, grounding the geometry of autopoietic coherence.

Key mathematical structures:

  1. Mass-curvature coupling through the recurged Einstein equation $R_{ij} - \frac{1}{2}g_{ij}R = 8\pi G_s T^{\text{rec}}_{ij}$
  2. Attractor potential $\Phi(p,t)$ derived from nonlocal mass distribution
  3. Recursive force $F_i = -\nabla_i\Phi$ driving coherence flow

Within this framework, semantic mass mediates between:

  • Constraint geometry (encoded in $g_{ij}$)
  • Coherence field dynamics (governed by $\mathcal{L}$ in §6)
  • Recursive stress-energy (mediated by $T^{\text{rec}}_{ij}$ from §4)

This construction completes the geometric architecture of Recurgent Field Theory: concentrations of semantic mass curve the semantic manifold, giving rise to attractor basins that direct the flow of recursive processes and counteract entropic dissipation. The field dynamics exhibit phase transitions at critical values of $M(p, t)$, beyond which autopoietic regeneration cycles begin.

5.1 Semantic Mass

The semantic mass $M(p, t)$ is formally defined as the product of recursive depth, constraint density, and attractor stability at each point $p \in \mathcal{M}$ and time $t$: \(M(p, t) = D(p, t) \cdot \rho(p, t) \cdot A(p, t)\) where:

  • $D(p, t)$ denotes the recursive depth field, quantifying the maximal recursion depth sustainable at $p$ before coherence degrades,
  • $\rho(p, t)$ is the constraint density, given by $\rho(p, t) = 1/\det(g_{ij}(p, t))$, encoding the tightness of local semantic geometry,
  • $A(p, t)$ is the attractor stability field, measuring the local tendency of a semantic state to return after perturbation.

Semantic mass $M(p, t)$ acts as the principal source term for semantic curvature, anchoring persistent structures within the semantic manifold. Regions of high $M$ function as stable attractors, exerting a stabilizing influence on the evolution of coherence fields and modulating the propagation of recursive processes. The persistence of high-mass structures follows from their recursive depth, constraint density, and local stability, independent of their propositional content.

5.2 Recurgent Einstein Equation

The coupling between recursive stress and semantic curvature is governed by the recurged Einstein field equation: \(R_{ij} - \frac{1}{2}g_{ij}R = 8\pi G_s T^{\text{rec}}_{ij}\) where:

  • $R_{ij}$ is the Ricci curvature tensor of the semantic manifold,
  • $R$ is the scalar curvature,
  • $g_{ij}$ is the metric tensor,
  • $T^{\text{rec}}_{ij}$ is the recursive stress-energy tensor (see Section 4.3),
  • $G_s$ is the semantic gravitational constant.

This equation expresses the principle that recursive tension and constraint, as encoded in $T^{\text{rec}}_{ij}$, generate curvature in semantic space, shaping the geometry of meaning in direct analogy to the role of mass-energy in general relativity.

5.3 Attractor Potential Field $\Phi(p, t)$

The attractor potential field $\Phi(p, t)$ is defined as the integral over semantic mass, weighted by geodesic distance: \(\Phi(p, t) = -G_s \int_{\mathcal{M}} \frac{M(q, t)}{d(p, q)} \, dV_q\) where:

  • $d(p, q)$ is the geodesic distance between points $p$ and $q$ in the semantic manifold,
  • $M(q, t)$ is the semantic mass at $q$,
  • $dV_q$ is the volume element.

The gradient of this potential gives the recursive force: \(F_i(p, t) = -\nabla_i \Phi(p, t)\) which directs the flow of coherence and draws new semantic structures into existing attractor basins. This formalism describes how regions of high semantic mass modulate the dynamics of meaning, pulling recursive processes toward stable configurations.

5.4 Potential Energy of Coherence

The potential energy associated with the scalar coherence magnitude $C_{\mathrm{mag}}$ is given by: \(V(C_{\mathrm{mag}}) = \frac{1}{2}k(C_{\mathrm{mag}} - C_0)^2\) where:

  • $C_{\mathrm{mag}} = \sqrt{g^{ij}(p, t) C_i(p, t) C_j(p, t)}$ is the scalar magnitude of the coherence field,
  • $C_0$ is the equilibrium coherence level of the attractor,
  • $k$ is the coherence rigidity parameter, quantifying the stiffness of the attractor basin.

This quadratic potential models the energetic landscape of semantic attractors:

  • Soft attractors (e.g., metaphoric or fluid conceptual structures) correspond to small $k$,
  • Hard attractors (e.g., axiomatic, rigid, or dogmatic structures) correspond to large $k$.

The parameter $k$ modulates the resistance of an attractor to perturbation and the rate at which coherence returns to equilibrium following displacement.

Next: Recurgent Field Equation and Lagrangian Mechanics