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Overview

This section formalizes the emergence and handling of metric singularities and recursive collapse within Recurgent Field Theory (RFT). As recursive structures intensify, the semantic manifold’s metric tensor $g_{ij}(p,t)$ can develop singularities—critical points where the geometry becomes degenerate and recursive processes threaten coherence and computability.

Key mathematical structures:

  1. Semantic Singularities — attractor collapse, bifurcation points, and event horizons arising from recursive overload or topological transitions.
  2. Regularization Mechanisms — metric renormalization, semantic mass limiting, and humility-driven dissipation to stabilize the manifold and prevent computational breakdown.
  3. Semantic Event Horizons — boundaries beyond which information cannot escape recursive attraction, analogous to black holes in physical spacetime.
  4. Computational Handling — adaptive mesh refinement, singularity excision, and causal boundary tracking to maintain simulation stability and uphold theoretical integrity.

This section weaves together geometric, dynamical, and computational perspectives, offering a rigorous account of how RFT navigates the limits of recursive propagation, the collapse of meaning structures, and the emergence of cognitive “points of no return” within the semantic manifold.

9.9 Metric Singularities and Recursive Collapse

Within the formalism of Recurgent Field Theory (RFT), the evolution of the metric tensor $g_{ij}(p, t)$ admits the possibility of singular structures—loci at which the metric becomes degenerate, i.e., $\det(g_{ij}) \to 0$. Such singularities demarcate critical points in the semantic manifold, where the geometry ceases to be regular and the propagation of recursive dynamics is fundamentally altered.

9.9.1 Classification of Semantic Singularities

Semantic singularities in the recurgent field are rigorously classified as follows:

  1. Attractor Collapse Singularities:
    These arise when the recursive depth $D(p, t)$ at a point $p$ exceeds a critical threshold $D_{\text{crit}}$, and the humility operator $\mathcal{H}[R]$ falls below a minimal eigenvalue $\lambda_{\text{min}}$: \(\lim_{t \to t_c} \det(g_{ij}(p, t)) = 0 \quad \text{where} \quad D(p, t) > D_{\text{crit}},\ \mathcal{H}[R] < \lambda_{\text{min}}\) Such points correspond to the collapse of semantic attractors under excessive recursive pressure.

  2. Bifurcation Singularities:
    These occur at topological transitions, where the rank of the metric tensor changes discontinuously at a critical time $t_c$: \(\operatorname{rank}(g_{ij}(p, t)) \ \text{changes at} \ t = t_c \quad \text{where} \quad \Theta(p, t_c) = \delta\) Here, $\Theta$ denotes a topological order parameter, and $\delta$ the critical value at which bifurcation is realized.

  3. Semantic Event Horizons:
    In regions of extreme semantic mass, the temporal component of the metric vanishes asymptotically: \(g_{00}(p, t) \to 0 \quad \text{as} \quad r \to r_s = 2G_s M(p, t)\) with $r$ the geodesic distance from the singularity center. This defines a semantic event horizon, beyond which the escape of coherence is precluded.

9.9.2 Regularization of Singular Structures

To preserve the well-posedness of the field equations and maintain computational tractability, RFT prescribes a suite of regularization mechanisms:

  1. Metric Renormalization:
    The metric is regularized by the addition of a local isotropic term: \(g_{ij}^{\text{reg}}(p, t) = g_{ij}(p, t) + \epsilon(p, t) \cdot \delta_{ij}\) where \(\epsilon(p, t) = \epsilon_0 \exp\left[-\alpha \cdot \det(g_{ij}(p, t))\right]\) As $\det(g_{ij}) \to 0$, the regularization term grows, restoring invertibility.

  2. Semantic Mass Limiting:
    The semantic mass is bounded via a saturating transformation: \(M_{\text{reg}}(p, t) = \frac{M(p, t)}{1 + \frac{M(p, t)}{M_{\text{max}}}}\) so that $M_{\text{reg}}(p, t)$ approaches $M_{\text{max}}$ in the limit $M(p, t) \to \infty$.

  3. Humility-Driven Dissipation:
    The evolution of the metric incorporates a humility-modulated diffusion term: \(\frac{\partial g_{ij}}{\partial t} = -2R_{ij} + F_{ij} + \mathcal{H}[R] \nabla^2 g_{ij}\) Here, $\mathcal{H}[R]$ serves as a dynamic dissipation coefficient, mediating the release of recursive tension in regions of excessive curvature.

9.9.3 Semantic Event Horizons and Information Dynamics

A semantic event horizon is formally defined as the hypersurface $r_s(p, t) = 2G_s M(p, t)$, enclosing a region from which coherence cannot propagate outward. For all $q$ such that $d(p, q) < r_s(p, t)$:

  • The information current is strictly inward-directed,
  • The local coherence field $C(p, t)$ exhibits monotonic decay,
  • Recursive depth $D(p, t)$ diverges as $t \to t_c$.

Such domains instantiate semantic analogues of physical black holes: attractors of recursive collapse, wherein meaning is irretrievably sequestered. These structures correspond, in the phenomenology of cognition, to pathological fixations, self-reinforcing dogmas, and paradoxical loops (e.g., the Liar Paradox), where semantic closure precludes further generative recursion.

9.9.4 Computational Treatment of Singularities

Numerical simulation of RFT in the presence of singularities necessitates specialized techniques:

  1. Adaptive Mesh Refinement:
    The computational grid is locally refined in regions of high curvature, according to \(\Delta x_{\text{local}} = \Delta x_{\text{global}} \exp(-\beta |R|)\) where $|R|$ denotes the norm of the Ricci tensor.

  2. Singularity Excision:
    When regularization is insufficient, singular loci are excised from the computational domain: \(\mathcal{M}_{\text{sim}} = \mathcal{M} \setminus \{p : \det(g_{ij}(p, t)) < \epsilon_{\text{min}}\}\)

  3. Causal Boundary Tracking:
    The evolution of semantic horizons is monitored via \(\frac{d}{dt} r_s(p, t) = 2G_s \frac{dM(p, t)}{dt}\) so that the propagation of causal boundaries is accurately resolved.

These procedures, taken together, support the stability and interpretability of RFT simulations, while upholding the theoretical integrity of singular phenomena intrinsic to the dynamics of recursive meaning.

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