Semantic Collapse Geometry:
A Unified Topological Model Linking Gödelian Logic, Attractor Dynamics, and Prime Number Gaps
Abstract
Modern mathematics and complexity science lack a unified framework to describe how meaning, structure, and discontinuity emerge across logic, number theory, and topology. In particular, the fragmentation between Gödelian incompleteness, prime number distribution, and attractor dynamics leaves the nature of emergence and undecidability conceptually disconnected. This work addresses that gap by introducing Semantic Collapse Geometry (SCG)—a topological formalism that extends Semantic Meme Field Theory (SMFT) to map meaning-generation as a process of collapse through singularities in semantic space.
We formalize semantic collapse events as topological singularities and attractors, drawing rigorous analogies between logical undecidability, prime gaps, and bifurcation behavior in semantic fields. Using tools from variational geometry, homological topology, and analytic number theory, we construct a unified geometry where Gödelian obstructions, Riemann zeta patterns, and semantic curvature gaps are facets of the same underlying structure.
Key results include the definition of semantic primes (irreducible attractors of meaning), the equivalence between logical incompleteness and geometric obstruction, and the derivation of predictive equations for collapse bifurcations, semantic trace curvature, and event spacing. These insights are supported by visualizations of attractor landscapes and semantic flow discontinuities.
Our framework offers a new bridge across foundational domains of mathematics—recasting logic, topology, and number theory within a single collapse-driven paradigm. Beyond theory, SCG suggests practical modeling strategies for complex adaptive systems, organizational dynamics, and epistemology centered on observer entanglement and meaning formation.
This paradigm opens a path toward novel mathematical structures, new computational tools, and cross-disciplinary collaborations that reconceive emergence, uncertainty, and lawfulness across scales and domains.
