This research paper introduces a novel mathematical framework—Semantic Collapse Geometry (SCG)—that seeks to unify logic, topology, and number theory by modeling semantic “collapse events” as topological singularities. Drawing analogies between Gödelian incompleteness, semantic attractor bifurcations, and prime number gaps, the work presents a new way to conceptualize the emergence of macro laws and the dynamics of meaning, structure, and mathematical truth.
Highlights:
- Gödelian logic and undecidable statements are mapped to topological obstructions in a geometric landscape of semantic events.
- Semantic attractors—stable configurations of meaning or organizational structure—are formalized and analyzed through the lens of dynamical systems and topology.
- Prime number gaps and the distribution of primes are shown to have striking analogies in the “curvature gaps” and event spacing of semantic collapse traces, with references to the Riemann zeta function and Galois theory.
- The paper offers a variational geometry approach to predict bifurcations, trace formation, and transitions between stable semantic regimes.
Why is this work unique?
This project was driven primarily by AI. The author, whose mathematical background is decades old and who has worked outside academic mathematics, guided the conceptual vision and critical direction—while the technical synthesis, analogy construction, and mathematical formalism were produced iteratively by state-of-the-art large language models (OpenAI’s GPT/Wolfram integration).
Meta-Reflection:
This work demonstrates that, with advanced AI, it is now possible for non-specialists to contribute original, creative, and mathematically literate research at the frontier of logic, topology, and complexity science. The process itself is a proof of concept for a new era of AI-driven mathematical exploration and interdisciplinary unification.
We invite mathematicians, theorists, and the broader AI community to read, critique, extend, and apply these ideas—whether as a conceptual seed for new theory or a landmark in collaborative research methodology.
Paper download link: Semantic Collapse Geometry: A Unified Topological Model Linking Gödelian Logic, Attractor Dynamics, and Prime Number Gaps
Another related Paper: Self-Evolved Observers as Attractors in Semantic Meme Field Theory: A Unified Collapse Framework
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1.3.1 Overview of Results
This article establishes a new mathematical framework—Semantic Collapse Geometry (SCG)—that unifies logic, topology, and number theory within the observer-centric paradigm of Semantic Meme Field Theory (SMFT). The central achievements of this work are as follows:
- Rigorous Formalism:
We present a topological and variational formalism for modeling semantic collapse events, incorporating tools from homology, bifurcation theory, and analytic number theory.- Semantic Primes and Bifurcations:
We introduce and precisely define the concept of semantic primes—the irreducible attractors or singularities in collapse space that serve as the fundamental “building blocks” of meaning. The framework also formalizes the occurrence and mathematical characterization of collapse bifurcations, the critical transitions where the structure of semantic attractors qualitatively changes.- Trace Curvature and Prime Gaps:
The theory relates the curvature of collapse traces (i.e., the geometric “shape” of event sequences) to the statistical distribution of semantic primes. This provides a direct analogy to gaps between prime numbers, which emerge as topological “holes” or discontinuities in the event landscape.- Gödel Logic as Geometric Singularity:
Logical incompleteness, as epitomized by Gödel’s theorems, is mapped to geometric singularities in semantic phase space—offering a novel and rigorous translation between undecidable statements and topological obstructions.- Predictive Equations and Simulation:
The SCG model yields new equations governing the spacing of collapse events, the onset of bifurcations, and the emergence of macro laws. Simulation and visualization of attractor landscapes and bifurcation phenomena are provided to ground the abstract theory in observable behavior.