The Fractal-Information Geometry of Language: Multiscale Self-Similarity, Statistical Manifolds, and Empirical Signatures Across Natural, Synthetic, and Manipulative Discourse

Natural Language Structures and Structure of Logics: Traditional Computing Language largely reflects a one-dimensional, linear process of discrete symbolic tokens (so-called semantic signatures) is presented in natural language using linear (logistically rewritten) systemic processes or in the context of self-attention algorithms. However, human language has structural constraints, longer-range dependences, and a multi-tiered hierarchical organization that functions synchronously across acoustic/phonetic, morphological, syntactic, semantic and pragmatic scales. This paper formulates a unified mathematical and empirical framework titled Fractal-Information Geometry of Language (FIGL). By integrating fractal geometry (power-law scaling, Hurst exponents, and multifractal singularity spectra) with information geometry (Riemannian statistical manifolds, Fisher information metrics(FIMs), and Kullback-Leibler divergences), we model language as a dynamic, continuous statistical manifold embedded with multiscale self-similarities. We evaluate the empirical viability of FIGL across three core dimensions: Typological dynamics: Comparing analytical-isolating structures (English) against rich, low-resource agglutinative morphologies (Azerbaijani). Generative attribution: Differentiating human-written discourse from large language model (LLM) synthetic outputs. Pragmatic intent: Dissecting structural phase transitions between manipulative and non-manipulative discourse. Our findings demonstrate that human natural text operates near a critical phase-transition state characterized by broad multifractal spectra, non-Gaussian information-geometric curvature, and scale-invariant long-range mutual information. In contrast, LLMs display localized fractal breakdown, over-smoothing on Riemannian probability manifolds, and truncated long-range correlation lengths. Furthermore, agglutinative languages like Azerbaijani concentrate high fractal complexity at sub-word morphological scales, shifting the information-geometric metric tensor compared to English. Finally, manipulative discourse exhibits characteristic fractal compression and geometric curvature collapse. We outline open mathematical problems and present a futuristic research outlook for geometric NLP architecture.