Scalable GRAVITAS: Complexity Reduction for Semantic Geometric Analysis

The GRAVITAS framework models semantic spaces as dynamic Riemannian manifolds, but its practical application is hindered by the O(N³) computational complexity of Christoffel symbols. This paper introduces a comprehensive mathematical framework that reduces this complexity to effective linear O(N) through ten complementary methods, including the Semantic Tensor Decomposition Theorem, Conformal Semantic Geometry, and Warped Product Manifold Solutions. We prove that semantic spaces exhibit structures enabling reduction to an effective dimension d_eff ≤ C log₂(N) with bounded error.

Building on these reductions, we refine five geometric invariants for diagnostics: the discrete Yamabe-type invariant ˆσ, angular spectral contrast ρ_spec, asymptotic volume ratio θ, renormalization critical exponent ν, and adimensional screening length S. We provide formal derivations, discrete estimators, corrections (e.g., Green's function asymptotics, p-Laplacian notation), and per-invariant recommendations (preprocessing, regularization, bootstrap tests).

Experimental validation on synthetic manifolds (sphere, torus, warped product) shows ˆσ and ρ_spec as stable and discriminative; θ usable but sensitive; ν estimator-dependent; and S undetected (power-law preferred). We include calibration procedures, reporting templates, and a reproducible pipeline (code on GitHub).

This work transforms GRAVITAS into a scalable paradigm for large-scale semantic processing, with implications for NLP, information retrieval, and geometric AI.