GRAVITAS: A Dynamic Semantic Geometry for Information Retrieval

GRAVITAS (Geometric Retrieval via Adaptive Tensorial Information-density Attractors in Semantic space) is a mathematical framework that reconceives information retrieval as geodesic navigation on a dynamically curved Riemannian manifold. In contrast to the standard paradigm — in which documents and queries are compared by Euclidean or cosine proximity in a fixed embedding space — GRAVITAS promotes the semantic representation space to a smooth Riemannian manifold (M, g, φ) whose metric g is not prescribed a priori but is determined self-consistently by the distribution of information content, encoded in a semantic scalar field φ. The governing principle is an analogy with Einstein general relativity, transposed to Riemannian positive-definite signature: semantic density curves the representation space, and curvature in turn guides retrieval trajectories. Core Field Equations The coupled system governing geometry and scalar field on M: Gᵢⱼ = κ\, Sᵢⱼ, Δg φ = -V'(φ) where Gᵢⱼ = Rᵢⱼ - 1/2 R\, gᵢⱼ is the Einstein tensor of the semantic manifold, κ > 0 is the semantic plasticity constant, and V(φ) is a potential encoding domain-specific structure. Semantic density tensor, derived variationally from the canonical scalar Lagrangian LS = 1/2 gⁱʲ ∂ᵢ φ\, ∂ⱼ φ - V(φ): Sᵢⱼ = -∂ᵢ φ\, ∂ⱼ φ + gᵢⱼ \!( 1/2 |∇φ|² - V(φ) ) Trace-reversed form, explicit Ricci tensor for N ≥ 3: Rᵢⱼ = κ \!( -∂ᵢ φ\, ∂ⱼ φ + 2\,V(φ)/N-2\, gᵢⱼ ) Scalar curvature: R = κ \!( 2N\,V(φ)/N-2 - |∇φ|² ) Principal Results C1 — Dimensional non-degeneracy (Proposition 1.4) The Einstein tensor Gᵢⱼ vanishes identically in dimension $N = 2$, since Rᵢⱼ = R/2 gᵢⱼ forces Gᵢⱼ = 0. The minimum viable intrinsic dimension for the framework is therefore $N = 3$, which is not restrictive in practice since modern embedding models operate in dimensions D ~ 384–$768$. C2 — Variational derivation of Sᵢⱼ (Proposition 2.1) The tensor Sᵢⱼ is derived from the canonical scalar Lagrangian LS via the standard metric variational definition: Sᵢⱼ ≡ -2√ g\, δ\!(√ g\;LS)δ gⁱʲ with careful treatment of sign conventions appropriate to Riemannian as opposed to Lorentzian signature. C3 — On-shell conservation (Proposition 2.4) ∇ⁱ Sᵢⱼ = 0 holds whenever φ satisfies Δg φ = -V'(φ), matching the contracted Bianchi identity ∇ⁱ Gᵢⱼ = 0 and ensuring internal consistency without additional postulates. C4 — Curvature sign analysis (Proposition 3.4) For any unit vector v ⊥ ∇φ: Rᵢⱼ vⁱ vʲ = 2κ\, V(φ)/N-2 > 0 whenever V(φ) > 0 Potential-dominated regions act as geometric attractors focusing retrieval trajectories. Gradient-dominated regions encode semantic boundaries. The scalar curvature is positive whenever V(φ) > N-2/2N |∇φ|². C5 — Geodesic retrieval (Section 4) A directed query (q, v₀, T) launches a geodesic trajectory through the curved semantic space. Documents are ranked by proximity to the geodesic endpoint rather than by flat-space distance to the query point. C6 — Graph discretization (Proposition 9.4) Density-corrected graph Laplacians approximate the Laplace–Beltrami operator with bias–variance bound: O\!(ε + √log KK\,εN+2) C7 — Local nonlinear existence (Theorem C.16) Under non-resonance and regularity hypotheses, the gauge-fixed Gravitas system admits a locally unique solution near the flat background via the Banach-space implicit function theorem. Approximation Hierarchy Six levels from the full nonlinear theory to practical gradient-flow surrogates: - Level 0 — Full nonlinear coupled system - Level 1/2 — Conformal ansatz, metric reduced to a single scalar mode - Level 1 — Weak-field linearization, Poisson equation for semantic potential - Level 2 — Graph discretization via density-corrected Laplacians - Level 3 — Screened semantic diffusion, resolvent-based score propagation - Level 4 — Gradient-flow surrogates Additional Algorithmic Extensions - Low-rank neural metric parametrizations with Woodbury inversion - Gravitational attention mechanism augmenting dot-product attention with geodesic distance bias - Hamiltonian geodesic integration via generalized leapfrog schemes - Screened diffusion with proved energy dissipation (Proposition 9.11) and spectral filtering (Proposition 9.8)