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, ) 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. 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: G₈₉ = \, S₈₉, g = -V' () where G₈₉ = R₈₉ - 12 R\, g₈₉ is the Einstein tensor of the semantic manifold, > 0 is the semantic plasticity constant, and V () is a potential encoding domain-specific structure. Semantic density tensor, derived variationally from the canonical scalar Lagrangian LS = 12 g^ij ᵢ \, ⱼ - V (): S₈₉ = -ᵢ \, ⱼ + g₈₉ \! (12 ||² - V () ) Trace-reversed form, explicit Ricci tensor for N 3: R₈₉ = \! (-ᵢ \, ⱼ + 2\, V () N-2\, g₈₉) Scalar curvature: R = \! (2N\, V () N-2 - ||²) Principal Results C1 — Dimensional non-degeneracy (Proposition 1. 4) The Einstein tensor G₈₉ vanishes identically in dimension N = 2, since R₈₉ = R2 g₈₉ forces G₈₉ = 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 S₈₉ (Proposition 2. 1) The tensor S₈₉ is derived from the canonical scalar Lagrangian LS via the standard metric variational definition: S₈₉ -2 g\, \! (g\;LS) g^{ij} with careful treatment of sign conventions appropriate to Riemannian as opposed to Lorentzian signature. C3 — On-shell conservation (Proposition 2. 4) ⁱ S₈₉ = 0 holds whenever satisfies g = -V' (), matching the contracted Bianchi identity ⁱ G₈₉ = 0 and ensuring internal consistency without additional postulates. C4 — Curvature sign analysis (Proposition 3. 4) For any unit vector v: R₈₉ vⁱ vʲ = 2\, V () N-2 > 0 whenever V () > 0 Potential-dominated regions act as geometric attractors focusing retrieval trajectories. Gradient-dominated regions encode semantic boundaries. The scalar curvature is positive whenever V () > N-22N ||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\! (+ K{K\, ^{N+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)