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SuperLocalMemory V3: Information-Geometric Foundations for Zero-LLM Enterprise Agent Memory

Varun Pratap Bhardwaj

Abstract

Persistent memory is a central capability for AI agents, yet the mathematical foundations of memory retrieval, lifecycle management, and consistency remain unexplored. Current systems employ cosine similarity for retrieval, heuristic decay for salience, and provide no formal contradiction detection. We establish information-geometric foundations through three contributions. First, a retrieval metric derived from the Fisher information structure of diagonal Gaussian families, satisfying Riemannian metric axioms, invariant under sufficient statistics, and computable in O(d) time. Second, memory lifecycle formulated as Riemannian Langevin dynamics with proven existence and uniqueness of the stationary distribution via the Fokker-Planck equation, replacing hand-tuned decay with principled convergence guarantees. Third, a cellular sheaf model where non-trivial first cohomology classes correspond precisely to irreconcilable contradictions across memory contexts. On the LoCoMo benchmark, the mathematical layers yield +12.7 percentage points over engineering baselines across six conversations, reaching +19.9 pp on the most challenging dialogues. A four-channel retrieval architecture achieves 75% accuracy without cloud dependency. Cloud-augmented results reach 87.7%. A zero-LLM configuration satisfies EU AI Act data sovereignty requirements by architectural design. To our knowledge, this is the first work establishing information-geometric, sheaf-theoretic, and stochastic-dynamical foundations for AI agent memory systems.

SuperLocalMemory V3: Information-Geometric Foundations for Zero-LLM Enterprise Agent Memory

Abstract

Persistent memory is a central capability for AI agents, yet the mathematical foundations of memory retrieval, lifecycle management, and consistency remain unexplored. Current systems employ cosine similarity for retrieval, heuristic decay for salience, and provide no formal contradiction detection. We establish information-geometric foundations through three contributions. First, a retrieval metric derived from the Fisher information structure of diagonal Gaussian families, satisfying Riemannian metric axioms, invariant under sufficient statistics, and computable in O(d) time. Second, memory lifecycle formulated as Riemannian Langevin dynamics with proven existence and uniqueness of the stationary distribution via the Fokker-Planck equation, replacing hand-tuned decay with principled convergence guarantees. Third, a cellular sheaf model where non-trivial first cohomology classes correspond precisely to irreconcilable contradictions across memory contexts. On the LoCoMo benchmark, the mathematical layers yield +12.7 percentage points over engineering baselines across six conversations, reaching +19.9 pp on the most challenging dialogues. A four-channel retrieval architecture achieves 75% accuracy without cloud dependency. Cloud-augmented results reach 87.7%. A zero-LLM configuration satisfies EU AI Act data sovereignty requirements by architectural design. To our knowledge, this is the first work establishing information-geometric, sheaf-theoretic, and stochastic-dynamical foundations for AI agent memory systems.
Paper Structure (125 sections, 7 theorems, 62 equations, 5 figures, 9 tables)

This paper contains 125 sections, 7 theorems, 62 equations, 5 figures, 9 tables.

Table of Contents

  1. Introduction
  2. The mathematical poverty of current memory systems.
  3. Three open problems.
  4. The formal absence.
  5. Our approach.
  6. Empirical evidence: mathematics improves retrieval.
  7. The scale argument.
  8. Regulatory context.
  9. Reproducibility.
  10. Contributions.
  11. Paper organization.
  12. Background
  13. Complementary Learning Systems Theory
  14. Information Geometry and the Fisher Information Metric
  15. Hyperbolic Geometry and the Poincaré Ball
  16. ...and 110 more sections

Key Result

Theorem 6.1

Let $\mathcal{S}_{\mathrm{diag}} = \{ \mathcal{N}(\mu, \mathrm{diag}(\sigma^2)) : \mu \in \mathbb{R}^d,\, \sigma \in \mathbb{R}^d_{>0} \}$ be the family of $d$-dimensional Gaussian distributions with diagonal covariance. Then the Fisher--Rao distance $d_{\mathrm{FR}}: \mathcal{S}_{\mathrm{diag}} \ti

Figures (5)

  • Figure 1: The SLM-V3 architecture. Left: Ingestion pipeline processes content through entropy gating, fact extraction, entity resolution, graph construction, and sheaf consistency checking ($H^1 \neq 0$ detects contradictions). Center: Four-channel retrieval with Fisher-information-weighted scoring, BM25 keyword matching, entity graph traversal, and temporal reasoning, fused via weighted reciprocal rank fusion. Right: Three mathematical layers providing geometric retrieval, algebraic consistency, and self-organizing lifecycle dynamics. Bottom: Three operating configurations spanning a privacy--capability gradient.
  • Figure 2: Competitive landscape of agent memory systems (March 2026) evaluated on LoCoMo. All systems above SLM-V3 require cloud LLM dependency. SLM-V3 Mode A Retrieval (74.8%) is the highest reported score achievable without cloud dependency during retrieval. Stars ($\star$) denote zero-LLM configurations.
  • Figure 3: Mathematical layers improve retrieval quality by an average of $+12.7$ pp across six conversations. The improvement is largest on harder conversations (conv-44: $+19.9$ pp, conv-49: $+18.8$ pp), suggesting that information-geometric retrieval provides the greatest benefit where heuristic similarity measures struggle most.
  • Figure 4: Per-category ablation on conv-30. Multi-hop questions show the largest sensitivity to component removal: cross-encoder removal drops multi-hop from 50% to 15%, BM25 removal drops it to 23%, and disabling all math layers drops it to 38%. Open-domain questions are more robust across configurations.
  • Figure 5: Ablation study on conv-30 (81 scored questions). Cross-encoder reranking is the single largest contributor ($-30.7$ pp when removed), followed by Fisher metric ($-10.8$ pp) and BM25 ($-6.5$ pp). The three mathematical layers contribute $-7.6$ pp in aggregate.

Theorems & Definitions (27)

  • Definition 2.1: Statistical Manifold
  • Definition 2.2: Fisher Information Matrix
  • Definition 2.3: Fisher--Rao Distance
  • Definition 2.4: Poincaré Ball Model
  • Definition 2.5: Möbius Addition
  • Definition 2.6: Rate--Distortion Function
  • Definition 2.7: Modern Hopfield Energy
  • Theorem 6.1: Fisher--Rao Metric Properties
  • proof : Proof sketch
  • Corollary 6.2: Fisher--Rao Dominates Cosine in Heteroscedastic Settings
  • ...and 17 more