SuperLocalMemory V3 introduces three mathematical pillars — each a world first in agent memory systems. These are described in our paper (arXiv:2603.14588 | Zenodo).

The problem: Cosine similarity treats embeddings as direction vectors. Two memories with the same meaning but different confidence look identical.

Our solution: We use the Fisher-Rao geodesic distance — the natural metric on statistical manifolds. Each memory embedding is modeled as a diagonal Gaussian distribution with learned mean and variance. Distance is measured along the geodesic (shortest path on the manifold), not through Euclidean space.

What this means in practice:

• A high-confidence memory and a low-confidence memory about the same topic are distinguished
• Retrieval improves as the system learns — variance shrinks with repeated access (Bayesian conjugate update after 3+ accesses)
• Graduated ramp from cosine to full Fisher-Rao distance over the first 10 accesses per memory
• Computation complexity: Theta(d) time — same order as cosine similarity

Benchmark impact: Removing the Fisher metric causes -10.8pp on conv-30 ablation. Across 6 conversations (n=832 questions), the three mathematical layers collectively contribute +12.7pp average improvement, reaching +19.9pp on the most challenging dialogues (conv-44).

Code: src/superlocalmemory/math/fisher.py

The problem: As memories accumulate, contradictions emerge. "Alice moved to London in March" vs "Alice lives in Paris as of April." Pairwise checking is O(n2) and misses transitive contradictions.

Our solution: We model the knowledge graph as a cellular sheaf — an algebraic structure from topology. Each edge carries a restriction map that relates adjacent memories. Computing the first cohomology group H1(G,F) reveals global inconsistencies:

• H1 = 0 — All memories are globally consistent
• H1 ≠ 0 — Contradictions exist, even if every local pair looks fine

This catches contradictions that no pairwise method can detect. Runs in O(|E| * d) time — subquadratic in N when the context graph is sparse.

What this means in practice:

• The system detects when new information contradicts existing knowledge
• Contradictions are flagged with severity scores (>0.45 threshold triggers a SUPERSEDES edge)
• Knowledge graph maintains algebraic consistency as memories accumulate

Benchmark impact: Removing sheaf consistency causes -1.7pp on single-conversation ablation. The effect is subtle on individual conversations but becomes critical at scale (at N=100,000 memories, expected contradiction count exceeds ~5,000).

Code: src/superlocalmemory/math/sheaf.py

The problem: Memory systems need lifecycle management — old, unused memories should be archived. Current systems use hardcoded thresholds (e.g., "archive after 30 days"). This doesn't adapt to usage patterns.

Our solution: Memory lifecycle evolves via stochastic gradient flow on a Riemannian manifold. The potential function encodes access frequency, trust score, and recency. The dynamics provably converge to a stationary distribution — the mathematically optimal allocation of memories across lifecycle states.

Four lifecycle states:

• Active — Frequently used, instantly available
• Warm — Recently used, included in searches
• Cold — Older, retrievable on demand
• Archived — Compressed, restorable when needed

What this means in practice:

• No manual thresholds — the system self-organizes
• Frequently accessed memories stay active longer
• Low-trust memories decay faster (coupled with Fisher-Rao via information geometry)
• Mathematically guaranteed convergence — not heuristic

Code: src/superlocalmemory/dynamics/fisher_langevin_coupling.py

Evaluated on LoCoMo conv-30 (81 scored questions). Each row removes one component.

Key findings:

• Cross-encoder reranking is the single largest contributor (−30.7pp when removed)
• Fisher-Rao metric alone: −10.8pp — the largest single mathematical layer effect
• All three math layers collectively: −7.6pp
• Bootstrap 95% CI for full system: [53.4, 74.0]; for cross-encoder removed: [17.1, 45.7] — non-overlapping, confirming statistical significance

Mathematical layers provide the greatest benefit precisely where heuristic methods struggle — the harder the conversation, the bigger the improvement.

For the full mathematical treatment including proofs, theorems, and detailed experimental methodology:

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

Varun Pratap Bhardwaj, Independent Researcher, 2026

arXiv:2603.14588 | Zenodo DOI: 10.5281/zenodo.19038659

@article{bhardwaj2026slmv3,
 title={Information-Geometric Foundations for Zero-LLM Enterprise Agent Memory},
 author={Bhardwaj, Varun Pratap},
 journal={arXiv preprint arXiv:2603.14588},
 year={2026},
 url={https://arxiv.org/abs/2603.14588}
}

Part of Qualixar · Created by Varun Pratap Bhardwaj