Novel Contributions

Mathematical Foundations

Three techniques from differential geometry, algebraic topology, and stochastic analysis — each a first in agent memory systems.

Why Mathematics?

Current memory systems rely on cosine similarity for retrieval, pairwise checking for consistency, and hardcoded thresholds for lifecycle. These approaches work — but they have known limitations. Cosine ignores confidence. Pairwise checking misses transitive contradictions. Hardcoded thresholds do not adapt.

We asked: what if we replaced these heuristics with principled mathematics? The answer turned out to be three techniques from fields not previously applied to agent memory.

Fisher-Rao Information Geometry

Retrieval Layer

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

Our approach: We model each memory embedding as a diagonal Gaussian distribution with learned mean and variance. Similarity is measured along the Fisher-Rao geodesic — the natural metric on statistical manifolds. This is not an arbitrary choice: the Fisher-Rao metric is the unique Riemannian metric that is invariant under sufficient statistics.

What this means in practice:

High-confidence memories and low-confidence memories about the same topic are distinguished
Retrieval improves as the system learns — variance shrinks with repeated access (Bayesian conjugate updates)
After 10 accesses, the system transitions from cosine similarity to full Fisher-Rao distance (graduated ramp)

Measured impact: Removing Fisher-Rao drops multi-hop accuracy by 12 percentage points (50% → 38%). Across six conversations, the three mathematical layers collectively contribute +12.7pp average improvement.

Sheaf Cohomology for Consistency

Consistency Layer

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

Our approach: We model the knowledge graph as a cellular sheaf — an algebraic structure from topology that assigns vector spaces to nodes and edges, with restriction maps encoding how local data relates across the graph. 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

What this means in practice: The system detects contradictions that no pairwise method can find. When detected, contradictions are resolved automatically (newer supersedes older with SUPERSEDES edges) or surfaced to the user.

Why sheaf theory: Sheaves are the natural mathematical language for "local-to-global" problems. Checking consistency of a knowledge graph from local edge data is exactly this kind of problem.

Riemannian Langevin Dynamics

Lifecycle Layer

The problem: Memory systems need lifecycle management — old, unused memories should be archived. Current systems use hardcoded thresholds ("archive after 30 days"). This does not adapt to usage patterns and requires manual tuning.

Our approach: Memory lifecycle evolves via stochastic gradient flow on the Poincaré ball. The potential function encodes access frequency, trust score, and recency. The dynamics are a discretized Langevin SDE on a Riemannian manifold with provable convergence to a stationary distribution.

Four lifecycle states:

Active — Near the origin. Frequently used, instantly available
Warm — Intermediate. Recently used, included in searches
Cold — Further out. Older, retrievable on demand
Archived — Near the boundary. Compressed, restorable

What this means in practice: No manual thresholds. Frequently accessed memories stay active longer. Low-trust memories decay faster. The system self-organizes toward the mathematically optimal distribution.

Ablation Results

Each row disables one component. The difference shows what each layer contributes.

Configuration	Micro Avg	Multi-Hop	Open Domain
Full system (all layers)	62.3%	50%	78%
− Math layers	59.3%	38%	70%
− Entity channel	56.8%	38%	73%
− BM25 channel	53.2%	23%	71%
− Cross-encoder	31.8%	17%	—

LoCoMo conv-30, 81 scored questions, Mode A (zero-LLM). Mathematical layers contribute +12pp on multi-hop reasoning (50% vs 38%).

Full Mathematical Treatment

Proofs, theorems, and detailed experimental methodology in the V3 paper.

Read the Paper (arXiv) View the Code (GitHub)

Every algorithm is open source under MIT license. Take it, use it, extend it.