SLM-V3 replaces cosine similarity with Fisher-information-weighted retrieval (Theorem 6.1), heuristic decay with Riemannian Langevin dynamics on Poincaré ball (Theorem 6.3), and silent inconsistency with sheaf cohomology ($H^1 \neq 0$ = contradiction) — achieving +12.7 pp over engineering baseline on LoCoMo, with +19.9 pp on the hardest conversations, all without cloud dependency.
Agent memory systems universally suffer three mathematically unaddressed problems: (1) uncertainty-blind retrieval — cosine similarity treats all embedding dimensions equally, ignoring per-dimension statistical precision; (2) unprincipled lifecycle — fixed TTL/exponential decay with manual half-lives ignores the geometry of memory space; (3) silent inconsistency — no system provides formal contradiction detection across multi-session, multi-partner, multi-temporal sources. Exhaustive venue search (NeurIPS/ICML/ICLR/ACL/EMNLP/AAAI 2020–2026 + arXiv) found no prior work connecting information geometry to agent memory.
Fisher-information-weighted retrieval (§5.2): Each memory modeled as diagonal Gaussian $\mathcal{N}(\mu, \mathrm{diag}(\sigma^2))$ on a statistical manifold. Retrieval scoring: $s_{\text{FR}}(q, m_i) = \exp\left(-\frac{1}{T}\sum_k \frac{(\mu_{q,k} - \mu_{i,k})^2}{\sigma_{i,k}^2}\right)$. High-variance dimensions contribute less (uncertain); low-variance dimensions contribute more (confident). Graduated ramp from cosine to Fisher over first 10 accesses. Theorem 6.1: Fisher–Rao metric satisfies identity, symmetry, triangle inequality, sufficient-statistic invariance (Čencov's uniqueness), and $\Theta(d)$ computability.
Riemannian Langevin lifecycle dynamics (§5.3): SDE on Poincaré ball: $d\xi = -\lambda_\xi^{-2}\nabla_E U(\xi)dt + \sqrt{2T}\lambda_\xi^{-1}dW + \frac{1}{2}T(d-2)\lambda_\xi^{-1}\xi\,dt$. Potential $U(\xi) = \alpha\|\xi\|^2 - \beta \cdot n_{\text{access}} - \gamma \cdot r(\xi, c)$ drives important memories toward origin, pushes unimportant ones toward boundary (natural forgetting). Theorem 6.3: unique stationary distribution $\rho_\infty(\xi) \propto (1-\|\xi\|^2)^{-d} \exp(-U(\xi)/T)$ via Fokker–Planck + Foster–Lyapunov.
Sheaf-cohomological consistency (§5.4): Graph $G = (V, E)$ where $V$ = memory contexts, $E$ = shared entities. Coboundary operator $(\delta f)(u,v) = \rho_{u \to v}(f(u)) - f(v)$. Contradiction score $\kappa = \|\delta f\|^2 / (\|f\|^2 + \epsilon)$; when $\kappa > \tau = 0.45$, system creates supersedes edges. $H^1(\mathcal{F}) \neq 0$ means irreconcilable contradictions.
Four-channel retrieval: Semantic (Fisher–Rao, weight 1.2), BM25 keyword (weight 1.0), entity graph spreading activation (weight 1.3), temporal reasoning (weight 1.0). Fused via weighted reciprocal rank fusion: $\text{WRRF}(m) = \sum_i w_i / (k + r_i(m))$.
Three operating modes: Mode A (zero-LLM, 768d local embeddings, EU AI Act compliant), Mode B (local Ollama), Mode C (cloud 3072d embeddings + gpt-4.1-mini).
核心技术壁垒: The Fisher metric's per-dimension uncertainty weighting — when two memories are equidistant from a query under cosine but one has high-variance dimensions (many similar items) and the other low-variance (few items, high confidence), Fisher promotes the higher-confidence memory. Cosine cannot distinguish them. This advantage grows with memory store density (Proposition 7.2 proves Fisher breaks the cosine concentration barrier).
| Metric | Value |
|---|---|
| Average improvement over engineering baseline | +12.7 pp (6 conversations, 832 questions) |
| Max improvement (hardest conversation, conv-44) | +19.9 pp |
| Mode A Retrieval (zero-LLM + external synthesis) | 74.8% on LoCoMo |
| Mode C (cloud-augmented, conv-30 only) | 87.7% |
| Cross-encoder ablation impact | −30.7 pp (single largest contributor) |
| Fisher metric ablation impact | −10.8 pp |
| Sheaf consistency ablation impact | −1.7 pp |
| EverMemOS (SOTA, cloud-dependent) | 92.3% |
Sheaf consistency operates at store time (creates supersedes edges on contradiction); Langevin dynamics run as background process. Fisher–Rao and reranking operate at retrieval time.
| Check | Status | Location |
|---|---|---|
| Fisher–Rao metric properties (Theorem 6.1) | Proven via product decomposition + Čencov | §6.1, Appendix A.1 |
| Langevin stationary distribution (Theorem 6.3) | Proven via Fokker–Planck + Foster–Lyapunov | §6.2, Appendix A.2 |
| Progressive-disclosure depth (Theorem 6.5) | Proven via rate-distortion | §6.3, Appendix A.3 |
| Bounded effective memory count (Theorem 6.6) | Proven via Hopfield capacity | §6.3, Appendix A.4 |
| Cosine concentration barrier (Proposition 7.1) | Proven via spherical cap area | §7.6 |
| Fisher breaks concentration (Proposition 7.2) | Proven | §7.6 |
Important gap: Hopfield retrieval channel and native hyperbolic embeddings are theoretically developed but NOT implemented or evaluated in the current system.
LoCoMo Benchmark (Table 3, §7.2): SLM-V3 Mode A Retrieval 74.8%, outperforming Mem0 (64.2%) without cloud dependency. Open-domain 85.0% (highest of any system). Mode C reaches 87.7% on conv-30 (81 questions) — comparable to Zep v3 (85.2%).
Fisher vs Cosine (Table 5, §7.4): Across 6 conversations, mathematical layers add +12.7 pp average. Improvement largest on hardest conversations: conv-44 (+19.9 pp), conv-49 (+18.8 pp). Consistent with theory: Fisher advantage grows with retrieval difficulty.
Ablation (Table 4, §7.3): Cross-encoder removal: −30.7 pp (dominant). Fisher: −10.8 pp. BM25: −6.5 pp. All math layers aggregate: −7.6 pp. Sheaf: −1.7 pp. Temporal: −0.2 pp. Bootstrap 95% CI for full system: [53.4, 74.0]. Cross-encoder CI [17.1, 45.7] — no overlap (statistically significant).
Scale Analysis (§7.6): At $N = 10^5$, cosine $\varepsilon$-cap contains ~100 vectors, overwhelming typical $K = 20$ retrieval budget (SNR → 0). Fisher–Rao provides strictly finer ranking under heteroscedasticity.
| Step | Claim | Evidence | Depends on |
|---|---|---|---|
| 1 | Agent memory systems universally use cosine/heuristic-decay/no-consistency-check | Literature survey: MemGPT, Mem0, Zep, MemOS, SimpleMem, A-MEM | — |
| 2 | Fisher–Rao metric is the unique statistically principled retrieval metric | Čencov's theorem + Theorem 6.1 | Step 1 |
| 3 | Langevin dynamics on Poincaré ball provide self-organizing lifecycle with proven convergence | Theorem 6.3 (Fokker–Planck proof) | Step 1 |
| 4 | Sheaf cohomology formally detects contradictions ($H^1 \neq 0$) | §5.4 coboundary construction | Step 1 |
| 5 | Mathematical layers improve retrieval by +12.7 pp average, +19.9 pp on hardest tasks | Table 5 (6 conversations, 832 questions) | Steps 2, 3, 4 |
| 6 | Advantage grows with scale (cosine concentration barrier) | Propositions 7.1, 7.2 | Step 2 |
MIT license. Builds on SuperLocalMemory V2 infrastructure. 111 ACI containers for evaluation (2 vCPU + 4 GB each). SQLite with 21 tables, WAL mode. nomic-embed-text-v1.5 (768d) for Mode A, text-embedding-3-large (3072d) for Mode C. bge-reranker-v2-m3 for cross-encoder.
[实现未公开] — No specific repository URL for V3 codebase in the paper.
关键实现细节: