Semantic Level of Detail (SLoD)

Multi-scale knowledge representation via heat kernel diffusion on hyperbolic manifolds.

Eduard Izgorodin — arXiv:2603.08965

The problem

AI memory systems store knowledge as flat vector collections, discarding the hierarchical structure inherent in semantic information. A software project has architecture-level concepts, module-level patterns, and line-level details — but a vector database treats them all the same.

The idea

SLoD borrows Level of Detail from computer graphics — where 3D engines render geometry at variable resolution depending on distance — and builds an analogous operator for semantic data.

The key insight: hyperbolic space is the natural substrate. The Poincare ball has exponential volume growth, embedding tree-structured hierarchies with distortion O(log n) — provably optimal. Heat kernel diffusion on this space provides a continuous zoom parameterized by scale sigma:

sigma -> 0    Fine detail (individual facts, specific memories)
sigma -> inf  Global summary (high-level themes, abstract concepts)

Components

Component	What it does	Paper reference
SLoD Operator	Heat kernel weights + Frechet mean on Poincare ball	Algorithm 1, Def. 2
Hierarchical Coherence	Bounded error O(sigma), distortion O(log n)	Theorems 1-2
Boundary Scanner	Detects natural abstraction levels from spectral gaps	Algorithm 2, Prop. 3
Multi-Center Extension	Mixture representation when single summary is lossy	Def. 3, Corollary

Boundary detection

Three complementary signals detect where the representation undergoes qualitative transitions:

• V(sigma) — representation velocity: how fast does the summary move in hyperbolic space?
• D_w(sigma) — weight divergence: Jensen-Shannon between consecutive weight distributions
• C_k(sigma) — neighborhood churn: do the nearest neighbors change?

Peaks in the composite score reveal natural abstraction boundaries — no manual tuning required.

Theoretical guarantees

Theorem 1 (Hierarchical Coherence): For a tree with n nodes embedded in the Poincare ball:

d_H(Phi_s1, Phi_s2) <= C * |s2 - s1| * log(n)

Nearby scales produce semantically related representations.

Theorem 2 (Approximation Error): Memories within cognitive distance R can be approximated with error O(sigma).

Proposition 3 (Spectral Boundaries): JSD peaks near sigma* ~ 1/lambda_k when spectral gap ratio exceeds threshold R.

Experimental validation

Experiment	Data	Key result	Status
Boundary Recovery	HSBM (1024 nodes, 3 levels)	ARI macro=1.00, meso=0.91 at r=200	Complete (8/8 pass)
WordNet Consistency	WordNet 3.0 (82K synsets, depth 19)	Kendall tau=0.79, Recall@2=0.75	Complete

Citation

@article{izgorodin2026slod,
  title   = {Semantic Level of Detail: Multi-Scale Knowledge Representation
             via Heat Kernel Diffusion on Hyperbolic Manifolds},
  author  = {Izgorodin, Edward},
  year    = {2026},
  eprint  = {2603.08965},
  archivePrefix = {arXiv}
}

Paper

• arXiv: 2603.08965
• Format: 11 pages (9 body + references), 34 citations

Related

• Tensor-Hyperbolic Graphs — the library implementing SLoD geometry
• Manifesto — why hyperbolic geometry matters for memory
• Scientific sources — bibliography underpinning the approach