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Semantic Level of Detail: Multi-Scale Knowledge Representation via Heat Kernel Diffusion on Hyperbolic Manifolds
Edward Izgorodin
Intelligence
Status: succeeded | Model: google/gemini-3.1-flash-lite-preview | Prompt: intel-v1 | Confidence: 96%
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Summary
The paper introduces Semantic Level of Detail (SLoD), a framework for continuous resolution control in knowledge graphs using heat kernel diffusion on the Poincaré ball. By leveraging the hyperbolic geometry of the Poincaré ball, SLoD enables a 'zoom' operator that aggregates semantic information across scales. The authors demonstrate that spectral gaps in the graph Laplacian induce emergent scale boundaries, allowing for automatic detection of abstraction levels without manual tuning, validated on synthetic hierarchies and the WordNet noun hierarchy.
Entities (6)
Relation Signals (4)
SLoD → evaluatedon → WordNet
confidence 98% · On the full WordNet noun hierarchy (82K synsets), detected boundaries align with true taxonomic depth
SLoD → utilizes → Poincaré ball
confidence 98% · We introduce Semantic Level of Detail (SLoD)... via heat kernel diffusion on the Poincaré ball
Graph Laplacian → induces → Scale boundaries
confidence 97% · spectral gaps in the graph Laplacian induce emergent scale boundaries
Heat kernel diffusion → computes → SLoD representation
confidence 95% · The SLoD representation at scale σ with focus x0 is the weighted Fréchet mean
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Abstract
Abstract:AI memory systems increasingly organize knowledge into graph structures -- knowledge graphs, entity relations, community hierarchies -- yet lack a principled mechanism for continuous resolution control: where do the qualitative boundaries between abstraction levels lie, and how should an agent navigate them? We introduce Semantic Level of Detail (SLoD), a framework that answers both questions by defining a continuous zoom operator via heat kernel diffusion on the Poincaré ball $\mathbb{B}^d$. At coarse scales ($\sigma \to \infty$), diffusion aggregates embeddings into high-level summaries; at fine scales ($\sigma \to 0$), local semantic detail is preserved. We prove hierarchical coherence with bounded approximation error $O(\sigma)$ and $(1+\varepsilon)$ distortion for tree-structured hierarchies under Sarkar embedding. Crucially, we show that spectral gaps in the graph Laplacian induce emergent scale boundaries -- scales where the representation undergoes qualitative transitions -- which can be detected automatically without manual resolution parameters. On synthetic hierarchies (HSBM), our boundary scanner recovers planted levels with ARI up to 1.00, with detection degrading gracefully near the information-theoretic Kesten-Stigum threshold. On the full WordNet noun hierarchy (82K synsets), detected boundaries align with true taxonomic depth ($\tau = 0.79$), demonstrating that the method discovers meaningful abstraction levels in real-world knowledge graphs without supervision.
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- Source: https://arxiv.org/abs/2603.08965v1
- Canonical: https://arxiv.org/abs/2603.08965v1
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Semantic Level of Detail: Multi-Scale Knowledge Representation via Heat Kernel Diffusion on Hyperbolic Manifolds Edward Izgorodin Mnemoverse.AI Funchal, Madeira, Portugal Corresponding author: izgorodin@me.com Abstract AI memory systems increasingly organize knowledge into graph structures—knowledge graphs, entity relations, community hierarchies—yet lack a principled mechanism for continuous resolution control: where do the qualitative boundaries between abstraction levels lie, and how should an agent navigate them? We introduce Semantic Level of Detail (SLoD), a framework that answers both questions by defining a continuous zoom operator via heat kernel diffusion on the Poincaré ball dB^d. At coarse scales (σ→∞σ→∞), diffusion aggregates embeddings into high-level summaries; at fine scales (σ→0σ→ 0), local semantic detail is preserved. We prove hierarchical coherence with bounded approximation error O(σ)O(σ) and (1+ε)(1+ ) distortion for tree-structured hierarchies under Sarkar embedding. Crucially, we show that spectral gaps in the graph Laplacian induce emergent scale boundaries—scales where the representation undergoes qualitative transitions—which can be detected automatically without manual resolution parameters. On synthetic hierarchies (HSBM), our boundary scanner recovers planted levels with ARI up to 1.00, with detection degrading gracefully near the information-theoretic Kesten–Stigum threshold. On the full WordNet noun hierarchy (82K synsets), detected boundaries align with true taxonomic depth (τ=0.79τ=0.79), demonstrating that the method discovers meaningful abstraction levels in real-world knowledge graphs without supervision. 1 Introduction Equipping AI agents with persistent, navigable memory is an open problem. Modern approaches increasingly move beyond flat retrieval: graph-based methods such as GraphRAG organize knowledge into entity graphs with community structure, while persistent memory systems like MemGPT and MemoryOS maintain hierarchical stores. Yet a fundamental question remains unanswered: where do the meaningful abstraction boundaries lie within a knowledge graph, and how should an agent transition between them? Current graph-based systems rely on discrete community detection with manually tuned resolution parameters (e.g., Leiden γ), offering no continuous mechanism to zoom between granularity levels. A software project has architecture-level concepts, module-level patterns, and line-level details; an agent reasoning over such a graph needs to dynamically select the appropriate resolution for each step—a capability that existing systems lack. We draw inspiration from computer graphics, where Level of Detail (LOD) allows rendering engines to represent geometry at variable resolution depending on the viewer’s distance [19]. We ask: can we build an analogous LOD operator for semantic data? Our key insight is that hyperbolic space provides the natural substrate for such an operator. The Poincaré ball dB^d has exponential volume growth, which allows it to embed tree-structured hierarchies with distortion (1+ε)(1+ ) for any ε>0 >0 [25]. By defining a heat kernel on dB^d, we obtain a family of smoothing operators parameterized by a continuous scale σ that naturally respects hierarchical structure. A second contribution addresses the natural question: how should an agent choose σ? Rather than requiring manual scale selection, we show that spectral gaps in the graph Laplacian induce emergent scale boundaries—values of σ where the representation undergoes qualitative transitions—which can be detected automatically via a boundary scanner algorithm. This connects SLoD to the literature on multi-scale community detection [5]. Contributions. We present: (1) A mathematical formulation of Semantic LOD as heat kernel diffusion on the Poincaré ball; (2) Theoretical guarantees on hierarchical coherence and approximation quality; (3) An efficient tangent-space aggregation algorithm (Algorithm 1); (4) An emergent scale selection procedure with formal guarantees tied to spectral structure (Algorithm 2); (5) A multi-center extension for scales where a single summary is lossy; (6) Empirical validation on synthetic hierarchies (HSBM, 1024 nodes) and a real-world DAG (WordNet, 82K synsets). 2 Preliminaries 2.1 The Poincaré Ball Model Let d=x∈ℝd:‖x‖<1B^d=\x ^d:\|x\|<1\ be the open unit ball equipped with the Riemannian metric: gx=(21−‖x‖2)2gEg_x= ( 21-\|x\|^2 )^2g_E (1) where gEg_E is the Euclidean metric. The induced geodesic distance is: dℍ(x,y)=arcosh(1+2‖x−y‖2(1−‖x‖2)(1−‖y‖2))d_ H(x,y)=arcosh\! (1+2 \|x-y\|^2(1-\|x\|^2)(1-\|y\|^2) ) (2) Exponential and Logarithmic Maps. For x∈dx ^d and v∈Txdv∈ T_xB^d: Expx(v) _x(v) =x⊕(tanh(λx‖v‖2)v‖v‖) =x ( \! ( _x\|v\|2 ) v\|v\| ) (3) Logx(y) _x(y) =2λxarctanh(‖−x⊕y‖)−x⊕y‖−x⊕y‖ = 2 _xarctanh(\|-x y\|) -x y\|-x y\| (4) where λx=21−‖x‖2 _x= 21-\|x\|^2 is the conformal factor and ⊕ denotes Möbius addition [8]. 2.2 Heat Kernel on Hyperbolic Space The heat kernel Kσ(x,y)K_σ(x,y) on ℍdH^d is the fundamental solution of the heat equation ∂σu=Δℍu _σu= _Hu, where Δℍ _H is the Laplace–Beltrami operator. For ℍdH^d with curvature κ=−1κ=-1: Kσ(x,y)=1(4πσ)d/2e−(d−1)2σ/4∫dℍ(x,y)∞se−s2/(4σ)(coshs−coshdℍ(x,y))1/2sK_σ(x,y)= 1(4πσ)^d/2e^-(d-1)^2σ/4 _d_ H(x,y)^∞ s\,e^-s^2/(4σ)( s- _ H(x,y))^1/2\,ds (5) for d odd; analogous integral formulas exist for even dimensions [10]. The key property is that KσK_σ is a function of geodesic distance alone (isotropy) and satisfies: σ→0:Kσ(x,y)→δ(x,y),σ→∞:Kσ→uniformσ→ 0:K_σ(x,y)→δ(x,y), σ→∞:K_σ (6) This makes σ a natural “zoom” parameter. The non-enhancement of local extrema under the heat semigroup [17] guarantees that SLoD progressively simplifies representations without introducing spurious detail. 2.3 Fréchet Mean on Hadamard Manifolds The Poincaré ball is a Hadamard manifold (complete, simply connected, nonpositive curvature). On such spaces, the Fréchet functional F(z)=∑iwidℍ2(z,vi)F(z)= _iw_i\,d_ H^2(z,v_i) is strictly convex, ensuring a unique minimizer whenever the weights sum to 1 [26, 1]. This is a consequence of the CAT(0) comparison inequality: geodesic triangles in dB^d are “thinner” than their Euclidean counterparts, yielding the variance inequality ∑iwidℍ2(z,vi)−∑iwidℍ2(μ,vi)≥dℍ2(z,μ) _iw_i\,d_ H^2(z,v_i)- _iw_i\,d_ H^2(μ,v_i) _ H^2(z,μ) for Fréchet mean μ. The mean can be computed via Riemannian gradient descent with guaranteed convergence [2], and the computation is differentiable [18]. Remark 1 (No bifurcation on dB^d). Unlike Euclidean or spherical spaces, the Fréchet mean on dB^d is always unique. Scale boundaries therefore manifest not as mean bifurcation, but as rapid displacement of the unique mean and sensitivity changes in the Hessian of the Fréchet functional. This observation shapes our boundary detection approach in Section 5. 3 Semantic Level of Detail 3.1 Problem Formulation Given a knowledge corpus embedded as =v1,…,vN⊂dV=\v_1,…,v_N\ ^d, we seek a family of operators Φσ:2d→d _σ:2^B^d ^d such that: 1. Coarse scale (σ large): Φσ() _σ(V) captures the global semantic theme. 2. Fine scale (σ small): Φσ() _σ(V) preserves local semantic detail. 3. Continuity: The map σ↦Φσ()σ _σ(V) is smooth. 4. Hierarchy-aware: Nearby scales produce semantically related representations. 3.2 Diffusion-Based Aggregation We define the SLoD operator via a two-step process: weighting by the heat kernel, then aggregation via the Fréchet mean. Definition 1 (Heat Kernel Weights). For a focus point x0∈dx_0 ^d and scale σ>0σ>0, the weight of viv_i is: wi(σ,x0)=Kσ(x0,vi)∑j=1NKσ(x0,vj)w_i(σ,x_0)= K_σ(x_0,v_i) _j=1^NK_σ(x_0,v_j) (7) Definition 2 (Semantic LOD Operator). The SLoD representation at scale σ with focus x0x_0 is the weighted Fréchet mean: Φσ(,x0)=argminy∈d∑i=1Nwi(σ,x0)⋅dℍ2(y,vi) _σ(V,x_0)= *arg\,min_y ^d _i=1^Nw_i(σ,x_0)·d_ H^2(y,v_i) (8) Since the Fréchet mean has no closed form on dB^d, we compute it iteratively in tangent space: Algorithm 1 SLoD via Tangent-Space Aggregation 0: Embeddings =vii=1N⊂dV=\v_i\_i=1^N ^d, focus x0∈dx_0 ^d, scale σ, iterations T 1: Compute weights wi←Kσ(x0,vi)/∑jKσ(x0,vj)w_i← K_σ(x_0,v_i)/ _jK_σ(x_0,v_j) 2: Initialize μ(0)←x0μ^(0)← x_0 3: for t=0t=0 to T−1T-1 do 4: ui←Logμ(t)(vi)u_i _μ^(t)(v_i) for all i Map to tangent space 5: u¯←∑iwi⋅ui u← _iw_i· u_i Weighted average 6: μ(t+1)←Expμ(t)(η⋅u¯)μ^(t+1) _μ^(t)(η· u) Map back; η is step size 7: Project: ‖μ(t+1)‖←min(‖μ(t+1)‖,1−ϵ)\|μ^(t+1)\|← (\|μ^(t+1)\|,1-ε) 8: end for 9: return μ(T)μ^(T) Practical Approximation. For large N, we approximate via the graph Laplacian: construct a k-N graph on V with edge weights Wij=exp(−dℍ2(vi,vj)/2τ2)W_ij= (-d_ H^2(v_i,v_j)/2τ^2), compute the normalized Laplacian L=I−D−1/2WD−1/2L=I-D^-1/2WD^-1/2, and use: (σ)=exp(−σL)x0w(σ)= (-σ L)\,e_x_0 (9) where x0e_x_0 is the indicator vector for the nearest neighbor of x0x_0. This admits efficient computation via Chebyshev polynomial approximation [11]. Remark 2 (Practical computation). In our implementation, exp(−σL) (-σ L) is never formed as a dense matrix. We compute k smallest eigenvalues λi\ _i\ and eigenvectors ϕi\ _i\ of L via the Lanczos algorithm (scipy.sparse.linalg.eigsh), then evaluate Kσ(x0,j)=∑i=1ke−σλiϕi(x0)ϕi(j)K_σ(x_0,j)= _i=1^ke^-σ _i _i(x_0)\, _i(j) directly. This avoids the O(N3)O(N^3) cost of matrix exponentiation at the cost of truncation error bounded by ∑i>ke−σλi _i>ke^-σ _i, which decays exponentially with the spectral gap. 4 Theoretical Analysis Theorem 1 (Hierarchical Coherence). Let T be a weighted tree with n nodes embedded in dB^d via a Sarkar-type embedding with distortion δ=(1+ε)δ=(1+ ) for ε>0 >0. For any node v at depth dvd_v and σ1<σ2 _1< _2: dℍ(Φσ1(v,v),Φσ2(v,v))≤C⋅|σ2−σ1|⋅(1+ε)d_ H\! ( _ _1(V_v,v), _ _2(V_v,v) )≤ C·| _2- _1|·(1+ ) (10) where vV_v is the set of descendants of v and C depends only on the curvature. Proof sketch. The Fréchet mean on dB^d is Lipschitz-continuous in the weights (a consequence of strict convexity, cf. Afsari 1). The heat kernel weights (7) are Lipschitz in σ with constant controlled by the maximum geodesic distance, which is O(1+ε)O(1+ ) for Sarkar embeddings. The result follows by composition. ∎ Theorem 2 (Scale-Dependent Approximation). For query radius R and scale σ, memories within cognitive distance R can be approximated by the SLoD representation with error: ‖Φσ(∩BR(x0),x0)−Φ0(∩BR(x0),x0)‖≤O(σ) \| _σ(V∩ B_R(x_0),x_0)- _0(V∩ B_R(x_0),x_0) \|≤ O(σ) (11) Proof sketch. As σ→0σ→ 0, the heat kernel converges to a delta distribution. The error is controlled by the variance of the weight distribution, which decays linearly in σ for the hyperbolic heat kernel near a point. ∎ Proposition 1 (Computational Complexity). Algorithm 1 runs in O(TNd+NklogN)O(TNd+Nk N) time: O(NklogN)O(Nk N) for graph construction and O(TNd)O(TNd) for T Fréchet mean iterations over N points in d dimensions. With Chebyshev approximation (9) of order M, diffusion costs O(MNk)O(MNk). Scalability. The direct spectral pipeline (graph Laplacian eigendecomposition with k=50k=50 modes via Lanczos iteration) processes N=1,024N=1,024 nodes in 0.20.2–0.40.4 s. For the full WordNet hierarchy (N=82,115N=82,115), Poincaré embedding takes 9898 s and kkNN graph construction 123123 s on commodity hardware. The computational bottleneck is the embedding step, not the spectral analysis itself. The direct spectral pipeline—operating on the graph Laplacian without embedding—avoids this bottleneck entirely and scales as O(Nk2)O(Nk^2) via the Lanczos algorithm. Why Hyperbolic? In Euclidean space ℝdR^d, embedding a tree with n leaves requires distortion Ω(nd) ( [d]n) [9]. The Poincaré ball achieves (1+ε)(1+ ) for any ε>0 >0 regardless of d [25]. This exponential gap means that SLoD on dB^d preserves hierarchical relationships that would be destroyed in ℝdR^d. 5 Emergent Scale Selection The SLoD operator provides representations at any scale σ, but practical systems need to identify which scales matter. We show that the spectral structure of the underlying graph induces natural scale boundaries that can be detected automatically. 5.1 Spectral Motivation The heat kernel’s spectral decomposition on the graph Laplacian L with eigenvalues 0=λ1≤λ2≤⋯≤λN0= _1≤ _2≤·s≤ _N gives: Kσ(i,j)=∑k=1Ne−σλkϕk(i)ϕk(j)K_σ(i,j)= _k=1^Ne^-σ _k _k(i) _k(j) (12) At diffusion time σ, modes with λkσ≫1 _kσ 1 are exponentially suppressed. A spectral gap between consecutive eigenvalues λk _k and λk+1 _k+1 creates a range 1/λk+1<σ<1/λk1/ _k+1<σ<1/ _k where exactly k modes dominate. Transitions between these regimes are the natural scale boundaries. Proposition 2 (Spectral Scale Boundaries). Let L have a gap ratio rk=λk+1/λk>Rr_k= _k+1/ _k>R for threshold R>1R>1. Then: (i) The weight distribution divergence Dw(σ)=JSD((σ)∥(σ+δ))D_w(σ)=JSD(w(σ)\|w(σ+δ)) achieves a local maximum in [1/λk+1,1/λk−1][1/ _k+1,1/ _k-1]. (i) The effective dimensionality (number of eigenvalues of KσK_σ exceeding threshold ε ) drops from k to k−1k-1 as σ crosses σ∗≈1/λkσ^*≈ 1/ _k. (i) The representation velocity V(σ∗)=dℍ(Φσ∗+δ,Φσ∗)/δV(σ^*)=d_ H( _σ^*+δ, _σ^*)/δ satisfies V(σ∗)≥c⋅(λk+1−λk)V(σ^*)≥ c·( _k+1- _k) for a geometry-dependent constant c. Proof sketch. At σ=1/λkσ=1/ _k, the k-th mode’s contribution e−λkσ=e−1e^- _kσ=e^-1 transitions from significant to negligible. When rk>Rr_k>R, this transition is sharp: the mode is still contributing e−1/Re^-1/R at σ=1/λk+1σ=1/ _k+1 but has decayed to e−Re^-R at σ=1/λk−1σ=1/ _k-1. The JSD between weight distributions before and after this transition is bounded below by the squared difference in effective support, yielding (i). Parts (i) and (i) follow from the spectral decomposition (12) and the Lipschitz property of the Fréchet mean. ∎ 5.2 Boundary Detection Algorithm We combine three complementary signals into a composite boundary score: Definition 3 (Boundary Indicators). For scale σ with step δ: V(σ) V(σ) =dℍ(Φσ+δ,Φσ)δ(representation velocity) = d_ H( _σ+δ, _σ)δ (representation velocity) (13) Dw(σ) D_w(σ) =JSD((σ)∥(σ+δ))(weight divergence) =JSD(w(σ)\|w(σ+δ)) (weight divergence) (14) Ck(σ) C_k(σ) =1−|NNk(σ)∩NNk(σ+δ)||NNk(σ)∪NNk(σ+δ)|(neighborhood churn) =1- |N_k(σ)∩ N_k(σ+δ)||N_k(σ)∪ N_k(σ+δ)| (neighborhood churn) (15) where NNk(σ)N_k(σ) is the k-nearest-neighbor set of Φσ _σ in V. Jensen–Shannon divergence [16] is ideal here: it is bounded in [0,ln2][0, 2], symmetric, always finite, and its square root is a true metric [6]. Algorithm 2 SLoD-BoundaryScan 0: Embeddings ⊂dV ^d, focus x0x_0, scale grid Σ=σ1,…,σT =\ _1,…, _T\ (log-spaced), neighborhood size k, threshold multiplier α 0: Boundary set Σ∗=σ1∗,…,σm∗ ^*=\σ^*_1,…,σ^*_m\, multimodality scores K∗(σj∗)\K^*(σ^*_j)\ 1: Compute graph Laplacian L and its eigenvalues λ1,…,λK _1,…, _K Spectral prior 2: ←1/λk:λk+1/λk>RC←\1/ _k: _k+1/ _k>R\ Candidate scales from gaps 3: for t=1t=1 to T do 4: t←w_t← DiffusionWeights(,x0,σt)(V,x_0, _t) via (9) 5: mt←m_t← Algorithm 1(,x0,σt)(V,x_0, _t) SLoD summary 6: end for 7: for t=1t=1 to T−1T-1 do 8: Vt←dℍ(mt+1,mt)/(σt+1−σt)V_t _ H(m_t+1,m_t)/( _t+1- _t) 9: Dt←JSD(t∥t+1)D_t (w_t\|w_t+1) 10: Ct←C_t← NeighborhoodChurn(mt,mt+1,,k)(m_t,m_t+1,V,k) 11: end for 12: St←α1V^t+α2D^t+α3C^tS_t← _1 V_t+ _2 D_t+ _3 C_t Composite score, ⋅ · = normalized 13: Σ∗← ^*← PeakPick(S,threshold=median+α⋅MAD)(S,threshold=median+α·MAD) 14: for each σ∗∈Σ∗σ^*∈ ^* do 15: K∗(σ∗)←|k:e−σ∗λk>ε|K^*(σ^*)←|\k:e^-σ^* _k> \| Effective dimensionality 16: end for 17: return Σ∗,K∗(σ∗) ^*,\K^*(σ^*)\ Complexity. The spectral prior (step 1) costs O(NK2)O(NK^2) via partial eigendecomposition. The scan (steps 3–9) costs O(T⋅(Nd+Nk))O(T·(Nd+Nk)). Total: (+)O(TNd+NK^2), linear in N for fixed T,K,dT,K,d. 5.3 Multi-Center Extension When K∗(σ)>1K^*(σ)>1 at a detected boundary, the weight distribution becomes multi-modal and a single Fréchet mean is a poor summary. We extend SLoD to a mixture representation. Definition 4 (Multi-Center SLoD). For σ with K∗(σ)=K>1K^*(σ)=K>1, define: ΦσMC(,x0)=(μj,πj)j=1K ^MC_σ(V,x_0)=\( _j, _j)\_j=1^K (16) where μj\ _j\ are cluster centers and πj\ _j\ mixture weights obtained by weighted Riemannian k-means on the top-M weight points: assign each viv_i to argminjdℍ(vi,μj) *arg\,min_jd_ H(v_i, _j), then update each μj _j as the weighted Fréchet mean of its assigned points. On Hadamard manifolds, each Fréchet mean update minimizes a geodesically convex objective, so within-cluster cost monotonically decreases and the algorithm converges [2]. For probabilistic modeling, the wrapped normal distribution on ℍdH^d [21] provides a reparametrizable density for EM-based fitting. Corollary 1 (Multi-Center Distortion Bound). If K∗(σ)=K^*(σ)=K and each cluster contains a subtree of T, then each cluster center achieves Sarkar distortion (1+ε)(1+ ) over at most n/Kn/K nodes, reducing the maximum geodesic diameter by a factor of K compared to the single-center representation. 6 Experimental Design We validate SLoD through two experiments: boundary recovery on synthetic hierarchies (6.1) and hierarchical consistency on a real-world DAG (6.2). 6.1 Experiment 1: Boundary Recovery on Synthetic Hierarchies Data. Hierarchical Stochastic Block Model (HSBM) with planted 3-level structure: 1024 nodes → 2 macro → 8 meso → 64 micro communities. Edge probabilities: pwithin=40/Np_within=40/N, pmeso=8/Np_meso=8/N, pmacro=2/Np_macro=2/N, pbetween=0.5/Np_between=0.5/N. Vary inter-level ratio r∈20,40,60,80,100,150,200r∈\20,40,60,80,100,150,200\ to test detection sensitivity across the Kesten–Stigum phase transition. Protocol. Extract the largest connected component (LCC) and compute the normalized graph Laplacian with eigendecomposition. Evaluate K∗(σ)K^*(σ) across a log-spaced grid and perform spectral clustering at each K∗K^*-crossing to obtain partitions. Additionally, embed in 10B^10 via hyperbolic Sammon MDS and run the full Algorithm 2 pipeline to validate end-to-end consistency. Compare detected boundaries Σ∗ ^* against planted levels. Metrics. (a) Scale detection precision/recall: a detected boundary is a true positive if within tolerance τ of a planted scale. (b) Adjusted Rand Index (ARI) [13] between induced partition at each σ∗σ^* and planted partition. (c) Variation of Information (VI) [20] as a true metric on partitions. Baselines. (i) Louvain community detection (resolution limit test); (i) Greedy modularity maximization; (i) Spectral k-sweep (eigengap heuristic) [29]; (iv) Leiden with CPM quality function [28] at multiple resolutions (γ∈0.001,…,0.5γ∈\0.001,…,0.5\). Additional multi-scale baselines—Markov Stability [5, 15] and nested SBM [24]—are left for future systematic comparison. Results. Table 1 reports ARI from spectral clustering on the direct graph Laplacian (no embedding step), validating the spectral mechanism of Proposition 2. We sweep r∈20,40,60,80,100,150,200r∈\20,40,60,80,100,150,200\ on HSBM graphs (N=1024N=1024). Table 1: Adjusted Rand Index at macro (K∗=2K^*=2) and meso (K∗=8K^*=8) scales across signal-to-noise ratios. r 20 40 60 80 100 150 200 ARI (macro, K∗=2K^*=2) 0.04 0.67 0.84 0.93 0.97 1.00 1.00 ARI (meso, K∗=8K^*=8) 0.09 0.15 0.24 0.43 0.63 0.79 0.91 Figure 1: Effective dimensionality K∗(σ)K^*(σ) as a function of diffusion scale for varying signal strength r. Horizontal dashed lines mark planted hierarchy levels (K=2K=2 macro, K=8K=8 meso, K=64K=64 micro). At high r, K∗K^* exhibits clear transitions at the planted levels; at low r (below the Kesten–Stigum threshold), transitions blur. Three key findings emerged. (1) K∗(σ)K^*(σ) as continuous LOD: The effective dimensionality K∗(σ)K^*(σ) transitions correspond precisely to planted hierarchy levels: K∗K^* crosses k=2k=2 at the macro boundary and k=8k=8 at the meso boundary. (2) Kesten–Stigum phase transition: ARI exhibits a sharp jump between r=20r=20 (macro ARI 0.040.04) and r=40r=40 (macro ARI 0.670.67), corresponding to the information-theoretic detection threshold where SNR=(din−dout)2/(2(din+dout))≥1SNR=(d_in-d_out)^2/(2(d_in+d_out))≥ 1. Below this threshold, no algorithm can recover the planted partition. (3) Baseline comparison: At r=200r=200, spectral clustering on heat kernel weights achieves ARI 0.910.91 (meso), versus Louvain 0.490.49, greedy modularity 0.580.58, and Leiden CPM 0.840.84 (best γ=0.005γ=0.005). Louvain and Leiden modularity both suffer from resolution limits (24–25 flat communities). Critically, Leiden CPM requires selecting a different γ per hierarchy level, whereas K∗(σ)K^*(σ) detects all levels simultaneously from a single spectral sweep. Validation. Multi-seed stability (n=10n=10, ARI macro 0.944±0.0220.944± 0.022), scale invariance (N∈512,1024,2048N∈\512,1024,2048\, meso K∗K^* boundary CV <5%<5\%; macro CV ≈18%≈ 18\%), and kkNN pipeline consistency (Δ <0.06<0.06 vs. direct graph) were confirmed across 8 independent checks (8/8 PASS). BoundaryScan peak positions align with spectral gap predictions within 3–4%. The full embedding pipeline (Sammon MDS into 10B^10) shows that Spearman ρ degrades with graph density (ρ=0.783ρ=0.783 at r=20r=20, ρ=0.726ρ=0.726 at r=200r=200), confirming that the direct spectral approach is more robust for dense graphs (Figure 1). Figure 2: Phase transition in boundary recovery. (a) ARI at macro (K∗=2K^*=2) and meso (K∗=8K^*=8) scales vs. hierarchy strength r; shaded region marks the Kesten–Stigum threshold. (b) Spectral gap λ3/λ2 _3/ _2 grows with r, confirming increasing signal separability. (c) Signal-to-noise ratio at macro and meso levels; the dashed line marks SNR=1\,=1. 6.2 Experiment 2: Hierarchical Consistency on WordNet Data. WordNet 3.0 noun hierarchy [7]: ∼ 82K synsets, DAG structure, max depth 13–20 hops, rooted at entity.n.01. Protocol. Embed the transitive closure in 10B^10 using Poincaré embeddings [22] (low dimension to maximize hyperbolic advantage). Run Algorithm 2 from 100 random leaf nodes. For each detected σ∗σ^*, extract the nearest ancestor at the corresponding depth. Metrics. (a) Correlation between detected boundary order and ancestor depth (Kendall’s τ). (b) Hit@L: fraction of boundaries within L hops of a true depth boundary. (c) Mean Reciprocal Rank for ancestor retrieval across scales. Baselines. Same as Experiment 1, plus Euclidean LOD (PCA-based coarsening) and flat-embedding retrieval (FAISS). Results. We embedded the full WordNet noun hierarchy (N=82,115N=82,115 synsets, 75,850 hypernym edges, max depth 19) into 10B^10 using Poincaré embeddings [22] (300 epochs, contrastive loss with negative sampling, RiemannianAdam). Embedding quality gates confirmed hierarchy preservation (HP=0.994\,=0.994) and sibling proximity (SP=0.931\,=0.931). Depth correlation (DC=0.113\,=0.113) is expected to be low for DAG structures where multiple hypernym paths yield ambiguous depth values. The kkNN graph (k=50k=50) was constructed using approximate nearest neighbors in the tangent space with exact Poincaré distance refinement, yielding ∼6 6M edges in under 60 seconds. Table 2 summarizes results from BoundaryScan applied at 100 stratified leaf nodes. Table 2: Experiment 2 metrics on full WordNet noun hierarchy (N=82,115N=82,115). Metric Value Interpretation Kendall τ 0.79 strong scale↔ ordering Recall@1 0.56 56% of true depths detected ± 1 level Recall@2 0.75 75% detected ± 2 levels Precision@0.5 0.79 79% of detections near a true depth MRR (focus) 0.076 moderate retrieval in large candidate pool Avg. peaks/node 3.9 ∼ 3–4 scale boundaries per leaf Figure 3: Detected boundary scale σ∗σ^* vs. true ancestor depth for 100 stratified leaf nodes in WordNet (N=82,115N=82,115). Strong positive correlation (Kendall τ=0.79τ=0.79) confirms that larger diffusion scales correspond to shallower (more abstract) ancestors. Three key findings emerged. (1) Scale–depth correspondence: Kendall’s τ=0.79τ=0.79 confirms that BoundaryScan correctly orders detected boundaries: larger diffusion scales σ∗σ^* consistently correspond to shallower (more abstract) ancestors. This validates Proposition 2 on real-world data with ∼82 82K nodes and DAG structure. (2) Boundary precision vs. recall trade-off: BoundaryScan detects 56% of true depth levels within ±1± 1 hop (Recall@1) and 75% within ±2± 2, averaging 3.9 peaks per focus node against hierarchies of depth 10–19. Precision@0.5=0.79\,=0.79 indicates that most detected boundaries correspond to genuine ancestor depths. BoundaryScan is conservative: it finds real boundaries but misses subtle ones at deeper levels. (3) Retrieval in large candidate pools: MRR=0.076\,=0.076 reflects the difficulty of ancestor retrieval when thousands of nodes share the same depth level. The Fréchet mean at σ∗σ^* acts as a community centroid rather than a pointer to a specific ancestor—a finding consistent with the diffusion interpretation where heat kernel weights average over neighborhoods. Depth-limited subtree. On a depth-limited subtree (d≤7d≤ 7, N=29,368N=29,368), BoundaryScan achieves τ=0.81τ=0.81 with higher recall (Recall@2=0.86\,=0.86) and perfect precision (Precision@0.5=1.0\,=1.0), suggesting that the method performs well on intermediate-scale subgraphs where hierarchy is shallower but denser. 7 Related Work Hyperbolic Embeddings. Poincaré embeddings [22] and hyperbolic neural networks [8] demonstrated the advantages of hyperbolic geometry for hierarchical data. Chami et al. [3] extended graph neural networks to dB^d. Our work builds on this foundation by introducing a diffusion-based multi-scale operator on hyperbolic embeddings, rather than using them as a static representation. AI Agent Memory. MemGPT [23], Mem0, MemoryOS, and graph-based systems like GraphRAG represent the current landscape of agent memory. Recent surveys [30] document a shift from flat-vector to graph-structured approaches, yet all rely on discrete community detection or fixed-tier hierarchies for multi-scale access. Leiden clustering, for instance, requires manual resolution sweeps over γ with no guarantee that the chosen granularity matches the task, and produces discrete partitions rather than a continuous zoom. None of these systems provide a resolution operator with formal coherence guarantees or automatic boundary detection. Multi-Scale Analysis on Graphs. Graph wavelets [11] and diffusion maps [4] define multi-scale analysis via spectral methods. Markov Stability [5] uses diffusion time as a resolution parameter for community detection. Heat Kernel Signatures [27] established heat diffusion as a provably informative multi-scale descriptor for 3D shapes. Our approach transfers the HKS idea from shape analysis to semantic spaces and combines it with hyperbolic geometry specifically for knowledge hierarchies—a combination not previously explored. Causal Emergence. Hoel et al. [12] showed that macro-scale descriptions can carry more effective information than micro-scale ones. Klein & Hoel [14] developed spectral methods for identifying emergence scales in networks and showed that causal emergence is most pronounced in high-degeneracy networks (biological, social) and less so in low-degeneracy ones (technological, hierarchical). The connection between SLoD’s spectral boundaries and causal emergence in evolved memory graphs is a promising direction for future work. LOD in Computer Graphics. Level of Detail techniques [19] are fundamental to real-time rendering. The analogy to semantic data was, to our knowledge, not previously explored in the machine learning literature. 8 Discussion and Conclusion We introduced Semantic Level of Detail, a framework for continuous multi-scale knowledge representation via heat kernel diffusion on the Poincaré ball. Two experiments establish SLoD as a validated framework: (1) synthetic hierarchies confirm that BoundaryScan recovers planted boundaries with ARI up to 1.00, bounded by the Kesten–Stigum information-theoretic limit; (2) the full WordNet noun hierarchy (82K synsets) demonstrates that boundary–depth correspondence holds on real-world DAGs (τ=0.79τ=0.79). A notable finding from Experiment 2 is that the Fréchet mean at detected boundary scales acts as a community centroid rather than a pointer to a specific ancestor. This is consistent with the diffusion interpretation: heat kernel weights average over neighborhoods, producing a representative embedding for the abstraction level rather than retrieving a named entity. This behavior is desirable for agent memory systems, where the appropriate level of abstraction matters more than a specific node identity. Limitations and Future Work. Five limitations remain. Tree assumption. Theorems 1–2 assume tree-structured hierarchies with Sarkar embedding distortion (1+ε)(1+ ). Real knowledge graphs are DAGs or denser. The spectral gap mechanism (Proposition 2) is purely graph-theoretic and applies to any graph, but the coherence bound degrades from O(1+ε)O(1+ ) to O(δ(G))O(δ(G)) for general graphs, where δ(G)δ(G) can be Ω(n) ( n). Experiment 2 confirms that BoundaryScan works on a sparse DAG (WordNet, 82K nodes, τ=0.79τ=0.79), but extension to dense knowledge graphs remains open. Approximation gaps. The pipeline introduces three approximations: graph Laplacian vs. continuous heat kernel, truncated eigendecomposition (k=50k=50 of N=1024N=1024), and Sammon MDS embedding (ρ=0.78ρ=0.78–0.730.73 depending on graph density). No end-to-end error bound exists for the composed pipeline. At fine scales (σ<2σ<2), eigenvalue truncation captures less than 5% of total energy, making boundary detection unreliable. Notably, the direct spectral pipeline avoids the embedding approximation entirely, which is why Table 1 reports results on the graph Laplacian. Baselines. Experiment 1 compares against Louvain, greedy modularity, and Leiden CPM (all outperformed at meso scale) and spectral k-sweep. A systematic comparison against multi-scale methods—Markov Stability [5] and nested SBM [24]—is needed to establish SLoD’s competitive position on shared benchmarks. Static graphs. The current formulation assumes a fixed graph. For production AI memory systems where knowledge evolves continuously, incremental eigendecomposition and online boundary tracking are needed. A promising direction is coupling SLoD with Hebbian co-activation learning, where usage patterns modify graph weights and periodically trigger spectral re-analysis—analogous to biological wake/sleep consolidation cycles. Causal emergence and evolved memory graphs. Our experiments use static hierarchies (HSBM, WordNet) with low degeneracy, where causal emergence in Hoel’s sense is less pronounced [14]. However, production memory graphs evolve through Hebbian co-activation, developing the high-degeneracy topology where causal emergence is maximal. We hypothesize that (H1) causal emergence appears above a critical degeneracy threshold as the graph matures, (H2) emergence peaks align with BoundaryScan boundaries, and (H3) this alignment strengthens with usage intensity. 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