latent-latency — Инструмент для топологической химии

--- name: latent-latency description: Latent-Latency Skill version: 1.0.0 --- # Latent-Latency Skill **Trit**: 0 (ERGODIC - mediates space ↔ time) **Bundle**: core **Status**: ✅ New --- ## The Fundamental Duality ``` LATENT (Space) ↔ LATENCY (Time) ↓ ↓ Compression Speed ↓ ↓ Representation Response ↓ ↓ dim(z) τ_mix ``` **Core Theorem**: Good latent representations minimize latency. ``` t_response ∝ 1 / compression_ratio(z) ``` ## Spectral Gap Bridge The **spectral gap** (λ₁ - λ₂) connects both domains: | Domain | Spectral Gap Role | |--------|-------------------| | **Latent** | Separation of clusters in representation space | | **Latency** | Mixing time τ_mix = O(log n / gap) | From Ramanujan graphs (optimal expanders): ``` gap ≥ d - 2√(d-1) [Alon-Boppana bound] τ_mix = O(log n) [Logarithmic mixing] ``` ## Mathematical Foundation ### Latent Space Dynamics ```python # Encoder: Observable → Latent z = encode(x) # dim(z) << dim(x) # Decoder: Latent → Reconstructed x̂ = decode(z) # Bidirectional loss L = ||x - x̂||² + β·KL(q(z|x) || p(z)) ``` ### Latency Dynamics ```python # Fokker-Planck: Distribution evolution ∂p/∂t = ∇·(∇L(θ)·p) + T∆p # Mixing time from Hessian τ_mix ≈ 1 / λ_min(H) # Gibbs equilibrium p∞(θ) ∝ exp(-L(θ)/T) ``` ### The Bridge Equation ``` τ_latency = f(dim_latent, spectral_gap, temperature) Specifically: τ_response = (dim(z) / gap) × log(1/ε) Where: - dim(z) = latent dimension - gap = spectral gap of computation graph - ε = target accuracy ``` ## MCP Energy-Latency Tradeoff From [MCP_OPTIMAL_TRANSITIONS.md](./mcp-tripartite/MCP_OPTIMAL_TRANSITIONS.md): | MCP Server | Latency | Latent Cost | Energy | |------------|---------|-------------|--------| | `gay` | ~10ms | 0.1KB context | LOW | | `tree-sitter` | ~50ms | 1KB context | LOW | | `exa` | ~1s | 3KB context | HIGH | | `firecrawl` | ~2s | 10KB context | HIGH | **Optimal triad**: `gay → tree-sitter → marginalia` (560ms, 5 energy) ## Worlding Skill Integration From [worlding_skill_omniglot_entropy.py](../ies/worlding_skill_omniglot_entropy.py): ```python class BidirectionalCharacterLearner: def __init__(self, char_dim: int = 28, latent_dim: int = 64): self.char_dim = char_dim self.latent_dim = latent_dim # Compression ratio: 784 → 64 def encode_character(self, image: np.ndarray) -> np.ndarray: """READ: Image → Latent Code (learn what the character means)""" # Latency: O(dim_latent) pass def generate_character(self, latent_code: np.ndarray) -> np.ndarray: """WRITE: Latent Code → Image (learn how to express the character)""" # Latency: O(dim_output) pass ``` **Compression**: 784 → 64 = 12.25× compression **Expected Latency Reduction**: ~12× for downstream tasks ## Fokker-Planck Convergence Training latency depends on reaching Gibbs equilibrium: ``` Stopped Early: t < τ_mix → Poor latent representation Fully Converged: t > τ_mix → Optimal latent representation ↓ Minimal inference latency ``` From [fokker-planck-analyzer](./fokker-planck-analyzer/SKILL.md): ```python def check_convergence(trajectory, temperature): # Mixing time from loss landscape geometry τ_mix = 1 / λ_min(Hessian(loss)) # Check if training exceeded mixing time if training_steps > τ_mix: return "CONVERGED: Good latent representation" else: return f"EARLY STOP: Need {τ_mix - training_steps} more steps" ``` ## GF(3) Decomposition | Skill | Trit | Role | |-------|------|------| | `fokker-planck-analyzer` | -1 | Verifies convergence (latency) | | `latent-latency` | 0 | Mediates space ↔ time | | `compression-progress` | +1 | Generates compressed representations | **Conservation**: (-1) + (0) + (+1) = 0 ✓ ## Practical Applications ### 1. Optimize Inference Latency ```python def optimize_latent_for_latency(model, target_latency_ms): """ Find optimal latent dimension for target latency. Relationship: latency ∝ dim(z) / spectral_gap """ current_dim = model.latent_dim current_latency = measure_latency(model) # Target dimension target_dim = int(current_dim * (target_latency_ms / current_latency)) # Retrain with smaller latent space return retrain_model(model, latent_dim=target_dim) ``` ### 2. Predict Mixing Time ```python def predict_mixing_time_from_latent(latent_structure): """ Estimate training latency from latent space properties. """ # Spectral gap of latent similarity graph gap = spectral_gap(latent_similarity_matrix(latent_structure)) # Mixing time bound n = latent_structure.n_samples τ_mix = np.log(n) / gap return τ_mix ``` ### 3. Ramanujan-Optimal Routing ```python def route_with_ramanujan(nodes, message): """ Route through network with optimal latency. Ramanujan graphs achieve t_mix = O(log n). """ # Build routing graph with Ramanujan property G = build_lps_graph(nodes, degree=7) # (7+1)-regular assert spectral_gap(G) >= 7 - 2*np.sqrt(6), "Not Ramanujan!" # Route via non-backtracking walk path = non_backtracking_path(G, source, target) # Expected latency: O(log n) hops return path ``` ## Detection Latency SLA From security applications: ``` Detection latency = O(log N) / gap For Ramanujan (gap = 1/4): N = 1000 nodes → detection in ~37ms N = 1M nodes → detection in ~74ms ``` ## Commands ```bash # Analyze latent-latency tradeoff just latent-latency-analyze model.pt # Optimize for target latency just latent-optimize --target-ms=100 # Measure spectral gap of latent space just latent-spectral-gap embeddings.npy # Predict mixing time just predict-mixing-time --hessian=H.npy # Route with Ramanujan optimality just ramanujan-route --nodes=1000 ``` ## DuckDB Schema ```sql CREATE TABLE latent_latency_metrics ( model_id VARCHAR PRIMARY KEY, latent_dim INT, spectral_gap FLOAT, mixing_time_estimate FLOAT, inference_latency_ms FLOAT, compression_ratio FLOAT, is_converged BOOLEAN, created_at TIMESTAMP DEFAULT NOW() ); -- Query: find optimal models SELECT model_id, latent_dim, inference_latency_ms FROM latent_latency_metrics WHERE is_converged = true ORDER BY inference_latency_ms ASC LIMIT 10; ``` ## Triads ``` fokker-planck-analyzer (-1) ⊗ latent-latency (0) ⊗ compression-progress (+1) = 0 ✓ ramanujan-expander (-1) ⊗ latent-latency (0) ⊗ agent-o-rama (+1) = 0 ✓ spi-parallel-verify (-1) ⊗ latent-latency (0) ⊗ gay-mcp (+1) = 0 ✓ ``` ## References - Fokker-Planck equation for neural network training - Ramanujan graphs and optimal expanders (Lubotzky-Phillips-Sarnak) - Variational autoencoders and latent space geometry - MCP optimal transitions (plurigrid/asi) ## See Also - `fokker-planck-analyzer` - Convergence verification - `langevin-dynamics` - SDE-based learning - `ramanujan-expander` - Spectral gap optimization - `compression-progress` - Intrinsic motivation - `mcp-tripartite` - Energy-latency tradeoffs --- **Skill Name**: latent-latency **Type**: Theoretical Bridge **Trit**: 0 (ERGODIC - space ↔ time mediation) **Core Equation**: τ_response = dim(z) / gap × log(1/ε) **Status**: ✅ Available ## Scientific Skill Interleaving This skill connects to the K-Dense-AI/claude-scientific-skills ecosystem: ### Graph Theory - **networkx** [○] via bicomodule - Universal graph hub ### Bibliography References - `general`: 734 citations in bib.duckdb ## SDF Interleaving This skill connects to **Software Design for Flexibility** (Hanson & Sussman, 2021): ### Primary Chapter: 10. Adventure Game Example **Concepts**: autonomous agent, game, synthesis ### GF(3) Balanced Triad ``` latent-latency (+) + SDF.Ch10 (+) + [balancer] (+) = 0 ``` **Skill Trit**: 1 (PLUS - generation) ### Secondary Chapters - Ch6: Layering - Ch7: Propagators ### Connection Pattern Adventure games synthesize techniques. This skill integrates multiple patterns. ## Cat# Integration This skill maps to **Cat# = Comod(P)** as a bicomodule in the equipment structure: ``` Trit: 0 (ERGODIC) Home: Prof Poly Op: ⊗ Kan Role: Adj Color: #26D826 ``` ### GF(3) Naturality The skill participates in triads satisfying: ``` (-1) + (0) + (+1) ≡ 0 (mod 3) ``` This ensures compositional coherence in the Cat# equipment structure.