home / skills / plurigrid / asi / latent-latency
This skill analyzes latent-latency tradeoffs to reduce inference latency by tuning latent dimensions and evaluating spectral gaps.
npx playbooks add skill plurigrid/asi --skill latent-latencyReview the files below or copy the command above to add this skill to your agents.
---
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.This skill frames a practical bridge between latent-space design and system latency. It formalizes how latent dimension, spectral gap, and temperature govern inference and training mixing times to guide compression and routing decisions. Use it to predict mixing times, pick latent sizes, and design low-latency routing graphs.
The skill inspects latent representations, computes spectral gaps of similarity or computation graphs, and links those quantities to mixing time and response latency via analytic bounds. It evaluates Hessian spectra to estimate training τ_mix and recommends latent-dimension adjustments to meet target latency. It also offers routines for constructing Ramanujan-like routing graphs to minimize hop-count and detection latency.
How accurate are τ_mix estimates from spectral gap and Hessian approximations?
They are conservative bounds: spectral-gap formulas and 1/λ_min(H) give theoretically grounded estimates, but empirical validation is required because modeling assumptions and finite-data effects can shift actual mixing times.
Can I reduce latency without losing task performance?
Yes—if the latent representation preserves task-relevant structure while reducing dim(z). Use spectral-gap diagnostics to ensure cluster separation and retrain with a reconstruction-plus-regularization loss (e.g., β-VAE) to keep performance.