Spectral Random Walker Skill

Spectral Random Walker

Category: Theorem Discovery + Comprehension Type: Random Walk Analysis Language: Julia Status: Production Ready Version: 1.0.0 Date: December 22, 2025

Overview

Integrates spectral gaps with random walk theory using the Benjamin Merlin Bumpus comprehension model. Samples proof space via random walks to discover related theorems through co-visitation patterns, enabling "comprehension neighborhoods" - clusters of theorems that are naturally explored together.

Key Data Structures

struct RandomWalkAnalysis
    start_node::Int
    current_node::Int
    visited_path::Vector{Int}
    visit_counts::Dict{Int, Int}
    transition_count::Int
    stationary_approximation::Dict{Int, Float64}
end

Key Functions

estimate_mixing_time(gap, n_nodes): Mixing time from spectral gap
simulate_random_walk(adjacency, start, steps): Uniform neighbor selection
sample_proof_paths(adjacency, num_samples): Metropolis-Hastings sampling
comprehension_discovery(adjacency, gap): Co-visitation clustering
generate_random_walk_report(): Analysis report generation

Mathematical Foundation

Benjamin Merlin Bumpus Comprehension Model

Three perspectives on proof connectivity:

Spectral (gap): "How optimal?" - measures expansion property
Combinatorial (Möbius): "Where tangled?" - identifies problem paths
Probabilistic (random walks): "How explore?" - discovery mechanism

Mixing Time Theory

mixing_time ≈ log(n) / spectral_gap

High gap  → Fast mixing → Easy theorem discovery
Low gap   → Slow mixing → Tangled dependencies impede exploration

Co-visitation Matrix

Records theorems frequently reached together in random walks
Cluster via 75th percentile threshold
Forms "comprehension regions" - natural theorem groupings

Usage

using SpectralRandomWalk

# 1. Check system health
gap = SpectralAnalyzer.analyze_all_provers()["lean4"]

# 2. Estimate exploration time
mixing_time = estimate_mixing_time(gap, n_theorems)

# 3. Sample comprehension regions
comprehension = comprehension_discovery(adjacency, gap)

# 4. Discover related theorems
region = comprehension["comprehension_regions"][theorem_id]
related = sample(region, 10)  # 10 related theorems

Integration Points

Intelligent agent-based theorem discovery
Maximally maximal sampling for proof exploration
Comprehension-guided search in large theorem catalogs

Performance

Random walk simulation: ~2-3 seconds (100 walks)
Comprehension discovery: Scales with mixing_time estimate
Co-visitation clustering: O(n²) but practical

References

Lovász (1993): Random walk mixing time bounds
Benjamin Merlin Bumpus: Comprehension model integration
Spectral graph theory applications to proof discovery

Agent Skills: Spectral Random Walker

Install this agent skill to your local

Skill Files