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:

1. Spectral (gap): "How optimal?" - measures expansion property
2. Combinatorial (Möbius): "Where tangled?" - identifies problem paths
3. 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