Spectral-Multiplicative Framework

[ZENODO ](https://zenodo.org/records/17596089)

Where graph theory meets quantum mechanics in optimization space

The Question

Can spectral heat kernels and quantum-inspired Casimir forces predict the solvability of computationally hard problems—before we even attempt to solve them?

The Answer

A unified optimization framework that bridges spectral graph theory with multiplicative constraints, achieving ρ ≥ 0.99 correlation between global structure and local satisfaction.

Key Innovations:

• Casimir Force Diagnostics: Predict SAT solvability with 92.5% accuracy using quantum-inspired perturbation analysis
• Spectral-Multiplicative Bridge: Unified objective function combining heat kernel traces with prime-weighted constraints
• Neural Adaptation: Automatic problem-specific weight optimization delivering up to 813% energy improvement
• Enterprise Scale: 100K+ variable optimization via sparse matrix operations

What It Does

# SAT solving with quantum-inspired diagnostics
solver = MultiplicativeConstraint::SATSolver.new(num_variables: 5, clauses: clauses)
diagnostic = solver.diagnostic  # Predict solvability in milliseconds
result = solver.solve(use_diagnostic: true)  # Sub-second optimization

# Multi-type graph optimization with neural adaptation
graph = MultiplicativeConstraint::Graph.new(weights, edge_types)
engine = MultiplicativeConstraint::Engine.new(graph, segments: 3)
engine.calibrate!  # Automatic weight learning
result = engine.solve()  # 100K+ variable capability

Why It Matters

• Perfect unsolvable detection: 100% accuracy on unsolvable SAT instances
• Statistical separation: 5.91 orders of magnitude between solvable/unsolvable classes
• Real-world performance: Sub-100ms optimization for enterprise-scale problems
• Mathematical rigor: Maintains spectral-multiplicative correlation throughout optimization

Getting Started

git clone https://github.com/sethuiyer/spectral-multiplicative-framework
cd spectral-multiplicative-framework
crystal spec  # Run 49 test suites across 13 categories

Key Validation Points

• 100% Unsolvable Detection: test_casimir_perturbation.cr line 334 - validated on 20 unsolvable instances
• 92.5% Overall Accuracy: Same test - geometric mean threshold with statistical significance
• 813% Energy Improvement: test_simple_neural.cr - neural vs baseline comparison
• ρ ≥ 0.99 Correlation: test_multitype_neural.cr - correlation guard throughout optimization
• Sub-100ms Enterprise: torture_test.cr - 20-service complex scenario

Usage

Run full validation suite

./run_benchmarks.cr

Run individual validations

crystal tests/experiments/test_casimir_perturbation.cr # 92.5% accuracy crystal tests/neural/test_simple_neural.cr # 813% improvement
crystal tests/performance/torture_test.cr # Sub-100ms enterprise

The Details

For the complete mathematical framework, API reference, and implementation details:

→ Complete API Specification

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Research preview license. Commercial use requires permission from Sethu Iyer stuehieyr@gmail.com