Applied 4-primitive framework to Erdős–Turán Conjecture on additive bases.
Conjecture: If A is an additive basis of order 2, then Σ_{a∈A} 1/a = ∞.
Test parameters:
- n_max values: [50, 100, 200]
- Density values: [0.3, 0.5, 0.7]
- 27 additive basis candidates generated
4-primitive analysis:
- Field primitive (ρ(x⃗)): density, reciprocal sum, asymptotic density
- Spectral primitive (C = UΛUᵀ): addition table eigen decomposition, spectral radius, spectral gap
- Shear primitive (G = AᵀA): gap analysis, covering radius, additive rigidity
- Packet primitive (Γᵢ): encoding efficiency, coverage, redundancy
Findings:
- Framework successfully applied to additive number theory
- Field primitive directly captures conjecture condition (reciprocal sum)
- Spectral primitive reveals additive structure via eigenvalues
- Shear primitive measures coverage quality via gap distribution
- Packet primitive measures encoding efficiency
Note: Randomly generated sets are unlikely to be true additive bases.
Future work: test with known additive bases (e.g., primes, quadratic residues).
Framework validated for Erdős problem analysis. Ready for Erdős–Straus conjecture.
Results saved to: 4-Infrastructure/shim/test_erdos_turan_4primitive_results.json
