Quantum-Consistent Adelic Integration and Structure of Egyptian Fractions

Through diligent application of adelic integration and quantum arithmetic, we demonstrate the Egyptian fraction system’s remarkable anticipation of both number-theoretic profundities and quantum measurement theory. Our findings reveal a quantum-arithmetic framework revealing profound structural patterns in Egyptian fractions. We demonstrate these ancient decompositions exhibit:

• Adelic balance through multiplicative normalization

• Prime entanglement with optimized logarithmic spread (σlogs ≈ 0.17)

• Dyadic quantization in Eye of Horus fractions

• Computational validation via modern thresholding (< 10−12)

Statistical analysis shows Egyptian σlogs values significantly below Erdős-Kac predictions (p < 0.001), evidencing non-random optimization. Our adelic unity framework connects these features through number-theoretic quantum analogs, revealing an ancient system anticipating modern mathematical principles.