Olivier MEHAYE

The TCAP Transformation (Ξ) VII – Ξ⁺ and Twin Primes: Diamond Confluence, Fragmentation, and Direct Preimage Structure

This work studies the arithmetic dynamics induced by the TCAP variant Ξ⁺ and its unexpected connection with twin primes. For an integer n = p₁p₂⋯pₖ with ordered prime factors, Ξ⁺ is defined by: Ξ⁺(n) = ∑ᵢ₌₁ᵏ pᵢ + ∑ᵢ₌₁ᵏ⁻¹ pᵢpᵢ₊₁. A central identity is: Ξ⁺(p²) = p(p + 2), which pro...

The TCAP Transformation (Ξ) VII – Ξ⁺ and Twin Primes: Diamond Confluence, Fragmentation, and Direct Preimage Structure

This work studies the arithmetic dynamics induced by the TCAP variant Ξ⁺ and its unexpected connection with twin primes. For an integer n = p₁p₂⋯pₖ with ordered prime factors, Ξ⁺ is defined by: Ξ⁺(n) = ∑ᵢ₌₁ᵏ pᵢ + ∑ᵢ₌₁ᵏ⁻¹ pᵢpᵢ₊₁. A central identity is: Ξ⁺(p²) = p(p + 2), which pro...

TCAP Transformation Ξ V: Controlled Atomization and Deterministic Prime Certification

TCAP Transformation Ξ V: Controlled Atomization and Deterministic Prime Certification This fifth paper develops a deterministic prime-selection and certification dynamics from the TCAP transformations Ξ and Ξ⁺. The starting difficulty is structural: Ξ and Ξ⁺ are defined from orde...