Computational Validation of Twin Primes at the 10²⁰ Scale

This preprint documents computational tests performed in Magma V2.29-8 for the identification and validation of twin-prime pairs at the 10²⁰ scale. The study examines arithmetic scans of odd candidates around 10²⁰, using IsPrime(p) and IsPrime(p+2) as the computational primality criteria and requiring p+2-p=2. Two test windows are documented: a central window containing 1,000 odd candidates and an expanded window containing 10,000 odd candidates. The recorded executions identified two twin-prime pairs in the 1,000-candidate window and ten twin-prime pairs in the 10,000-candidate test, with corresponding observed rates of 0.2% and 0.1%, respectively. The document provides the Magma procedures used to construct and validate the candidates, lists the recorded twin-prime pairs, documents a candidate that was rejected by the primality tests, and records a separate blind-generation experiment in which the Shuffle operation produced a Magma argument-type error. That failed Shuffle execution is explicitly excluded from the quantitative results. The document also distinguishes the arithmetic scans from an experimental implementation of the proposed Phi-harmonic formula. The reported 10,000-candidate test is characterized strictly as an arithmetic scan of the defined 10²⁰ region and is not presented as an experimental validation of the Phi-harmonic generation formula. The preprint is intended as a technical record of the computational procedures, recorded results, validation criteria, and execution conditions used in the Magma tests. Keywords: twin primes; prime numbers; primality testing; computational number theory; large primes; twin prime conjecture; Magma.