Hu_2026_Prime_3_Cycle_Twin_Primes_V2

This paper establishes a line of argument independent of traditional analytic number theory, starting from the structural foundation of prime generation, and does not rely on density estimates or asymptotic analysis from analytic number theory. The core argument adopts proof by contradiction and is divided into five layers of logic: First, prime generation is based on the periodic cycle of prime 3, and gaps are infinitely regenerated with the cycle; Second, the Prime Fingerprint Cipher determines that each individual prime uniformly traverses the four candidate tail digits, and the crowding rate decreases as primes grow larger; Third, the cake model is used to establish a criterion for finite companion infinity, and this paper adopts the strictest finite companion assumption as the premise for the argument; Fourth, assuming twin primes are permanently closed, it necessarily follows that a macroscopic reverse-growth intrinsic property emerges, and regardless of whether the crowding distribution is uniform, this property will ultimately propagate to the single-prime level, contradicting the fundamental conclusion of the infinitude of primes; Fifth, the argument structure is isomorphic to Euclid's proof of the infinitude of primes, employing pure proof by contradiction rather than positive enumeration. **Keywords:** twin primes; periodic cycle of prime 3; fingerprint cipher; reverse growth; companion infinity; proof by contradiction