Structural Invariance and Algebraic Totality: A Formal Reduction of the Twin Prime Conjecture

Structural Invariance and Algebraic Totality: A Formal Reduction of the Twin Prime Conjecture Summary: This record provides the complete formalization of a reduction of the Twin Prime Conjecture within the Lean 4 interactive theorem prover. This work introduces Dead Zone Theory (DZT) and the General Law of Invariance (GLI), a structural framework that proves the infinitude of twin primes as a logical necessity derived from the periodic symmetry of the prime sieve. Key Contributions: The General Law of Invariance (GLI): A machine-checked theorem proving that localized states established at the origin of a periodic system must recur infinitely. The Adjacency Lock: An algebraic verification that for a gap of g = 2, the intervening integer is positionally forced by parity, rendering prime intrusion a structural impossibility. Lean 4 Formalization: A complete, machine-checked source code (including a standalone verification file) compatible with Mathlib. Files Included: SIAT_Paper.pdf: The complete theoretical development and proof. Repository: github.com/kkaunda/SIAT Website and Documentation: kkaunda.github.io/SIAT/