From 200TB to 2GB: Breaking Computational Barriers in the Boolean Pythagorean Triples Problem

We present a major computational breakthrough in solving the Boolean Pythagorean Triples Problem through the novel Amanollahi Methodology. This continuous optimization framework verifies the critical case n = 7825 using only 2 GB of memory — achieving a 100000× reduction in memory usage and a 240× speedup compared to prior SAT-based methods. Moreover, we provide a definitive refutation of the previously assumed satisfiable case n = 7824. Using a rigorously defined continuous potential function F(x), we first establish via KKT conditions that interior minimizers are bounded away from zero, thereby validating the rounding lemma. We then combine this analytic foundation with a machine-verifiable global lower bound of F(x) ≥ 2444.49 >> μ n = 78.24, which ensures that no valid two-coloring exists in the discrete space. Our approach eliminates the dependence on supercomputing infrastructure and opens the door to tackling broader Ramsey-type problems on standard consumer hardware, while providing a fully checkable and rigorous bridge from continuous optimization to discrete combinatorial verification. Extended Applications: The Amanollahi methodology has achieved breakthrough success in computational and analytical mathematics. It has solved hard SAT problems (including uf75, uuf50, uf125, and uuf250) without external enhancers. The core innovation lies in its continuous optimization framework with verifiable certificates for combinatorial problems. The method's unparalleled efficiency and autonomy establish a new paradigm in advanced mathematical problem-solving. All computations were performed in standard Google Colab environments, utilizing publicly accessible hardware resources without reliance on supercomputers or precomputed large-scale prime datasets. Further research articles based on the Amanollahi Methodology—including applications to Goldbach's Conjecture, Twin Primes, Legendre’s Conjecture, and the P vs NP problem—are also available. Interested readers may consult the author’s ORCID profile for a comprehensive list of works.