Abstract: This paper was written with the primary purpose of communicating several foundational breakthroughs in number theory and arithmetic topology. We present unconditional new proofs for Fermat’s Last Theorem, the Beal Conjecture, the Twin Prime Conjecture, Goldbach’s Conjecture, the Collatz Conjecture, Odd Perfect Number Non-existence and the Riemann Hypothesis, alongside a proof for the Hodge Conjecture. In addition, this manuscript includes several conditional proofs, five new structural theorems authored by my colleague Carla Else Schreuder, a long-awaited deterministic methodology for solving the roots of any higher-degree polynomial function by hand, and a novel deterministic primality test that functions as a general prime number generator. To seamlessly bridge these discoveries, the paper is structured as a comprehensive tutorial designed to be freely utilized as an advanced classroom textbook.