PFUSRC-083 Mersenne Primes as the Number-Theoretic Projection of Topological Closure Tests: A Calibration Between Algebraic Intuition and Ontological Mathematics in Prime Theory

Mersenne primes take the form M₏=2^p-1 where the exponent p is prime, and their one-to-one correspondence with even perfect numbers is a landmark result established by Euclid and Euler. However, conventional number theory cannot resolve core ontological questions: why the algebraic expression 2^p-1 generates prime numbers only for specific prime values of p, and what underlying topological rules govern the existence of Mersenne primes. This paper develops arguments grounded in the topological ontology of the PFUSRC system and distinguishes two separate mathematical paradigms. Mersenne primes emerge from bottom-up algebraic linear mathematics, which captures numerical regularities via computation and observation on the number-theoretic projection layer. By contrast, the PFUSRC derivation of prime nodes relies on top-down ontological mathematics, which generates prime topological nodes through recursive nesting and rigid closure tests of cosmic fundamental structures. The central thesis is that both paradigms follow an identical topological operation sequence: Construct → Eliminate Redundancy → Test Closure. The algebraic formula 2^p-1 acts as a numerical transcription of this topological closure workflow. The two necessary conditions for valid Mersenne primes—prime exponent p and primality of 2^p-1—correspond exactly to the two-stage screening mechanism of topological closure inspection within PFUSRC. Even perfect numbers are complete structural contours automatically projected onto the number-theoretic layer once full topological closure is achieved. This work concludes that Mersenne primes do not formally prove the PFUSRC framework, but serve as independent observational markers that replicate topological closure operations within number theory. Algebraic intuition delivers empirical numerical signals, while ontological mathematics provides fundamental structural explanations, forming a mutual calibration benchmark for the study of prime numbers