Inferring the Goldbach Conjecture under the Truth as Recursive Meta-Nesting Function Paradigm and an Isomorphic Analysis with Perelman's Proof

Abstract

In this paper we apply the ``Truth as Recursive Meta‑Nesting Function'' paradigm to the Goldbach Conjecture. This paradigm, established in \cite{zhu2026truth}, starts from the two irreducible root consensuses of causality and self‑consistency, constructs a truth space \(\Omega\) in a categorical framework, and proves that every cognitive state is mapped to an infinitely nested recursive element by a truth function satisfying a recursive equation. We perform an isomorphic analysis between the core ingredients of the Goldbach Conjecture — the distribution of primes, sieve methods, Chen's theorem, recursive search algorithms — and the key concepts of Perelman's proof of the Poincaré Conjecture (Ricci flow, entropy functional, singularity analysis, surgery, finite‑time extinction). Notably, Goldbach's own recursive construction of primes using Fermat numbers provides a historical precursor to the recursive nesting idea. We then give a detailed mathematical construction: the set of even numbers is modelled as an object in a cognitive category, its truth function is constructed, and we prove that every even number greater than 2 can necessarily be expressed as a sum of two primes. Thus the Goldbach Conjecture is established as a necessary consequence of the recursive meta‑nesting structure. This work not only provides a new philosophical foundation for the Goldbach Conjecture but also demonstrates the universality of the recursive meta‑nesting function paradigm across different mathematical fields.

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