This paper, grounded in the Differential Algebraic Finite Representation Theory originated by Shifa Liu 21–24, systematically extends this theoretical framework to seven core mathematical conjectures: the Riemann Hypothesis, the Birch and Swinnerton-Dyer Conjecture, the Hodge Conjecture, the Poincare Conjecture/Geometrization Conjecture, ´the Navier-Stokes existence and smoothness problem, the Yang-Mills mass gap problem, and the Beal Conjecture. We demonstrate that all these conjectures can be reformulated within the unified differential algebraic framework (𝐶0, 𝑂2) and its associated differential algebraic closure Q¯2. For each conjecture, we provide a constructive finite representation with complete 20+ step proofs, meaning that the central objects of the conjecture are shown to be definable within Q¯2 through a finite sequence of algebraic and differential operations starting from algebraic numbers. This allows us to establish a bi-spectral theorem adapted to each conjecture and to articulate a functorial form of the Unified Rank Correspondence, revealing deep structural analogies. Furthermore, we design constructive verification algorithms for each conjecture, demonstrating that, in principle, the truth of these conjectures can be reduced to verifying properties of finitely presented differential algebraic objects. This work unveils a profound unity among these fundamental problems and offers a novel, constructive methodology for their potential resolution.
shifa liu (Wed,) studied this question.
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