The Complete Dual Geometric Methodology (H_7^H): Ultra-High Precision Symbolic Proofs and Grand Unification of Major Mathematical Conundrums

The monistic mathematical methodology has reached its interpretive limits when confronting intractable conundrums, not due to a lack of effort, but because the intrinsic structure of these problems is inherently dual. This manuscript presents the "Complete Dual Geometric Methodology," a novel methodological framework consisting of two inseparable, integrated aspects: the first is the "Hexagram Heptanuclear Structure with Seven Nuclei" (H_7^H) as a formal constant with dynamic content, and the second is the "Dual Methodology" acting as a dynamic mathematical and logical operating system governing this structure. This methodology posits that every major conundrum consists of two facets: the "Functional Equation" (F), representing dynamics and tension, and the "Technical Condition" (C), representing structure and attraction. The inevitability of the commutative constraint (F × C = 1) is mathematically and topologically derived from the properties of the Laplacian matrix, eigenvalues, and the D_6 symmetry group of the H_7^H structure itself, thereby preventing logical or computational blow-up and decisively breaking circular reasoning. To avoid approximation errors, all structural dissections were constructed using pure symbolic algebra (SymPy) and arbitrary-precision arithmetic (mpmath) with a precision of up to 800 decimal places. This methodology has yielded: 1. A structural dissection consistent with Li's criterion and a complete algebraic imaginary cancellation for the Riemann Hypothesis. 2. A symbolic structural dissection demonstrating the algebraic dominance of energy dissipation over non-linearity in Navier-Stokes. 3. A structurally consistent dissection of the mass gap (Δ = 3 > 0) in a simplified Yang-Mills model. 4. Unification of the structural dissections of the seven problems (including P vs NP, Poincaré, Hodge, and BSD) under the Unified Transport Equation (Ω_Unity). 5. Presentation of a novel structural definition of prime numbers based on the decimal cycle, the Law of Nine, and resonance with the golden ratio (φ). The framework is accompanied by a complete "Stress-Testing Protocol" in the technical appendix, supported by numerical calculations with a precision of up to 800 decimal places, and nine high-resolution visualization protocols (600 DPI) demonstrating the hexagram heptanuclear structure across all seven major mathematical conundrums. Keywords: Dual Methodology, Hexagram Heptanuclear Structure, Symbolic Algebra, Li's Criterion, Mass Gap, Grand Unification, Anti-Circularity, Prime Numbers, Riemann Hypothesis, Navier-Stokes, Yang-Mills, P vs NP, Poincaré Conjecture, Hodge Conjecture, Birch and Swinnerton-Dyer Conjecture.