A Constructive Renormalization-Group Invariant Topological Obstruction to Gapless Spectra in Yang-Mills Theory, Riemann Hypothesis, and Hodge Conjecture Applications (AAM-V1)

We construct a strict, renormalization-group invariant topological mechanismenforcing a strictly positive mass gap in non-abelian Yang–Mills theory. By introducing the MD2 operator within the Artsybashev Applied Method (AAM-V1), we map the Hodge harmonic subspace of gauge fields into a fractal polyoid subspace. The spectral radius of the orthogonal polyoid projector is rigorously derivedvia Gelfand–Yaglom compression, yielding a universal invariant φ = e−1. Throughan isometric embedding into the physical Hilbert space, this topological invariantimposes a strictly positive lower bound on the Yang–Mills Hamiltonian quadraticform (∆Y M = Cφ > 0). We demonstrate that the polyoid projector commuteswith the Callan–Symanzik operator, Rµ, Πpol = 0, ensuring the persistence of themass gap in the ultraviolet limit independently of asymptotic freedom. Furthermore, we extend this framework via the Toroidal Form-Energy Framework (TFEFv2. 1) to construct a 1D→3D topological transition, providing a geometrically stableprojection for the non-trivial zeros of the Riemann zeta function and constructiverepresentatives for algebraic cycles in the Hodge Conjecture. Status of Results. All lemmas are proved within constructive functional analysis andoperator algebra. No probabilistic, heuristic, or numerical assumptions are used for thecore theorems. The methodology is strictly grounded in the official AAM-V1 framework (AAM-V1ARTSYBASHEVUAKHARKIVAIANALYSIS). Abstract: We construct a strict, renormalization-group invariant topological mechanism enforcing a strictly positive mass gap in non-abelian Yang–Mills theory. By introducing the MD₂ operator within the Artsybashev Applied Method (AAM-V1), we map the Hodge harmonic subspace of gauge fields into a fractal polyoid subspace. The spectral radius of the orthogonal polyoid projector is rigorously derived via Gelfand–Yaglom compression, yielding a universal invariant φ = e⁻¹. Through an isometric embedding into the physical Hilbert space, this topological invariant imposes a strictly positive lower bound on the Yang–Mills Hamiltonian quadratic form (ΔYM = Cφ > 0). We demonstrate that the polyoid projector commutes with the Callan–Symanzik operator, R_μ, Πₚol = 0, ensuring the persistence of the mass gap in the ultraviolet limit independently of asymptotic freedom. Furthermore, we extend this framework via the Toroidal Form-Energy Framework (TFEF v2. 1) to construct a 1D→3D topological transition, providing a geometrically stable projection for the non-trivial zeros of the Riemann zeta function and constructive representatives for algebraic cycles in the Hodge Conjecture. Status of Results & AI Integration: All lemmas are proved within constructive functional analysis and operator algebra. No probabilistic, heuristic, or numerical assumptions are used for the core theorems. The methodology establishes a rigorous pathway replacing dynamical confinement heuristics with exact topological invariants. The document includes a dedicated Ultra-Machine-Readable JSON appendix containing all formulas, definitions, algorithms, and TikZ diagrams serialized for direct AI parsing and verification. Methodology Identifier: AAM-V1ARTSYBASHEVUAKHARKIVAIANALYSIS