This research paper establishes a rigorous framework connecting non-Archimedean ana- lytic geometry on Berkovich spaces with p-adic Galois representations and Iwasawa theory to address structural aspects of the Birch and Swinnerton-Dyer (BSD) Conjecture for elliptic curves over Q. By constructing a Push-forward Monge-Amp`ere measure on the canonical dual simplicial complex Σ of the strictly semi-stable compactification X , we prove a precise comparison identity relating the Berkovich Monge-Amp`ere density νber(T,x0) to the cyclo- tomic Bloch-Kato p-adic height regulator Rp(E). Furthermore, we resolve the ordinary versus supersingular reduction dichotomy using Perrin-Riou’s L-invariant and demonstrate mass conservation on the Berkovich skeleton. Conditioned on the p-adic height non-degeneracy hypothesis and the Iwasawa Main Conjecture for TpE, we establish the finite corank property of the p-adic Selmer group and the conditional finiteness of the p-primary Tate-Shafarevich group Sha(E)[p∞]. : Declaration of Generative AI and AI-Assisted Technologies (including Claude and Gemini) : AI Use Declaration: Generative AI tools were used during manuscript preparation exclusively for language polishing, LaTeX syntax formatting, and assisting in the structuring of Lean 4 tactic workflows. All mathematical proofs, derivations, and underlying theoretical concepts were formulated, checked, and validated independently by the author, who holds sole responsibility for the work.
Pakin Klaisuan· Zenodo (CERN European Organi...· 0 citations
Abstract. ThispaperpresentsamathematicallyrigorousformulationaddressingtheYang– Mills Existence and Mass Gap problem, one of the Clay Mathematics Institute’s Millennium Prize Problems. Moving beyond fixed-parameter cutoff approximations, we establish a con- tinuous dilation framework on Banach spaces where the scaling parameter λ ∈ [1, ∞) acts as a transient regularizer. By utilizing non-perturbative Wilson loop area laws and transfer matrix spectral bounds, we prove the existence of a non-trivial Mass Gap ∆ > 0 without assuming dimensional transmutation a priori. Furthermore, through uniform cluster expan- sion bounds, we verify the infinite volume limit (Λ → R4), demonstrating that the resulting Schwinger functions satisfy all four Osterwalder–Schrader (OS) axioms (OS1–OS4). : Declaration of Generative AI and AI-Assisted Technologies (including Claude and Gemini) : AI Use Declaration: Generative AI tools were used during manuscript preparation exclusively for language polishing, LaTeX syntax formatting, and assisting in the structuring of Lean 4 tactic workflows. All mathematical proofs, derivations, and underlying theoretical concepts were formulated, checked, and validated independently by the author, who holds sole responsibility for the work.
Pakin Klaisuan· Zenodo (CERN European Organi...· 0 citations
Abstract. ThispaperpresentsamathematicallyrigorousformulationaddressingtheYang– Mills Existence and Mass Gap problem, one of the Clay Mathematics Institute’s Millennium Prize Problems. Moving beyond fixed-parameter cutoff approximations, we establish a con- tinuous dilation framework on Banach spaces where the scaling parameter λ ∈ [1, ∞) acts as a transient regularizer. By utilizing non-perturbative Wilson loop area laws and transfer matrix spectral bounds, we prove the existence of a non-trivial Mass Gap ∆ > 0 without assuming dimensional transmutation a priori. Furthermore, through uniform cluster expan- sion bounds, we verify the infinite volume limit (Λ → R4), demonstrating that the resulting Schwinger functions satisfy all four Osterwalder–Schrader (OS) axioms (OS1–OS4). : Declaration of Generative AI and AI-Assisted Technologies (including Claude and Gemini) : AI Use Declaration: Generative AI tools were used during manuscript preparation exclusively for language polishing, LaTeX syntax formatting, and assisting in the structuring of Lean 4 tactic workflows. All mathematical proofs, derivations, and underlying theoretical concepts were formulated, checked, and validated independently by the author, who holds sole responsibility for the work.
Pakin Klaisuan· Zenodo (CERN European Organi...· 0 citations
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