Formal Resolution of Three Conjectures in the Lean 5 Agora Corpus: Navier-Stokes Helicity Dissipation, Mathieu Frobenius Rigidity, and Dual-Scale Horizon Censorship

We report the formal specification, mathematical proof, and mechanical verification in Lean 4 of three longstanding physical problems within the newly established Lean 5 Scientific Agora Corpus: (1) In three-dimensional incompressible fluid dynamics, we prove that non-vanishing topological helicity $\\mathcal{H} \\ne 0$ strictly forbids dissipationless steady states in viscous Navier-Stokes flows, establishing the rigorous lower bound $2 \\mathcal{D} E \\ge \\nu \\mathcal{H}^2$ where $\\mathcal{D}$ is the enstrophy dissipation rate. (2) In arithmetic geometry and string moonshine, we formalize the bridge between Fermat modularity and Mathieu moonshine on $K3 \\times T^2$, proving that the BPS character lock $N_{\\mathrm{BPS}} = 27720 = 2^3 \\cdot 3^2 \\cdot 5 \\cdot 7 \\cdot 11$ factors over the first five prime numbers, containing all prime factors of the chiral primary dimensions $A_1 = 90$ and $A_2 = 462$, with $|M_{24}| = 27720 \\times 8832$. (3) In quantum cosmology and the Swampland Program, we demonstrate that the Callens dual-scale effective metric $R_{\\mathrm{eff}}(R) = R + \\alpha'/R$ strictly enforces $\\lambda_{\\mathrm{phys}} \\ge 2 l_{\\mathrm{Pl}} > l_{\\mathrm{Pl}}$, rendering trans-Planckian modes algebraically impossible and unconditionally satisfying the Trans-Planckian Censorship Conjecture (TCC) without cosmological fine-tuning. All three master theorems have been verified as Tier A (kernel-checked with no axioms beyond the three standard axioms propext, Classical.choice, Quot.sound), with strictly zero sorry statements in the Lean 4 kernel, and integrated into an autonomous exploration pipeline inspired by Karpathy's AutoResearch. Scope note (added at publication, 2026-09-18). The Lean results this paper reports are arithmetic instances — integer or rational models — of results from the cited literature: kernel-checked, but thin. Where the title or abstract says “formal resolution” or “we prove”, it refers to these instances, not to the physical problems themselves. The tier of each claim is listed in the repository’s docs/VERIFIED_FOUNDATION.md §2. Epistemic tiers. Claims are labelled Tier A (checked by the Lean 4 kernel; axioms propext, Classical.choice, Quot.sound only), Tier L (literature, cited) or Tier C (conjecture). Only Tier A statements are machine-checked; physical interpretation is not. Lean artifact: SocrateAI-Scientific-Agora-LeanMaster, tag v3.5.0 (Lean 4 v4.33.1, Mathlib). Release gates on this tag: all 8 libraries build; 456 theorems pass #print axioms with standard axioms only; no sorry/admit. AI-assisted tooling (Anthropic Claude models) was used in preparing the formalization and the manuscript, under the direction of the author. Companion records (LeanMaster v3.5.0): The Dual-Scale String: T-Duality, K3 × T², and Their Formalization in Lean 4 — A Student's Companion — 10.5281/zenodo.22823716 Dual-Scale Generalized Geometry and Non-Perturbative Moduli Stabilization on K3 × T² — 10.5281/zenodo.22823718 Mathieu M24 Moonshine Rigidity, Mukai Lattices, and Holographic BPS Dyons on K3 × T² — 10.5281/zenodo.22823720 The Frontier Triad: Swampland Distance Bounds, Tachyon Condensation, and Non-Perturbative Vacuum Decay on K3 × T² — 10.5281/zenodo.22823722 Formal Resolution of Three Conjectures in the Lean 5 Agora Corpus: Navier-Stokes Helicity Dissipation, Mathieu Frobenius Rigidity, and Dual-Scale Horizon Censorship — 10.5281/zenodo.22823724 Formal Resolution of Five Frontier Problems in the Lean 5 Agora Corpus: Mukai Monodromy, Kolmogorov Turbulence, Flux Swampland, Courant Torsion, and Golay Holography — 10.5281/zenodo.22823726 Formal Resolution of Three Advanced Frontier Problems in the Lean 5 Agora Corpus: Kummer Surface Modularity, Non-Perturbative SYM Instantons, and Holographic Entanglement Strong Subadditivity — 10.5281/zenodo.22823729 The Dual-Scale String Theory: Mechanized Foundations, Singularity Resolution, Mathieu Moonshine, and a Zero-Free-Parameter Cosmological Conjecture on K3 × T² — 10.5281/zenodo.22823731 Lattices, T-Duality, and Double Field Theory on K3 × T²: A Lean 4 Companion Formalization — 10.5281/zenodo.22823733