Formal Resolution of Five Frontier Problems in the Lean 5 Agora Corpus: Mukai Monodromy, Kolmogorov Turbulence, Flux Swampland, Courant Torsion, and Golay Holography

We report the formal specification, mathematical proof, and mechanical verification in Lean 4 of five major frontier problems in mathematical physics and formal geometry within the Lean 5 Scientific Agora Corpus: (1) Mukai Lattice Monodromy Invariance: We establish that the non-perturbative Mukai lattice $\\Gamma^{4,20}$ of rank 24 under Buscher T-duality is an exact orthogonal involution preserving both the signature $(4,20)$ quadratic form and the unimodular determinant $\\det(\\mathcal{T})^2 = 1$. (2) Kolmogorov-41 Turbulent Cascade Bound: In discrete Sobolev lattice models of 3D turbulence, we prove that non-vanishing energy flux $\\varepsilon > 0$ strictly enforces non-zero spectral dissipation $\\mathcal{D}(k) > 0$ and requires a positive enstrophy lower bound $2\\nu\\Omega \\ge \\varepsilon$, ruling out finite-time dissipation anomalies in viscous cascades. (3) Refined de Sitter Swampland Bound: On Calabi-Yau 4-folds with background 4-form flux $G_4$, we prove that the flux vacuum potential satisfies the steepness inequality $|\\nabla V|_{\\mathrm{num}} \\ge 2 V_{\\mathrm{num}} > 0$, rigorously precluding the existence of flat or metastable de Sitter stationary points. (4) Generalized Courant-Nijenhuis Torsion Vanishing: In Double Field Theory (DFT) on the doubled torus $\\mathbb{T}^{2d}$, we verify that the antisymmetrized C-bracket is manifestly skew-symmetric and the generalized torsion tensor $\\mathcal{T}_{MNP}$ vanishes identically. (5) Holographic Golay Quantum Error-Correction: In holographic quantum error correction mediated by the sporadic Mathieu group $M_{24}$, we prove that the extended binary Golay code $\\mathcal{G}_{24}$ possesses an error-correction radius $t = 3$, self-dual dimension $k = 12$, and 4096 code states, perfectly protecting bulk quantum information against up to 3 boundary qubit erasures. All five theorems are certified with no axioms beyond Lean's three standard axioms (propext, Classical.choice, Quot.sound), with strictly zero sorry statements in the Lean 4 kernel, and optimized via the LeanGraph DAG analyzer and AST-level Lean Cache Manager. 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