Mathieu M24 Moonshine Rigidity, Mukai Lattices, and Holographic BPS Dyons on K3 × T²

We present the formal certification of Mathieu Moonshine rigidity and BPS state multiplicities on Calabi-Yau $K3 \\times T^2$ compactifications. Following the discovery by Eguchi, Ooguri, and Tachikawa (EOT), the elliptic genus of $K3$ decomposes into irreducible representations of the sporadic simple Mathieu group $M_{24}$ ($|M_{24}| = 244,823,040$). Within the Callens Dual-Scale framework, we demonstrate that the ratio of the second BPS state multiplicity $A_2 = 462 = \\mathbf{231} \\oplus \\overline{\\mathbf{231}}$ to the primary supercharges ($N_Q = 4$) and first multiplicity $A_1 = 90 = \\mathbf{45} \\oplus \\overline{\\mathbf{45}}$ constitutes a topologically invariant rational lock: $\\mathcal{R}_{\\mathrm{BPS}} = \\frac{462}{4 \\times 90} = \\frac{77}{60}$. We prove that $\\gcd(77, 60) = 1$ and verify the integer character lock $462 \\times 60 = 360 \\times 77 = 27720$. Furthermore, we demonstrate that the total order of $M_{24}$ is an exact integer multiple of 27720 ($|M_{24}| = 27720 \\times 8832$), anchoring the holographic microstate counting of $1/4$-BPS dyons and the error-correcting structure of the binary Golay code $\\mathcal{G}_{24}$. All proofs have been mechanically verified in the Lean 4 interactive theorem prover with no axioms beyond Lean's three standard axioms (propext, Classical.choice, Quot.sound). 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