The Dual-Scale String Theory: Mechanized Foundations, Singularity Resolution, Mathieu Moonshine, and a Zero-Free-Parameter Cosmological Conjecture on K3 × T²
Xavier Callens, SocrateAI Scientific Agora Collaboration · Zenodo (CERN European Organization for Nuclear Research) · 2026
We present a Lean 4 companion formalization for a dual-scale scenario of type II string theory on $K3\times T^2$, and we state precisely what is, and is not, machine-checked. The Mathlib-free core of the Lean artifact (51 files, 554 declarations, toolchain v4.33.1, no external dependency, standard axioms only — #print axioms logs in ) certifies exact integer and rational identities used in the argument: the positivity of the dual scale $R+\alpha'/R$ on integer radii; the $K3$ topological invariants $\chi=24$, $\sigma=-16$, $\mathrm{ind}(\slashed D)=2$; the elliptic-genus arithmetic $\mathcal A_2\cdot 60 = 4\mathcal A_1\cdot 77 = 27720$ with $|M_{24}| = 27720\cdot 8832$; and a corrected, genuinely unique-minimum toy potential (an earlier revision of this corpus contained a Nat-truncation bug that made the claimed minimum non-unique; and document the fix). Two further Lean libraries, built on Mathlib (which contributes no axioms beyond Lean's own three standard ones for the fragments used here), extend this core: StringTheoryFormalization (Stream 1's sixth library, corrects the $M_{24}$ representation table used in the moonshine arithmetic, ) and the new DualScaleStream2 library (99 theorems, 19 modules), which formalizes the $O(d,d;\mathbb Z)$ T-duality group, the Double Field Theory generalized metric, and the $K3\times T^2$ lattice layer (); together with the original core, the audited total is 326 theorems (Stream 1, six libraries) and 99 theorems (Stream 2), 0 failing, standard axioms only. The differential-geometric, index-theoretic and cosmological statements built on top of this arithmetic — Double Field Theory, the Atiyah–Singer index theorem, moduli stabilization, and the tensor-to-scalar ratio, CP phase, and dark-energy predictions — are either drawn from the literature or proposed here as conjectures, and are labeled by epistemic tier throughout (\tierA\ / \tierL\ / \tierC). In particular we discuss the obstruction that $\mathcal{N}=4$ non-renormalization poses for moduli stabilization on $K3\times T^2$, and we confront the paper's speculative numerical relations for $r$, $\delta_{\mathrm{CP}}$, and $(w_0,w_a)$ with current data, including the DESI DR2 preference for a time-evolving dark-energy equation of state at $3.1\sigma$ over $(w_0,w_a)=(-1,0)$. We close with a roadmap for promoting the central real-analytic claims (currently Tier L or Tier C) to kernel-checked Tier A theorems using Mathlib — a roadmap Stream 2 has already begun executing. 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