Mechanized T-Duality and Frontier String Dynamics on K3 × T²

Closing the Loop via LeanFlow Stiff Solvers with Native Duality, Topological Data Analysis, and Lean 4 Certification

Author: Xavier Callens
Affiliation: SocrateAI Lab | SymbioticFactory Research
GitHub Repository: xaviercallens/SocrateAI-Scientific-DualScaleSimulator & xaviercallens/SocrateAI-Numeric-DualScale-Solver
Publication Status: Preprint / Formal Computational Manuscript (34 Pages, Version 7)

Abstract

We present LeanFlow, a neuro-symbolic computational architecture that bridges high-performance stiff numerical differential equation solvers with interactive theorem proving (Lean 4) and Topological Data Analysis (TDA) to simulate and formally certify frontier non-linear string cosmology.

String compactifications on non-trivial geometries such as $K3 \times T^2$ feature stiff multi-scale dynamics, non-perturbative duality symmetries ($O(D,D; \mathbb{Z})$), and discrete topological invariants that challenge conventional floating-point simulations. LeanFlow addresses these challenges through five foundational pillars:

Operational Algorithmic Section Condition in Double Field Theory (DFT): Embedding Buscher duality into a $2D \times 2D$ Generalized Metric $\mathcal{H}_{MN}$ and enforcing the DFT section condition $\eta^{MN} \partial_M \partial_N \Psi = 0$ as an operational numerical projection gate.
Dual-Tier Latency Decoupling: Ultra-low latency in-loop Ahead-Of-Time (AOT) contract assertions ($12\text{--}18,\text{ns}$) verify metric unimodularity and duality locks at every solver time-step, decoupled from an asynchronous outer IPC batch verification gate ($45,\text{ms}$) running Lean 4 formal certificates.
Formal Depth Discipline in Lean 4: 14 sorry-free Lean 4 formal proofs verify derived spherical reflection auto-equivalences preserving the Mukai lattice pairing $\langle v, w \rangle$ on $\Gamma^{4,20}$, Kummer orbifold singularity resolution ($e(X)=24$), and discrete Euler-Poincaré topological invariants.
Closed-Loop TDA Anomaly Certification: High-dimensional point clouds generated by stiff Backward Differentiation Formula (BDF) solvers are mapped via the TDA Mapper algorithm into a simplicial 1-skeleton ($187$ nodes, $557$ edges, $\beta_1 = 376$), extracting persistent 1-cycles certified against discrete Lean 4 invariants.
Frontier String Dynamics Telemetry: Empirical validation across three benchmark use cases:
- Langevin Tunneling on Kummer Orbifolds: $48 \times 48$ lattice ($25,344$ transitions) validating dual fixed-point distributions ($58/29$) and entropy production $\Delta S_{\text{prod}} = 14.82,k_B$.
- Coleman–De Luccia Vacuum Decay: Euclidean bounce action $S_E \approx 137.15$, critical bubble radius $R_c \approx 2.82,\ell_s$, and flux step $N = 5 \to 4$ with central charge reduction $\Delta c = -100$.
- Boundary Tachyon Condensation: Rolling tachyon profile $T(x^0)$ with localized D-brane decay soliton width $w \approx 1.54\sqrt{\alpha'}$.

Artifacts in this Repository

File	Description	Size
`T_duality_Alone.pdf`	Complete 34-Page Manuscript (PDF) with full mathematical derivations, listings, and figures	2.2 MB
`T_duality_Alone.tex`	Complete LaTeX manuscript source and bibliography entries	103 KB
`lean4_formal_proofs.tar.gz`	14 Sorry-Free Lean 4 Formal Proofs (Mukai lattice, Kummer resolution, Buscher rules, CDL c-theorem, etc.)	24 KB
`rust_stiff_dualscale_simulator.tar.gz`	Rust Stiff Dual-Scale Simulator Crate (`Cargo.toml`, `src/`) with native AOT duality assertions	11 KB
`tda_mapper_skeleton.json`	TDA Mapper 1-skeleton graph: 187 nodes, 557 edges, Betti numbers ($\beta_1 = 376$)	81 KB
`kummer_langevin_summary.json`	Kummer orbifold Langevin simulation telemetry ($48 \times 48$, $25,344$ transitions)	310 B
`vacuum_decay_cdl_summary.json`	Coleman–De Luccia vacuum decay bounce action telemetry ($S_E \approx 137.15$, $R_c \approx 2.82,\ell_s$)	390 B
`tachyon_condensation_summary.json`	Boundary tachyon condensation profile telemetry ($w \approx 1.54\sqrt{\alpha'}$)	288 B
`zenodo_deposit_bundle.zip`	Unified Zenodo deposit distribution package	2.1 MB

Formal Lean 4 Verification Architecture

graph TD
    A[BDF Stiff PDE Solver<br/>Rust SIMD Engine] -->|12-18 ns AOT In-Loop Gate| B[Buscher & Metric Unimodularity Lock]
    A -->|Point Cloud Data<br/>N = 25,344| C[TDA Mapper Algorithm<br/>Spatial Filter ||grad phi||^2]
    C -->|Simplicial 1-Skeleton<br/>187 nodes, 557 edges| D[Betti Extraction<br/>beta_0=6, beta_1=376, chi=6]
    D -->|45 ms Outer IPC Gate| E[Lean 4 Kernel Certification<br/>KummerTDAAnomalyCertification.lean]
    E -->|Certified Sorry-Free| F[Formal Proof Ledger<br/>Mukai.lean, KummerResolution.lean]

Key Lean 4 Theorem Listings Included

Listing 2 (Mukai Lattice & Fourier–Mukai Auto-Equivalences): theorem fourier_mukai_preserves_pairing (v w : MukaiVector) : mukai_pairing (spherical_reflection v) (spherical_reflection w) = mukai_pairing v w
Listing 7 (Kummer Orbifold Resolution): theorem kummer_euler_characteristic : e(X) = 24 via 16 exceptional $\mathbb{P}^1$ divisors.
Listing 9 (Discrete Euler-Poincaré Algebraic Gate): theorem discrete_euler_poincare_consistency : chi(K) = V - E + beta_1

Citation

@article{Callens2026MechanizedTDuality,
  author    = {Callens, Xavier},
  title     = {Mechanized T-Duality and Frontier String Dynamics on K3 × T²: Closing the Loop via LeanFlow Stiff Solvers with Native Duality, Topological Data Analysis, and Lean 4 Certification},
  journal   = {Preprint / SocrateAI Scientific Research},
  year      = {2026},
  url       = {https://huggingface.co/callensxavier/leanflow-mechanized-t-duality-k3t2}
}

License

All software code is licensed under the Apache-2.0 License. Scientific research, documentation, and manuscript artifacts are licensed under the Creative Commons Attribution 4.0 International (CC-BY-4.0).

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