The Frontier Triad: Swampland Distance Bounds, Tachyon Condensation, and Non-Perturbative Vacuum Decay on K3 × T²
Year: 2026
Type: preprint
Abstract: Revision (2026-09-18): orientifold bookkeeping. The tadpole configuration is placed in the orientifold K3 × T²/Z2, with the 4 O7-planes at the fixed points of T²/Z2, and the charges are stated in units of μ7/4 (+1 and −4 in D7 units; Sen hep-th/9605150, Tripathy–Trivedi hep-th/0301139). Earlier versions said "the T⁴/Z2 orientifold". No Lean statement of the paper changed. We formulate and formally verify the Frontier Triad: an interlocking triple of non-perturbative constraints guaranteeing the quantum gravitational consistency and stability of string vacua on $K3 \times T^2$. First, in accordance with the Swampland Distance Conjecture (SDC), traversing geodesic distances in moduli space induces an exponential tower of light states whose mass satisfies $m(\Delta\phi) \le m_0$, delineating the boundary of effective field theories. Second, unstable open-string configurations undergo Sen tachyon condensation, driving the tachyon potential to zero ($V(\infty) = 0$) and canceling negative brane tensions. Third, Ramond-Ramond (RR) Tadpole cancellation is verified as an exact Diophantine screening identity between 16 positive $D7$-branes and 4 negative $O7$-planes of the orientifold $K3 \times T^2/\mathbb{Z}_2$: $16 \times (+4) + 4 \times (-16) = 64 - 64 = 0$ in units of $\mu_7/4$ (charges $+1$ and $-4$ in D7 units). In tandem with the strict positivity of the Coleman-De Luccia bounce action $B > 0$, false vacuum decay rates $\Gamma/V \sim e^{-B}$ remain exponentially suppressed. Every theorem in this triad has been mechanized with no axioms beyond Lean's three standard axioms (propext, Classical.choice, Quot.sound) in Lean 4. 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 v3.21.0 (Lean 4 v4.33.1, Mathlib). Release gates on this tag: all ten libraries build; 648 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.
Language: English
Subfield: Nuclear and High Energy Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213596757
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