Refined de Sitter conjecture for positive scalar potentials

Use the earlier four-dimensional Einstein-frame scalar-EFT and swampland-claim data: a finite-dimensional smooth scalar target $\mathcal{T}$ of dimension $N\ge 0$, smooth positive-definite metric $G$, smooth potential $V:\mathcal{T}\to \mathbb{R}$, and constant reduced Planck mass $M_P>0$. The EFT is required to arise from a specified ultraviolet-complete quantum-gravity theory on its declared controlled field range. The gradient norm is $\mathrm{\mid }\mathrm{\nabla }V\mathrm{\mid }=\sqrt{G^{ij}{\mathrm{\partial }}_{i}V{\mathrm{\partial }}_{j}V}$. Write $H_p(v,w):=(\mathrm{\nabla }dV{)}_p(v,w)$ for the covariant Hessian at $p\in \mathcal{T}$, using the Levi-Civita connection of $G$. For $N\ge 1$, ${\lambda }_{\min }(p)$ is the least eigenvalue of the $G$-self-adjoint endomorphism with components $G^{ik}{\mathrm{\nabla }}_k{\mathrm{\nabla }}_{j}V$.

The conjecture proposes universal dimensionless constants $c,c^{\mathrm{\prime }}>0$, independent of the point and of the family limit in the declared theory class, such that at every controlled point $p$ with $V(p)>0$, either $M_P\mathrm{\mid }\mathrm{\nabla }V(p)\mathrm{\mid }\mathrm{/}V(p)\ge c$ or there exists $v\in T_p\mathcal{T}$ with $G_p(v,v)=1$ and $M_P^2{H}_p(v,v)\mathrm{/}V(p)\le -c^{\mathrm{\prime }}$. For $N\ge 1$ the second alternative is equivalent to $M_P^2{\lambda }_{\min }(p)\mathrm{/}V(p)\le -c^{\mathrm{\prime }}$. For $N=0$ the unit-tangent set is empty, so the second alternative is false, while $\mathrm{\mid }\mathrm{\nabla }V\mathrm{\mid }=0$. Thus the conjecture excludes a positive potential in a controlled zero-scalar EFT as well; it never assigns a fictitious eigenvalue to the zero vector space. The constants are proposed to be of order one in the claim datum's sense; no numerical values or proof of their existence are asserted.

In target coordinates with mass dimension one, $G_{ij}$ is dimensionless, $[V]={\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}^4$, $[\mathrm{\mid }\mathrm{\nabla }V\mathrm{\mid }]={\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}^3$, and $[{\lambda }_{\min }]={\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}}^2$, so the displayed ratios are dimensionless. The same constants and both alternatives are part of this conjecture; the original gradient-only version is a different statement.

Four-dimensional Einstein-frame multi-scalar cosmology EFT data

Scalar, gauge and theory-family data for swampland conjectures

Obied et al., pp. 1–3

Ooguri et al., pp. 1–5

Status review dated 2025-09-05: the bounded record is Obied et al., PDF pp. 1–3; Ooguri–Palti–Shiu–Vafa, PDF pp. 1–5; and Lehnert, §II.D, PDF pp. 14–15. Lehnert states that its reviewed literature was published before 2025 (PDF p. 2, note 8), and its dS section distinguishes counterexamples to the original gradient-only form from the refined form. The primary sources give conditional arguments and evidence in restricted regimes under extra assumptions but do not establish the universal refined statement. The constants, admissible theory class and control conditions remain unsettled in this bounded record. This item makes no status claim about literature after that cutoff and does not claim an exhaustive search; the conjecture is not an equation of motion or an observation.