Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-11-05 (gpt-6.1-sol)

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Local persistence of an uplifted minimum in a finite-dimensional scalar EFT

Physical scope

Conditional theorem for a finite-dimensional canonical-scalar chart of a two-derivative four-dimensional Einstein-frame EFT. It does not identify the positive profile with an anti-D3, T-brane, D-term, or localized string source and does not bound omitted corrections.

Statement

Use natural units $ℏ = c = 1$ and one four-dimensional Einstein-frame normalization. Let $n \geq 1$ , let $x \in R^{n}$ be canonical real scalar coordinates of mass dimension one, and let $B_{r} (x_{0}) = {x : ∥ x - x_{0} ∥_{2} \leq r}$ with $r > 0$ . Let $U \subset R^{n}$ be open with $B_{r} (x_{0}) \subset U$ , and let $V_{0}, f \in C^{2} (U)$ , where $V_{0}$ has units of energy density and $f$ is a dimensionless real source profile. Write $I_{n}$ for the identity matrix, $A ⪰ B$ when $v^{T} (A - B) v \geq 0$ for every $v \in R^{n}$ , and $∥ A ∥_{o p} : = \sup_{∥ v ∥_{2} = 1} ∥ A v ∥_{2}$ for the Euclidean operator norm. Suppose $\nabla V_{0} (x_{0}) = 0$ , $D^{2} V_{0} (x) ⪰ μ I_{n}$ on the whole ball for $μ > 0$ , $f (x) \geq f_{\min} > 0$ , $∥ D^{2} f (x) ∥_{o p} \leq L$ , and ${osc}_{B_{r}} f : = \sup_{B_{r}} f - \inf_{B_{r}} f \leq C$ for $L, C \geq 0$ . For a source coefficient $D \geq 0$ with energy-density units, put $V_{D} = V_{0} + D f$ . If $D L < μ$ and $D C < μ r^{2} / 2$ , then $V_{D}$ has a unique interior minimizer $x_{D}$ in $B_{r} (x_{0})$ . If also $V_{0} (x_{0}) + D f_{\min} > 0$ , then $V_{D} (x_{D}) > 0$ . This is a conditional result about the stipulated finite-dimensional EFT potential, not a derivation of a source profile or a claim that a string compactification realizes it.

Statement references

Kachru, Kallosh, Linde, and Trivedi, §III.B

McAllister and Quevedo, §4.3

Facts & Assumptions

Given: A finite-dimensional canonical Euclidean scalar chart, a closed ball of positive radius, and the hypotheses in the Statement.

[F1]

Compactification-specific uplift source data in a 4D moduli EFT defines a source contribution $Δ V_{s r c} = D f$ with a real coefficient and profile, fixed in the same four-dimensional Einstein-frame normalization as the base potential. It assigns no universal volume scaling.

[F2]

Canonical moduli mass matrix and EFT scale-control data defines finite-dimensional retained scalar EFT data and requires any physical mass/cutoff comparison to use the actually present omitted spectrum in one common Einstein frame. This theorem makes no mass/cutoff comparison and uses only the definition's finite scalar EFT interface.

[F3]

Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line: in Euclidean $R^{n}$ with $n \geq 1$ , every closed bounded subset is compact. The ball $B_{r} (x_{0})$ is closed and bounded.

[F4]

A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value: a continuous real-valued function on a nonempty compact metric space attains its maximum and minimum. We use only attainment of $V_{D}$ on $B_{r} (x_{0})$ .

[A1]

The Euclidean norm and Hessian operator norm in the Statement use the canonical local scalar metric. In these units $μ$ has mass dimension two, $L$ has inverse-mass-squared dimension, $C$ is dimensionless, and both $D L$ and $μ$ have mass dimension two while $D C$ and $μ r^{2}$ have mass dimension four.

[A2]

The bounds on $f$ , $D^{2} f$ , and $osc f$ hold uniformly on the entire closed ball, not just at its center.

Proof

Given: [F1]–[F4], the regularity and uniform bounds [A1]–[A2], and the strict inequalities in the Statement.

1.1givenF1F2F3F4A1algebra

By [F1], $D f$ is the declared source contribution in the same Einstein-frame potential; [F2] places the scalar chart in the finite retained EFT, and neither supplier adds hypotheses to the analytic estimates. The closed ball is compact by [F3] and nonempty because it contains $x_{0}$ . Since $V_{D} = V_{0} + D f$ is continuous on the ball, [F4] gives $x_{D} \in B_{r} (x_{0})$ at which $V_{D}$ attains its minimum.

1.2givenA1A2algebra

For any $x \in B_{r} (x_{0})$ , the segment $γ (t) = x_{0} + t (x - x_{0})$ stays in the ball by convexity. The function $g (t) = V_{0} (γ (t))$ satisfies $g^{'} (0) = 0$ and $g^{''} (t) = (x - x_{0})^{T} D^{2} V_{0} (γ (t)) (x - x_{0}) \geq μ ∥ x - x_{0} ∥_{2}^{2}$ . Integrating twice on $[0, 1]$ yields $V_{0} (x) \geq V_{0} (x_{0}) + \frac{μ}{2} ∥ x - x_{0} ∥_{2}^{2}$ .

1.3givenA1A2algebra

For each vector $v \in R^{n}$ and $x$ in the ball, $v^{T} D^{2} f (x) v \geq - ∥ D^{2} f (x) ∥_{o p} ∥ v ∥_{2}^{2} \geq - L ∥ v ∥_{2}^{2}$ . Hence $D^{2} V_{D} (x) = D^{2} V_{0} (x) + D D^{2} f (x) ⪰ (μ - D L) I ≻ 0$ . So $V_{D}$ is strictly convex on the convex ball and can have at most one minimizer there.

2.1givenA2step 1.1step 1.2step 1.3algebra

If $∥ x - x_{0} ∥_{2} = r$ , then [A2] gives $f (x) - f (x_{0}) \geq - C$ . Using step 1.2 and $D \geq 0$ , $V_{D} (x) - V_{D} (x_{0}) \geq \frac{μ r^{2}}{2} - D C > 0.$ The boundary therefore has strictly larger value than the center. The minimizer from step 1.1 cannot lie on the boundary, so it is interior; by strict convexity from step 1.3 it is unique.

3.1givenstep 1.2algebra∎

Applying step 1.2 to $x_{D}$ gives $V_{0} (x_{D}) \geq V_{0} (x_{0})$ . Since $f (x_{D}) \geq f_{\min}$ and $D \geq 0$ , $V_{D} (x_{D}) = V_{0} (x_{D}) + D f (x_{D}) \geq V_{0} (x_{0}) + D f_{\min}$ . Under the final positivity hypothesis in the Statement this is strictly positive.

Remarks

The extra dependency Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line supplies compactness of the closed Euclidean ball; the extreme-value supplier alone assumes compactness and does not establish it. No step uses the diagrammatic shape of the basin or any empirical observation.

Local persistence of an uplifted minimum in a finite-dimensional scalar EFT

Statement

Statement references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources