ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-11-04 (gpt-6-luna)

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Exact parameter point in the one-modulus KKLT EFT

Physical scope

A single exact parameter substitution in the conditional one-modulus, two-derivative four-dimensional N=1 Einstein-frame EFT in Planck units. It is not a flux choice, measured value, or full string compactification.

Example

In the one-modulus EFT of the cited theorem, take $a = A = 1$ and $W_{0} = - 5 / (3 e)$ , with $M_{P} = 1$ and the real axion branch $θ = 0$ . These parameters give the exact supersymmetric point $τ = 1$ , $W = - 2 / (3 e)$ , and $V = - 1 / (6 e^{2})$ . This is a parameter example within the stipulated EFT only; it asserts no flux realization, instanton or condensate sector, or controlled string vacuum.

Statement references

Supersymmetric AdS critical point in the one-modulus KKLT EFT

Facts & Assumptions

Given: The model and conventions of Supersymmetric AdS critical point in the one-modulus KKLT EFT, the real exponential $e = \exp (1)$ , and the exact parameter assignments in the Example.

[F1]

Supersymmetric AdS critical point in the one-modulus KKLT EFT gives the unique-branch criterion $0 < - W_{0} / A < 1$ and the formulas for $W$ and $V$ at its supersymmetric point; its statement and proof remain conditional on the upstream readiness record described there.

[F2]

The real exponential function and the number $e$ by a power series defines $e = \exp (1)$ by the real exponential series.

[F3]

Series, partial sums, convergence and the sum, divergence, and the tail series defines the sum of a convergent series as the limit of its partial sums.

[F4]

Limits preserve non-strict inequalities states that an eventual non-strict inequality between convergent real sequences is preserved by their limits.

[F5]

The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ gives $e > 0$ and $e^{- 1} = 1 / e$ .

[A1]

The values $a = A = 1$ and $W_{0} = - 5 / (3 e)$ are stipulated EFT parameters, not inferred from flux quantization or an instanton determinant.

Verification

1.1givenF2F3F4F5A1algebra

At $x = 1$ , the first two terms of the positive series for $e$ have partial sum $1 + 1 = 2$ , so [F2]–[F4] give $e \geq 2 > 0$ . Hence $0 < 5 / (3 e) \leq 5 / 6 < 1$ , and the assigned $W_{0}$ satisfies the theorem's strict ratio condition.

2.1givenF1A1step 1.1algebra

With $a = 1$ , $A = 1$ , and $τ = 1$ , the defining equation becomes $- W_{0} / A = 5 / (3 e) = e^{- 1} (1 + 2 / 3) = e^{- a τ} (1 + 2 a τ / 3)$ . Thus $D_{T} W = 0$ at $τ = 1$ ; by [F1] it is the unique point on the stated branch.

3.1givenF1F5A1step 2.1algebra∎

Substituting $a = A = τ = 1$ into [F1] gives $W = - 2 / (3 e)$ and $V = - 1 / (6 e^{2}) < 0$ . This is the exact conditional EFT value for the selected parameter point.

Unresolved supplier

The verification uses the current KKLT theorem conditionally. That theorem remains escalated on its declared physical supplier readiness inputs, as detailed in its own unresolved-supplier section.