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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Supersymmetric AdS critical point in the one-modulus KKLT EFT

Physical scope

Conditional one-complex-modulus, two-derivative four-dimensional N=1 Einstein-frame EFT in Planck units, on the real axion branch and with no D-term or other potential contribution. It proves an EFT critical point only; flux quantization, instanton or condensate realization, correction control, scale separation, and stability are not established.

Statement

In Planck units $M_{P} = 1$ , consider the one-chiral-field, two-derivative four-dimensional $N = 1$ supergravity effective field theory in Einstein frame with $T = τ + i θ$ , $τ > 0$ , $K = - 3 \log (T + \overset{ˉ}{T})$ , and $W = W_{0} + A e^{- a T}$ , where $a, A > 0$ and $W_{0} < 0$ is real. Define $D_{T} W = \partial_{T} W + K_{T} W$ . On the branch $θ = 0$ , the equation $D_{T} W = 0$ has a finite solution with $τ > 0$ exactly when $0 < - W_{0} / A < 1$ , and that solution is unique. At it,

W_{0} = - A e^{- a τ} (1 + \frac{2 a τ}{3}), W = - \frac{2 a τ}{3} A e^{- a τ}, V = - 3 e^{K} ∣ W ∣^{2} = - \frac{a^{2} A^{2} e^{- 2 a τ}}{6 τ} < 0.

Under the one-overall-modulus coordinate map $T = - i ρ$ , this is the same critical locus as $D_{ρ} W = 0$ for $K (ρ, \overset{ˉ}{ρ}) = - 3 \log [- i (ρ - \overset{ˉ}{ρ})]$ and $W = W_{0} + A e^{i a ρ}$ . This is a conditional four-dimensional EFT critical point, not a claim that flux quantization, instanton zero modes, a localized uplift, or all string corrections realize a controlled compactification.

Statement references

Kachru, Kallosh, Linde, and Trivedi, §II.C

Louis, §8.2

McAllister and Quevedo, §4.1

Facts & Assumptions

Given: Work on the real branch $θ = 0$ , with $T = τ > 0$ , in the stated one-field Einstein-frame EFT and in Planck units.

[F1]

Compactification-derived four-dimensional moduli EFT data defines a two-derivative four-dimensional Einstein-frame moduli EFT by its retained chiral coordinates, $K, W, f_{A B}$ , Planck normalization, truncation domain, and omitted sectors. Its data tuple by itself asserts no critical point, uplift, or scale-separated truncation.

[F2]

GKP single overall Kähler-modulus coordinate and normalization gives the local coordinate $ρ = b / \sqrt{2} + i e^{4 u_{E}}$ and the map $T = - i ρ$ , so $b = 0$ gives $ρ = i τ$ and $T = τ$ . This supplier is restricted to its fixed-reference-metric, constant-warp, large-radius reduction and does not set a global axion period.

[F3]

Four-Dimensional $N = 1$ Matter-Coupled Supergravity adopts the local two-derivative four-dimensional Einstein-frame old-minimal $N = 1$ framework and defines $D_{T} W = \partial_{T} W + K_{T} W$ when $M_{P} = 1$ . It is an adopted model input, not a theorem about all four-dimensional theories.

[F4]

Four-Dimensional $N = 1$ F- and D-Term Scalar Potential gives the conditional F-term potential $V_{F} = e^{K} (K^{T \overset{ˉ}{T}} ∣ D_{T} W ∣^{2} - 3 ∣ W ∣^{2})$ in Planck units. Its auxiliary-square proof establishes this formula under the old-minimal Einstein-frame and Planck conventions adopted by its Postulate; this item uses that formula only as a conditional potential interface.

[F5]

The real exponential function and the number $e$ by a power series defines $\exp (x) = \sum_{n = 0}^{\infty} x^{n} / n!$ and $e = \exp (1)$ .

[F6]

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

[F7]

Limits preserve non-strict inequalities: if two real sequences converge and one is eventually no greater than the other, their limits satisfy the same non-strict inequality.

[F8]

The exponential function is smooth and $(\exp)^{'} = \exp$ : the real exponential is smooth and $\exp^{'} (x) = \exp (x)$ .

[F9]

The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ : $\exp (x) > 0$ and $\exp (- x) = 1 / \exp (x)$ for every real $x$ .

[F10]

Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function: sums, scalar multiples, and products of continuous real functions are continuous.

[F11]

Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ : the derivative of a product of differentiable real functions is given by the product rule.

[F12]

The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ : if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ .

[F13]

The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ : for a real function continuous on $[u, v]$ and differentiable on $(u, v)$ , some $c \in (u, v)$ satisfies $f (v) - f (u) = f^{'} (c) (v - u)$ .

[F14]

Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a, b]$ takes every value between $f (a)$ and $f (b)$ : a continuous real function on a closed interval takes each value between its endpoint values.

[F15]

The natural logarithm as the inverse of the exponential function defines $\log : (0, \infty) \to R$ as the inverse of the real exponential, so $\exp (\log x) = x$ for $x > 0$ .

[F16]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t gives $\log^{'} (x) = 1 / x$ for $x > 0$ .

[F17]

The exponential addition formula $\exp (x + y) = \exp (x) \exp (y)$ gives $\exp (u + v) = \exp (u) \exp (v)$ for real $u, v$ .

[A1]

$A, a$ are fixed positive dimensionless constants, $W_{0}$ is a fixed negative real parameter, and $K, W$ are exactly the functions in the Statement; no equation for their string-theoretic origin is assumed.

[A2]

The only potential term in the retained one-field model is the ungauged F-term potential in [F4]; all other fields are held at their integrated-out EFT values, and this item makes no assertion that their mass hierarchy or backreaction is controlled.

Proof

1.1givenF1F3F8F12F15F16A1algebra

Within the compactification-derived four-dimensional Einstein-frame EFT data [F1], adopt the local old-minimal $N = 1$ derivative [F3]. Use the Wirtinger operators $\partial_{T} = \frac{1}{2} (\partial_{τ} - i \partial_{θ})$ and $\partial_{\overset{ˉ}{T}} = \frac{1}{2} (\partial_{τ} + i \partial_{θ})$ on the open half-plane $τ > 0$ . The logarithm argument is $T + \overset{ˉ}{T} = 2 τ > 0$ , so [F15]–[F16] and the real chain rule apply. They give $K_{T \overset{ˉ}{T}} = 3 / (4 τ^{2}) > 0$ . Since $W$ is holomorphic in the adopted chiral-field model, its $T$ derivative on the real axis equals its real-axis derivative. On $θ = 0$ , $T = \overset{ˉ}{T} = τ$ , so differentiation gives $K_{T} = - 3 / (2 τ)$ , $W = W_{0} + A e^{- a τ}$ , and $\partial_{T} W = - a A e^{- a τ}$ ; thus $D_{T} W = - a A e^{- a τ} - (3 / (2 τ)) (W_{0} + A e^{- a τ})$ .

1.2F5F6F7F8F9F10F11F12F13algebra

Define $f : [0, \infty) \to R$ by $f (x) : = e^{- x} (1 + 2 x / 3)$ . For each $c \geq 0$ and $ε > 0$ , continuity of $\exp$ at $- c$ gives $δ > 0$ such that $∣ t + c ∣ < δ$ implies $∣ \exp (t) - \exp (- c) ∣ < ε$ ; if $x \geq 0$ and $∣ x - c ∣ < δ$ , then $t = - x$ satisfies $∣ t + c ∣ = ∣ x - c ∣ < δ$ . Hence $x \mapsto \exp (- x)$ is continuous at $c$ . The affine factor is continuous by [F10], so [F10] makes $f$ continuous on $[0, \infty)$ ; also $f (0) = 1$ . For $x > 0$ , [F5]–[F7] give $e^{x} \geq x^{3} / 6$ : the partial sums include the nonnegative terms through $x^{3} / 3!$ , and their limit preserves this bound. Hence $0 < f (x) \leq 6 / x^{3} + 4 / x^{2} \leq 10 / x^{2}$ for $x \geq 1$ , so $f (x) \to 0$ as $x \to \infty$ . Differentiating by [F8], [F11], and [F12] gives $f^{'} (x) = - e^{- x} (1 + 2 x) / 3 < 0$ ; [F13] therefore makes $f$ strictly decreasing.

2.1step 1.1step 1.2A1algebra

Multiplying $D_{T} W = 0$ from step 1.1 by $- 2 τ / 3$ and setting $x = a τ > 0$ gives $- W_{0} / A = f (x)$ , with $f$ as defined in step 1.2.

2.2givenA1step 1.2F10F13F14choose

Because $W_{0} < 0$ and $A > 0$ , the ratio $r : = - W_{0} / A$ is positive. If a solution $x > 0$ exists, step 1.2 gives $r = f (x) \in (0, 1)$ . Conversely, if $r \in (0, 1)$ , continuity, $f (0) = 1$ , and $f (x) \to 0$ give $x_{-} < x_{+}$ with $0 < x_{-} < x_{+}$ , $f (x_{-}) > r > f (x_{+})$ ; [F14] supplies a root on $[x_{-}, x_{+}]$ , and strict decrease from step 1.2 makes it unique. Thus the stated interval is necessary and sufficient.

2.3givenF2F3F12F15F16A1step 1.1algebra

Under $T = - i ρ$ , one has $ρ = i T$ and $e^{- a T} = e^{i a ρ}$ . On $b = 0$ , $ρ = i τ$ , so the logarithm argument is $- i (ρ - \overset{ˉ}{ρ}) = 2 τ > 0$ . Using [F16] and $\partial_{ρ} = \frac{1}{2} (\partial_{Re ρ} - i \partial_{Im ρ}) = - i \partial_{T}$ gives $\partial_{ρ} K = - 3 / (ρ - \overset{ˉ}{ρ}) = 3 i / (2 τ)$ and $\partial_{ρ} W = i a A e^{- a τ}$ . Therefore $D_{ρ} W = i (a A e^{- a τ} + 3 W / (2 τ)) = - i D_{T} W$ , using step 1.1; the two zero loci agree on this branch.

3.1step 2.1A1algebra

Substituting the solution relation from step 2.1 into $W = W_{0} + A e^{- a τ}$ gives $W = - (2 a τ / 3) A e^{- a τ}$ .

4.1givenF4F9F15F17A1step 3.1algebra

At $D_{T} W = 0$ , the F-term expression [F4] reduces to $V = - 3 e^{K} ∣ W ∣^{2}$ , and [F9], [F15], and three uses of [F17] give $e^{K} = \exp (- 3 \log (2 τ)) = \exp (\log (2 τ))^{- 3} = (2 τ)^{- 3}$ . Using step 3.1 yields $V = - a^{2} A^{2} e^{- 2 a τ} / (6 τ) < 0$ .

5.1givenF4A2step 1.1algebra∎

At this point, differentiating the positive square term $e^{K} K^{T \overset{ˉ}{T}} ∣ D_{T} W ∣^{2}$ in [F4] gives zero because each product-rule term contains $D_{T} W$ or its conjugate. Also $\partial_{T} (e^{K} ∣ W ∣^{2}) = e^{K} (K_{T} W + \partial_{T} W) \overset{ˉ}{W} = e^{K} D_{T} W \overset{ˉ}{W} = 0$ , since $W$ is holomorphic. The conjugate derivative vanishes as well, so both real first derivatives of $V$ vanish; the solution is a critical point, not a stability claim.

Unresolved supplier

The one-field calculation is complete, but its assigned Step 1 readiness remains escalated. The current loadStep1Record/step1Decision survey found:

def-cmp4-compactification-derived-moduli-eft-data has an existing escalated record whose stored dependency hash is stale. Its current reason identifies missing Step 1 records for the batch-78 O3/O7 chiral-coordinate and K/f carriers, unresolved orientifold/parity and $N = 2$ period inputs, and malformed derivative notation in a local metric supplier.
def-cmp4-gkp-single-overall-kahler-modulus-coordinate has an existing escalated record with a stale stored hash. Its current reason identifies escalated IIB pseudoaction and five-form inputs; it also cites the conventions item as missing.
post-four-dimensional-n1-matter-coupled-supergravity and pthm-four-dimensional-n1-fd-scalar-potential have existing escalated records whose stored hashes are stale and whose reasons say the conventions carrier is absent. That carrier, def-supergravity-conventions-and-normalizations, is now present at level 5 with a current ready record. These escalations are retained for owner refresh; they are not overwritten here.

The exact suppliers’ current statements and available proofs were inspected, and this proof uses their interfaces conditionally. Because their current readiness records remain escalated or stale, this item cannot be marked ready. The result asserts no flux realization, controlled correction series, uplift, or full-field stability.

Supersymmetric AdS critical point in the one-modulus KKLT EFT

Statement

Statement references

Facts & Assumptions

Proof

Unresolved supplier

Depends on

Dependency tree · 2 levels

Sources