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

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Unique negative stationary point and positive-definite coordinate Hessian of the exact two-variable function

Physical scope

Conditional exact finite-dimensional two-radial-coordinate four-dimensional Einstein-frame EFT function in Planck units. The theorem proves the unique negative-energy radial stationary point and its coordinate Hessian signature only; it does not derive the function from a string compactification, test axions or other fields, or establish physical mass, correction-control, or scale-separation claims.

Statement

In Planck units, let $τ > 0$ and $V > 0$ be independent dimensionless radial coordinates, and let $A_{1}, B_{1}, C_{1}, a$ be fixed positive dimensionless constants. For the exact scalar function

V_{3 t e r m} (τ, V) = \frac{A_{1} \sqrt{τ} e^{- 2 a τ}}{V} - \frac{B_{1} τ e^{- a τ}}{V^{2}} + \frac{C_{1}}{V^{3}},

there is exactly one stationary point in $(0, \infty)^{2}$ . Its value is negative and the Hessian matrix of $V_{3 t e r m}$ in the two radial coordinates is positive definite. The theorem concerns this exact finite-dimensional function. A type-IIB large-volume interpretation requires a separate $K, W$ -to-coefficient map, a Kähler-cone check, analysis of axion and other retained directions, and quantitative control of omitted corrections; none is asserted here.

Statement references

Balasubramanian, Berglund, Conlon, and Quevedo, §2 and §3.1

Louis, §8.3

McAllister and Quevedo, §4.2

Facts & Assumptions

Given: The exact function in the Statement, with independent positive real coordinates $τ, V$ and fixed constants $A_{1}, B_{1}, C_{1}, a > 0$ .

[F1]

Calabi–Yau threefold intersection and Kähler-volume coordinates defines the dimensionless total Calabi–Yau volume $V = \frac{1}{6} κ_{a b c} v^{a} v^{b} v^{c}$ and dual divisor coordinates $τ_{a} = \frac{1}{2} κ_{a b c} v^{b} v^{c}$ in its declared $α^{'} = 1$ convention. The theorem uses only the positive quadrant and does not assert that every point lies in a given Kähler cone.

[F2]

Compactification-derived four-dimensional moduli EFT data defines the declared four-dimensional Einstein-frame EFT data and Planck units; the data tuple alone does not establish a critical point or a valid compactification truncation. Its current Step 1 readiness record is missing, so this interface is used conditionally.

[F3]

Canonical moduli mass matrix and EFT scale-control data defines physical scalar masses using the field-space metric and distinguishes them from coordinate-Hessian eigenvalues. Its current Step 1 readiness record is missing; this theorem makes no physical mass or omitted-mode claim.

[F4]

Admissible nonperturbative sectors contributing to a moduli superpotential states sector-specific conditions for a candidate $A e^{- a T}$ term and does not infer it from divisor rigidity alone. Its current Step 1 readiness record is missing; the theorem does not assert that such a sector exists in a compactification.

[F5]

GKP no-scale cancellation in the classical IIB flux potential proves conditionally that the single-overall-modulus tree-level flux F-term leaves that modulus unfixed on the stated flux F-flat locus. Its actual hypotheses exclude nonperturbative, $α^{'}$ , loop, uplift, and multifield no-scale conclusions. Its current Step 1 readiness record is missing; it is used only as single-modulus background context, not as a two-modulus identity or as proof of the displayed function.

[F6]

Four-Dimensional $N = 1$ F- and D-Term Scalar Potential states the conditional Einstein-frame F-term potential and its auxiliary-square proof is complete under the old-minimal conventions adopted by its Postulate. This item uses only that formula as model context; the exact function here is an input, not a derivation from $K, W$ .

[F7]

The real exponential function and the number $e$ by a power series defines the real exponential by its convergent power series.

[F8]

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

[F9]

The exponential function is smooth and $(\exp)^{'} = \exp$ gives $(e^{x})^{'} = e^{x}$ and smoothness.

[F10]

Continuity and derivatives of positive-base real powers gives continuity and $(x^{α})^{'} = α x^{α - 1}$ for $x > 0$ and $α \in R$ .

[F11]

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 $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ gives the derivative of a composition of differentiable real functions at a point where both limit-point hypotheses hold.

[F12]

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 $g (c) \neq 0$ gives the product and quotient rules for differentiable real functions.

[F13]

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 gives continuity of sums, products, and scalar multiples of continuous real functions.

[F14]

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 function continuous on a closed interval and differentiable on its interior, its endpoint difference equals its derivative at some interior point times the interval length.

[F15]

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.

[F16]

The intrinsic Hessian of a smooth function at a critical point defines the intrinsic Hessian from the second coordinate derivatives at a critical point and makes it independent of chart.

[F17]

Critical-point Hessian matrices transform by congruence under chart changes gives the exact chart-change law $H_{y} = J^{T} H_{x} J$ at a critical point.

[A1]

The scalar function displayed in the Statement is exact on $(0, \infty)^{2}$ , and all four coefficients are independent positive constants; no parameter relation is assumed.

[A2]

No $K, W$ -to-coefficient derivation, flux choice, tadpole solution, instanton determinant, axion potential, or estimate for omitted terms is part of the hypotheses.

[A3]

Positivity of this coordinate Hessian is a property of the exact two-variable function. It is not by itself a full compactification stability, canonical-mass, or EFT scale-separation claim.

Proof

1.1givenF1F2F3F4F5F6F7F8F9F10F11A1algebra

For scope, [F1] fixes the Calabi–Yau volume-coordinate convention, [F2] records the four-dimensional Einstein-frame EFT data, and [F3] distinguishes a coordinate Hessian from physical scalar masses. [F4]–[F6] provide only conditional one-sector/single-modulus model context: they do not derive the exact two-variable function, which is input [A1]. Put $x = a τ > 0$ , $q = V e^{- x} τ^{- 1 / 2} > 0$ , $c = C_{1} τ^{- 3 / 2} > 0$ , and $E = e^{- 3 x} q^{- 3} > 0$ . The inverse change is $τ = x / a$ , $V = q e^{x} τ^{1 / 2}$ , so $(τ, V) \mapsto (τ, q)$ is a smooth coordinate change with $\partial q / \partial V = e^{- x} τ^{- 1 / 2} > 0$ . Substitution into the exact function gives $V = E p$ with $p = A_{1} q^{2} - B_{1} q + c$ .

1.2F10F11F12F14A1algebra

Define $L (x) = C_{1} a^{3 / 2} x^{- 3 / 2}$ and $R (x) = (B_{1}^{2} / A_{1}) 4 x (x - 1) / (4 x - 1)^{2}$ on $[1, \infty)$ . Their derivatives on $x > 1$ are $L^{'} (x) = - (3 / 2) C_{1} a^{3 / 2} x^{- 5 / 2} < 0$ and $R^{'} (x) = (B_{1}^{2} / A_{1}) 4 (2 x + 1) / (4 x - 1)^{3} > 0$ , so [F14] makes $L$ strictly decreasing and $R$ strictly increasing. Also $L (1) = C_{1} a^{3 / 2} > 0 = R (1)$ , while $L (x) \to 0$ and $R (x) \to B_{1}^{2} / (4 A_{1}) > 0$ as $x \to \infty$ . The first limit follows from $x^{- 3 / 2} \leq x^{- 1}$ for $x \geq 1$ ; the second follows by dividing numerator and denominator by $x^{2}$ .

2.1step 1.1F10F11F12algebra

Holding $τ$ fixed, $E_{q} = - 3 E / q$ and $p_{q} = 2 A_{1} q - B_{1}$ , hence $\partial_{q} V = E (p_{q} - 3 p / q)$ . Since $E, q > 0$ , $\partial_{q} V = 0$ is equivalent to $A_{1} q^{2} - 2 B_{1} q + 3 c = 0$ .

2.2step 1.1F9F10F11F12algebra

Holding $q$ fixed, $E_{τ} = - 3 a E$ and $c_{τ} = - 3 c / (2 τ)$ , so $\partial_{τ} V = E (- 3 a p - 3 c / (2 τ))$ . Thus $\partial_{τ} V = 0$ is equivalent to $p = - c / (2 a τ) = - c / (2 x)$ .

2.3step 1.2F13F14F15A1choose

The continuous difference $L - R$ is positive at $1$ and is negative for some sufficiently large finite $x$ , by step 1.2. Applying [F15] on that finite interval gives a root in $(1, \infty)$ . Since $L - R$ is strictly decreasing by step 1.2, this root is unique.

3.1step 2.1step 2.2A1algebra

Combining $p = A_{1} q^{2} - B_{1} q + c = - c / (2 x)$ with the equation in step 2.1 gives $B_{1} q = c (2 - 1 / (2 x))$ and $A_{1} q^{2} = c (1 - 1 / x)$ . Positivity of $A_{1}, q, c$ forces $x > 1$ . Dividing the square of the first equality by the second yields $C_{1} a^{3 / 2} x^{- 3 / 2} = \frac{B_{1}^{2}}{A_{1}} \frac{4 x (x - 1)}{(4 x - 1)^{2}} .$ This scalar equation is necessary for stationarity.

4.1step 1.1step 2.1step 2.2step 3.1A1algebra

Conversely, let $x > 1$ solve the scalar equation in step 3.1, put $τ = x / a$ , $c = C_{1} a^{3 / 2} x^{- 3 / 2}$ , and set $q = (c / B_{1}) (2 - 1 / (2 x)) = (c / B_{1}) (4 x - 1) / (2 x) > 0$ . The scalar equation says $c = (B_{1}^{2} / A_{1}) 4 x (x - 1) / (4 x - 1)^{2}$ , so direct substitution gives $A_{1} q^{2} = \frac{A_{1} c^{2}}{B_{1}^{2}} \frac{(4 x - 1)^{2}}{4 x^{2}} = c \frac{x - 1}{x} .$ Also $B_{1} q = c (2 - 1 / (2 x))$ , whence $p = A_{1} q^{2} - B_{1} q + c = - c / (2 x)$ and $A_{1} q^{2} - 2 B_{1} q + 3 c = 0$ . Steps 2.1–2.2 then give both partial derivatives zero. For this $x$ , $q$ is uniquely fixed, and the inverse coordinate change in step 1.1 uniquely fixes $V$ . Thus stationary points correspond one-to-one to roots of the scalar equation.

5.1step 2.1step 2.2step 4.1step 2.3F10F11F12algebra

At a stationary point, $h_{q} : = p_{q} - 3 p / q = 0$ and $h_{τ} : = - 3 a p + c_{τ} = 0$ . Differentiating $V_{q} = E h_{q}$ and $V_{τ} = E h_{τ}$ , the terms involving $E_{q}$ or $E_{τ}$ vanish there. Since $c_{τ} = - 3 c / (2 τ)$ and $c_{τ τ} = 15 c / (4 τ^{2})$ , this gives $V_{τ τ} = E (- 3 a c_{τ} + c_{τ τ}) = E (\frac{9 a c}{2 τ} + \frac{15 c}{4 τ^{2}}) .$ For the other entries, $V_{q q} = E (p_{q q} - 3 p_{q} / q + 3 p / q^{2})$ and $V_{τ q} = - 3 a E p_{q}$ . Using $p_{q q} = 2 A_{1}$ , $p_{q} = 2 A_{1} q - B_{1}$ , and $p_{q} = 3 p / q$ at stationarity reduces the first expression to $E 2 (B_{1} - A_{1} q) / q$ and the second to $3 a E (B_{1} - 2 A_{1} q)$ .

6.1step 5.1step 3.1A1algebra

From step 3.1, $A_{1} q / B_{1} = 2 (x - 1) / (4 x - 1)$ , so $0 < A_{1} q / B_{1} < 1$ , $1 - A_{1} q / B_{1} = (2 x + 1) / (4 x - 1)$ , and $1 - 2 A_{1} q / B_{1} = 3 / (4 x - 1)$ . Using also $A_{1} q^{2} = c (x - 1) / x$ and $B_{1} q = c (4 x - 1) / (2 x)$ gives $V_{τ τ} = \frac{3 E a^{2} c (6 x + 5)}{4 x^{2}}, V_{q q} = \frac{E A_{1} (2 x + 1)}{x - 1}, V_{τ q} = \frac{9 a E c}{2 x q} .$ Thus both diagonal entries are positive, and direct multiplication yields $\begin{aligned} \det D_{(τ, q)}^{2} V & = \frac{E^{2} a^{2} c A_{1}}{4 x^{2} (x - 1)} (3 (6 x + 5) (2 x + 1) - 81 x) \\ = \frac{E^{2} a^{2} c A_{1}}{4 x^{2} (x - 1)} 3 (12 x^{2} - 11 x + 5) . \end{aligned}$ By the scalar equation in step 3.1, $c A_{1} / [4 x^{2} (x - 1)] = B_{1}^{2} / [x (4 x - 1)^{2}]$ . Hence $\det D_{(τ, q)}^{2} V = E^{2} \frac{B_{1}^{2} a^{2}}{x (4 x - 1)^{2}} 3 (12 x^{2} - 11 x + 5) > 0,$ since $12 x^{2} - 11 x + 5 = 12 (x - 11 / 24)^{2} + 119 / 48$ .

7.1step 1.1step 6.1F16F17algebra

The symmetric $2 \times 2$ Hessian in step 6.1 is positive definite: for coordinates $(u, v)$ its quadratic form is $V_{τ τ} (u + V_{τ q} v / V_{τ τ})^{2} + (\det D^{2} V / V_{τ τ}) v^{2}$ , positive unless $(u, v) = (0, 0)$ . The change from $(τ, V)$ to $(τ, q)$ is a smooth chart change with invertible Jacobian by step 1.1, so [F16]–[F17] preserve the Hessian's positive-definite signature at this critical point.

8.1step 1.1step 2.2F3F7F8A1A3algebra∎

At the stationary point, step 2.2 gives $p = - c / (2 x)$ , so $V = E p = - E c / (2 x) < 0$ . This is the negative value of the exact radial function; [F3] and [A3] do not permit interpreting the coordinate Hessian alone as a physical mass or full compactification stability result.

Unresolved supplier

The exact radial-function proof is complete, but readiness remains escalated on current supplier records:

def-cmp-cy3-kahler-moduli-and-divisor-volumes has a stored ready decision, but step1Decision reports “Item or dependency changed; record current readiness”; its recorded hash does not match the current closure. This is a stale record, not a pending-precheck issue.
def-cmp4-compactification-derived-moduli-eft-data, def-cmp4-moduli-mass-matrix-and-evolution-cutoff, def-cmp4-nonperturbative-moduli-superpotential-sector, and pthm-r2-gkp-no-scale-flux-potential have existing escalated records with stale stored hashes. Their current reasons identify unresolved O3/O7/Kähler-coordinate and $N = 2$ period inputs, IIB pseudoaction/five-form inputs, and an unused extreme-value dependency routed to the separate uplift-stability page.
pthm-four-dimensional-n1-fd-scalar-potential has an existing escalated record with a stale hash; its stored reason says the supergravity-conventions carrier is absent. The carrier, def-supergravity-conventions-and-normalizations, is now present at level 5 and ready-current. The existing escalation is retained for owner refresh.

The current supplier statements and available proofs were inspected. The exact function is proved without deriving it from those physical inputs; they remain conditional model context. This theorem makes no K,W-to-coefficient, Kähler-cone, axion, full-compactification mass, or omitted-correction-control claim. It cannot be marked ready until its direct readiness records are current.

Unique negative stationary point and positive-definite coordinate Hessian of the exact two-variable function

Statement

Statement references

Facts & Assumptions

Proof

Unresolved supplier

Depends on

Dependency tree · 2 levels

Sources