ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 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.

DGKT massive IIA flux family with a smeared O6 truncation

Physical scope

One source-specific, classical, four-dimensional N=1 constant-mode DGKT solution with smeared/averaged O6 charge. It is not a localized global ten-dimensional solution or a universal theorem about massive IIA.

Example

In the DGKT source model of DGKT classical smeared four-dimensional effective solution, use its $ℓ_{s} = 1$ string-unit convention and choose $f_{0} = 1$ , $h_{3} = - 2$ , $m_{0} > 0$ , and, for every bulk index $i$ and every blow-up index $A$ , set $m_{i} = n_{A} = 0, f_{4}^{i} = - N, N \in N, c_{4} : = \frac{κ^{1 / 3}}{\sqrt{2}} (2 π)^{3}, e_{i} = c_{4} f_{4}^{i} = - \overset{ˉ}{e}, \overset{ˉ}{e} : = c_{4} N, f_{A} \in Z_{< 0}, \overset{ˉ}{e} ≫ ∣ f_{A} ∣ ≫ ∣ m_{0} ∣ .$ This is the $T^{6} / Z_{3}$ orientifold with a further freely acting $Z_{3}$ quotient described by DGKT, with all three bulk and all nine blow-up Kähler moduli included. Its source-reported classical stationary point has positive internal volume, weak coupling at large $\overset{ˉ}{e}$ , and $(H_{A d S} R_{K K})^{2} \sim {\overset{ˉ}{e}}^{- 1},$ so the four-dimensional classical constant-mode truncation is parametrically scale separated. Here $H_{A d S}^{2} : = ∣ Λ_{4} ∣ / (3 M_{P}^{2})$ defines the AdS curvature scale in the source's four-dimensional Einstein frame. This is a smeared/averaged, leading-supergravity example; it does not certify a global exact localized O6 solution or a full string-theory vacuum.

Facts & Assumptions

Given: The source model and flux data stated above.

[A1]

Massive IIA Romans-mass and NS three-form flux quantization: $f_{0} = (2 π ℓ_{s}) F_{0}^{G r a \tilde{n} a} \in Z$ and $n_{H} = (2 π ℓ_{s})^{- 2} \int_{Σ_{3}} H_{3} \in Z$ in the declared free-period convention. In the DGKT convention of [A2], the positively oriented dual cycle has $n_{H} = - h_{3}$ because $H_{3} = - p β^{0}$ and $p = (2 π ℓ_{s})^{2} h_{3}$ .

[A2]

DGKT classical smeared four-dimensional effective solution adopts the DGKT source-model branch, its source-normalized fluxes, solution formulas and signs, and source-reported scaling exponents. Its source-model tadpole input is $f_{0} h_{3} = - 2$ ; this is part of the smeared/averaged reduction data, not a consequence of a localized-current theorem.

[A3]

DGKT classical smeared four-dimensional effective solution: on the stated DGKT source cover, the signed source term is $N_{D 6} - 2 N_{O 6} = - 2$ . This is a different cover convention from the separate $T^{3} / Z_{2}$ count in the generic tadpole item.

Verification

Given: The conditional inputs [A1]–[A3], $κ, β, m_{0} > 0$ , and the integer family in the Example.

1.1A1A2A3algebra

By [A1] and [A2], the selected periods have $f_{0} = 1$ and $n_{H} = - h_{3} = 2$ , so they satisfy the free-period quantization input. Their dimensionful flux contribution to the integrated source equation is $\frac{F_{0}^{G r a \tilde{n} a}}{2 π ℓ_{s}} \int_{Σ_{3}} H_{3} = f_{0} n_{H} = - f_{0} h_{3} = 2.$ The source DGKT smeared/averaged model has signed source charge $- 2$ by [A3] and adopts $f_{0} h_{3} = - 2$ in [A2]; hence its own source-model tadpole sum is $- 2 - f_{0} h_{3} = 0$ for this choice. This is a check of the DGKT model input, not an application of the separate localized-current theorem, and it does not assert a localized ten-dimensional solution.

1.2A2givenalgebra

The bulk choice is on the actual lattice in [A2]: $f_{4}^{i} = - N$ is integral and $e_{i, p h y s} = ℓ_{s}^{3} c_{4} f_{4}^{i}$ , hence $e_{i} = - c_{4} N = - \overset{ˉ}{e}$ in string units. No integrality of the basis coefficient $e_{i}$ is assumed. Since $m_{i} = n_{A} = 0$ , [A2] gives ${\hat{e}}_{i} = e_{i} = - \overset{ˉ}{e}$ and ${\hat{f}}_{A} = f_{A}$ . Thus the three ${\hat{e}}_{i}$ have one common sign, $m_{0} {\hat{e}}_{1} {\hat{e}}_{2} {\hat{e}}_{3} = - m_{0} {\overset{ˉ}{e}}^{3} < 0$ , and every $m_{0} {\hat{f}}_{A} = m_{0} f_{A} < 0$ . All source sign conditions hold.

1.3A2givenalgebra

The bulk formula in [A2] now gives $b_{i} = 0$ and $v_{i} = \frac{1}{\overset{ˉ}{e}} \sqrt{\frac{5 {\overset{ˉ}{e}}^{3}}{3 κ m_{0}}} = \sqrt{\frac{5}{3 κ m_{0}}} {\overset{ˉ}{e}}^{1 / 2} > 0.$ For every one of the nine exceptional divisors it gives $b_{A} = 0$ and $v_{A} = \sqrt{- \frac{10 f_{A}}{3 β m_{0}}} > 0.$ Thus $t_{i} = i v_{i}$ and $t_{A} = - i v_{A}$ : the positive $v_{A}$ denotes a magnitude, not the exceptional imaginary coordinate. The branch therefore fixes all twelve geometric Kähler coordinates. Its dilaton formula has positive right-hand side and fixes $e^{D}$ ; the source equation $ξ = e_{0} / p$ fixes the RR axion. As $h^{2, 1} = 0$ in the adopted model, no complex-structure moduli are omitted.

2.1A2givenstep 1.3algebra∎

Take the allowed integer flux sequence $N \to \infty$ with $c_{4}$ fixed, so $\overset{ˉ}{e} = c_{4} N \to \infty$ . With $κ, β, m_{0}, p, f_{A}$ fixed, the bulk term $κ v_{1} v_{2} v_{3}$ grows as ${\overset{ˉ}{e}}^{3 / 2}$ while the nine blow-up terms remain fixed; so $V = A {\overset{ˉ}{e}}^{3 / 2} - B$ with $A : = κ (5 / (3 κ m_{0}))^{3 / 2} > 0$ and $B : = (β / 6) \sum_{A} v_{A}^{3} > 0$ fixed. It is positive for every integer $N$ with $(c_{4} N)^{3 / 2} > B / A$ , and $V \sim {\overset{ˉ}{e}}^{3 / 2}$ . The exact formula $e^{D} = 5 ∣ p ∣ / (4 \sqrt{2} ∣ m_{0} ∣ V)$ in [A2] gives $e^{D} \sim {\overset{ˉ}{e}}^{- 3 / 2}$ , and $e^{ϕ} = e^{D} \sqrt{V}$ gives $g_{s} = e^{ϕ} \sim {\overset{ˉ}{e}}^{- 3 / 4}$ . In the source's four-dimensional Einstein frame, $∣ Λ_{4} ∣ / M_{P}^{4} \sim {\overset{ˉ}{e}}^{- 9 / 2}$ , $M_{P} ℓ_{s} \sim {\overset{ˉ}{e}}^{0}$ , and $(R_{K K} / ℓ_{s})^{2} \sim {\overset{ˉ}{e}}^{7 / 2}$ . Thus $(H_{A d S} R_{K K})^{2} = \frac{1}{3} \frac{∣ Λ_{4} ∣}{M_{P}^{4}} (M_{P} R_{K K})^{2} \sim {\overset{ˉ}{e}}^{- 1},$ where $H_{A d S}^{2} : = ∣ Λ_{4} ∣ / (3 M_{P}^{2})$ and $Λ_{4}$ is an energy density. The hierarchy $∣ {\hat{e}}_{i} ∣ ≫ ∣ {\hat{f}}_{A} ∣ ≫ ∣ m_{0} ∣$ is the source's dimensionless string-unit control regime; the calculation remains within the constant-mode approximation.

DGKT massive IIA flux family with a smeared O6 truncation

Example

Facts & Assumptions

Verification

Source references

Depends on

Dependency tree · 2 levels

Sources