ExampleConstruction: Literature-sourcedVerification: 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.

Local A1-Fibered Calabi–Yau Engineering of Pure $N = 2$ SU(2)

Physical scope

The source-reported local Type IIA A1-fibered pure N=2 SU(2) branch, with a static single-Abelian D2 probe and the stated scale limits. It is not a globally compact vacuum, and its remaining threshold conditions are explicit inputs rather than a reconstructed full spectrum.

Example

Let $Y_{ϵ}$ be the source-specified resolved local Type IIA Calabi–Yau threefold, with a holomorphic map $π_{ϵ} : Y_{ϵ} \to B$ to $B = P_{b}^{1}$ . A selected smooth K3 fiber contains an exceptional holomorphic curve $C_{f} ≅ P_{f}^{1}$ in the chosen local resolution of the quotient surface germ $C^{2} / {I, - I}$ , where $- I$ acts by $(z_{1}, z_{2}) \mapsto (- z_{1}, - z_{2})$ ; the exceptional curve is the inverse image of the quotient point. This is the local $A_{1}$ surface singularity and its resolution. A K3 fiber means a compact simply connected smooth complex surface with trivial canonical bundle. Let $C_{b} ≅ B$ be the selected base curve in the local threefold. Write $A_{f} (ϵ)$ and $A_{b} (ϵ)$ for the areas induced by the string-frame Kähler form, and set $t_{f} = A_{f} / α^{'}$ and $t_{b} = A_{b} / α^{'}$ . Use natural units, $ℓ_{s} = \sqrt{α^{'}} = ϵ ↓ 0$ , a fixed dimensionless string coupling $g_{s t r} = e^{Φ_{\infty}} > 0$ , and fixed positive energy parameters $a$ and $Λ$ , where $a$ is the selected Coulomb-branch scalar coordinate and $Λ$ the strong scale. Suppose the source-normalized limits satisfy $t_{f} / (ϵ a) \to c_{f} > 0$ and $e^{- t_{b}} / (ϵ^{4} Λ^{4}) \to c_{b} > 0$ . The selected field theory is pure four-dimensional $N = 2$ $S U (2)$ gauge theory, meaning a theory with eight real supercharges, gauge group $S U (2) = {U \in M_{2} (C) : U^{*} U = I, \det U = 1}$ , and no matter hypermultiplets. The source model assigns oppositely oriented D2-branes on $C_{f}$ to $W^{\pm}$ BPS states, with $M_{W} = ∣ Z (W^{\pm}) ∣$ ; take a static fiber probe $ι_{f} : W_{f} = R_{t} \times C_{f} ↪ X^{10}$ , where $X^{10}$ is the ten-dimensional background spacetime, with induced string-frame metric $- d t^{2} \oplus h_{f}$ , constant $Φ = Φ_{\infty}$ , vanishing pulled-back $B_{2}$ and $C_{3}$ , zero worldvolume curvature $F_{A}$ , and no backreaction. For the base-wrapped sector take a static probe $ι_{b} : W_{b} = R_{t} \times C_{b} ↪ X^{10}$ with induced metric $- d t^{2} \oplus h_{b}$ and the same constant-dilaton, zero-pullback, zero-curvature, and no-backreaction assumptions. Adopt the source's six-dimensional gauge zero-mode sector on $M_{4} \times B$ and the fixed strong scale $Λ$ . The field-theory energy cutoff is $E_{F T} : = \max {a, Λ}$ ; assume $M_{P l} (ϵ) / E_{F T} \to \infty$ and $M_{q} (ϵ) / E_{F T} \to \infty$ for every other unwanted threshold $M_{q}$ not calculated here. Under the stated DBI normalization, the fiber D2 rest mass tends $c_{f} a / ((2 π)^{2} g_{s t r})$ , $t_{f} \to 0$ , $t_{b} \to \infty$ , the string scale and base-wrapped D2 threshold diverge, and the bare string-scale gauge coupling tends zero. This source-specific local limit does not assert a compact completion satisfying global flux and brane consistency conditions.

Example references

Type IIA Local Calabi–Yau Geometric Engineering Limit

Special Kähler period data in Calabi–Yau threefold reductions

Supergravity Conventions And Normalizations

Type Iia Two Derivative Supergravity Action

Type II Dp Brane DBI WZ Action

The Axiom of Choice

Katz–Klemm–Vafa, §2 and §2.1

Mayr, §2

Polchinski, Chaudhuri, and Johnson, Lecture II §2.5

Facts & Assumptions

Given: The local fibration and source-reported pure $N = 2$ SU(2) dictionary in the Example; $ℓ_{s} = ϵ ↓ 0$ ; fixed $g_{s t r} > 0$ , $a > 0$ , and $Λ > 0$ ; the two stated source-normalized area limits; and the stated probe and six-dimensional gauge-sector hypotheses.

[F1]

Type IIA Local Calabi–Yau Geometric Engineering Limit defines the local smooth/resolved geometry, dimensionless curve areas, charge map, field-theory data, and explicit decoupling requirements. It does not assert existence of a family satisfying them.

[F2]

Special Kähler period data in Calabi–Yau threefold reductions states compact special-Kähler period data under smooth proper compact Kähler CY3, locally constant Hodge-number, locally free relative-filtration, flat rational polarization, and marked torsion-free degree-three lattice hypotheses. Its carrier exists, but its current Step 1 decision is escalated: that record cites an escalated four-dimensional $N = 2$ postulate whose own blocker says the conventions carrier is absent, although that conventions carrier and its current ready record are present. This example uses only the actual compact-family statement as a conditional scope boundary, not as an accepted proof of a local period result; its local $a$ and charge dictionary are source-model inputs.

[F3]

Supergravity Conventions And Normalizations fixes $α^{'}$ as a length-squared parameter and $ℓ_{s} = \sqrt{α^{'}}$ in natural units; it identifies the dimensionless coupling $g_{s t r} = e^{Φ_{\infty}}$ separately from the string-frame metric.

[A1]

Adopt the Axiom of Choice (AC): every family of nonempty sets has a choice function, as defined in The Axiom of Choice. It is inherited through the compact period/Hodge supplier [F2]; no choice argument is used in the DBI or limit calculations.

[A2]

Type Iia Two Derivative Supergravity Action is the source-free leading bulk Type IIA model. It supplies the string-frame bulk context only; the D2 probe action is a separate physical input.

[A3]

Type II Dp Brane DBI WZ Action adopts a single-Abelian D $p$ -brane probe action with Dirac–Born–Infeld (DBI) term $- μ_{p} \int_{W} e^{- Φ} \sqrt{- \det (P [g^{(s)}] + F)}$ and Wess–Zumino (WZ) term, where $F_{A} = d A$ , $F = 2 π α^{'} F_{A} - P [B_{2}]$ , $μ_{p} = ((2 π)^{p} α^{' (p + 1) / 2})^{- 1}$ , and the constant-dilaton tension is $T_{p} = μ_{p} / g_{s t r}$ . Here $g^{(s)}$ denotes the string-frame metric and $g_{s t r} = e^{Φ_{\infty}}$ the dimensionless coupling; these are the same metric and coupling that the supplier now denotes separately by $G^{(s)}$ and $g_{s t r}$ . With $p = 2$ this is the D2 probe used here. The WZ contribution vanishes for the explicitly assumed zero pullbacks and zero curvature; the supplier's explicit local RR-potential crosswalk is unused because this WZ contribution is zero.

[A4]

Katz–Klemm–Vafa and Mayr supply the physical model dictionary: the fiber D2 orientations are the $W^{\pm}$ states, the base-wrapped D2 states belong to the extra $S U (2)_{B a s e}$ sector, the fiber area sets the W mass, the base volume controls the four-dimensional bare coupling, and the pure asymptotically-free SU(2) limit holds $Λ$ and the Coulomb parameter fixed. Here asymptotically free means that the gauge coupling decreases as energy increases. The source's $b = 0$ conformal case is excluded.

[A5]

For the base-coupling calculation, adopt the source's six-dimensional Yang–Mills zero-mode sector on $M_{4} \times B$ , with product metric and a gauge field $A_{μ} (x)$ independent of $B$ , where $M_{4}$ is the four-dimensional spacetime and the curvature is $F = d A + A \land A$ . Its action and normalization are $S_{6} = - \frac{1}{4 g_{6}^{2}} \int_{M_{4} \times B} d^{4} x d^{2} y \sqrt{- g_{4}} \sqrt{h_{b}} Tr (F_{μ ν} F^{μ ν}), g_{6}^{2} = {\hat{g}}_{6}^{2} ℓ_{s}^{2},$ where ${\hat{g}}_{6} = 1$ is the adopted source-matched dimensionless normalization, and the trace has the same fixed gauge-generator normalization as the four-dimensional coupling in the Definition. The coefficient obtained after integrating this zero mode is the bare coupling evaluated at $M_{s}$ ; this convention gives $g_{4}^{- 2} = t_{b}$ , matching the coupling normalization in the source scaling [A4]. This is the scoped source input behind Mayr's equation (2.6), not a consequence of the source-free ten-dimensional bulk action. A different fixed six-dimensional normalization would require changing the exponential fixed- $Λ$ scaling in the Statement consistently.

[A6]

The source's $M_{P l} (ϵ) / E_{F T} \to \infty$ limit and the condition $M_{q} (ϵ) / E_{F T} \to \infty$ for every remaining unmodeled threshold are adopted hypotheses. This example calculates the string scale and the named base/fiber D2 thresholds only; it does not reconstruct the full Kaluza–Klein spectrum or a compact global completion.

Verification

technique · conditional DBI determinant and dimensional-reduction calculations followed by direct asymptotic limits

Given: The local $A_{1}$ fibration, $A_{f} = ℓ_{s}^{2} t_{f}$ , $A_{b} = ℓ_{s}^{2} t_{b}$ , static D2 probe and source dictionary [A3]–[A6], with the units and local-model boundary in [F1]–[F3].

1.1F1F2A1A4given

By the source-specific input [A4], the two orientations of the exceptional fiber curve label $W^{\pm}$ in the selected pure SU(2) theory; this is a model assignment, not a consequence of broad CY data [F1]. The compact supplier [F2] applies only to its smooth proper compact-family hypotheses, so it supplies no unproved local period matching in this example.

1.2A2A3A4F3givenalgebra

For $p = 2$ , [A3] gives $μ_{2} = ((2 π)^{2} ℓ_{s}^{3})^{- 1}$ and $T_{2} = μ_{2} / g_{s t r}$ . Since $F_{A} = 0$ and the pulled-back $B_{2}$ and $C_{3}$ vanish, $F = 0$ and the WZ contribution is zero under [A3]. On the static worldvolume the induced metric is $- d t^{2} \oplus h_{f}$ , so $- \det P [g^{(s)}] = \det h_{f}$ and the DBI action per unit time is $- T_{2} \int_{C_{f}} d A_{s} = - T_{2} A_{f}$ . Thus the rest energy is $M_{W} = T_{2} A_{f} = t_{f} / ((2 π)^{2} g_{s t r} ℓ_{s})$ ; the source-free bulk action [A2] alone does not supply this localized probe action, and no higher-derivative correction enters this calculation.

2.1F3A4step 1.2givenalgebra

Since $ℓ_{s} = ϵ$ and $t_{f} / (ϵ a) \to c_{f} > 0$ , the formula in step 1.2 gives $M_{W} \to c_{f} a / ((2 π)^{2} g_{s t r})$ , which is finite and positive, while $M_{s} = ϵ^{- 1} \to \infty$ . Also $e^{- t_{b}} / (ϵ^{4} Λ^{4}) \to c_{b} > 0$ and $ϵ^{4} Λ^{4} \to 0$ , hence $e^{- t_{b}} \to 0$ and $t_{b} = - \log (e^{- t_{b}}) \to + \infty$ .

2.2A3A4step 1.2algebra

The base probe has the same stated DBI assumptions as the fiber probe, so its rest energy is $M_{b} = T_{2} A_{b} = t_{b} / ((2 π)^{2} g_{s t r} ϵ)$ . Because $t_{b} \to \infty$ and $ϵ ↓ 0$ , $M_{b} \to \infty$ ; this explicitly realizes the source's decoupling of the base-wrapped $S U (2)_{B a s e}$ states in the asymptotically-free branch.

3.1A4A5step 2.1algebra

For the zero mode independent of the base, integrating the declared six-dimensional gauge action [A5] over $B$ multiplies the four-dimensional gauge kinetic term by $A_{b}$ : $g_{4}^{- 2} = A_{b} / g_{6}^{2} = (ℓ_{s}^{2} t_{b}) / (ℓ_{s}^{2}) = t_{b}$ . This is the source normalization $g^{- 2} = t_{b}$ used in the fixed- $Λ$ scaling [A4]. Hence $g_{4} (M_{s}) \to 0$ as $t_{b} \to \infty$ . The fixed $Λ$ is a separate input; it is not the bare coupling at $M_{s}$ .

4.1F1A6step 2.1step 2.2step 3.1∎

The remaining $M_{P l}$ and unmodeled-threshold conditions are exactly the explicit hypotheses [A6] required by [F1]. The calculations above verify the listed string and base/fiber D2 scales only; they do not prove a compact completion, global flux and brane-charge cancellation, or a complete Kaluza–Klein (KK) mass spectrum for modes associated with the internal dimensions.

Remarks

The fixed quantity is the low-energy gauge-theory data, especially $Λ$ ; the bare coupling at the string scale runs to zero in this asymptotically-free branch. For a conformal $b = 0$ theory, Mayr notes that a large-base limit does not decouple the base-wrapped sector, so that branch is outside this example.

This diagram records only the specified holomorphic map to the base. The exceptional curve $C_{f}$ lies in the chosen resolved fiber over a selected base point; the diagram does not assert a product fibration or a compact global completion, and no proof step uses it.

Local A1-Fibered Calabi–Yau Engineering of Pure $N = 2$ SU(2)

Example

Example references

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

Local A1-Fibered Calabi–Yau Engineering of Pure N=2N=2 SU(2)

Example

Example references

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

Local A1-Fibered Calabi–Yau Engineering of Pure $N = 2$ SU(2)