ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passverified 2026-11-05 (gpt-6.1-sol)

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A 2026 G2 vertical-flux model with charge 657

Physical scope

Source-reported arithmetic for one fixed toric Calabi–Yau fourfold, base, G2 divisor, vertical flux, charge normalization, and source tadpole convention. The local supersymmetry/transversality and R3-normalized tadpole carriers are contextual comparisons, not premises of the source-unit arithmetic. No source-unit field crosswalk, universal charge bound, complete realistic vacuum, or resolution of the cited codimension-two (4,6) sectors is asserted.

Example

Li reports a fixed four-dimensional F-theory model with a toric Calabi–Yau fourfold over a toric threefold base. In the source's model, the base divisor $D^{19}$ has local surface geometry $F_{0} = P^{1} \times P^{1}$ and supports a rigid $G_{2}$ gauge factor. A vertical flux is a source-normalized four-form class in the source's integral span of products of divisor classes. Under the source's flux quantization, transversality, primitivity, and tadpole conditions, the choice $(a, b) = (329, 657)$ with flux parameters $(n_{S}, n_{F}) = (1, - 1)$ breaks this $G_{2}$ factor to the abelian gauge group generated by $T = 329 T_{1} + 657 T_{2}$ .

In the source's charge normalization, the adjoint charge set is $q = 1, 327, 328, 329, 330, 657$ together with the conjugate charges. The thesis reports that charge $657$ occurs in a nonzero vectorlike sector of this model; the reported multiplicity uses a separate source calculation. Thus $657$ is the largest charge in this model's displayed charge set, not a bound for all F-theory compactifications. The thesis treats any landscape-wide upper-bound interpretation as heuristic.

The model retains other gauge factors; the source adds flux to break the adjacent $S U (2)$ when decoupling it from the constructed $U (1)$ . The thesis also identifies codimension-two $(4, 6)$ sectors on other divisors as probably conformal and not fully understood. This example does not claim a complete realistic vacuum.

The flux and tadpole equations here use Li's own $G_{4}$ notation and normalization. They are not identified with the R3-normalized class $g_{4} = [G_{4} / (2 π ℓ_{p})^{3}]$ of the contextual local tadpole carrier without the separate physical field identification and source crosswalk.

Example references

F-theory flux transversality and supersymmetry conditions

Li, §2.2.8

Li, §§3.2.1–3.2.3

Li, §§5.4.1–5.4.2

Facts & Assumptions

Given: Li's fixed toric fourfold and base, its source-normalized vertical flux and charge conventions, the source-reported flux conditions, and the source data below. The local verification checks the source's stated arithmetic; it does not establish the physical field/source crosswalk.

[A1]

Li's source data: ${\hat{Y}}_{4}$ is its fixed resolved fourfold, the base divisor is denoted $D^{19}$ and is locally $F_{0}$ with a $G_{2}$ factor, and vertical flux lies in the source's integral span of products of divisor classes in $H_{v e r t}^{2, 2} ({\hat{Y}}_{4})$ . The flux-breaking ansatz uses coprime integers $(a, b)$ and parameters $n_{S}, n_{F}$ . For $b$ divisible by $3$ , the source permits integer $n_{S}, n_{F}$ and gives $ϕ_{1 S} = (a - 2 b / 3) n_{S}$ , $ϕ_{2 S} = (2 a - b) n_{S}$ . The source's generator is $T = a T_{1} + b T_{2}$ , and its adjoint charges are $a$ , $b$ , $3 a - b$ , $2 a - b$ , $a - b$ , and $3 a - 2 b$ , with conjugates. Li, §2.2.8 and §§5.4.1–5.4.2

[A2]

In Li's own conventions, Eq. (2.62) is $N_{D 3} + \frac{1}{2} \int_{{\hat{Y}}_{4}} G_{4} \land G_{4} = χ ({\hat{Y}}_{4}) / 24$ with $N_{D 3} \geq 0$ . For the selected model the thesis gives $χ ({\hat{Y}}_{4}) = 1,737,888$ . Eq. (5.20) gives $\frac{1}{2} \int_{{\hat{Y}}_{4}} G_{4} \land G_{4} = - 2 (a^{2} - a b + b^{2} / 3) n_{S} n_{F}$ , and Eq. (5.21) imposes $a^{2} - a b + b^{2} / 3 \leq 36206$ for the selected model. These are the thesis's source conventions; no physical identification with the R3-normalized class is assumed. Li, §2.2.8 and §5.4.2

[A3]

The thesis separately reports nonzero vectorlike multiplicity at charge $657$ in this model, based on a distinct source calculation; that multiplicity is not derived by the charge arithmetic below. Li, §5.4.2

Verification

1.1A1givenalgebra

Since $657 = 2 \cdot 329 - 1$ , $\gcd (329, 657) = 1$ , and $657 = 3 \cdot 219$ . Thus the source's coprimality and $3 ∣ b$ conditions hold. With $(n_{S}, n_{F}) = (1, - 1)$ , its flux formula gives $ϕ_{1 S} = 329 - 2 (657 / 3) = - 109$ and $ϕ_{2 S} = 2 \cdot 329 - 657 = 1$ , agreeing with the source's displayed choice.

2.1A2step 1.1algebra

The source's quadratic expression is $329^{2} - 329 \cdot 657 + 657^{2} / 3 = 108241 - 216153 + 143883 = 35971$ , which is at most $36206$ . Its source-normalized flux contribution is therefore $2 \cdot 35971 = 71942$ . The same thesis gives $χ ({\hat{Y}}_{4}) = 1737888$ , so $χ / 24 = 72412$ and the source's integrated equation leaves $72412 - 71942 = 470$ units of its residual space-filling-brane charge. This verifies only the thesis's arithmetic in its own conventions.

2.2A1A3step 1.1algebra

Substituting $(a, b) = (329, 657)$ into the six source charges gives $329$ , $657$ , $987 - 657 = 330$ , $658 - 657 = 1$ , $329 - 657 = - 328$ , and $987 - 1314 = - 327$ . Taking absolute values yields exactly $1, 327, 328, 329, 330, 657$ , whose maximum is $657$ . The thesis separately reports the nonzero vectorlike multiplicity in [A3]; this arithmetic does not prove that multiplicity or any universal upper bound.

3.1A1A2step 2.1step 2.2∎

The locally verified content is the fixed model and source-unit arithmetic in [A1]–[A2]. These source data do not identify Li's $G_{4}$ with an independently normalized R3 class. The calculation establishes no universal charge bound or complete vacuum.

Remarks

The proposed item identifier contains “e7”, but the cited thesis's §5.4 model breaks a $G_{2}$ factor on $D^{19}$ to $U (1)$ ; the display title and content follow the source. The source's reference to the maximal charge as a plausible landscape bound is heuristic, not a theorem. Its $(4, 6)$ sectors, other gauge factors, superpotential masses, and normalization conventions remain part of the scope.

Context comparison

The supersymmetry/transversality and R3 tadpole items in the references are contextual comparisons only. Their physical field and source conventions are not identified with Li’s conventions here, and neither is used to derive the arithmetic above.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Shing Yan Li, Applications of Flux Compactifications in F-Theory, MIT PhD thesis, arXiv:2608.14844v1