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

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Gts odd factor even total r3 parity

Example

Explicit CY4 degree-minus-one self-dual loop changes factor parity while an admitted total remains even. Compute the parity of every subsum and expose the unauthorized odd 5.10 application, without claiming a full counterexample.

Example references

Bojko, Wall-crossing for Calabi–Yau fourfolds: framework, tools, and applications, Assumptions 5.1 and 5.10; Definition 5.3; §6.2; §6.3; Assumption 6.9; Remark 6.10; Example 6.11; Theorem 6.12; Lemma 6.13; Equation (6.14); §7.2.

Bojko, Kuhn, Liu, and Thimm, Wall-crossing for equivariant DT4 invariants, §2.2.8, §3.2, Assumption 4.2.4, and Theorem 1.2.1; used only under their stated hypotheses.

Setup

The ambient category is ${Rep}_{C} (b \to a)$ , with dimension lattice $Γ = Z e_{a} \oplus Z e_{b}$ . Use the CY4 completion with one self-dual degree-minus-one loop at $a$ and the usual degree-minus-two dual to the degree-zero arrow. Modulo $2$ , the middle loop contributes $d_{a}^{2} \equiv d_{a}$ , while vertex and ordinary-arrow/dual terms are even. The diagnostic admitted monoid is $Γ_{e v e n} = {(d_{a}, d_{b}) \in N^{2} : d_{a} is even}$ ; this is a parity model, not an assertion that every source 5.1 decomposition hypothesis holds.

Use the source-style filtration $0 \subset Z e_{a} \subset Γ$ with $Z_{1} (e_{a}) = i$ and $Z_{2} (e_{b}) = - 1$ . It gives phase $1 / 2$ when $d_{b} = 0$ and phase $1$ when $d_{b} > 0$ , with weak see-saw equal to the maximum. For any representation $f : B \to A$ , $(B, im f)$ is a phase-one surjective-arrow subrepresentation and $(0, A / im f)$ has phase one-half, so this is an HN filtration after omitting zero terms. The canonical diagnostic framing is $F r (B, A) = A \oplus B$ , exact on the ambient representation category; no 5.3 uniformity or CY4 virtual class is asserted. Let the torus $T = G_{m}$ act trivially on the degree-zero quiver and loop.

Let $T_{0} = (C_{b} \overset{1}{\to} C_{a})$ and $S = (C_{b} \to 0)$ . Then $T_{0}$ is stable of odd self-Euler parity and $S$ is stable of even parity. The admitted total is $E = T_{0}^{\oplus 6} \oplus S^{\oplus 4}$ , of dimension $(d_{a}, d_{b}) = (6, 10)$ . The actual ambient subrepresentations of $E$ are pairs $(B^{'} \subseteq C^{10}, A^{'} \subseteq C^{6})$ with $π (B^{'}) \subseteq A^{'}$ , where $π$ is projection. In the diagnostic inherited exact structure, only sequences whose three class vectors have even $a$ -coordinate are admitted. The split summands $T_{0}$ and odd direct subsums are ambient objects, not admitted 5.10 classes. The physical torus is trivial and the calculation uses no orientation-dependent virtual class.

For every direct subsum $T_{0}^{\oplus u} \oplus S^{\oplus v}$ with $0 \leq u \leq 6$ and $0 \leq v \leq 4$ , its self-Euler parity is $u (m o d 2)$ . Thus odd subsums are exactly those with odd $u$ , while the total $u = 6$ is even. The example tests an odd-subsum proof shortcut; it does not claim the diagnostic datum satisfies all original 5.1(a) or 5.10 hypotheses.

Facts & Assumptions

Given: the exact data and all unresolved suppliers retain their stated status.

[F1]

Gts weak phase framing moduli datum: exact planned interface: “Specify original weak stability, exact phase category, actual ordinary-vs-admissible maps, original concentration opens and exact Fr_k, common-line framing, even emergent class domain, torus linearizations and orientations. No sum closure, cofinal-index availability, or abelian phase category is implicit.” The link records scope, not proof acceptance.

Context only — no mathematical supplier use

lem-gts-parity-safe-admitted-total-closed-orbit-reductivity name the held construction interfaces discussed by this diagnostic. Their conclusions are not assumed or used to establish the calculations below; this example does not prove those interfaces.

Verification

1.1givenF1algebra

The CY4 Euler contributions are $2 d_{a}^{2} + 2 d_{b}^{2}$ from vertices and $- 2 d_{a} d_{b}$ from the ordinary arrow and its dual; each is even. The single self-dual middle loop contributes $d_{a}^{2}$ . Hence $χ (d, d) \equiv d_{a} (m o d 2)$ . Therefore $T_{0}$ is odd, $S$ is even, and $E$ is even because $d_{a} (E) = 6$ .

1.2givenF1algebra

The representation $T_{0}$ has only the proper sink subobject $(C_{a}, 0)$ , of phase $1 / 2$ , while its quotient has phase $1$ ; so it is stable. The representation $S$ is simple. Their direct sum has surjective arrow $C^{10} \to C^{6}$ . For any proper subrepresentation with nonzero source space, its source quotient is nonzero: if the source spaces agreed, surjectivity would force the target spaces to agree too. Thus both it and the quotient have phase one. A source-zero subobject has phase $1 / 2$ and source-positive quotient. This verifies semistability directly.

2.1givenF1step 1.1algebra

In the stabilizer of this surjection, choose splittings $C^{10} = C^{6} \oplus C^{4}$ . The subgroup acting by $(\begin{array}{cc} I_{6} & 0 \\ N & I_{4} \end{array})$ , for arbitrary $N : C^{6} \to C^{4}$ , preserves the arrow. It is nonzero and unipotent because $N^{2} = 0$ . The full stabilizer consists of $g_{a} = A \in {G L}_{6}$ and $g_{b} = (\begin{array}{cc} A & 0 \\ N & D \end{array})$ with $D \in {G L}_{4}$ ; hence the displayed additive subgroup $Hom (C^{6}, C^{4})$ is normal, by block-matrix conjugation. It is therefore a nontrivial connected normal unipotent subgroup, so the stabilizer is nonreductive. Thus the total even parity does not alone imply a reductive stabilizer.

3.1givenF1step 1.2step 2.1algebra∎

This is a diagnostic against inferring odd-factor reductivity from total parity; the full source hypotheses are not all checked, so it is not a counterexample to the original theorem.

Depends on

Gts weak phase framing moduli datum

Used by

Nothing in the library uses this result yet.

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Sources

Y. Bojko, Wall-crossing for Calabi–Yau fourfolds: framework, tools, and applications (standard reference, not scraped)
Bojko, Kuhn, Liu, and Thimm, Wall-crossing for equivariant DT4 invariants (standard reference, not scraped)