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

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Gts weak phase ordinary low kernel

Example

Surjective-arrow weak maximum phase: ordinary intersections/kernels can lie LOW although ambient objects and image are critical. Distinguish diagnostic from a model satisfying all 5.10.

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

Let $Q$ be the one-arrow quiver $b \to a$ . A representation is a map $f : B \to A$ of finite-dimensional complex vector spaces. Use the source-style filtration $0 \subset Z e_{a} \subset Z e_{a} \oplus Z e_{b}$ , with $Z_{1} (e_{a}) = i$ and $Z_{2} (e_{b}) = - 1$ . It gives phase $ϕ (B, A) = 1$ for $B \neq 0$ and $1 / 2$ for $B = 0$ . Since source dimensions add in exact sequences, the phase satisfies weak see-saw as the maximum of the two phases. Use the semistability convention $ϕ (U) \leq ϕ (E / U)$ for every proper nonzero subrepresentation. Every representation has an HN filtration: $(B, im f)$ is the phase-one subobject and its quotient $(0, A / im f)$ has phase $1 / 2$ .

Set $E = (C^{2} \overset{I}{\to} C^{2})$ and $F = (C^{3} \overset{π}{\to} C^{2})$ with $π (k, w_{1}, w_{2}) = (w_{1}, w_{2})$ . Define $h_{B} (e_{1}) = k$ , $h_{B} (e_{2}) = w_{1}$ and $h_{A} (e_{1}) = 0$ , $h_{A} (e_{2}) = w_{1}$ . The commuting square is

The diagram records the actual linear maps and the equality $h_{A} f_{E} = f_{F} h_{B}$ ; the proof below uses the matrices, not the drawing. This is a diagnostic weak-phase example only; no claim is made that its CY4 master satisfies all original 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.

Verification

1.1givenF1algebra

If $f : B ↠ A$ is surjective and $B \neq 0$ , then $(B, A, f)$ has phase $1$ and is semistable. For a proper subrepresentation $(B^{'}, A^{'})$ , if $B^{'} = 0$ its phase is $1 / 2$ and the quotient has nonzero $B / B^{'}$ and phase $1$ . If $B^{'} \neq 0$ , its phase is $1$ ; if $B^{'} = B$ , surjectivity forces $A^{'} = A$ , so a proper subobject must instead have $B / B^{'} \neq 0$ , whose quotient phase is $1$ . Each case satisfies $ϕ (B^{'}, A^{'}) \leq ϕ ((B, A) / (B^{'}, A^{'}))$ .

2.1givenF1step 1.1algebra

Both $E$ and $F$ are surjective-arrow objects, so 1.1 makes them phase- $1$ semistable. The morphism commutes since $f_{F} h_{B} (e_{1}) = 0 = h_{A} f_{E} (e_{1})$ and $f_{F} h_{B} (e_{2}) = w_{1} = h_{A} f_{E} (e_{2})$ . Its image has $B$ -space $⟨ k, w_{1} ⟩$ , $A$ -space $⟨ w_{1} ⟩$ , and a surjective induced arrow; hence its image is also phase- $1$ semistable by 1.1.

3.1givenF1step 2.1algebra

The map $h_{B}$ is injective and $\ker h_{A} = C e_{1}$ . Therefore the ordinary kernel has $B = 0$ and $A = C e_{1}$ , so its phase is $1 / 2$ . Thus the ambient objects and image are critical while the ordinary kernel is low phase.

4.1givenF1step 3.1algebra∎

This calculation proves that weak-phase semistability alone does not place ordinary kernels in the exact phase category. It is diagnostic and does not verify the original CY4 obstruction, properness, or 5.10 hypotheses.

Depends on

Gts weak phase framing moduli datum

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

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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)