ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-11-04 (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.

An explicit Liouville four-point charge tuple in the strict domain

Example

Set $γ = 1$ , $μ = 1$ , $z = 1 / 2$ , and $α_{1} = α_{2} = α_{3} = α_{4} = 13 / 10$ in the probabilistic Liouville sphere convention. Then $Q = 5 / 2$ , each charge is below $Q$ , each pair sum is above $Q$ , and $0 < γ < \sqrt{2}$ . The insertion points $(0, 1 / 2, 1, \infty)$ are distinct. This is an exact example of parameters in the strict source domain; it gives no value for the four-point function and proves no bootstrap identity.

Example references

Probabilistic Liouville CFT on the sphere

Facts & Assumptions

Given: The rational parameters displayed in the Example.

[F1]

Probabilistic Liouville CFT on the sphere: $Q = 2 / γ + γ / 2$ , $Δ_{α} = \frac{α}{2} (Q - \frac{α}{2})$ , and the real-charge Seiberg domain is $α_{i} < Q$ and $\sum_{i} α_{i} > 2 Q$ .

Verification

technique · direct exact arithmetic

1.1F1givenalgebra

Substituting $γ = 1$ gives $Q = 2 + \frac{1}{2} = \frac{5}{2}$ and $Δ_{13 / 10} = \frac{13}{20} (\frac{5}{2} - \frac{13}{20}) = \frac{481}{400}$ . The associated central charge is $1 + 6 Q^{2} = 77 / 2$ .

2.1F1givenstep 1.1algebra

Each $α_{i} = \frac{13}{10} < \frac{25}{10} = Q$ , and each pair sum is $\frac{26}{10} > \frac{25}{10} = Q$ ; the total is $\frac{52}{10} > \frac{50}{10} = 2 Q$ . Also $1 < \sqrt{2}$ since both sides are positive and $1 < 2$ .

3.1givenalgebra∎

The four points $0, 1 / 2, 1, \infty$ are pairwise distinct and $0 < ∣ 1 / 2 ∣ < 1$ . Thus every displayed parameter and insertion inequality is strict.

This verifies parameter membership only; it does not use the escalated four-point theorem as a premise.

Depends on

Probabilistic Liouville CFT on the sphere