ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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.

Allowed and excluded order-two forward coefficients

Example

At truncation order $N = 2$ , let $(b_{0}, b_{1}, b_{2})$ denote the normalized forward on-shell amplitude coordinates and let $K_{2}$ be the finite feasible set defined by the order-two constraints. The strict region $U_{2} = {b_{0} > 0, b_{1} > 0, b_{0} - b_{1} > 0, b_{0} b_{2} - b_{1}^{2} > 0}$ is contained in $K_{2}$ . The strict region $V_{2} = {b_{0} > 0, b_{1} > 0, b_{0} - b_{1} > 0, b_{0} b_{2} - b_{1}^{2} < 0}$ is disjoint from $K_{2}$ . The vectors $b^{p a s s} = (1, \frac{1}{2}, \frac{1}{3}) \in U_{2}$ and $b^{f a i l} = (1, \frac{1}{2}, \frac{1}{8}) \in V_{2}$ witness that both regions are nonempty. The first region passes the finite necessary test; this example does not assert that any vector passing that test extends to full analytic amplitude data or a physical theory, and it assigns no sign to a generic off-shell Lagrangian coefficient.

Example references

Finite truncated forward-amplitude bootstrap data

Finite truncated forward-positivity constraints

Forward positivity for a gapped identical scalar

de Rham et al., Positivity Bounds for Scalar Theories, §2.2

Basile et al., Lectures in Quantum Gravity, §3.3.1

Facts & Assumptions

Given: The normalized on-shell coordinates $b_{0}, b_{1}, b_{2}$ , the definition of $K_{2}$ , and the finite-constraint proposition.

[F1]

Finite truncated forward-amplitude bootstrap data: for $N = 2$ , the applicable constraints are $b_{0} \geq 0$ , $(\begin{array}{cc} b_{0} & b_{1} \\ b_{1} & b_{2} \end{array}) ⪰ 0$ , $b_{1} \geq 0$ , and $b_{0} - b_{1} \geq 0$ .

[F2]

Finite truncated forward-positivity constraints: every amplitude-data vector under the stated gapped-scalar hypotheses lies in $K_{2}$ .

Verification

1.1F1algebra

For any $b \in U_{2}$ , the order-one Hankel quadratic form is $b_{0} u^{2} + 2 b_{1} u v + b_{2} v^{2} = b_{0} (u + (b_{1} / b_{0}) v)^{2} + ((b_{0} b_{2} - b_{1}^{2}) / b_{0}) v^{2}$ , which is positive for every nonzero $(u, v)$ because $b_{0} > 0$ and $b_{0} b_{2} - b_{1}^{2} > 0$ . The remaining order-two inequalities are $b_{0} > 0$ , $b_{1} > 0$ , and $b_{0} - b_{1} > 0$ by [F1]; hence $U_{2} \subseteq K_{2}$ . For $b^{p a s s} = (1, \frac{1}{2}, \frac{1}{3})$ , the order-one determinant is $\frac{1}{3} - \frac{1}{4} = \frac{1}{12} > 0$ , while $b_{0} = 1$ , $b_{1} = \frac{1}{2}$ , and $b_{0} - b_{1} = \frac{1}{2}$ are positive, so it lies in $U_{2}$ .

1.2F1algebra

For any $b \in V_{2}$ , choose $a = (- b_{1} / b_{0}, 1)$ . Its order-one Hankel quadratic form equals $b_{2} - b_{1}^{2} / b_{0} = (b_{0} b_{2} - b_{1}^{2}) / b_{0} < 0$ , so that matrix is not positive semidefinite and $b \notin K_{2}$ . In particular, for $b^{f a i l} = (1, \frac{1}{2}, \frac{1}{8})$ this vector is $a = (- \frac{1}{2}, 1)$ and the value is $\frac{1}{8} - \frac{1}{4} = - \frac{1}{8}$ .

2.1F2step 1.1step 1.2algebra∎

By [F2], any coefficient vector arising from the specified gapped amplitude data must satisfy every order-two constraint. Hence $b^{f a i l}$ is excluded under those hypotheses, while the finite test alone only labels $b^{p a s s}$ as allowed by $K_{2}$ ; it supplies no converse or physical realization.

Allowed and excluded order-two forward coefficients

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources