LemmaStatement: AI-adaptedProof: 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.

Crossing evenness at the scalar subtraction point

Statement

Let $m > 0$ , let $A (s)$ be the forward slice of identical-scalar amplitude data satisfying $A (s) = A (4 m^{2} - s)$ , and let $P (s)$ be a finite sum of isolated simple real pole principal parts in $0 < s < 4 m^{2}$ and their $s \leftrightarrow u$ crossing partners, satisfying $P (s) = P (4 m^{2} - s)$ . Put $B (v) = A (2 m^{2} + v) - P (2 m^{2} + v)$ wherever these expressions are defined. Then $B (- v) = B (v)$ . If $B$ is complex differentiable in a neighbourhood of $v = 0$ , then $B^{'} (0) = 0$ .

Statement references

Gapped identical-scalar forward-amplitude data

Adams et al., Causality, Analyticity and an IR Obstruction to UV Completion, §4

de Rham et al., Positivity Bounds for Scalar Theories, §§2.1–2.2

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

Facts & Assumptions

Given: $m > 0$ , the forward variable $s$ , the crossing relation $u = 4 m^{2} - s$ , and the crossing-symmetric pole-subtracted amplitude data.

[F1]

Gapped identical-scalar forward-amplitude data: at $t = 0$ , $u = 4 m^{2} - s$ , $A (s) = A (u)$ , $P (s) = P (u)$ , and $B (v) = A (2 m^{2} + v) - P (2 m^{2} + v)$ with $ν_{0} = 2 m^{2}$ .

[A1]

The displayed crossing equalities hold wherever both sides are defined.

Proof

1.1givenF1A1algebra

Set $s = 2 m^{2} + v$ . Then the crossed value is $u = 4 m^{2} - s = 2 m^{2} - v$ . By [A1], $A (2 m^{2} + v) = A (2 m^{2} - v)$ and $P (2 m^{2} + v) = P (2 m^{2} - v)$ . Subtracting the pole terms gives $B (v) = B (- v)$ .

2.1givenstep 1.1algebra∎

Differentiate $B (- v) = B (v)$ at $v = 0$ . The chain rule gives $- B^{'} (0) = B^{'} (0)$ ; since $2 \neq 0$ in $C$ , $B^{'} (0) = 0$ .

Depends on

Gapped identical-scalar forward-amplitude data