ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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.

One null ray with distinct momentum scales

Physical scope

Algebraic illustration of affine-parameter and momentum-scale freedom for a null test-particle ray; no scale is inferred from the unparameterized image.

Example

In a prescribed background, let $I \subseteq R$ be a nonempty open interval and let $x : I \to M$ be an affinely parametrized nonzero future-null free geodesic with momentum $P = d x / d λ$ . For any constant $a > 0$ , on the rescaled open interval $I / a : = {λ / a : λ \in I}$ the same ray image admits momentum $P_{a} = a P$ using the affine parameter $λ_{a} = λ / a$ .

Facts & Assumptions

Given: A null free trajectory from Free massive and null particle geodesics, a positive constant $a$ , and, at any chosen $λ_{0} \in I$ , an observer $n$ at the event $x (λ_{0})$ .

[F1]

A null free particle has $d x / d λ = P$ and nonzero future-null momentum (Free massive and null particle geodesics).

[F2]

Observer energy is $E_{n} (P) = - c g (P, n)$ (Mechanical momentum and observer energy).

Verification

1.1givenF1constructalgebra

Define $x_{a} (λ_{a}) = x (a λ_{a})$ . Then $d x_{a} / d λ_{a} = a P = : P_{a}$ , which remains nonzero future null. Since $D_{λ} P = 0$ , the constant rescaling also gives $D_{λ_{a}} P_{a} = 0$ ; the image of $x_{a}$ equals the image of $x$ . Thus both parameterizations satisfy the free null equations of [F1].

2.1step 1.1F2algebra∎

For any $λ_{0} \in I$ , the rescaled curve satisfies $x_{a} (λ_{0} / a) = x (λ_{0})$ and $P_{a} (λ_{0} / a) = a P (λ_{0})$ . For an observer $n$ at this common event, [F2] gives $E_{n} (P_{a}) = a E_{n} (P)$ . Hence the same unparameterized ray supports distinct positive momentum and energy scales; the path image alone does not select one.

One null ray with distinct momentum scales

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources