ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck pass

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.

The scaled Newtonian comparison system for the quadratic potential: explicit harmonic solutions, a compact convex containing ball and the hypotheses of the scaled trajectory theorem

Example

Work on the flat model with inertial chart $x$ , let $L > 0$ be a length, $Ω = R \times B_{2 L} (0)$ with spatial base $U = B_{2 L} (0)$ , and $K = {\overline{B}}_{L / 2} (0) \subset U$ . Set $C : = 1$ , $ε_{0} : = 1 / 16$ , $r \equiv 0$ and

$φ (x) : = \frac{∣ x ∣^{2}}{2 L^{2}}, Φ = ε c^{2} φ .$

Every unmarked norm $∣ \cdot ∣$ of a vector in $R^{3}$ below means $∥ \cdot ∥_{2}$ from the Euclidean inner product (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ). The gradient is the coordinate vector of partial derivatives (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case), and the Hessian bound below uses the induced operator norm $∥ D^{2} ψ ∥_{o p}$ of Scaled trajectory limit: weak-field geodesics converge to the Newtonian system, with rate O(epsilon) for identical initial data.

These are stationary weak-field data (Stationary weak-field data: the diagonal ansatz, its bounds, stationarity and the Einstein-family condition). The scaled spatial base is $U_{L} = B_{2} (0)$ and the scaled potential is

$ψ (X) = φ (L X) = \frac{1}{2} ∣ X ∣^{2}, \nabla ψ (X) = X, D^{2} ψ = I_{3} .$

For any $S_{*} > 0$ and initial vectors $X_{0}, W_{0} \in R^{3}$ satisfying $R_{0} : = ∣ X_{0} ∣ + ∣ W_{0} ∣ < 2$ , the Newtonian comparison system of Scaled trajectory limit: weak-field geodesics converge to the Newtonian system, with rate O(epsilon) for identical initial data

$X_{*}^{'} (S) = W_{*} (S), W_{*}^{'} (S) = - \nabla ψ (X_{*} (S)) = - X_{*} (S)$

has the solution

$X_{*} (S) = \cos S X_{0} + \sin S W_{0}, W_{*} (S) = - \sin S X_{0} + \cos S W_{0} .$

Its spatial range is contained in ${\overline{B}}_{R_{0}} (0) : = {X : ∥ X ∥_{2} \leq R_{0}}$ , with ${\overline{B}}_{0} (0) = {0}$ . Set $R^{'} : = \max {1 / 2, R_{0}} < 2$ and $K^{'} : = {\overline{B}}_{R^{'}} (0)$ . This is a compact convex subset of $U_{L}$ containing the comparison trajectory. For the trajectory estimate take $0 < ε \leq ε_{*}$ from Scaled trajectory limit: weak-field geodesics converge to the Newtonian system, with rate O(epsilon) for identical initial data. If (H1) holds, then $X_{ε} (S) \in {\overline{B}}_{1 / 2} (0) \subseteq K^{'}$ as well, so (H2) and (H3) hold with $K_{2} = 1$ . With matched scaled initial data, the conclusion (ii) of the scaled trajectory theorem gives $\sup_{[0, S_{*}]} D (S) \leq C_{3} ε S_{*} e^{2 S_{*}} = O (ε)$ .

Facts & Assumptions

Given: The flat model with inertial chart $x$ , $L > 0$ , $Ω = R \times B_{2 L} (0)$ , $U = B_{2 L} (0)$ , $K = {\overline{B}}_{L / 2} (0)$ , $C = 1$ , $ε_{0} = 1 / 16$ , the functions $φ$ and $r \equiv 0$ above, and $c > 0$ .

[L1]

The metric coefficients factor through $U$ and are Lorentzian for $0 < ε \leq 1 / 16$ : on $U$ , $0 \leq φ < 2$ , so $1 - 2 ε φ > 3 / 4 > 0$ and $1 + 2 ε φ > 0$ . On $K$ , $∣ φ ∣ \leq 1 / 8$ , $∣ \partial_{i} φ ∣ \leq 1 / (2 L)$ and $∣ \partial_{i} \partial_{j} φ ∣ \leq 1 / L^{2}$ ; $r = 0$ satisfies the remainder bounds (Stationary weak-field data: the diagonal ansatz, its bounds, stationarity and the Einstein-family condition clause (a)).

[L2]

Under $X \mapsto L X$ , the chain rule gives $\partial_{i} ψ (X) = L \partial_{i} φ (L X)$ and $D^{2} ψ (X) = L^{2} D^{2} φ (L X)$ ; for the stated potential, $\nabla ψ (X) = X$ and $D^{2} ψ = I_{3}$ (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder).

[L3]

Sine and cosine derivatives and bounds: $(\cos S)^{'} = - \sin S$ , $(\sin S)^{'} = \cos S$ , and $∣ \cos S ∣, ∣ \sin S ∣ \leq 1$ (The derivatives of sine and cosine are cosine and minus sine, Signs, monotonicity intervals, and ranges of sine and cosine).

[L4]

Closed Euclidean balls are compact and convex; for $v, w$ in the closed ball of radius $R^{'}$ , $∣ (1 - t) v + t w ∣ \leq (1 - t) ∣ v ∣ + t ∣ w ∣ \leq R^{'}$ for $0 \leq t \leq 1$ (Euclidean spheres and closed balls as subspaces of $R^{n}$ , For $n \geq 1$ , every Euclidean closed ball and every Euclidean sphere of positive radius is compact, A convex subset of $R^{m}$ contains every line segment between two of its points, The inner-product norm is definite, homogeneous, and satisfies the triangle inequality).

[L5]

The scaled trajectory theorem requires $K^{'} \subseteq U_{L}$ to be compact and convex, to contain every $X_{ε} (S)$ and the range of a $C^{1}$ comparison solution, and to satisfy $∥ D^{2} ψ ∥_{o p} \leq K_{2}$ on $K^{'}$ ; under (H1) and matched initial data it yields the stated $O (ε)$ bound (Scaled trajectory limit: weak-field geodesics converge to the Newtonian system, with rate O(epsilon) for identical initial data).

[L6]

The norm $∥ \cdot ∥_{2}$ is induced by the Euclidean inner product, the gradient is the coordinate vector of first partial derivatives, and for a totally differentiable map the total derivative has those partials as its coordinate entries (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, A total derivative computes every directional derivative, and its matrix is the Jacobian).

Verification

technique · verify the metric and derivative bounds on the stated spatial domain, solve the harmonic comparison system explicitly, and construct a compact convex subset of the scaled spatial domain containing both paths when (H1) holds

1.1L1L2L6algebra

(Data and scaled potential.) For every $x \in U$ , the metric is diagonal with entries $- (1 + 2 ε φ)$ and $(1 - 2 ε φ) δ_{i j}$ ; by [L1] it is Lorentzian on $Ω$ for $0 < ε \leq ε_{0}$ . The spatial derivative bounds on $K$ follow from [L1], so the tuple is stationary weak-field data. By [L2], $ψ (X) = ∣ X ∣^{2} / 2$ , $\nabla ψ (X) = X$ and $D^{2} ψ = I_{3}$ on $U_{L} = B_{2} (0)$ .

1.2L2L3algebra

(The comparison solution.) Define $X_{*}, W_{*}$ by the displayed formulas. Differentiating with [L3] gives $X_{*}^{'} = - \sin S X_{0} + \cos S W_{0} = W_{*}$ and $W_{*}^{'} = - \cos S X_{0} - \sin S W_{0} = - X_{*} = - \nabla ψ (X_{*})$ by [L2]. Since $\cos 0 = 1$ and $\sin 0 = 0$ , the initial values are $X_{*} (0) = X_{0}$ and $W_{*} (0) = W_{0}$ . These formulas are smooth on $R$ ; the range check below verifies that the position range stays in $U_{L}$ , where the comparison equation is defined.

1.3L2L3L4L5L6algebra

(Compact convex containing ball.) By [L3], $∣ X_{*} (S) ∣ \leq ∣ X_{0} ∣ + ∣ W_{0} ∣ = R_{0}$ for every $S$ . Thus its range lies in ${\overline{B}}_{R^{'}} (0)$ ; by [L4] this is compact and convex. As $R^{'} < 2$ , $K^{'} \subset U_{L}$ . If (H1) holds, then $∣ X_{ε} (S) ∣ \leq 1 / 2$ because $x_{ε} (t) \in K = {\overline{B}}_{L / 2} (0)$ , so $X_{ε} (S) \in K^{'}$ . Moreover $∥ D^{2} ψ ∥_{o p} = 1$ on $K^{'}$ by [L2]. Hence (H2)-(H3) hold with $K_{2} = 1$ under (H1).

2.1step 1.2step 1.3L5algebra∎

(Conclusion and limitation.) For $0 < ε \leq ε_{*}$ , if (H1) holds for the geodesic family with scaled initial data matching $(X_{0}, W_{0})$ , then $D (0) = 0$ . The scaled theorem's conclusion (ii) gives $\sup_{0 \leq S \leq S_{*}} D (S) \leq C_{3} ε S_{*} e^{2 S_{*}} = O (ε)$ . The example verifies the Newtonian trajectory and the domain hypotheses conditional on (H1); it asserts neither existence of such geodesics nor a physical orbit.

The scaled Newtonian comparison system for the quadratic potential: explicit harmonic solutions, a compact convex containing ball and the hypotheses of the scaled trajectory theorem

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources