LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passverified 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.

Smooth shell-tangent force initial-value problem

Statement

Let $(M, g, N)$ be a smooth time-oriented Lorentzian four-manifold and let $m, c > 0$ . Let $V \subset T M$ be an open set of future timelike vectors and let $f$ be a smooth vector field along the bundle projection $V \to M$ , with $f (p, V) \in T_{p} M$ and $g_{p} (V, f (p, V)) = 0$ for every $(p, V) \in V$ . Fix an initial parameter $s_{0} \in R$ . For every initial datum $(p_{0}, U_{0}) \in V$ satisfying $g_{p_{0}} (U_{0}, U_{0}) = - c^{2}$ , there is an open interval $J ∋ s_{0}$ and a unique local solution consisting of a $C^{2}$ curve $x : J \to M$ and its $C^{1}$ tangent field $U = d x / d s$ , with $x (s_{0}) = p_{0}$ , $U (s_{0}) = U_{0}$ , $d x / d s = U$ , and $D_{s} U = f (x, U) / m$ . The solution depends smoothly on its initial data and on any finite-dimensional parameters on which $f$ depends smoothly, on a common sufficiently short compact interval containing $s_{0}$ near the base datum. It remains on the future unit timelike shell. There is a unique maximal solution on an open interval containing $s_{0}$ , but this interval need not be all of $R$ .

Facts & Assumptions

Given: The prescribed background of Particle spacetime, observers and worldlines, its metric-compatible torsion-free connection Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields, and the open-domain force field of [A1].

[A1]

The force field is prescribed and smooth on an open set of future timelike velocities, is orthogonal to the velocity argument, and $m, c > 0$ .

[F1]

Picard–Lindelöf: “a unique local solution of the IVP exists on some interval around the initial time” (Picard-Lindelöf local existence and uniqueness for first-order systems).

[F2]

Smooth dependence: for a jointly smooth field, near base data “there are a compact time interval $I$ and a neighbourhood $W$ ” on which the solutions exist and depend smoothly on initial data and parameters (Smooth dependence of ODE solutions on parameters).

[F3]

Along a curve, $\frac{d}{d s} g (V, W) = g (D_{s} V, W) + g (V, D_{s} W)$ (Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed).

[F4]

Every IVP satisfying Picard–Lindelöf has a “unique maximal solution” on an “open interval containing the initial time” (Every Picard–Lindelöf initial value problem has one maximal solution on an open interval).

[F5]

A nonzero causal vector is future-directed exactly when its pairing with the time-orientation field is negative, and that pairing cannot vanish (Classification of future and past causal vectors by a timelike field).

[F6]

The covariant derivative along a curve is the pullback connection (Covariant derivative along a curve); chart connection coefficients are defined by $\nabla_{\partial_{b}} \partial_{c} = Γ^{a}_{b c} \partial_{a}$ (Christoffel symbols of an affine connection).

Proof

1.1givenF1F6algebra

In a chart write $U = U^{c} \partial_{c}$ . The pullback connection and its Leibniz rule in [F6] give $D_{s} U = ({\dot{U}}^{a} + Γ^{a}_{b c} {\dot{x}}^{b} U^{c}) \partial_{a}$ ; since ${\dot{x}}^{b} = U^{b}$ , the covariant system is ${\dot{x}}^{a} = U^{a}$ , ${\dot{U}}^{a} = - Γ^{a}_{b c} (x) U^{b} U^{c} + m^{- 1} f^{a} (x, U)$ . The right side is smooth on an open chart-state domain, hence locally Lipschitz; [F1] gives a unique local solution through each allowed initial datum, and uniqueness makes coordinate solutions agree on chart overlaps.

1.2givenA1F3F5algebra

Along this solution, metric compatibility [F3] gives $\frac{d}{d s} g (U, U) = 2 g (D_{s} U, U) = 2 m^{- 1} g (f (x, U), U) = 0$ by [A1]. Thus $g (U, U)$ stays equal to its initial value $- c^{2}$ . The vector $U$ never vanishes; by [F5] its continuous pairing with the time-orientation field cannot vanish, so the initially negative pairing remains negative and its future direction is preserved.

2.1step 1.1F1F2F4∎

The coordinate field is jointly smooth in initial coordinates and any smooth force parameters, so [F2] supplies smooth dependence on a common compact local interval. To obtain a maximal solution on $V$ , take the union of all local solutions through the datum on open intervals containing $s_{0}$ . Any two such intervals intersect in an interval containing $s_{0}$ . The set where their position-velocity pairs agree is closed by continuity and open by local uniqueness [F1] in a chart; since it contains $s_{0}$ , connectedness makes the pairs agree on the whole overlap. Their union is therefore a well-defined solution on the open interval union and is maximal by construction. This is the chartwise gluing of the Euclidean maximal-ODE result [F4]; no infinite lifetime follows.

Smooth shell-tangent force initial-value problem

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources