Physics theoremStatement: 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.

Controlled Newtonian reductions for test motion

Physical scope

Two explicitly separate conditional model branches: (i) flat inertial constant-mass test-particle kinematics under a low-speed family bound; (ii) massive free test-particle geodesics in a stationary compact weak-field family satisfying the exact hypotheses of the GR mathematics limit theorem. The flat-frame momentum and energy estimates are not transferred to curved coordinate components. No metric family is constructed from Newtonian data, and no empirical, self-force, or global-time claim is made.

Statement

The following are two separate conditional reductions, with no identification of the flat inertial quantities in clause (a) with the curved coordinate quantities in clause (b).

(a) Flat inertial kinematics. In a named inertial chart of the flat massive-particle model Adopted massive-particle momentum, observer measurement and isolated-collision conservation, use the mechanical energy and momentum of Mechanical momentum and observer energy. Let $C > 0$ , $0 < λ \leq λ_{*}$ , and $ρ = C^{2} λ_{*} < 1$ . For any massive state with constant $m > 0$ , speed $v$ , and $∣ v ∣ \leq C \sqrt{λ} c$ , the bounds of Uniform low-speed relativistic remainders hold with $ε = λ$ : the relativistic momentum and kinetic energy have the stated uniform Newtonian remainders, and the coordinate-time action density after subtracting the rest term differs from $\frac{1}{2} m ∣ v ∣^{2}$ by at most $m C^{4} c^{2} λ^{2} / [8 (1 - ρ)^{3 / 2}]$ . This is a kinematic assertion; it assumes no force law and proves no trajectory convergence.

(b) Curved weak-field free motion. Adopt the separate massive test-particle action on a prescribed smooth background (Massive spinless test-particle action). Let the stationary weak-field metric family, scale $L$ , spatial potential $Φ = ε c^{2} φ$ , massive geodesic, compact spatial region, speed bound, Newtonian comparison solution, and Hessian bound satisfy exactly hypotheses (H1)–(H3) of Scaled trajectory limit: weak-field geodesics converge to the Newtonian system, with rate O(epsilon) for identical initial data. If the massive particle's free worldline is the geodesic of that theorem, then for the scaled trajectories $(X_{ε}, W_{ε})$ and $(X_{*}, W_{*})$ ,

$D (S) : = ∥ X_{ε} (S) - X_{*} (S) ∥_{2} + ∥ W_{ε} (S) - W_{*} (S) ∥_{2} \leq (D (0) + C_{3} ε S_{*}) e^{(1 + K_{2}) S_{*}} (0 \leq S \leq S_{*}),$

with $C_{3}, K_{2}, S_{*}$ exactly as in that theorem. In particular, for identical scaled initial data, $\sup_{0 \leq S \leq S_{*}} D (S) \leq C_{3} ε S_{*} e^{(1 + K_{2}) S_{*}} = O (ε)$ on this fixed scaled interval. The corresponding coordinate-time horizon is $T_{ε} = L S_{*} / (\sqrt{ε} c)$ ; this is not a large-time limit at fixed $ε$ .

Facts & Assumptions

Given: The separate flat inertial state and low-speed family of clause (a), and the prescribed curved weak-field data, massive geodesic, compact-containment, scaled Newtonian comparison solution, and Hessian bound (H1)–(H3) of clause (b).

[A1]

The flat massive model adopts constant $m > 0$ , $P = m U$ , the inertial-frame particle energy and momentum, and the massive branch only (Adopted massive-particle momentum, observer measurement and isolated-collision conservation). This is separate from the curved-background model.

[F1]

The pointwise massive-particle definitions are $P = m U$ and, in an inertial chart, $p = m γ v$ , $E = γ m c^{2}$ (Mechanical momentum and observer energy). Use SI units in both branches: $m$ is kg; $c$ and $c_{*}$ are m s $^{- 1}$ ; $t, τ, T, T_{ε}$ are seconds; $x, L$ are metres; $v$ is m s $^{- 1}$ ; $p$ is kg m s $^{- 1}$ ; $E, L_{c}, L_{N}$ are joules; and action is J s. In (a), $λ, λ_{*}, ρ, C$ are dimensionless. In (b), $ε, φ, C$ are dimensionless, $Φ$ is m $^{2}$ s $^{- 2}$ , and the scaled variables $X = x / L$ , $W = v / (\sqrt{ε} c)$ , $S = t \sqrt{ε} c / L$ , as well as $S_{*}, D, C_{3}, K_{2}$ , are dimensionless (Massive spinless test-particle action, Scaled trajectory limit: weak-field geodesics converge to the Newtonian system, with rate O(epsilon) for identical initial data). The formal parameter $c$ in (a) or (b) retains the unit m s $^{- 1}$ ; varying it in a model family is not a claim that the exact SI defining value varies in nature.

[L1]

For $∣ v ∣ \leq C \sqrt{λ} c$ and $C^{2} λ_{*} = ρ < 1$ , the uniform momentum, kinetic-energy and fixed-time action-density remainders hold with constants depending only on $C, ρ, m, c$ (Uniform low-speed relativistic remainders).

[A2]

The massive spinless test-body model adopts $S_{m} [x] = - m c^{2} \int d τ$ on a prescribed smooth time-oriented background, with fixed-endpoint stationary paths as its free trajectories; its ideal-clock increment is the stipulated proper time (Massive spinless test-particle action, Ideal clocks on a prescribed Lorentzian background). It excludes self-gravity, radiation reaction, spin and finite-size structure. These are draft physical postulates in batch 22 and are used at their inspected scopes; in the curved branch the local time orientation is established in [L3].

[L2]

Under that adopted action, stationary massive paths satisfy $D_{τ} U = 0$ and are affinely parametrized timelike geodesics; only local solutions and open maximal intervals are asserted (Free massive and null particle geodesics). This is a current in-run draft result, so its use here is conditional on its exact statement and proof status.

[L3]

For $0 < ε \leq ε_{*}$ in the stationary weak-field data, Stationarity, timelikeness and uniform inverse-metric bounds for stationary weak-field data gives $g (ε)_{00} \leq - 1 / 2$ at every event whose spatial coordinate lies in $K$ . The image $Γ_{ε} = {(c t, x_{ε} (t)) : 0 \leq t \leq T_{ε}}$ is compact and contained in $Ω$ by (H1). Since $g (ε)_{00}$ is continuous, $O_{ε} : = {p \in Ω : g (ε)_{00} (p) < 0}$ is an open neighborhood of $Γ_{ε}$ . The smooth coordinate field $\partial_{0}$ is timelike throughout $O_{ε}$ , so choosing $N_{ε} : = \partial_{0}$ defines a time orientation there. Thus the restricted background $(O_{ε}, g (ε), N_{ε})$ meets the time-oriented-background hypothesis; no orientation of the whole $Ω$ is needed (Stationarity, timelikeness and uniform inverse-metric bounds for stationary weak-field data, Particle spacetime, observers and worldlines).

[L4]

For stationary weak-field data with the prescribed $C^{2}$ metric bounds, compact spatial containment, the uniform speed hypothesis, a Newtonian comparison solution on a fixed scaled interval, and $∥ D^{2} ψ ∥_{o p} \leq K_{2}$ , the published mathematics theorem gives the displayed finite-time estimate; it neither constructs those geodesics nor metric families (Scaled trajectory limit: weak-field geodesics converge to the Newtonian system, with rate O(epsilon) for identical initial data).

Proof

technique · apply the uniform kinematic estimate in the flat branch and the exact geodesic comparison theorem in the separate curved branch

1.1givenA1A2F1L1algebra

In clause (a), the named inertial chart has $η = diag (- 1, 1, 1, 1)$ and $x^{0} = c t$ . For a future-timelike history with coordinate velocity $v$ , the proper-time definition gives $c^{2} (d τ / d t)^{2} = - η ((c, v), (c, v)) = c^{2} - ∣ v ∣^{2}$ , so $d τ / d t = \sqrt{1 - ∣ v ∣^{2} / c^{2}}$ . Specializing the adopted action [A2] to this flat, time-oriented branch yields $S_{m} = \int L_{c} (v) d t$ with $L_{c} (v) = - m c^{2} \sqrt{1 - ∣ v ∣^{2} / c^{2}}$ ; hence the coordinate-time action density used by [L1] is exactly the postulated density, and subtracting the rest term replaces it by $L_{c} + m c^{2}$ . By [A1] and [F1] the inertial quantities are $p = m γ v$ and $E = γ m c^{2}$ . The family speed bound is [L1]'s hypothesis with $ε = λ$ and $ρ = C^{2} λ_{*}$ . Substitution into [L1] proves the momentum, kinetic-energy and action-density error bounds for every member of the family. No equation of motion or trajectory estimate is used.

1.2givenA2L2L3L4

In clause (b), [L3] supplies a time-oriented open background neighborhood of the full contained worldline segment, which meets the background hypothesis of [A2]. The postulated stationary free path is therefore a timelike geodesic by [L2]. The theorem assumes this geodesic and the Newtonian comparison solution exist on the stated interval, and by hypothesis they satisfy exactly (H1)–(H3) recorded in [L4]. Applying [L4] gives the displayed bound with the same constants and scaled variables, without changing the metric assumptions or asserting that such a family can be constructed from arbitrary Newtonian data.

2.1givenstep 1.2algebra∎

When $D (0) = 0$ , the bound from step 1.2 reduces to $C_{3} ε S_{*} e^{(1 + K_{2}) S_{*}}$ ; with $C_{3}, K_{2}, S_{*}$ fixed this is $O (ε)$ as $ε ↓ 0$ . The coordinate-time identity is the stated rescaling $t = L S / (\sqrt{ε} c)$ , so only a fixed scaled interval is covered. Clauses (a) and (b) have different geometries and no flat momentum or energy formula was used in the curved branch.

Remarks

The flat inertial low-speed family and the curved weak-field geodesic family are separate hypotheses. In particular, the curved statement is not a Newtonian limit for every solution of Einstein's equations, does not establish existence of an Einstein metric family from Newtonian initial data, and does not imply geodesic completeness.

Controlled Newtonian reductions for test motion

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources