ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-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.

Boosted uniform electric field and the mixed magnetic field

Physical scope

An exact algebraic example for one uniform static electric field ( $E \equiv E_{0} e_{2}$ , $B \equiv 0$ on $R^{3} \times R$ , $E_{0} > 0$ in $V m^{- 1}$ ) and one passive boost along $e_{1}$ with $0 \leq β < 1$ : the boosted components, the resulting mixed magnetic field and the two field invariants. Excluded: real sources of the uniform field, boundary or finite-support configurations, measurements and any empirical claim.

Example

Let $E_{0} > 0$ carry the unit $V m^{- 1}$ and consider the constant fields of The electric and magnetic fields as fixed-time spatial vector fields

$E : R^{3} \times R \to R^{3}, E \equiv E_{0} e_{2}, B \equiv 0,$

which form a source-free smooth solution of the vacuum Maxwell equations The microscopic vacuum Maxwell equations: smooth and weak branches (all first derivatives vanish and $div E = div B = 0$ , $curl E = curl B = 0$ ). Let $0 \leq β < 1$ , $γ : = (1 - β^{2})^{- 1 / 2}$ and $v : = β c e_{1}$ , and let $x^{'} = Λ_{β} x + b$ be the boosted inertial chart of Covariance, boosts and the two field invariants. Then the components of the same field in $x^{'}$ are

$E^{'} = γ E_{0} e_{2}, B^{'} = - \frac{γ β E_{0}}{c} e_{3} = - \frac{γ ∣ v ∣ E_{0}}{c^{2}} e_{3},$

so the moving observer reads an electric field of magnitude $γ E_{0}$ along $e_{2}$ and, for $β > 0$ , a nonzero magnetic field along $- e_{3}$ : a purely electric uniform field acquires a magnetic component in the boosted description. The two invariants check directly: with the spatial Euclidean structure of the chart (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The cross product in $R^{3}$ )

$∣ B^{'} ∣^{2} - \frac{∣ E^{'} ∣^{2}}{c^{2}} = γ^{2} (\frac{β^{2} E_{0}^{2}}{c^{2}} - \frac{E_{0}^{2}}{c^{2}}) = - \frac{E_{0}^{2}}{c^{2}} = ∣ B ∣^{2} - \frac{∣ E ∣^{2}}{c^{2}}, E^{'} \cdot B^{'} = 0 = E \cdot B,$

using $γ^{2} (1 - β^{2}) = 1$ (Rational powers $a^{r}$ of a positive base, Laws of rational exponents). At $β = 0$ the formulas reduce to $E^{'} = E$ , $B^{'} = 0$ . The computation is exact for the constant field; the example reports no measurement, identifies no source producing the field, and asserts nothing about finite-energy or bounded configurations.

Facts & Assumptions

Given: The constant $E_{0} > 0$ in $V m^{- 1}$ , the fields $E \equiv E_{0} e_{2}$ and $B \equiv 0$ on the whole-space laboratory region $Ω \times I = R^{3} \times R$ , the boost parameter $0 \leq β < 1$ , $γ = (1 - β^{2})^{- 1 / 2}$ , $v = β c e_{1}$ , and the boosted inertial chart $x^{'} = Λ_{β} x + b$ .

[F1]

The fields are those of The electric and magnetic fields as fixed-time spatial vector fields: $E$ carries $V m^{- 1}$ and $B$ the tesla, jointly $C^{k}$ means $C^{k}$ in the chart variables, and the spatial structure is the $δ$ -orthonormal basis $(e_{1}, e_{2}, e_{3})$ with Euclidean inner product and right-handed cross product (The electric and magnetic fields as fixed-time spatial vector fields (c), Framework-local SI units, the vacuum constants and the exact/measured status of their values (b), The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The cross product in $R^{3}$ ).

[F2]

Smooth branch: for jointly $C^{1}$ fields with the additional derivatives required by each equation existing and continuous and continuous sources, the smooth branch of The microscopic vacuum Maxwell equations: smooth and weak branches consists of $div E = ρ / ε_{0}$ , $div B = 0$ , $curl E = - \partial_{t} B$ and $curl B = μ_{0} J + c^{- 2} \partial_{t} E$ ; for $ρ = 0$ and $J = 0$ these are the source-free equations. The divergence and curl are those of Divergence and curl of a $C^{1}$ vector field, with partial derivatives at fixed time.

[F3]

Boost components and invariants (Covariance, boosts and the two field invariants (a), (b), (c)): for $0 \leq β < 1$ and the chart $x^{'} = Λ_{β} x + b$ of the statement, the dictionary components of the same $F$ satisfy $E_{2}^{'} = γ (E_{2} - β c B_{3})$ , $E_{3}^{'} = γ (E_{3} + β c B_{2})$ , $B_{2}^{'} = γ (B_{2} + β E_{3} / c)$ , $B_{3}^{'} = γ (B_{3} - β E_{2} / c)$ , with the components along $e_{1}$ unchanged, and $∣ B^{'} ∣^{2} - ∣ E^{'} ∣^{2} / c^{2} = ∣ B ∣^{2} - ∣ E ∣^{2} / c^{2}$ , $E^{'} \cdot B^{'} = E \cdot B$ ; at $β = 0$ the boost is the identity so $E^{'} = E$ , $B^{'} = B$ .

[F4]

Powers: for $a > 0$ and rationals $r, s$ one has $a^{r} > 0$ , $a^{r + s} = a^{r} a^{s}$ ; with $a : = 1 - β^{2} > 0$ this gives $γ^{2} (1 - β^{2}) = 1$ (Rational powers $a^{r}$ of a positive base, Laws of rational exponents).

[F5]

Orthonormality: $∣ e_{2} ∣ = ∣ e_{3} ∣ = 1$ , $e_{1}, e_{2}, e_{3}$ are pairwise orthogonal, and $e_{2} \cdot e_{3} = 0$ (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields (b)).

Verification

Proof technique: direct; substitute the constant data into the sourced smooth branch to verify the source-free solution, apply the axis-boost components of Covariance, boosts and the two field invariants (a), and evaluate the two invariants with the orthonormal-basis identities.

1.1givenF1F2algebra

Source-free solution: the field pair $(E, B)$ is constant, hence $C^{\infty}$ with every first partial derivative continuous and equal to $0$ , in particular jointly $C^{k}$ for every $k \geq 1$ [F1]. With the divergence and curl of [F2], $div E = \partial_{1} E_{1} + \partial_{2} E_{2} + \partial_{3} E_{3} = 0$ , $div B = 0$ , $curl E = (\partial_{2} E_{3} - \partial_{3} E_{2}, \partial_{3} E_{1} - \partial_{1} E_{3}, \partial_{1} E_{2} - \partial_{2} E_{1}) = 0$ and $curl B = 0$ , and $\partial_{t} E = \partial_{t} B = 0$ ; hence the four source-free smooth-branch equations of [F2] with $ρ = 0$ , $J = 0$ hold on $R^{3} \times R$ .

1.2givenF1F3algebra

Boosted components: substitute $E_{1} = 0$ , $E_{2} = E_{0}$ , $E_{3} = 0$ and $B_{1} = B_{2} = B_{3} = 0$ in the formulas of [F3]: $E_{1}^{'} = E_{1} = 0$ , $E_{2}^{'} = γ (E_{0} - β c \cdot 0) = γ E_{0}$ , $E_{3}^{'} = γ (0 + β c \cdot 0) = 0$ , $B_{1}^{'} = B_{1} = 0$ , $B_{2}^{'} = γ (0 + β \cdot 0 / c) = 0$ and $B_{3}^{'} = γ (0 - β E_{0} / c) = - γ β E_{0} / c$ ; therefore $E^{'} = γ E_{0} e_{2}$ and $B^{'} = - (γ β E_{0} / c) e_{3} = - (γ ∣ v ∣ E_{0} / c^{2}) e_{3}$ , since $∣ v ∣ = β c$ by $v = β c e_{1}$ and the right-handed basis [F1]. This is the asserted pair of components; the magnetic component is nonzero exactly when $β > 0$ , and the direction is $- e_{3}$ .

2.1givenstep 1.2F3F4F5algebra

Invariants: by [F3] the invariants agree with those computed from $(E, B)$ , and directly from step 1.2, $∣ E^{'} ∣^{2} = γ^{2} E_{0}^{2}$ , $∣ B^{'} ∣^{2} = γ^{2} β^{2} E_{0}^{2} / c^{2}$ and $E^{'} \cdot B^{'} = 0$ by [F5] ( $e_{2} ⊥ e_{3}$ ), while $∣ E ∣^{2} = E_{0}^{2}$ , $∣ B ∣^{2} = 0$ and $E \cdot B = 0$ ; hence $∣ B^{'} ∣^{2} - ∣ E^{'} ∣^{2} / c^{2} = γ^{2} E_{0}^{2} (β^{2} - 1) / c^{2} = - E_{0}^{2} / c^{2}$ by $γ^{2} (1 - β^{2}) = 1$ [F4], which equals $∣ B ∣^{2} - ∣ E ∣^{2} / c^{2}$ , and $E^{'} \cdot B^{'} = 0 = E \cdot B$ . At $β = 0$ one has $γ = 1$ and the formulas of [F3] give $E^{'} = E$ , $B^{'} = B = 0$ , so the boosted description is the laboratory description and no magnetic field appears.

3.1step 1.1step 1.2step 2.1F1F3∎

Closing: step 1.1 verifies that the constant data solve the source-free smooth branch; step 1.2 gives the exact boosted components and step 2.1 the two invariant checks, with the $β = 0$ reduction. The computation is exact algebra for the constant field and uses only the dictionary and boost formulas of [F3]; no measurement, source model, finite-energy, boundary or empirical claim is made, and no choice principle is used.

Boosted uniform electric field and the mixed magnetic field

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources