ExampleConstruction: AI-adaptedVerification: AI-adaptedjudge pass (gpt-6.1-sol)

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.

Dimensional check of every Maxwell and Lorentz term in SI units

Physical scope

A dimensional consistency check of the adopted SI Maxwell equations and Lorentz coupling of the classical framework. It is exact unit arithmetic, not a measurement, not a proof of the equations and not a claim about the value of any constant; it inherits the framework's adopted equations only for the list of terms checked.

Example

This example is conditional on the units of Framework-local SI units, the vacuum constants and the exact/measured status of their values, the definitions of divergence and curl in Divergence and curl of a $C^{1}$ vector field, the equations of The microscopic vacuum Maxwell equations: smooth and weak branches and the coupling of The Lorentz coupling: external force on a test charge and the continuum force and power densities; all equalities below are equalities of units, using $1 C = 1 A s$ , $1 V = 1 J C^{- 1}$ , $1 J = 1 N m$ , $1 W = 1 V A = 1 J s^{- 1}$ , $1 F = 1 C V^{- 1}$ and $1 T = 1 N A^{- 1} m^{- 1}$ .

(a) $div E$ has unit $V m^{- 2}$ and $ρ / ε_{0}$ has unit $C m^{- 3} / (F m^{- 1}) = C V m^{- 2} C^{- 1} = V m^{- 2}$ , so the two sides of $div E = ρ / ε_{0}$ agree.

(b) $curl B$ has unit $T m^{- 1}$ ; $μ_{0} J$ has unit $N A^{- 2} \cdot A m^{- 2} = N A^{- 1} m^{- 2} = T m^{- 1}$ ; $c^{- 2} \partial_{t} E$ has unit $m^{- 2} s^{2} \cdot V m^{- 1} s^{- 1} = V s m^{- 3} = N A^{- 1} m^{- 2} = T m^{- 1}$ , so both source terms of $curl B = μ_{0} J + c^{- 2} \partial_{t} E$ have the unit of $curl B$ .

(c) $curl E$ has unit $V m^{- 2} = T s^{- 1}$ and $\partial_{t} B$ has unit $T s^{- 1}$ , so $curl E = - \partial_{t} B$ is homogeneous; $div B = 0$ is a homogeneous equation.

(d) In the coupling of The Lorentz coupling: external force on a test charge and the continuum force and power densities, the adopted right-hand side $q Φ (U)$ has unit $C \cdot T \cdot m s^{- 1} = C N A^{- 1} s^{- 1} = N$ (using $1 T = 1 N A^{- 1} m^{- 1}$ and $1 C = 1 A s$ ) and the left-hand side $m d U / d τ$ has unit $k g m s^{- 2} = N$ , so the two sides of $m d U / d τ = q Φ (U)$ agree; in the coordinate expression $q (E + r^{'} \times B)$ on the right of the postulate's displayed nonrelativistic acceleration equation (that whole equation is not adopted as exact), $q E$ has unit $C \cdot V m^{- 1} = J m^{- 1} = N$ and $q (r^{'} \times B)$ has unit $C m s^{- 1} T = C N A^{- 1} s^{- 1} = N$ , so its two terms add with the unit of a force; and $ρ E + J \times B$ has unit $C m^{- 3} V m^{- 1} = N m^{- 3}$ , the unit of a force density, while $J \cdot E$ has unit $A m^{- 2} \cdot V m^{- 1} = W m^{- 3}$ , the unit of a power density.

(e) The energy density $\frac{1}{2} (ε_{0} E^{2} + B^{2} / μ_{0})$ has unit $J m^{- 3}$ and the flux $E \times B / μ_{0}$ has unit $W m^{- 2}$ , consistent with $J m^{- 3} \cdot m s^{- 1} = W m^{- 2}$ .

(f) The check is exact arithmetic of the adopted units; it is not a measurement, it does not prove the equations, and no numerical value of $μ_{0}$ is used or asserted.

Facts & Assumptions

Given: the unit conventions of Framework-local SI units, the vacuum constants and the exact/measured status of their values, the component definitions of Divergence and curl of a $C^{1}$ vector field, and the adopted equations of The microscopic vacuum Maxwell equations: smooth and weak branches and The Lorentz coupling: external force on a test charge and the continuum force and power densities.

[F1]

The SI unit table: $c$ in $m s^{- 1}$ , $E$ in $V m^{- 1}$ , $B$ in T with $1 T = 1 N A^{- 1} m^{- 1}$ , $ρ$ in $C m^{- 3}$ , $J$ in $A m^{- 2}$ , force in N, $ε_{0} = (μ_{0} c^{2})^{- 1}$ in $F m^{- 1}$ with $μ_{0}$ in $N A^{- 2}$ , energy density in $J m^{- 3}$ and energy flux in $W m^{- 2}$ . (Framework-local SI units, the vacuum constants and the exact/measured status of their values)

[F2]

The divergence is $\sum_{i} \partial_{i} F_{i}$ and each curl component is a difference of two first partial derivatives, so divergence and curl divide the field unit by a length. (Divergence and curl of a $C^{1}$ vector field)

[F3]

The four Maxwell equations are $div E = ρ / ε_{0}$ , $div B = 0$ , $curl E = - \partial_{t} B$ and $curl B = μ_{0} J + c^{- 2} \partial_{t} E$ . (The microscopic vacuum Maxwell equations: smooth and weak branches)

[F4]

The adopted coupling is $m d U / d τ = q Φ (U)$ , with the coordinate expression $q (E + r^{'} \times B)$ displayed in the nonrelativistic acceleration equation, which is not adopted as exact, and with the force density $ρ E + J \times B$ and the power density $J \cdot E$ . (The Lorentz coupling: external force on a test charge and the continuum force and power densities)

Verification

technique · direct unit arithmetic for each displayed term

1.1F1F2F3

Reading the units of $E$ , $ρ$ and $ε_{0}$ from [F1] and the component formula for the divergence from [F2]: $div E$ has unit $V m^{- 2}$ , while $ρ / ε_{0}$ has unit $C m^{- 3} / (F m^{- 1}) = C m^{- 3} \cdot m F^{- 1} = C V m^{- 2} C^{- 1} = V m^{- 2}$ , using $1 F = 1 C V^{- 1}$ ; the two sides of the first equation of [F3] therefore carry the same unit.

1.2F1F2F3

Again by [F1] and [F2], $curl B$ has unit $T m^{- 1}$ ; $μ_{0} J$ has unit $N A^{- 2} \cdot A m^{- 2} = N A^{- 1} m^{- 2} = T m^{- 1}$ ; and $c^{- 2} \partial_{t} E$ has unit $m^{- 2} s^{2} \cdot V m^{- 1} s^{- 1} = V s m^{- 3}$ , which equals $N A^{- 1} m^{- 2} = T m^{- 1}$ because $1 V = 1 J C^{- 1}$ , $1 J = 1 N m$ and $1 C = 1 A s$ ; hence all three terms of the fourth equation of [F3] carry the unit $T m^{- 1}$ .

1.3F1F2F3

By [F1] and [F2], $curl E$ has unit $V m^{- 2}$ and $\partial_{t} B$ has unit $T s^{- 1}$ ; since $V m^{- 2} = J C^{- 1} m^{- 2} = N m C^{- 1} m^{- 2} = N A^{- 1} s^{- 1} m^{- 1} = T s^{- 1}$ by the conversion identities, the third equation of [F3] is homogeneous, and the second equation of [F3] is homogeneous with $div B$ in $T m^{- 1}$ and a zero right-hand side of that same dimension.

1.4F1F4

By [F1] and [F4]: the adopted term $q Φ (U)$ has unit $C \cdot T \cdot m s^{- 1} = C N A^{- 1} s^{- 1} = N$ and $m d U / d τ$ has unit $k g m s^{- 2} = N$ ; in the displayed coordinate expression, $q E$ has unit $C \cdot V m^{- 1} = C \cdot J C^{- 1} m^{- 1} = N$ and $q (r^{'} \times B)$ has unit $C \cdot m s^{- 1} \cdot N A^{- 1} m^{- 1} = C N A^{- 1} s^{- 1} = N$ ; $ρ E$ has unit $C m^{- 3} \cdot V m^{- 1} = C V m^{- 4} = N m^{- 3}$ and $J \times B$ has unit $A m^{- 2} \cdot N A^{- 1} m^{- 1} = N m^{- 3}$ , so the force density $ρ E + J \times B$ has unit $N m^{- 3}$ ; and $J \cdot E$ has unit $A m^{- 2} \cdot V m^{- 1} = W m^{- 3}$ because $1 W = 1 V A$ .

1.5F1

By [F1]: $ε_{0} E^{2}$ has unit $F m^{- 1} \cdot V^{2} m^{- 2} = F V^{2} m^{- 3} = C V m^{- 3} = J m^{- 3}$ and $B^{2} / μ_{0}$ has unit $N^{2} A^{- 2} m^{- 2} \cdot A^{2} N^{- 1} = N m^{- 2} = J m^{- 3}$ , so the energy density $\frac{1}{2} (ε_{0} E^{2} + B^{2} / μ_{0})$ has unit $J m^{- 3}$ ; the flux $E \times B / μ_{0}$ has unit $V m^{- 1} \cdot N A^{- 1} m^{- 1} \cdot A^{2} N^{- 1} = V A m^{- 2} = W m^{- 2}$ .

2.1step 1.1step 1.2step 1.3step 1.4step 1.5F1F3F4∎

Every term listed in (a)-(e) of the example has been assigned the same unit as the term it is compared with, so each displayed Maxwell equation and coupling expression is dimensionally homogeneous in the adopted SI conventions; the computation used only the unit table and the component definitions, and no numerical value of $μ_{0}$ , no measurement and no claim that the equations hold was used.

Dimensional check of every Maxwell and Lorentz term in SI units

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources