Local conservation of the adopted matter stress-energy tensors

Clause (a). The adopted scalar model includes the field equation $\mathrm{\square }\phi =V^{\mathrm{\prime }}(\phi )$ on $\mathrm{\Omega }$; [F1] then gives ${\mathrm{\partial }}_a{T}^{ab}=K\cdot 0\cdot {\mathrm{\partial }}^b\phi =0$ on $\mathrm{\Omega }$. By [F5] the vanishing is independent of the inertial chart in which the components are computed.

Clause (b). The adopted dust model is exactly the pair of equations ${\mathrm{\partial }}_a(\rho n^a)=0$ and $n^a{\mathrm{\partial }}_a{n}^b=0$ with $\rho >0$; the forward direction of [F2] gives ${\mathrm{\partial }}_a{T}^{ab}=0$ for $T=\rho n\otimes n$, and the reverse direction gives that conservation with $\rho >0$ implies both equations.

Clause (c). The adopted fluid model is the pair of equations obtained by setting both bracket expressions of [F3] to zero; by the equivalence in [F3] this is exactly ${\mathrm{\partial }}_a{T}^{ab}=0$ for $T=(\epsilon +P)n\otimes n+P{\eta }^{-1}$, in both directions.

Clause (d). The definition Stress-energy tensor, force density and local balance on flat spacetime states explicitly in its clause (d) that it asserts no conservation of any prescribed symmetric field, and the present theorem derives conservation only from the adopted model equations of clauses (a)–(c); accordingly for a $C^1$ symmetric field prescribed on $\mathrm{\Omega }$ that is not obtained from one of the adopted models together with its field equations, no conservation is asserted, and any conservation statement about such a field would be an additional explicit premise.

Clause (e), conditional homogeneous-dust reading. Consider a homogeneous dust realization with comoving identical particles of rest mass $m>0$ and rest number density $\nu >0$, so that the adopted reading gives $\rho =\nu m{c}^2$, and consider a rest rectangular parcel of rest volume $V_0$. Take the tensor $T=\rho n\otimes n$ in the rest frame, where $n=e_0$, so that $T^{00}=\rho$ and all other components vanish. The passive boost with parameter $\beta$ along the parcel's motion direction has ${\mathrm{\Lambda }}^0{}_0=\gamma$ and is proper orthochronous (The passive Lorentz boost with parameter beta), so the rank-two law of [F5] gives $T^{\mathrm{\prime }00}={\mathrm{\Lambda }}^0{}_a{\mathrm{\Lambda }}^0{}_b{T}^{ab}=({\mathrm{\Lambda }}^0{}_0{)}^2\rho ={\gamma }^2\rho$; thus $T^{\mathrm{\prime }00}={\gamma }^2\nu m{c}^2=(\gamma m{c}^2)(\gamma \nu )$. In a frame related to the rest frame by that boost the simultaneous volume of the parcel is $V_0\mathrm{/}\gamma$ by the length-contraction part of [F4] (parallel length $1\mathrm{/}\gamma$, transverse lengths unchanged), so the particle number density in that frame is ${\nu }^{\mathrm{\prime }}=N\mathrm{/}(V_0\mathrm{/}\gamma )=\gamma \nu$; by the massive-particle reading [F4] each particle carries energy $\gamma m{c}^2$. Hence the two factors in ${\gamma }^2\rho$ correspond respectively to particle energy and volume/density in the boosted frame. This is a conditional interpretation of the factorization within the homogeneous particle realization; it is not a derivation of the continuum model from microscopic dynamics.

Empty-domain branch. If $\mathrm{\Omega }=\mathrm{\varnothing }$, Adopted continuum matter models, evolution equations and observer readings supplies only the unique empty fields described by C^k tensor fields on affine Minkowski space and their inertial components. Their component derivatives and products are empty maps; every equality asserted at a point of $\mathrm{\Omega }$ is therefore vacuous, including the two directions of the dust and fluid equivalences. There is no nonempty parcel in $\mathrm{\Omega }$ to which clause (e)'s conditional realization can apply. This proves the separate mathematical empty-domain clause without invoking the named nonempty-domain suppliers on that domain.

In the nonempty adopted-model branch, clauses (a), (b), (c) and (d) are steps 1.1–1.4 and clause (e) is step 1.5; the separate mathematical empty-domain clause is step 1.6. Every conservation conclusion is a conditional deduction from the adopted model equations; the theorem asserts no interaction, force law, cross section, measurement or empirical fact, contains no empirical premise and makes no claim about arbitrary prescribed tensors. All choices are finite; no choice principle is used.