Perfect entropy advection and restricted circulation

Given: The stated smooth isolated perfect-fluid model on $O$, with $n>0$, $T>0$, a smooth closed curve with compact image, and a local flow of $u$ defined near that image for sufficiently small $\lambda$.

Smooth perfect local material balance model (draft physical postulate): the selected model adopts $J^a=n{u}^a$, ${\mathsf{T}}^{ab}=\epsilon u^a{u}^b+p{\mathrm{\Delta }}^{ab}$, and the local balances ${\mathrm{\nabla }}_a{J}^a=0$, ${\mathrm{\nabla }}_a{\mathsf{T}}^{ab}=0$ on the specified smooth fixed-composition branch. This is an adopted physical model, not a deduction from geometry.

Perfect stress and particle balances with entropy identity (draft mathematical supplier): for a smooth state and perfect stress/current with ${\mathrm{\nabla }}_a{\mathsf{T}}^{ab}=f^b$ and ${\mathrm{\nabla }}_a{J}^a=0$, it states $T{\mathrm{\nabla }}_a(s{u}^a)=-u_b{f}^b$ and $nTD{\sigma }_p=-u_b{f}^b$. At $f=0$, $n,T>0$, it yields the two vanishing identities used here. These identities apply to the stated state and balance hypotheses.

Enthalpy vorticity and restricted material circulation (draft mathematical supplier): with the same smooth state and isolated perfect balances, $H=(\epsilon +p)\mathrm{/}n$, $P=(H\mathrm{/}c)u^{\mathrm{♭}}$, and the stated compact closed curve/local-flow hypotheses, it gives ${\iota }_{u}d(H{u}^{\mathrm{♭}})=Td{\sigma }_p$ and $d\mathcal{C}\mathrm{/}d\lambda =c^{-1}{\int }_{{\gamma }_{\lambda }}Td{\sigma }_p$. Its Statement also records circulation constancy when ${\sigma }_p$ is spatially constant on the swept region, but its proof derives the derivative identity and stops at the resulting ${\mathcal{C}}^{\mathrm{\prime }}=0$; this theorem obtains constancy from [F7] and does not infer it from $D{\sigma }_p=0$ alone. The compact-curve and local-flow hypotheses here are exactly those of the derivative identity.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): the supplier chain explicitly assumes countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$; this theorem carries that exact premise and invokes no stronger choice principle.

Rest fluid state, frame and geometric current definitions: the selected C2 state defines $n=1\mathrm{/}v$, $s=n{\sigma }_p$, $T=e_{{\sigma }_p}$, and ${\sigma }_p$ as entropy per marker. These state coordinates are used with the stated $C^2$ regularity.

Thermodynamics-GR fields, currents, reference normalization and units: use $x^0=ct$ with $c>0$ in SI units and dimensionless unit observers in signature $(-,+,+,+)$. Only its local unit and observer conventions are used here.

A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$; consequently two such functions with the same derivative differ by a constant (published mathematical corollary): a continuous real function on an order-convex interval whose derivative vanishes at every interior point is constant on the interval. The local circulation function is differentiable on its flow interval, so this applies when its derivative vanishes throughout that interval.

The curve is smooth, closed, and compactly contained in $O$; the local flow is parameterized by ${\mathrm{\partial }}_{\lambda }{\mathrm{\Phi }}_{\lambda }=u$ and is defined on a neighborhood of the curve for small $\lambda$. This is a local condition, not completeness of the flow.