zero_transport_cancels
plain-language theorem explainer
If the transport component of a ContributionFields structure vanishes at every lattice site, then the summed totalTransport contribution is zero. Researchers deriving exact J-cost derivative identities for discrete Navier-Stokes vorticity models cite this cancellation to isolate viscous and stretching terms. The proof is a direct unfolding of the total and totalTransport definitions followed by simplification under the pointwise zero hypothesis.
Claim. Let $c$ be a contribution field on a finite lattice with $N$ sites. If the transport component satisfies $c(i)=0$ for every site $i$, then the total transport contribution equals zero: $T(c)=0$, where $T(c)$ denotes the sum of the transport values over all sites.
background
The module supplies exact bookkeeping objects for the J-cost monotonicity program on finite vorticity fields. ContributionFields is the structure holding three scalar fields (transport, viscous, stretching) over a Fin $N$ lattice window. The total function sums any such field, and totalTransport extracts the sum of the transport component alone. This setup supports the exact decomposition of the J-cost derivative into the three contribution pieces, with hard PDE inequalities kept separate for later addition.
proof idea
The proof is a one-line wrapper. It unfolds the definitions of totalTransport and total, then applies simp using the hypothesis that every transport entry is zero.
why it matters
This lemma closes a basic cancellation case inside the exact derivative identity for J-cost in the discrete vorticity setting. It supports the module's goal of packaging transport, viscous, and stretching contributions so that monotonicity arguments can proceed term by term. The result sits in the bookkeeping layer that precedes any continuous-limit or inequality work on the Navier-Stokes side of the Recognition framework.
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