closed_iff_net_zero
plain-language theorem explainer
The equivalence states that a ledger constraint L on a state x is closed precisely when its net balance at x is zero. Researchers modeling discrete strain or balance in RRF would cite this to equate two views of equilibrium without extra checks. The proof reduces directly via simplification on the definitions of closed and net together with the subtraction-zero identity.
Claim. Let $L$ be a ledger constraint given by integer-valued debit and credit maps on a state space. For any state $x$, the constraint $L$ is closed at $x$ if and only if the net of $L$ at $x$ equals zero.
background
The RRF Core Strain module treats strain as the measure of deviation from equilibrium, with the governing law that strain tends to zero (corresponding to the recognition functional J tending to zero). A LedgerConstraint is the structure whose debit and credit functions on states must satisfy balance; the module supplies the abstract interface for such functionals on states drawn from discrete models. Upstream results include the discrete 2D Galerkin state from fluids, the finite vorticity lattice state, the local non-sealed RRF field interface, and concrete ledger instances in Recognition and Cycle3 that set debit and credit to constant values.
proof idea
The proof is a one-line simp that unfolds the definitions of isClosed and net, then applies the lemma that a difference is zero exactly when the two terms coincide.
why it matters
This equivalence supplies a direct test for closed ledgers inside the strain interface, supporting the RRF law that strain vanishes at balance. It belongs to the abstract ledger machinery that feeds equilibrium and minimizer results in the same module, though no downstream uses are yet recorded. The result aligns with the Recognition framework emphasis on strain to zero as the fundamental consistency condition.
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